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Response Surface Designs for Linear Mixed Models

Permanent Link: http://ufdc.ufl.edu/UFE0021748/00001

Material Information

Title: Response Surface Designs for Linear Mixed Models
Physical Description: 1 online resource (139 p.)
Language: english
Creator: Saha, Sourish Chandra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Response surface methodology (RSM) is a set of tools that includes setting up a series of experiments that produce reliable measurements of the response of interest, fitting and evaluating a given model, and determining the settings of the factors that yield optimum value of the predicted response. In the area of RSM, one of the main considerations is the choice of an experimental design. Extensive studies have been undertaken in the design area with regard to response surface models, the same is not true with regard to such models in the presence of a random block effect in the fitted model. Designs for the latter type depend on certain unknown parameters concerning the model's variance components. Hence, the construction of such designs requires some prior knowledge of the unknown parameters. The design dependence problem for mixed response surface models is addressed by applying quantile dispersion graphs (QDGs), which is a powerful graphical tool for comparing designs for such models. The generation of D-optimal designs sequentially for mixed response surface models is also discussed. The use of QDGs to compare designs for correlated response surface models with an unknown dispersion matrix is presented followed by summary and possible areas for further research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sourish Chandra Saha.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Khuri, Andre I.
Local: Co-adviser: Ghosh, Malay.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021748:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021748/00001

Material Information

Title: Response Surface Designs for Linear Mixed Models
Physical Description: 1 online resource (139 p.)
Language: english
Creator: Saha, Sourish Chandra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Response surface methodology (RSM) is a set of tools that includes setting up a series of experiments that produce reliable measurements of the response of interest, fitting and evaluating a given model, and determining the settings of the factors that yield optimum value of the predicted response. In the area of RSM, one of the main considerations is the choice of an experimental design. Extensive studies have been undertaken in the design area with regard to response surface models, the same is not true with regard to such models in the presence of a random block effect in the fitted model. Designs for the latter type depend on certain unknown parameters concerning the model's variance components. Hence, the construction of such designs requires some prior knowledge of the unknown parameters. The design dependence problem for mixed response surface models is addressed by applying quantile dispersion graphs (QDGs), which is a powerful graphical tool for comparing designs for such models. The generation of D-optimal designs sequentially for mixed response surface models is also discussed. The use of QDGs to compare designs for correlated response surface models with an unknown dispersion matrix is presented followed by summary and possible areas for further research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sourish Chandra Saha.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Khuri, Andre I.
Local: Co-adviser: Ghosh, Malay.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021748:00001


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4ef9ee60e728af27b3155a37e198ccc7f6f768d7







RESPONSE SURFACE DESIGNS FOR LINEAR MIXED MODELS


By
SOURISH C. SAHA



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007



































In loving memory of my father









ACKENOWLED GMENTS

I am grateful to my advisor, Dr. Andrii I. K~huri, for his excellent supervision during

my doctoral research. This dissertation would not have been possible without his guidance

and tremendous patience. I would also like to thank all my committee members: Dr.

Malay Ghosh, Dr. Ronald Randles, Dr. Trevor Park and Dr. Murali Rao. Special thanks

go to my wife, Soumita Lahiri.










TABLE OF CONTENTS


page


ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

ABSTRACT

CHAPTER

1 INTRODUCTION

1.1 Literature Review.
1.2 Dissertation Objectives.


2 COMPARISON OF DESIGNS ON THE BASIS OF THE PREDICTION
CRITERION RISING QITANTILE DISPERSION GRAPHS .....


VARIANCE
. 1:3


2.1 Introduction.
2.2 Model and Notation
2.3 Quantile Dispersion Graphs .....
2.4 Confidence Interval for if.. ..
2.5 Exaniple.. .
2.6 A Brief Review of Split Plot Designs .....


. 1
. 17
.. 22
. 24


:3 COMPARISON OF DESIGNS ON THE BASIS OF THE POWER CRITERION
RISING QITANTILE DISPERSION GR APHS


:3.1 Introduction.
:3.2 Development of Design Criteria
:3.3 Design Comparisons using QDGs
3.3.1 Quantile Dispersion Graphs
3.3.2 A Conservative Confidence Region for (o,2 ty).
:3.:3.3 An Alternative Space for 6.
:3.4 A Numerical Example

4 SEQUENTIAL GENER ATION OF D-OPTIMAL DESIGNS.

4.1 Introduction.
4.2 Model and Notation
4.3 Methodology
4.4 Equivalence Theorem.
4.5 Sequential Generation of D-optinial designs
4.6 A Numerical Example











5 COMPARISON OF DESIGNS FOR CORRELATED RESPONSE MODELS
WITH AN UNKNOWN DISPERSION MATRIX ... .. .. 68

5.1 Introduction ........ ... .. 68
5.2 Correlated Response Models ..... ... .. 69
5.3 Development of a Multivariate Test ...... .. 70
5.4 Desigfn Criteria ......... . . 73
5.4.1 Wilks' likelihood ratio ........ .. .. 74
5.4.2 Hotelling-Lawley's trace . ...... .. 75
5.4.3 Pillai's trace ......... .. 75
5.5 Quantile Dispersion Graphs . .... .. 76
5.5.1 Simultaneous Confidence Intervals on the elements of E .. .. .. 76
5.5.2 An Alternative Space for a = Gr .... .. 77
5.6 Example ......... . .... .. 78

6 SUMMARY AND FUTURE RESEARCH ..... .. . 85

APPENDIX

A R CODE USED IN CHAPTER 2 ....... ... .. 87

B R CODE USED IN CHAPTER 3 ....... ... .. 95

C R CODE USED IN CHAPTER 4 ....... ... .. 107

D R CODE USED IN CHAPTER 5 ....... ... .. 113

E SAS CODE: TO OBTAIN CONFIDENCE INTERVAL FOR rl .. .. .. .. 128

F PRICE'S CONTROLLED RANDOM SEARCH PROCEDURE .. .. .. .. 131

REFERENCES ....._.__ . ... . 133

BIOGRAPHICAL SK(ETCH ....._._. .. .. 139










LIST OF TABLES


Table page

2-1 Design settings and response values (shear strength in psi) .. .. 27

2-2 Design settings for D1, D2, D:4, D4 and Dg . . 28

:3-1 The experimental design along with the response values .. .. . .. 45

:3-2 Design settings for D1, D2 and D:4 . ... .. .. 47

:3-3 Number of points in each design ........ ... .. 48

4-1 Initial design along with response values ...... .. . 64

4-2 Augmented design along with updated estimates .. ... .. 65

5-1 Design 1 original variables along with responses .... .. 79

5-2 Design 1 and 2 Coded Variables ........ ... .. 80










LIST OF FIGURES


Figure

2-1 Plots of the design points for D1, D2, D3, D4, and Ds

2-2 QDGs for D1 and D2 ......

2-3 QDGs for D1 and D3 -*****

2-4 QDGs for D1 and D4 ......

2-5 QDGs for D1 and Ds.. ..

2-6 QDGs for D3 and D4 *****

2-7 QDGs for D4 and Ds.. ..

3-1 Confidence region for (of 9 ....

3-2 Plots of the design points (in Block 1) for designs D1,


page

29

30

31

32

33



35

48

49

50

51

52

53

54

55

64

81


D2, and D3 ....


3-3 QDGs of the noncentrality parameter (ncp) for D1 and D2


QDGs of the power function for D1 and D2

QDGs of the noncentrality parameter (ncp)

QDGs of the power function for D1 and D3

QDGs of the noncentrality parameter (ncp)

QDGs of the power function for D2 and D3

Initial Design .....

Plot of the design points ......

QDGs of the power function for D1 and D2

QDGs of the power function for D1 and D2

QDGs of the power function for D1 and D2


.o .1 .


for Dr and


(based on Wilks' Likelihood Ratio).

(based on Hotelling-Lawley's Trace)

(based on Pillai's Trace).










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RESPONSE SITRFACE DESIGNS FOR LINEAR MIXED MODELS

By

Sourish C. Saha

December 2007

Cl.! ny~: Dr. Andrid I. K~huri
Major: Statistics

Response surface methodology (R SM) is a set of tools that includes setting up a series

of experiments that produce reliable measurements of the response of interest, fittingf and

evaluating a given model, and determining the settings of the factors that yield optiniun

value of the predicted response. In the area of R SM, one of the main considerations is

the choice of an experimental design. Extensive studies have been undertaken in the

design area with regard to response surface models, the same is not true with regard to

such models in the presence of a random block effect in the fitted model. Designs for

the latter type depend on certain unknown parameters concerning the model's variance

components. Hence, the construction of such designs requires some prior knowledge of

the unknown parameters. The design dependence problem for mixed response surface

models is addressed by applying quantile dispersion graphs (QDGs), which is a powerful

graphical tool for comparing designs for such models. The generation of D-optinial designs

sequentially for mixed response surface models is also discussed. The use of QDGs to

compare designs for correlated response surface models with an unknown dispersion

matrix is presented followed by suninary and possible areas for further research.









CHAPTER 1
INTRODUCTION

The tools required for the adequate selection of a design and the subsequent fittingf

and evaluation of the hypothesized model, using the data generated by the design, have

been developed in an area of experimental design known as response surface methodology

(RSM). The formal development of this area started with the work of Box and Wilson

(1951). Many books and papers discussing RSM have been published since the appearance

of that seminal paper. The articles by Hill and Hunter (1966), Myers et al. (1989)

and Mead and Pike (1975) provide a broad review of RSM. The books by Box and

Draper (2007), Myers and Montgomery (1995), and K~huri and Cornell (1996) give a

comprehensive coverage of the various techniques used in RSM. One of the objectives of

response surface methodology is the selection of a response surface design according to a

certain optimality criterion. The design selection involves the specification of the settings

of a group of input or control variables that can be used as experimental runs in a given

experiment. The choice of a design pIIl we an important role in the precision of estimating

the unknown parameters of a given model.

Response surface models are widely used to describe an empirical relationship

between the response and a set of control variables. The experimental trials performed

in a response surface investigation are often conducted under homogeneous conditions.

However, there are situations where this assumption is invalid. Heterogeneity of

experimental conditions can be caused by extraneous sources of variation. In such

situations, the experimental trials should be performed in groups, or blocks, within each of

which homogeneous conditions can be maintained. Although extensive studies have been

undertaken in the design area with regard to response surface models, the same cannot

be said with regard to such models in the presence of a random block effect in the fitted

model. Designs for the latter type depend on certain unknown parameters concerning the

model's variance components. Thus the construction of such designs requires some prior










knowledge of the unknown parameters. As a result, the researcher is often presented with

a dilemma since the purpose of a design is to estimate the unknown parameters of the

model using the data generated by the design. In this dissertation, we shall address the

design dependence problem for a particular type of response surface models, namely, the

ones with a random block effect.

1.1 Literature Review

Several authors addressed the design problem for response surface models with a fixed

block effect (for example, Atkinson and Doney (1989) and Trinca and Gilmour (2000)).

Cook and Nu la t -1!, ;i!! (1989) proposed an exchange algorithm for the computation of

D-optimal designs in the presence of a fixed block effect. The choice of design when the

block effect is random was discussed by C'I. 1.; (1995), Atkins and C'I. 1.; (1999), and

Goos and Vandebrock (2001a) who considered D-optimal response surface designs. The

observations obtained within a block were assumed to be correlated, whereas observations

in different blocks were considered uncorrelated.

The most common algorithms for constructing exact optimal designs are exchange-type

algorithms. They start with an initial N~-point design, after which the points are added

or deleted either sequentially or simultaneously in order to improve the value of a given

design criterion. Some of the well-known examples of exchange-type algorithms are those

of Federov (1972), the DET1\AX algorithm (1l~lchell (1974)), the modified Fedorov

algorithm (Cook and Nu lat I~-1,l!, (1989)), the K(L-exchange (where K( and L denote the

Akti and Irix points respectively) and the BLK(L (where BL stands for block, and K( and

L denote the Akti and Irix points in a block respectively) algorithms (Atkinson and Doney

(1989), Atkinson and Doney (1992)), and the coordinate exchange algorithm ( \1. i-n r and

T u lii -1!, ;i!! (1995)). The main advantage of these algorithms is that different design

problems are reduced to a common optimization structure. Whether or not the design

problems involve quantitative or qualitative factors or mixture variables, the optimal

design is found by selecting design points from a list of candidate points. The algorithms









can therefore handle any form of design region. An excellent treatment of the construction

of D-optimal designs is given in Atkinson and Doney (1992).

Goos and Vandebroek (2001a) presented a point-exchange algorithm to construct

D-optimal designs under random block effects. Their algorithm is based on a direct

search method, so it only provides locally optimal solutions. In the generic algorithm

described in their paper, more than one starting design was used in order to increase

the chances of attaining a globally optimal solution. The initial design was improved by

sequentially adding candidate points with the largest prediction variance. Optimal designs,

in general, depend on the extent to which observations within a block are correlated.

They showed that optimal designs do not depend upon correlations in any of the three

situations: (i) block size greater than the number of model parameters, (ii) minimum

support design (or saturated design where the number of distinct design points is equal

to the number of model parameters), (iii) orthogonally-blocked first-order designs. They

also showed that when the block size is homogeneous, orthogonally-blocked designs will

be better than designs that are not blocked orthogonally with respect to any generalized

optimality criterion. However, the authors pointed out that orthogonal blocking may not

yield optimal designs when the block sizes are heterogeneous. When the block effects are

assumed to be fixed, Goos and Vandebroek (2001a) showed that orthogonal blocking is

indeed an optimal strategy for homogeneous as well as heterogeneous block sizes. Ch.! as~

(1995) and Atkins and ('I!. ng~ (1999) resorted to the approximate theory of optimal

designs to deal with the special feature of within-block correlation.

1.2 Dissertation Objectives

The objectives of this dissertation are:

1. To demonstrate techniques for comparison of designs for response surface models

with a random block effect based on two criteria, namely, the prediction variance and the

power of a statistical test.










2. T> develop methodology for generation of optimal designs for the aforementioned

models based on a sequential algorithm.

3. T> propose methodology for comparison of designs for correlated response surface

models with an unknown dispersion matrix.









CHAPTER 2
COMPARISON OF DESIGNS ON THE BASIS OF THE PREDICTION VARIANCE
CRITERION USING QUANTILE DISPERSION GRAPHS

2.1 Introduction

K~huri (1992) discussed the analysis of a response surface model in the presence of

a random block effect, but did not consider the design aspect. Giovannitti-Jensen and

Myers (1989) introduced the so-called variance dispersion graphs (VDGs) to assess the

overall prediction capability of a response surface design inside a region of interest, R. The

VDGs are two-dimensional plots of the maximum and minimum values of the prediction

variance on the surfaces of several concentric spheres inside R. In an effort to provide

more information concerning the prediction variance on such spheres, K~huri et al. (1996)

proposed the use of the quantile plots of the prediction variance on the surfaces of the

spheres. Such plots describe the entire distribution of the prediction variance and thereby

give more complete assessment of the effect of the design on the prediction variance.

For response surface models with a random block effect, the prediction variance

depends, unfortunately, on an unknown parameter, namely, the ratio of two unknown

variance components, as will be seen later. Consequently, the assessment of the design

effect on the prediction variance depends on the value of this unknown ratio. This

dependence problem is a common feature of designs for variance component estimation,

and for designs for nonlinear models, including generalized linear models. There are

several approaches to deal with this dependence problem in general. These approaches

include the use of

(i) locally optimal designs by specifying initial values of the unknown parameters.

(ii) the sequential method which allows the experimenter to obtain updated estimates of

the unknown parameters in successive stages following the use of the initial values in the

first stage.

(iii) the B li- I o methodology which requires the specification of some prior distributions

on the unknown parameters.









(iv) quantile dispersion graphs (QDGs), which describe the dispersion in the quantile

values of a particular design criterion function over a certain parameter space associated

with the unknown parameter. The use of this approach was first introduced in K~huri

(1997) for the comparison of designs for a random two-way model (see also Lee and K~huri

(1999, 2000)).
2.2 Model and Notation

K~huri (1992) considered the following model of order d (> 1) in k input variables

xl, x2., *** The experimental runs in the associated design are arranged in b blocks of

sizes nl, n2, ***, nb:


Y = P01, + XP + ZY + E~, (2-1)


where y is a vector of a (CL, us.) observations on the response, rSo and p

(P1, Pa2, **, Pp3)' are unknown parameters associated with the polynomial portion of the

model, y = (yl, y2, **, Yb) Where Ti denotes the effect of the Ith block (1 = 1, 2, ..., b), and E:

is a random error vector. Here, X and Z are known matrices of orders n x p and n x b and

ranks p and b, respectively. The rows of X consist of the values of f'(z,), a vector whose

elements are powers and crossproducts of powers of xl, x2,., *,k of degree d evaluated at

me, the vector of design settings at the Uth experimental run (u = 1, 2, .. ., n). The matrix
Z is of the form:


Z = ~ e-(1,, 1,, .., 1 b).(2-2)


Note that the columns of Z sum to 1,. The random vector y is assumed to be distributed

as N(0, o Ib) independently of e, which is assumed to follow the normal dlistribu~tion

NV(0, of2,).

Let us rewrite the model (2-1) as


y = W7 + Zy + E~, (2-3)









where 7 = (Po, p')' and W = [1, : X]. Hence,


E(y) = p, = p01, + XP = WT


Var~y = =e), + aZZ' = e)A,


(2-4)


(2-5)


where


A = diag(Al, A2, ..., Ab),


(2-6)


Al = Inc + r n,, (1 = 1, 2, ..., b), and rl = a?/a, .

If '7 = e /f is known, then the best i;nea~r urnbiasedl estimator (B3LUE) of r is the

generalized least squares estimator i given by


;i = (W'A- W)- W'A- y.


(2-7)


Its variance-covariance matrix is


Var(i) = (W'A- W)-la2.


(2-8)


Let y(2) = g'(z)i denote the predicted response at a point z in the experimental region

R, where g'(z) = [1, f'(z)]. The prediction variance, V, [ G;(2)], is of the form


V, [ G;(2)] = g'(z)(W'A-W- W~)- gx2a .


(2-9)


We define the scaled prediction variance as


V, [ G;(2)] = n[g'(2)(W'A- W)-1 g(x)].


(2-10)


It is to be noted that the division of the prediction variance by a,2 makes this quantity

scale free, and the multiplication by a puts the prediction variance in terms of the

weighted design moment matrix, W W This scaling convention is very common in










response surface methodology (see, for example, Myers and Montgomery (1995, p. 295)

and K~huri and Cornell (1996, p. 209)).

2.3 Quantile Dispersion Graphs

K~huri (1997) introduced a graphical technique to compare designs for analysis of

variance estimates of variance components. These graphs describe the dispersion in

the quantile values of a particular design criterion function over a certain parameter

space associated with the unknown parameter (or parameters). We shall use the scaled

prediction variance given in (2-10) as a design criterion for comparing designs for model

(2-1). These graphs offer a more comprehensive assessment of competing designs than

what is offered when comparing designs using a single-number optimality criteria. Unlike

locally optimal designs, this graphical technique takes into consideration the design

dependence problem.

Note that V, [ G;(2)] depends on z (through g(z)), D (the design under consideration)

through the matrix W, and rl. Let us therefore denote nV, I [G(2)]/of2 by AD~z ,), Where

n is the total number of experimental runs. If design D is used to fit the model, the

prediction capability of D can be assessed by considering the quantiles of the distribution

of aD z, rl) OVer COnCentriC SUTrfCeS Within the experimental region R obtained by

shrinking the boundary of R using a shrinkage factor. To address the problem of unknown

rl, we consider several values of rl that belong to a parameter space C. We choose C to

be a (1 a~)100; confidence interval on rl. Such a confidence interval can be constructed

using response data obtained under an initial design. A method by Harville and Fenech

(1985) is used to construct such a confidence interval, as will be shown in Section 2.4.

In order to study the performance of a design D throughout R, we consider several

concentric surfaces denoted by Rx located within R. These surfaces are obtained by

reducing the boundary of R using a shrinkage factor A. The prediction capability of D

can then be evaluated by considering the distribution of aD z, rl), aS determined by its

quantiles on R for a given rl in C. Small values of aD(z, r) are obviously desirable.









For a given design D and a given value of rl in C, let QD(p, rl, A) denote the pea quantile of

the distribution of the values of aD z, rl) On Rx foT a Specified A. The dependence of these

quantiles on rl is investigated by computing QD p, l, X) fOr SeV6Tra VaueS Of rl Selected from

C. We then calculate

Qg'"(p, A) = mmn QD~p rl X
rl6C
(2-11)
Q0a (p A) =max QD~o rl X -
rl6C

By plotting QB""(p, A) and Qg'"(p, X) against p for each D and A we obtain the so-called

quantile dispersion 7glell (QDGs). Such plots can be constructed for two given designs,

D1 and D2, and for each of several values of A. The plots provide a comprehensive

assessment of the prediction capabilities of D1 and D2 throughout the region R. Clearly,

D1 is preferred over D2 if the QDGs for D1 show smaller values of Qgaz(p, X) and
r~iQB "(p, A) than those for D2. Furthermore, the~ closeness o QB(p A)D to Qg"(p o

a given design indicates robustness, or lack of sensitivity, that is induced by the design of

the scaled prediction variance to changes in the parameter values over the parameter space

C.

2.4 Confidence Interval for rl

Harville and Fenech (1985) derived a confidence interval for a variance ratio for an

unbalanced mixed linear model. We now briefly outline their procedure for constructing

such an interval for rl. Let us consider the general model (2-3). Define r = rank(W, Z) -

rank(W), p* = rank(W), and f = n rank(W, Z). Fr-om (2-5) it follows that


Var(y) = of (I + rlZZ'). (2-12)









In order to derive the confidence interval for rl, let


S7 = y 'Pw y



SSE = y'y S7 SY

m = -tr(Ml)

Pw = W (W'W) -W'

M =Z'(I Pw) Z

q = Z'(I Pw)y,

where y is any solution to the equation Ml~ = q, which is obtained from the normal

equations that can be derived by treating y, like 7, as a vector of unknown parameters

and applying the method of ordinary least squares (see Harville and Fenech (1985, Eq.

2.3)). An interesting property of the matrix MI is that rank(Ml) = r (see Marsaglia and

Styan (1974)). Also note that Sy = R(y|7) = R(y, 7) R(7) and S7 = R(7), where

R(y|7) denotes the reduction in the sum of squares due to fitting y after 7, R(y, 7) is
the total regression sum of squares obtained from both the fixed and random effects in

the model, R(7) is the regression sum of squares due to fitting a model with just the fixed

effects, and SSE is the residual sum of squares for model (2-3). Applying Formula (79) of

Searle (1971, p. 445), we have:


E[Sy/r.] = e)+trMa



E[SSE/f = *,

It is well known from the theory of distribution of quadratic forms that SSE/, X 1

Suppose that we are interested in testing the null hypothesis that rl = 0. If this hypothesis

is true, then Sy/(rr, + mcT?) ~ X H~owe~ver if the null1 hypothesis is false, then the









disotribu~tion of Sq./(af ma2) is HOt X,, but is a linear combination of two or more

independently distributed chi-squared random variables. In what follows, we obtain a

pivotal quantity which can be used for constructing a confidence interval for rl.

Recall that r = rank(Ml). Let Al,..., ar denote the nonzero eigenvalues of MI and let

D = diag(A,,..., ar) with 0 < a, < ... < a,. Define N as a b x r matrix such that

N'N = I, and MIN = ND, that is, the columns of N are orthonormal eigfenvectors of

MI corresponding to the eigfenvalues Al, ..., A,. Treating y like 7 as a vector of unknown

parameters, Harville and Fenech (1985, p. 140) concluded the following: (i) a necessary

and sufficient condition for a linear function g'y to be estimable is the existence of a

vector h such that g' = h'Ml, and (ii) there exists an r x 1 (where r = rank(Ml)) vector

of linearly independent estimable functions of y. One such r x 1 vector is t = (tt,..., t,)'

where


t = D1/2 ply = D-1/2 pll~y. (2-13)


Since 9 is a solution to the linear equation My~ = q, t = (tt, ..., tr)' can be estimated by

t =(t1, ..., t~r y, where


t = D1/2 19 = D-1/2 /Mly = D-1/2 /q, (2-14)


which will be used to obtain a pivotal quantity for constructing a confidence interval for rl.

Notingthat P W = WPw = P ,=P2, it can be shown that the distribution of q is

a multivariate normal with

E(q) = 0

Var (q) = a, Z' (I Pw) (I +pZ Z') (I Pw) Z









where,~ ~ 2/ ifw eclafI ZZ') is equal to Var(y). As a result, the distribution of t is a
multivariate normal with

E (t)= 0

Var(t) = a,2D-1/2N'1~ pl r12)ND-1/2



Let us now define G(rl; y) as


(1/ f)SSE(2-15)



It can be shown that t (I + rD)-1?/,2 = L Var(t)li andu hlenlcet Gc;y Fr )

Note that although the matrix N is not unique, Var ('t) = a, (I + rlD) is invariant to the

choice of N. As the distribution of G(rl; y) does not depend on any unknown parameters

except rl, it can be used as a pivotal quantity for constructing a confidence interval for rl.
Burdick et al. (1986) used the so-called bisection method to compute the aforementioned

exact interval. The exact two-sided (1- a)101 '. confidence interval on rl is given by [1*, u*],

where 1*, u* are, respectively, the roots of the following equations:


(2-16)
G(rl; y) = F/2rf

where Prob[F,,, > Fa/2;r,f] = /a/. Noute thlat G~(r; y) can be rewritten as


G~q; ) = () SSE(2-17)

The function G(rl; y) is convex and monotonically nonincreasing with respect to rl for

0 < rl < 00 and G(0; y) = fCE= t"( /(rSE) If we replac (AF-- +/- r)-l with (A-1 + )-l

and (A, I + rl)-l we: get ~the funlctions Gil(r; y) and G',(r; y), respectively, such that

Gl(rl; y) < G(rl; y) < G,(rl; y). This is true because 0 < Al < ... < a,. The bisection









method makes use of the bounds li, le, ul, u, (see Eqs. (2-19)), which are obtained by

equating Gl(rl; y) and G,(rl; y) with F1-a/2;r,f and Fa/2;r,f, TOSpectively and solving for rl.

Thus by replacing rl with li, le, ul, u, we get the following equations:





(2-18)
G1(U1; y)= Fa/2;r,f

G,(u,; y) = F/2rf

The above equations yield the following bounds

11 =f, 1/A

rSSEF1-a/2;r,f


ini = -p 1/a, (2-19)


U, =-1/.
rSSE a/2;,f r



For example, G1(li; y) = F1-a/2;r,f implies that (see Eq. (2-17))


( ) SSEi\U F1-a/2;r, f




rSSEF1-a/2;r,f

Since Gi(rl; y) < G(rl; y) < G,(rl; y), li < l* < 1, and ui < u* < n,, where 1*, u* are

the roots of Eqs. (2-16). Instead of solving the two non-linear equations, the bisection

method uses these lower and upper bounds to compute l* and u*. A demonstration of

these calculations is given in Burdick and Graybill (1992, Appendix B).










2.5 Example

A numerical example describing the analysis of a response surface model with a

random block effect was given in K~huri (1992). The example investigated the effects of

two factors, temperature (:ri) and curing time (2-2), on the shear strength of the bonding

of galvanized steel hars with a certain adhesive. The galvanized steel hars were selected

over a period of 12 d or; Each batch of steel hars was selected at random on a given date.

Hence, the 12 dates were treated as random blocks. Three levels were chosen for each of

the two factors according to a :32 factorial design. The same design was used on each of

these 12 dates, except that on certain d or; replications were taken at the design center in

order to test for lack of fit of the fitted model. We refer to this :32 factorial design as D1.

A complete second-degree model was fitted to the data. The actual data set, consisting of

n = 118 observations in 12 blocks is reproduced in Table 2-1. Note that :ri and 2-2 denote

the coded values of temperature and time, respectively, namely,

temperature 400



time :35



The settings of :ri and IT2 from the :32 design are -1, 0, 2 for :ri, and -1, 0, 1 for IT2*

We compare D1 with four other second-order designs. The designs were chosen so

that a second-degree model can he fitted. They include the designs D2, D:3, D4, and

Dg. The first one is similar to D1 with the only difference being that the coded settings

of :ri are equally spaced having the values -1, 0.5, 2. Designs D:3 and D4 are modified

versions of the so-called uniform shell designs (see Doehlert (1970) and Doehlert and K~lee

(1972)). The original uniform shell designs consist of points that are uniformly spaced on

concentric spherical shells. In two-dimensional spaces, these designs are obtained from the

points of a double simplex with a center point. The proposed modified design points for

D:3 and D4 Tesemble the points of a two-dimensional uniform shell design, but they are not










located on the same sphere. Therefore, we refer to D3 and D4 aS Semi-uniform shell 1 and

semi-uniform shell 2 designs, respectively. The fourth design is Ds, which is obtained by

randomly generating points in the design space of D1. The design settings for D1 Ds are

presented in Table 2-2 and the corresponding design points are shown in Figure 2-1.

The experimental region, R, is rectangular with -1 < xl < 2, -1 < x2 < 1. For each

design, we consider the distribution of aD(z, rl) on each of several concentric rectangles,

R which are obtained by a reduction of the boundary of R using a shrinkage factor A,

0.5 < A < 1. Thus Rx is determined by the inequalities


ai + (1 A) (bi ai) < xi < bi (1 A) (bi ai), i = 1, 2,


where ai and bi are the bounds on xi in R (i = 1, 2). In order to investigate the

dependence of aD(z, rl) on rl, a parameter space for rl is established. For this purpose,

we use the method described in Section 2.4 to construct a confidence interval for rl. Using

the data set in Table 2-1, we obtain the 95' confidence interval, C, namely (1*, u*)=

(0.0763243, 0.9667678). For each design and a selected value of rl in C, quantiles of the

distribution of aD(z, rl) are obtained for z E R where A is one of several values of

A chosen from the interval (0.5, 1]. The number of points chosen on each Rx was 2000

consisting of 500 on each side. The quantiles are calculated for p = 0(0.05)1. The

procedure is repeated for other values of rl in C. Then QB""(p, A) and Qgm"(p, A) are

calculated using the formulas in (2-11). The R software was used in conducting the

numerical investigation and obtaining the actual plots.

The QDGs for the comparison of design D1 with D2, D3, D4, and Ds are given in

Figures 2-2, 2-3, 2-4 and 2-5, respectively. The QDGs for the comparison of D3 With D4,

and D4 With Ds are presented in Figures 2-6 and 2-7, respectively. From Figure 2-2, it can

be seen that D2 perfOrms slightly better than D1 for A = 0.6, 0.7 and 0.8. The difference

in prediction capability is distinctly evident in favor of D2 for A = 0.9 and 1. Figures 2-3

and 2-4 -11---- -1 that the semi uniform shell designs, D3 and D4, perfOrm better than D1










for A = 0.6, 0.7 and 0.8. However, for A = 0.9 and 1 and high values of p, D1 has better

prediction capability than D:4 and D4. The QDGs for the comparison of design D1 with

Dg (see Figure 2-5) clearly depict better prediction capability with design D1. Figures

2-6 and 2-7 indicate that design D4 is better than D:4 and Dg. It should be noted that

changing the confidence coefficient for the confidence interval C front 0.95 to 0.90 or 0.99

did not change much the pattern of the QDGs. This indicates that the QDGs are not too

sensitive to changes in the bounds of the confidence interval on rl.

2.6 A Brief Review of Split Plot Designs

Split-plot designs, developed by Sir Ronald A. Fisher, are widely applied in many

fields including industry and agriculture due to many reasons including cost and time

savings. Often, industrial experiments involve one or more hard-to-change variables

which are not reset for every run of the experiment. The resulting experimental designs

are of the split-plot type and fall in the category of nmulti-stratunt designs (see, for

example, Trinca and Gilmour (2001)). 1\athew and Sinha (1992) derived optiniun tests

in unbalanced split-plot designs under mixed and random models. Robinson et al. (2004)

considered the analysis of split-plot experiments with non-nornial responses. Complete

randonlization for many industrial and agricultural experiments is not practical due to

time or cost constraints, or existence of some hard-to change factors. In situations like

these, restrictions on randonlization lead to split-plot designs, allowing certain factor levels

to be randomly applied to the whole plot units, while the remaining factor levels randomly

applied to the subplot units. The concept of split-pllni r ir hi-randonlization, or two-stage

randonlization is heavily used in industrial experimentation. Huang et al. (1998) and

Bingham and Sitter (1999, 2001) have derived nxininiun aberration two-level fractional

factorial split-plot designs that are used for screening experiments. Goos and Vandebrock

(2001b) considered an exchange algorithm for constructing D-optinmal split-plot designs. A

proper classical statistical analysis requires the use of generalized least squares estimation

and inference procedures and, hence, the estimation of the variance components in









the statistical model under investigation. Goos and Vandebrock (200:3) demonstrated

the use of an efficient algorithm to compute D-optimal split-plot designs assuming a

fixed number of whole plots and pre-specified whole plot sizes. The two-level factorial

and fractional factorial designs were shown to be D-optimal for estimating first-order

response surface models for specific numbers and sizes of whole plots. Goos et al. (2006)

discussed the practical inference from industrial split-plot designs. The analysis of the

data involved mixed model using the generalized least squares (GLS) estimation. The

authors compared the analysis with the one involving ordinary least squares (OLS) which

treats the experiment to be completely randomized. It was recommended that split-plot

experiments involving extremely low numbers of whole plots should be avoided. When the

observations are correlated only to a small extent, it was found that there does not exist

much difference between the analysis performed using OLS and the one carried out by

GLS.

The common assumption in a split-plot experiment is that the same error variance

exits in all subplot treatments. Curnow (1957) dealt with tests of significance for the

departure from equality of variances for different subplot treatments, and also, extended

Pitman (19:39) method to construct confidence limits for the variance ratio. The design

and analysis of split-plot designs for mixture experiments were considered by K~owalski

et al. (2002) (see also K~owalski (1999) and Cornell (1988)). A classic example of a split

plot design is given in Montgomery (2005, Section 14-4). He considered an experiment

done by a paper manufacturer who is interested in studying the effects of three different

pulp preparation methods and four different cooking temperatures on the tensile strength

of the paper. However, the linear model for the split plot design given in Montgomery

(2005, Eq. (14-15)) is different from the linear mixed model used by Goos and Vandebrock

(200:3) and Liang et al. (2006). The latter authors considered the following model


y = W7 + Zy + 2, (2-20)









where y is a vector of n observations, W is the design matrix, 7~ is the vector of fixed

effects, Z is a diagonal matrix given by diag(14?,..., 1,4) (14i is a vector of one's and n~

is the number of subplots within the ith whole plot), y is the vector of random effects

corresponding to the a whole plots. The random vector ;t is assumed to be distributed

a~s N(0, cs le) independently of 2, which? is assumecd to follow th~e normal distribution?

NV(0, aS I) where n = CEf n. Here, a2 and aS denote the variability among whole

plots and subplot units, respectively. Liang et al. (2006) used three-dimensional variance

dispersion graphs to compare competing split-plot designs based on the prediction variance

criterion. It can be shown that the prediction variance depends on the unknown variance

components. In order to apply the technique of quantile dispersion graphs approach, the

parameter space for the unknown variance components needs to be constructed. The

methodology discussed in this chapter can be adapted to compare split plot designs.









Table 2-1. Design settings and response values (shear strength in psi)


Block (Batch)
2 3 4
(July 16) (July 20) (Aug. 7)
1075 1172 1213
1790 1804 1961
1843 2061 2184
1121 1506 1606
2175 2279 2450, 2355
2420, 2240
2274 2168 2298
1691 1707 1882
2513 2392 2531
2588 2617 2609


Tenip .
.I'1
-1
0
2
-1
0

2
-1
0
2



Tenip .
.I'1
-1
0
2
-1
0

2
-1
0
2


Time
x 2
-1
-1
-1





1
1
1


Time

* 2
-1
-1


-1


1
1


1
(July 11)
1226
1898
2142
1472
2010, 1882
1915, 2106
2352
1491
2078
2531


7
(Aug. 20)
1281
1833
2116
1502
2471

2430
1645
2392
2517


5
(Aug. 8)
1282
1940
2095
1572
2291

2147
1741
2366 i
2431


11
(Oct. 3)
1305
2011
2192
1584
2052, 2032
2190
2201
1744
2392
2588


6
(Aug. 14)
1142
1699
1935
1608
2374

2413
18463
2392
2408


12
(Oct. 10)
1207
1742
1995
1486
2339

2216
1751
2390
2572


Block (Batch)
8 9
(Aug. 22) (Sep. 11)
1305 1091
1774 1588
2133 1913
1580 1343
2393 2205, 2268
2103
2440 2093
16388 1582
2413 2392
2604 2477


10
(Sep. 24)
1281
1992
2213
1691
2142

2208
1692
2488
2601


Note: The original design settings of .ri corresponding to -1, 0, 2 are 3750F, 4000F,

and 4500F, respectively; those for .z- corresponding to -1, 0, 1 are 30, 35, and 40 sec.,

respectively. (Source: K~huri (1992, Table 1))










Table 2-2. Design


settings for D1, D2,

Dy D
X1 1 2 1 2 X
-1 -1 -1 -1
0 -1 0.5 -1
2 -1 2 -1
-1 0 -1 0
0 0 0.5 0
2 0 2 0
-1 1 -1 1
0 1 0.5 1
2 1 2 1


D3, D4 and Do

Dj oDf
X1 2 1 2 X
-1 0 -1 -1
1 -1 0.5 1
1 1 2 -1
-0.5 1 0.5 -1
-0.5 -1 -1 1
2 02 1
0 0 0.5 0
0 0 0.5 0
(1 (1 (15


1
0.54
0.:3:3
0.09
-0.91
0.96
-0.38
-0.80
-0.10
(1 59


0.14
0.99
-0.77
0.5:3
-0.22
-0.30
-0.38
0.52
(1 (9


a: D1 is the original design shown in Table 1.

h: D2 is the same as D1, but the settings of :ri are -1, 0.5, 2.

c: D:3 is the semi-uniform shell 1 design.

d: D4 is the semi-uniform shell 2 design.

e. Dg is the randomly generated design.














| (0. 5,1) 1


I

(2,-11


I_ _


I ---C


C (0.5,of ~


(o, 1)


(- 1,1)



1


(-1.-1)


(2, 1)



(20)


(-1,0)



(-1,-1)


(2,0)


I(0,0)



(0o,-1)


0 5,-1)


(2,-1)


D1


(- ,1)


(1,1)



o ,0) | (,)


-~----- (2, 1)


(- 1, 1)


X1


__


*- .
*


Y


'r


* ----------~~-:-


--*--------


-
e----
(-1.-1)


D3


(0.54, O.14


(-0.91, os.'31"


-0.TT:1


D5


Figure 2-1. Plots of the design points for D1, D2, D3, D4, and Ds





















1= 0.16


1= 0.7


1 = 0;.8


M
s- D1


0.0 01.4 0.8


1 11 1


~ 1~LLLTI
)I


IrC




O




IrC


0.0 0.4l 0.8






h= 0.9




















0 0 0.4 0.8


0.0 0.4 0.8


9,= 1


0.0 0.4 0.8


Figure 2-2. QDGs for D1 and D2


















31= 0.6


h= 0.7


-r- Og








jj-L-4L-rWCrl;rzr~-~


0.0 06.4 0.8


0O.0 0.4 0.8


Illlll
0.0 0.4a 0.8


31= 1


0.0 0.4 0.8


0.0 0B.4 0.8


Figure 2-3. QDGs for D1 and De



















;;I= 0.6


;I= 0.7


8
c
co

=i
LT:
a
Y
II
k
a
D
9s
i
o
4
~St~
E


A= 0 g


c
C







E:


0.0 0~i 0.8


1 DA 0A$


1].O 01 ca8


A=1


9

a

d
B


B
4


rr


8








os


0A 04


0. A


P



Figulrec 2-4. QD)Cs for Di) anld D4


















h= 0.6


















0.0I 0.4 0.8


;a= 0.7


















0O.0 0._4 0.8


h= 0.8


















0.0 0P.4 0.8


1= 1


e:


5 si
6
4

i ~s
B M
B
~ X;
"rs;
O o
b
6
a


6



i~s

5;

t~s




~I


0.0 0. 0.8

9


0O.0 0.4

P


Figure 2-5. QDGs for D1 and Ds


















h= 0.7


111111 c







0.01 0.4 0.8


0.0 01.4 0.8


0.0 0.;4 0.8


h=1


0.0 0.4 0.8


0.0 01.4 0.8


Figure 2-6. QDGs for D3 and D4



















h= 0.6



















0.0I 0.4 0.8


;a= 0.7


0O.0 0._4 0.8


--D4















0.0 0P.4 0.8


1= 1


e:


5 si
6
4


B
B
P
%' rJ

O o

6
a


6
e
$$
5


H


8
u3 Pi~
"B
PP
r
6
~


0.0 0. 0.8

9


0O.0 0.4

P


Figure 2-7. QDGs for D4 and Ds









CHAPTER :3
COMPARISON OF DESIGNS ON THE BASIS OF THE POWER CRITERION USING
QUANTILE DISPERSION GR APHS

3.1 Introduction

The performance of a given test of hypothesis is usually measured by its power value

under some alternative hypothesis. The purpose of this chapter is to compare designs

on the basis of the power of an F-test concerning the parameters of a response surface

model with a random block effect. Unlike the prediction variance criterion, the power

criterion is not as commonly used for comparing designs. Box and Draper (1975) listed 14

properties that a design should have (see also, K~huri and Cornell (1996, pp. 5:3,27:3)). One

of these properties states that a design should be able to detect model inadequacy. By a

proper choice of design, it is possible to maximize the power of the lack of fit test to detect

departure from the fitted model. For example, multiresponse designs based on the power

criterion are given in Wijesinha and K~huri (1987a). These authors introduced two design

criteria to improve the power of the multivariate lack of fit test for a linear multiresponse

model (see also the references cited therein). The development of this test was discussed

in K~huri (1985). In the case of a single response, Atkinson (1972), Atkinson and Fedorov

(1975), and Jones and 1\itchell (1978) derived design criteria to enhance the detection of

model inadequacy.

As will be seen later in the case of a response surface model with a random block

effect, the power function depends on the unknown variance components of the model.

The construction of designs for such a model would therefore require some prior knowledge

of these unknown parameters. The power of a generalized lectst .squares (GLS) F-test

is used to assess the efficiency associated with a given design. Rao and Wang (1995)

studied several properties of the power of this test for regression models with a nested

error structure. Since the power, in the case of a single response model, is a monotone

increasing function of the noncentrality parameter, the latter quantity is also used as a

design criterion for comparing designs. For a given design, quantiles of these two criteria









functions are obtained within the so-called alternative space to be defined later. These

quantiles depend on two unknown variance components. The dispersion of the quantiles

over a confidence region of the unknown variance components is obtained resulting in the

so-called quantile dispersion graphs (QDGs) of the criterion function under consideration.

3.2 Development of Design Criteria

Consider the same response surface model described in C'!s Ilter 2. In order to develop

the design criterion, let us first rewrite the model as


y = W7 + E* (3-1)


where E* = Zy + E~. Suppose that it is of interest to test the hypothesis, Ho : L7 d = 0

against the alternative hypothesis, H, : L7 d = 6, where L is a known [q x (p + 1)]

matrix of rank q(< p+1), d is a known q x 1 vector and 6 is an alternative value of L d

other than 0. If 17 = ae?/af is known, then the corresponding test statistic is of the form


Fors(rl) = (L+i d)'[L(W'A- W)- L']- 1(L+ d)/qM~SE (3-2)


where M~SE denotes the error mean square for model (3-1), and is given by


M~SE = y'[A-l A- W(W'A- W)- W'A- ]y/(n p 1). (3-3)


In this case, the test statistic Fors(rl) has an exact F distribution under Ho with q

and n p 1 degrees of freedom. Under the alternative hypothesis, Fors(rl) is distributed

as a noncentral F with q and n p 1 degrees of freedom and a noncentrality parameter,

A, given by (see for example, Searle (1971, Section 3.6))


A = G'[L(W'A- W)-1L']- 6/a,2 (3-4)


The power of Fans(rl) is a monotonically increasing function of the noncentrality

parameter, A. Let us denote this power by ~, which is given by


S= P[Fors(rl) > Fa;q,n-p-1|A / 0] (3-5)









Note that knowledge of both a,2 and rl (or o and ,a) is,,, neede fothe computation of the

power for specified alternative values of 6. Rao and Wang (1995) considered the power of

a similar test for a regression model with a nested error structure.

3.3 Design Comparisons using QDGs

In this section, the QDG approach is described first followed by the construction of a

nonsrvantiven confidennc region for (f,a 9) and an alternative space for 6. The need for the

confidence region and the alternative space will be explained later.

3.3.1 Quantile Dispersion Graphs

In C'!s Ilter 2, we used QDGs to compare designs based on the prediction variance

criterion. In this section, the noncentrality parameter and the power function of the

F-test statistic in (3-2) are used as design criteria for comparing designs for a response

surface model with a random block effect. The noncentrality parameter and the power are

dependent on two parameters, namely (a, rl), which are unknown. To address the problem

of unknown (a, rl), we consider several of their values inside a certain parameter space
for (af 9 depnoted byi S In addition to the parameter space S, we also need to specify

an alternative space for 6, which is a subset of RV. Recall that 6 is the value specified in

the alternative hypothesis H, in Section 3.2. The alternative space is needed because any

consideration of the power of the F-test in (3-2) requires the specification of values of 6.

In order to study and compare the performances of two designs, D1 and D2, throughout

the alternative space, we consider surfaces of several concentric hyperspheres of radius

p which are located within the alternative space. The surface of a sphere of radius p is

denoted by T,,. The noncentrality parameter in (3-4) depends on 6, D (the design under
cno~rnsdration through the matrix W),TI and' (f,2 r). Let us therefore denote the value of

the noncentrality parameter by AD(6, a, l). If design D is used to fit the model, the

capability of D to produce high values of AD 6, Uf rl) can be assessed by considering the

distribution of AD(6, a, rl), determined in terms of its quantiles on the surface T,,, for

different values of p.









Hence, for each design D ndl a given value of (a,2 r) E S, we can calculate the pea

quantile of the distribution of AD 6, Uf rl) on the surface, T,, for p = 0(.05)1. The next

step is to vary (a, rl) over S and calculate

Qg'"n(p, p) = mmn QD~p a," rl p)
(3-6)
/Imaz(p, p) = a DP 2 9


wherel QD\p, e rl, p) denotes the pea quantile of AD 6, a rl) foT 6 T,.

Large values of AD 6, Uf rl) are desirable. By plotting QB""(p, p) and Qg'"(p, p)

against p we obtain the quantile dispersion graphs (QDGs) for the noncentrality

parameter. Such plots can be constructed for two given designs, D1 and D2, and for

each of several values of p. Design D1 is preferred over design D2, if the QDGs for D1
depict higher values of Q B"a(p, p) and QB ""(p, p) than those for D2. Furthermore, the

closeness of QB""(p, p) to Qg'"(p, p) for a given design D indicates robustness, or lack of

sensitivity, of the noncentrality parameter to the parameter values in the set S.

As was mentioned in Section 3.2, the power of Fans(rl) is a monotonically increasing

function of the noncentrality parameter A. Hence, as a second criterion for comparing

designs, we can consider the actual power value itself. For a given design D, high power

values are desirable for several values of p. The power function depends on A, which

depends on a,2 and rl. As a result, the use of the power function as a design criterion also

suffers from the design dependence problem. The QDG approach can then be used to

circumvent this dependence problem by replacing in (3-6) quantiles of A by those of the

power function in (3-5).

It should be emphasized again that the parameter space S is needed in order to assess

the dependence of the chosen criterion function (the noncentrality parameter or the power

function of the F-test in (3-2)) on the values of the unknown variance components, a,2 and

mi(or of2 and rl = "i). Such dependence affects the choice of design, which results in the
aforementioned design dependence problem. There are different approaches to setting up









the parameter space S. One approach is to set a,2 = 1 and then select a particular range
of valueso for m basedl on anyr prior krnowledge that mayr be avalable mrregrdingr e2 Another

approach is to construct a confidence region on (a, l). This region requires that values

of the response y be obtained at particular settings of the control variables. The second

approach will be adopted in this chapter.
3.3.2 A Cornsenrvative Confidennc Region for (a,2 r)

A (1 la)10' confidence interval on 17 = e2/(T,2 namely, l* < 17 < u*, can be
constructed as explained in Section 2.4. This interval is derived by using the random

variable G'(11;y) as a pivotal quantity. Wle also have X1-,/2;f < S E &41/; hr

f = n rank(W, Z), and W, Z are defined in Section 2.2. Thus, a < a,2 < b with

probability 1 -a, where a = SS(E 1 /2;S f nd b = S(SIE -0/2; f
Hence, by Bonferroni's inequality,


P[a < a,2 < b, l* < rl < u*] > 1 2a~, (3-7)


that is,

P [(,), r) E S] '> 1 2a~ (3-8)

where S = [a, b] x [1*, u*] is the shaded region in Figure 3-1. Note that x denotes Cartesian

product. In this way, an exact, but conservative, confidence region for (a, rl) can be
obtained.

3.3.3 An Alternative Space for 6

In order to construct an alternative space for 6 = (61, ..., 6,)', which is a subset of RV,

one can select 6 points that fall on the surface of a hypersphere of radius p contained in

RV. Let T, denote the surface of such a hypersphere. The points on T, depend on only

q 1 independent variables. Using, for example, the spherical coordinates associated with










61, ..., b,, we obtain the equations (see K~huri et al. (1996))


61 = p COS ~1

62 SR p i 1 COS 2~



6,_1 = p sin I1 sin 2~ sin q-3 i q-2~ COS q-_1

6, = p sin I1 sin 2~ sin q-3 si q-2~ sin q-1


where fi, ..., 4,_1 are angles chosen independently such that 0 < 44 < xr for i=

1, 2,..., q 2, and 0 < ~,_1 < 2xr. To generate points on T,, fi,..., ~,_1 are chosen

randomly from independent uniform distributions, namely, U(0, xr) for #1, ..., #q-2 and

U(0, 2xr) for ~,_1.

By varying the value of p, we can obtain several values of 6 within the alternative

space.

3.4 A Numerical Example

To illustrate the proposed methodology described in the preceding sections, we

consider the results of an experiment presented in K~huri and Cornell (1996, p. 328)

concerning the effects of storage conditions on the quality of a certain variety of apples

(see Table 3-1). The apples were harvested from four different orchards which were

selected at random. Thus, 'orchard' can be considered as a random block effect. The

control variables were X1 = number of weeks in storage after harvest and X2 = StOTrge

temperature (oC) after harvest. The response variable of interest was y = amount of

extractable juice (ml/100g). A 4 x 4 factorial design was used in orchards 1, 2 and 3, and

a 3 x 4 factorial design was used in orchard 4. The coded values of the control variables

were obtained as follows: xi = (Xi Xi)/si, where Xi and as are the mean and standard

deviation (with divisor n) of the settings of Xi(i = 1, 2), respectively. For the given data










set in Table 3-1, we use the coded values


XI 2.4
1.083
X2 11.0
9.772

The corresponding design, which we denote by D1, is shown in Table 3-2. This

table includes the settings of X1 and X2 in all the four blocks. We compare D1 with two

other designs, D2 and D3, chosen such that a second-degree model can be fitted in each

block. Design D2 is a Semicentral composite design (SCCD) consisting of a complete

22 faCtOrial design having two settings coded as -1 and +1 for each factor, 7 center

points (no = 7), and 4 axial points. The center points were chosen so that the total

number of design points in D1 and D2 1S Same (see Table 3-3). This design is labeled

as semicentral because the axial points are not symmetric with respect to the origin, a

requirement for a central composite design (see, for example, K~huri and Cornell (1996,

Section 4.5.3)). The same design was used in all four orchards. Design D3 COntainS

arbitrarily generated design points in the experimental region for design D1, that is, R =

{(xi, x2) : -1.29 I xi I 1.48, -1.13 I x2 < 1.43}. The same points were used in all four

orchards. Table 3-2 shows the coded design settings of D1, D2 and D3. The number of

points considered in each design is given in Table 3-3 and the corresponding design points

(in Block 1) are shown in Figure 3-2.

Recall that the hypothesis of interest is in general Ho : L7 d = 0 versus H,

L7 d = 6, where L is a known [q x (p + 1)] matrix of rank q(< p + 1) and d is a known

q x 1 vector. In all the design comparisons, the fitted models are of the second degree:


E[y(z)] = po + ixl +2 2x 11 22ix 12 1~ 2*ixl~









We now consider the following hypothesis: Ho : P11 = P22 = 0, that is,






Ho : 01 0 0 1 0 0 42 0 (3-9)
00 0 0 1 0 pll





Here, L is a 2 x 6 matrix of rank 2. The purpose of this hypothesis is to determine

whether or not a complete second-degree model is needed to represent the mean response.

In this case, q = 2. Therefore, Tp is a circle of radius p centered at (0,0). One thousand

6's are generated on this circle using the method described in Section 3.3.3. We have

selected 5 different values of p that are equally spaced between 0.01 and 1. They are 0.01,

0.2575, 0.505, 0.7525 and 1. For higher values of p, the power increased to values close

to 1 for all the designs under consideration and the quantile plots of the designs were not
disotinguishhale. For each design andl a fixdvarlu om,,f (f,2 9) selected from the parameter

space S, the quantiles of the noncentrality parameter and corresponding power values for

p = 0(0.05)1 are obtained on T, for a specified value of p. The procedure is repeated for
other valueso of (,a, ) in S, which is constructed by using the method described in Section

3.3.2. Using the data set in Table 3-1, we obtained the following 95' confidence intervals,

namely, (a, b) = (1.797, 3.935) for a, and (1*, u*) = (0.0, 3.205) for rl. At first, 100 equally-
snlpace values of ,a are chosen from the interval (a,b). For each such selected values of of,

100 equally-spaced values of rl are then chosen from the interval (1*, u*). Hence, the total
number, of valueso of (f,a 9) chosen inside S is 10,000. Finally, after calculating QB=(p p)

and Q~im(p, p) using (3-6), the QDG plots for the noncentrality parameter were obtained.

The process was repeated using other values of p. Similar QDG plots were obtained for

the power function. The R software was used for conducting the numerical investigation

and obtaining the actual plots.










The QDGs of the noncentrality parameter and power function for the comparison

of designs D1 and D2 are Shown in Figures :3-:3 and :3-4. High values of the noncentrality

parameter and power are obviously desirable. Both graphs show that the maximum

quantiles of D1 are above those of D2, eSpecially for large p. Hence, D1 is preferred over

D2. The QDGs for the comparison of designs D1 and D:3 (see Figures :3-5 and :3-6) indicate

that D1 performs far better than D:3, which is expected since D:3 consists of arbitrarily

generated design points. Finally, the QDGs for the comparison of designs D2 and D:3 (see

Figures :3-7 and :3-8) show that D2 performs better than D3. Thus, in conclusion, among

the :3 designs, best power values for testing Ho in (:39) can he achieved by using D1,

followed by D2, and then D3.










Table :3-1. The experimental design along with the response values


Original control variables
X1 X2
1 0
2 0
:3 0
4 0
1 4
2 4
:3 4
4 4
1 15
2 15
:3 15
4 15
1 25
2 25
:3 25
4 25
1 0
2 0
:3 0
4 0
1 4
2 4
:3 4
4 4
1 15
2 15
:3 15
4 15
1 25
2 25
:3 25
4 25
1 0
2 0
:3 0
4 0
1 4
2 4
:3 4
4 4
1 15


Coded control variables
xl 1 2
-1.29 -1.1:3
-0.37 -1.1:3
0.55 -1.1:3
1.48 -1.1:3
-1.29 -0.72
-0.37 -0.72
0.55 -0.72
1.48 -0.72
-1.29 0.41
-0.37 0.41
0.55 0.41
1.48 0.41
-1.29 1.4:3
-0.37 1.4:3
0.55 1.4:3
1.48 1.4:3
-1.29 -1.1:3
-0.37 -1.1:3
0.55 -1.1:3
1.48 -1.1:3
-1.29 -0.72
-0.37 -0.72
0.55 -0.72
1.48 -0.72
-1.29 0.41
-0.37 0.41
0.55 0.41
1.48 0.41
-1.29 1.4:3
-0.37 1.4:3
0.55 1.4:3
1.48 1.4:3
-1.29 -1.1:3
-0.37 -1.1:3
0.55 -1.1:3
1.48 -1.1:3
-1.29 -0.72
-0.37 -0.72
0.55 -0.72
1.48 -0.72
-1.29 0.41


Response
.9
74.1
7:3.0
74.5
72.5
75.0
74.0
72.1
7:3.2
72.0
70.5
69.5
70.7
74.2
69.9
70.1
635.3
7:3.9
75.2
74.5
7:3.5
7:3.5
7:3.1
74.5
74.0
74.2
71.2
71.9
72.5
72.3
72.9
67.8
6;6.9
7:3.5
75.5
72.9
72.5
74.5
75.1
74.2
7:3.3
71.1


Orchard
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
:3
:3
:3
:3
:3
:3
:3
:3
:3










Table 3-1. Continluue


Original control variables
X1 X2
2 15
3 15
4 15
1 25
2 25
3 25
4 25
1 0
2 0
3 0
1 4
2 4
3 4
1 15
2 15
3 15
1 25
2 25
3 25


Coded control variables


Response
Y
71.9
69.1
638.3
72.5
71.5
66.1
65.9
75.0
70.1
75.9
76.1
71.5
75.5
71.9
68.9
69.1
68.5
65.1
6;6.3


Orchard
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4


X1
-0.37
0.55
1.48
-1.29
-0.37
0.55
1.48
-1.29
-0.37
0.55
-1.29
-0.37
0.55
-1.29
-0.37
0.55
-1.29
-0.37
0.55


2 4
0.41
0.41
0.41
1.43
1.43
1.43
1.43
-1.13
-1.13
-1.13
-0.72
-0.72
-0.72
0.41
0.41
0.41
1.43
1.43


Source: K~huri and Cornell (1996, p. 328)










Table 3-2. Design settings for D1, D2 and D3


Block (orchard)









1











2
3









4


1
-1.29
-0.37
0.55
1.48
-1.29
-0.37
0.55
1.48
-1.29
-0.37
0.55
1.48
-1.29
-0.37
0.55
1.48
same as
same as
-1.29
-0.37
0.55
-1.29
-0.37
0.55
-1.29
-0.37
0.55
-1.29
-0.37
0.55


-12
-1.13
-1.13
-1.13
-1.13
-0.72
-0.72
-0.72
-0.72
0.41
0.41
0.41
0.41
1.43
1.43
1.43
1.43e
above
above
-1.13
-1.13
-1.13
-0.72
-0.72
-0.72
0.41
0.41
0.41
1.43
1.43


1
-1
-1
1
1
-1.29
1.48
0
0
0
0
0
0
0
0
0


-1
1
-1
1
0
0
-1.13
1.43
0
0
0
0
0
0
0


1
0.36
1.35
1.02
0.53
0.45
-0.77
-0.19
1.14
1.28
-0.77
0.13
1.40
1.27
0.01
-0.49


-0.13
0.71
0.51
1.17
0.35
1.08
0.28
1.14
-0.68
1.40
1.26
0.81
-0.70
-0.67
0.67


same as above same as above
same as above same as above









same as above same as above


Note: The design settings in Blocks 2, 3, and 4 are the same as those in Block 1, except

for design D1 with regard to Block 4. D1 is the original design. D2 1S the semicentral

composite design. D3 is the randomly generated design.











Table 3-3. Number of points in each design
Design Block 1 Block 2 Block 3
D1 16 16 16
D2 15 15 15
D3 15 15 15


Block 4 Total
12 60
15 60
15 60


Figure 3-1. Confidence region for (o 2, r7)





















_


) X1


* 4
*
**
e & )


(-1.29,1.43)


(1.48,1.43)


(1.48,1.43)


(-1.29,1.43)


**


**S


X1


R o)


* *......


(1.48,-1.13)


(-1.29,-1.13)


(-1.29,-1.13)


(1.48,-1.13)


X2


(-1.29,1.43)


(1.48,1.43)


Xi


(1.48,-1.13)


Figure 3-2. Plots of the design points (in Block 1) for designs D1, D2, and De




















radius = 01 01


radius = 0.2575


ra diusI~= 0.505


-e cn
-r- ~I


CM
1- ~D










)p)~
Ic~~-ZfCCC~


I I I I I I


cr)

B-


O


Illlll


I l l l l l +9d i


o-


0.0 0.4 0.8






dradus = 0.7525
















t I-ll lll C~



0.0 05.4 0.8


0.0 01.4 10.8


10.10 0.4 0.8


radius = 1


M


+*I
I,





+I+5

rc~Z~Z;~CCL



III I I I


o


0.0 0.4 10.8


Figure :3-:3. QDGs of the noncentrality parameter (ncp) for D1 and D2

















radius = 0.2575


ra diusI~= 0.505
















10.10 0.4 0.8


Illlll S
0.0 0.4 0.8





dradus = 0.7525


radius = 1


M
~++~~

1~+6.~,


Illll
0.0 05.4 0.8


IlllllI
0.0 0.4 10.8


Figure 3-4. QDGs of the power function for D1 and D2


radius = 01 01


]Itrtllll
0.0 01.4 10.8





















radius = i0 2575


radius = 0.505


O
n


0O.0; 0.4 0.8


P
N

rr)
r

O
r


m


0.0 0._4 0.8


10.0 0.4 0.8


rad us = 0 7525


r-adius = 1


- D


M
~- DB

+II
IC
I*;r


~2 -In-IC"


++ICb
4)+
4C"O
rr,


Y-CCIlll


o


0.0 0.4 0.8

P


Figure 3-5. QDGs of the noncentrality parameter (ncp) for D1 and D3


radius = 01 01


Illll


Illlll

















radius = 0 25T5


ra d i~s = 06.505


0.0 DA4 0.8


0.0: DA_ 0.8





radius = 0 7525


radius = 1


W
~9~1~6)1
11

rL++ ~~L
bb~i


oer i~--
o


0.0: DA_ 0.8


Illlll
0.0 0.4 0.8


Figure 3-6. QDGs of the power function for D1 and D3


radius = 0.01


111111

0.0 0.4 0.8

















radius = 0 01


radius = 0.2575


radius= 0.505

















0.0 01.4 0.8


-I lllll H-
0.0 0l.4 10.8





rad us = 0.7525

















0.0 0l.4 10.8


radius = 15


0.0 0.-4 0.8


Figure 3-7. QDGs of the noncentrality parameter (ncp) for D2 and De


0.0 0.-4 0.8



















radius= 0.01


r~adiurs= 0 L57.5


-eD3
-+- D2


dradus = 0.505


-e D2


D3


0.0 0 4 0.8


0 0 0._4 0.8


radius = 1


-e C~
r- M

11~,,,,~,+1


0.0% 0.4 0.8






rad us = 0 7525










I lll
0. .4 08
aP


IlllllI
0.0 0.4 0.8

P


Figure 3-8. QDGs of the power function for D2 and De









CHAPTER 4
SEQUENTIAL GENERATION OF D-OPTIMAL DESIGNS

4.1 Introduction

The purpose of this chapter is to show how to generate optimal designs for response

surface models with a random block effect. K~huri (1992) discussed the analysis of the

fixed polynomial terms and the random block effects in such models. In this chapter, the

same model is considered under the assumption that no interactions exist between the

blocks and the model's fixed polynomial effects. K~iefer and Wolfowitz (1960) showed that

D-optimal and G-optimal designs for classical linear models are identical according to

their Equivalence Theorem. The multivariate version of this theorem is given in Federov

(1972, p. 212) and K~iefer (1974, Section 5). Federov (1972) introduced an algorithm

for the construction of a D-optimal multiresponse design using a sequential procedure.

However, his algorithm requires that the variance-covariance matrix for the multiresponse

vector be known. Based on Fedorov's work, Wijesinha and K~huri (1987b3) proposed a

sequential algorithm to generate a D-optimal multiresponse design using an estimate

of that variance-covariance matrix. In this chapter, the response surface model with a

random block effect is treated as a model for a multivariate response as in Atkins and

C'I. ing (1999). This enables us to apply the multivariate version of the Equivalence

Theorem.

4.2 Model and Notation

Consider the following model of degree d (> 1) in k control variables xl, x2, ***, k*

The experimental runs are arranged in b blocks of equal sizes (m being the size of each

block). Hence the model can be written as:


y = P01, + XP + ZY + E: (4-1)


where y is a vector of a observations on the response, Po and p = (P1, Pa2, **, ppl

are unknown parameters associated with the polynomial portion of the model, y










(Y1, 72, ***, 7b) Where Ti denotes the effect of the Ith block (1 = 1, 2, ..., b), and E: is a

random error vector. The matrices X and Z are known of orders n x p and n x b,

respectively. The polynomial portion of the model is defined on a bounded region X c sRk

the k-dimensional Euclidean space. The matrix Z is of the form:


Z = Ib 1, ,. (4-2)


It, is assumed that y is distributed a~s N(0, (T2 b) independently of E,1 which is assumed to

follow the NV(0, a,21) distribution.

4.3 Methodology

('I!. ing (1995) and Atkins and ('I!. ng~ (1999) resorted to the approximate theory of

optimal designs to deal with the special feature of within-block correlation. In this section

we show how the approach of Atkins and ('I!. ng~ (1999) can be adopted to derive optimal

designs in the presence of a random block effect. Let us rewrite (4-1) as


y = W7 + Zy + E~, (4-3)


where 7 = (Po, p')' and W = [1, : X], where n = mb. Let us also partition W as

W1 1m i : X1

W2 im X2
W=


Wb / \Im :Xb

where Wi corresponds to the ith block (i = 1, ..., b). The vector of observations belonging

to the ith block, yi can be represented as


yi = Wir + 1myi + E~i (4-4)


Each set of m points in X can be represented as a point n in Xm = {('z:,z'2) *** m)'

zi E X, i = 1, 2, ..., m}. Also, each point in Xm can be regarded as a block. There is,









therefore, a one-to-one correspondence between blocks of size m in X and points in Xm

Thus, the b blocks in a given design can be represented by the points ul, u2 *** ub in

Xm. Note that the first k elements in ui define the first point in the ith block, the next k
elements define the 2"d point (i = 1, 2, ..., b), and' so on. Let f'l (my denote the je" row of

Xi where f(.) is a known vector function defined on X whose elements are powers and

cross-products of powers of the control variables of degree d, and my = (xjl, Zj2, ***, xjk) ,

i = 1, 2, ..., b, j = 1, 2, ..., m. Thus Xi (i = 1, 2, ..., b) can be written as


f (mi)






Note that po + f'(z)ff provides a polynomial representation of degree d of the mean

response at a point z in the experimental region X. Given that the set of m points,

(m'z~I, '2~) m in X TepreSeHOS a point na Em, let the matrix G(u) be defined as







where g'(z) = [1, f'(z)]. Note that G' is of order m x (p + 1). Let y, be the vector of

m observations in the ith block. We treat yi as the value of an m-variate response. Its

variance-covariance matrix is given by Var (y,), i = 1, 2, ..., b


Vnr (yi) = e)Im, + Jil = a, V (4-5)

where V = Im + rlJm and rl = a2 /-2 Ntef that 0,,- < 0.Dfiningr 9* = e/(si + ef )=

rl/(r + 1) makes rl* lie between 0 and 1. The parameter rl* is often referred to as the

intra-class correlation coefficient. Hence, the variance-covariance matrix of y, the vector of









all observations from the b blocks, is given by


Var(y) = a,2A, (4-6)

where A = Ib 0 V. If rl is known, then the best linear unbiased estimator (BLUE) of 7 in

model (4-3) is the generalized least squares estimator + given by


i = (W'A- W)- W'A- y, (4-7)

and its variance-covariance matrix is


Var(+i) = a,2(W'A- W)-1 = (W'E- W)-1 (4-8)

where E = a A.

Note that in practical situations rl is unknown. An estimate of rl, can be obtained by
replacingr the u~nkrnown variance components ,a and a,2 by their corresponding estimates,

such as ANOVA, mainly Henderson's M~ethod III, maximum likelihood (jl1b), or restricted

maximum likelihood (REML) estimates. The first method may yield negative estimates,

but does not need the normality assumption or any other distributional assumption. The

latter two estimates are nonnegative and have desirable properties such as consistency

and .I--phlli'lc normality. However, they need an underlying probability distribution for

the data, usually considered to be the normal distribution. These estimates can be easily

obtained by using PROC MIXED in SAS (2000) or the 1me function in R (Pinheiro and

Bates (2000)).

At any point ne E m, the predicted response vector is


y(U) = G'(u); (4-9)


The prediction variance-covariance matrix is

Var [y(u)] = G'(u)[W'A- W]- G(u) a,
(4-10)
= G'(u) [W'E- W]- G(u)









Ignoring the constant of the information matrix of the design is


G'(ui)
W'A- W= [G(ul) : ... : G(Ub) (b V- )

G'(ub) 41


i= 1

Let a denote the number of distinct values of ul, u2, ***, Ub. Also, let us denote

the number of replications at the ith of such distinct points (i = 1, 2, ..., s). Then,

CE =1 u = b. For the discrete design measure (b that assigns the weight us/b to ui

(assuming ul, u2,, **, U are distinct), the information matrix per block, $W'A- W
denoted by M1((b, V), can be written as


M((b, V) = u s iG(ur)V- G'(ui) (4-12)
i= 1

Hence, for any given design measure ((u) on Xm, the information matrix is


M~((, V ): = (u)G(u)V- G'(u) (4-13)


4.4 Equivalence Theorem

Assuming normality of the random effect and the error term, and linearity of the

fitted model, a confidence region for 7 can be constructed. A precise estimate of 7 can

be obtained by making the content (volume) of this region as small as possible. Since

this volume is proportional to |M1((, V)|1-1/2, one can achieve this objective by choosing

a design measure which maximizes |M1((, V)| where |.| denotes the determinant of a

matrix. Such a design measure is said to be D-optimal. On the other hand, a G-optimal

design minimizes the maximum of the variance of the predicted response over the region of

interest (G-optimality). The equivalence of the D-optimality criterion to the G-optimality

criterion was proved by K~iefer and Wolfowitz (1960) for the single-response case.









Federov (1972) described an algorithm for the construction of a D-optimal multiresponse

design assuming that the variance-covariance matrix for the multiresponse vector is

known. To circumvent the problem of unknown variance-covariance matrix, Wijesinha and

K~huri (1987b3) proposed a sequential algorithm using an estimate of that variance-covariance

matrix. The following theorem is based upon the multivariate version of the Equivalence

Theorem given in Wijesinha and K~huri (1987b).

The following assertions are equivalent:

(i) The design measure (* is D-optimal, that is,


|MC((*, V)| = sup |M1((, V)|
(6H

where H = class of all design measures on Xm and |M|I is the determinant of the matrix



(ii) (* is G-optimal, that is, it minimizes


max tr [V- G'(u)Ml- ((, V)G(u)]


with respect to all ( E H.

111i


max tr [V- G'(u)Ml- ((* V)G(u)] = p+ 1
UEXm

where p + 1 is the number of fixed unknown parameters in model (4-1).

4.5 Sequential Generation of D-optimal designs

An initial guessed value of rl is considered. In addition, an initial design is used to

obtain response values which will be used to provide an updated estimate of rl. The main

steps of the procedure are given below.

(i) An initial design (b, o CISISting of bo blocks each of size m is chosen such that

M1((bo, Vo) is nonsingular, where Vo = -fm + rlolm, rlo = initial guess of rl. This

measure is defined by assigning the weight 1/bo to each point in (bo*










(ii) A design (b, is COnStructed from (bo by augmenting it with U1 E Xm which satisfies


tr [Vo G'(6 1)M-~ ((bo Vo)G(6C1)] max tr[Vo G'(u)Ml-l ((bo Vo)G(u)]
UEXm

Th~e weight ~7is now assigned to each point in (by,. Note thaRt the first k elemnclts in

6L1 define the first point in the (bo +1)st block, the next k elements define the second point,

and the last k elements define the mth point.

(iii) Repeat steps (i) and (ii) several times until for the (1 + 1)st added block


max tr [V,1G'(u)Ml-l ((bi VI)G(u)] (p + 1) < 6
UEXm

for some 6 > 0 chosen a priori, where VI = Im + rllIm, rll = Ith estimate of rl and (bl iS the

design obtained in the stage 1.

4.6 A Numerical Example

In this section, an example is presented to illustrate the proposed methodology using

an arbitrarily chosen initial design consisting of 2 blocks with 5 points in each block.

The initial design points are given in Table 4-1. Two control variables, xl and x2, arT

considered, and the experimental region is X = {(xl, x2) -1 I 6 1 2 -1 I EIi.

order to fit a complete second-degree model with a random block effect we need n > p + b,

where n = mb is the total number of experimental runs, p is the number of fixed effects

(excluding the intercept) and b is the number of blocks. Here b = bo = 2, m = 5. In

order to generate the response values, initial guesses for the values of the fixed unknown

parameters and variance components are needed. These initial values are used to generate

the response values only, but are not used in the sequential generation of the D-optimal

design. Since O.2, of ar~e unkinow-n, we obtain their estimates, namely resctrictel maximu~m,

likelihood (REML) estimates using the 1me function in the R software from the data

generated by the initial design, then update these estimates as we proceed.

The maximization of the trace function with respect to a E Xs at every iteration

was performed using a computer program written in the R software (Venables and Ripley










(2002)) and is based on the controlled random search (CRS) procedure of Price (1977)

(also, see Appendix F). CRS is a direct search technique and is well suited for objective

functions that are not differentiable. The sequential procedure is computationally

intensive. Without a proper global optimization algorithm, the computation can be

very slow and sometimes may fail to converge. Conlon (1985) wrote a similar program in

FORTRAN implementing Price's algorithm for function minimization. More recently, Ali

et al. (1997) provided comparisons of modified control search algorithms.

We have used the algorithm described in Section 4.5 to compute an optimal design

sequentially starting with an initial design point and different values of the unknown

parameters. The number of iterations required for the convergence of our algorithm

corresponding to 3 initial values of rl*, namely, 0.10, 0.50, and 0.90 (with a,2 held fixed

as 1) are 87, 19 and 100 respectively. Wijesinha and K~huri (1987b, Section 4.1) made

a conjecture regarding the choice of initial design. They observed that the number of

iterations required to satisfy the stopping criterion is greatly reduced by the inclusion of

the boundary points of the experimental region as initial design points. When we included

boundary points as initial design points, the procedure attained convergence at a faster

ratte. In this chapter, w-e are reporting the resullts for .r*= 0.50 (i.e. (2 = ef =1). The

initial values of the fixed unknown parameters of a second-degree model were chosen as

follows: Po = 1, P1 = 2, P2 = 2, P11 = 1, P22 = 1, and P12 = 1. The initial design points

together with the augmented design points form an ap~prox~imate D-optimal design. In

Table 4-2, the augmented design points along with updated. estimates of cT,2 ( rl and 17*

are provided. The value of 6 was chosen as 0.001 and convergence was attained at the 19th

iteration.

















Table 4-1. Initial design along with response values

x1 2~ block y
-0.70 -0.80 1.00 -0.16
-0.20 -0.90 1.00 1.93
0.00 0.00 1.00 -0.02
0.50 0.80 1.00 4.48

0.20 0.90 1.00 4.00
-0.60 0.50 2.00 2.34
-0.70 0.70 2.00 3.90
0.00 0.00 2.00 2.41
0.80 -0.70 2.00 0.68
0.20 -0.90 2.00 1.42












Initial Design


a





a



rVq
Xa



V)
a
r



a
I


* *


Figure 4-1. Initial Design










Table 4-2. Augfmented design along with updated estimates


21 Z2 boky iteration maximum trace o; o, I r*
-0.94 -0.31 3 -1.58 1 16.17 1.09 0.32 0.30 0.23
0.86 0.01 3 1.49
-0.89 0.45 3 1.01
0.93 -0.95 3 -1.39
0.99 0.63 3 1.83
0.37 0.92 4 4.49 2 9.69 0.91 0.34 0.37 0.27
-0.95 -0.99 4 -0.49
-0.83 -0.91 4 -1.61
-0.85 -0.18 4 0.34
-0.06 -0.38 4 2.33
-0.72 0.77 5 4.02 3 11.48 0.89 0.03 0.03 0.03
-0.94 0.87 5 3.03
-1.00 0.90 5 3.50
-0.46 -0.30 5 1.29
0.65 -0.88 5 0.28
-0.87 -0.97 6 -0.42 4 8.00 0.73 0.00 0.00 0.00
0.85 0.82 6 3.45
0.76 0.89 6 4.12
0.01 -0.85 6 0.71
-0.76 0.90 6 3.10
0.12 0.97 7 5.27 5 8.01 0.60 0.03 0.05 0.04
0.99 -0.54 7 0.37
-0.84 -0.87 7 -1.74
-0.45 0.89 7 5.78
0.92 0.82 7 3.86
-0.97 0.20 8 1.35 6 6.77 0.64 0.00 0.00 0.00
0.94 -0.86 8 -0.59
-0.92 0.07 8 1.55
0.50 0.58 8 3.71
0.06 0.09 8 3.50
-0.48 0.95 9 4.23 7 7.20 0.60 0.04 0.07 0.07
0.96 -0.15 9 0.74
-0.04 0.31 9 2.97
0.75 -0.82 9 -0.04
0.92 -0.99 9 -1.17
0.96 -0.03 10 0.47 8 6.15 0.54 0.05 0.09 0.08
-0.19 0.99 10 5.03
0.17 0.57 10 5.62
0.82 -0.21 10 1.45
0.99 0.49 10 1.84
0.96 0.13 11 0.30 9 6.84 0.56 0.03 0.06 0.06
-0.99 -0.88 11 -2.09










Table 4-2. Continluue


2I1 Z2 loky iteration maximum trace o; o, 2I
-0.98 -0.21 11 -0.20
-0.74 -0.88 11 -1.15
-0.86 -0.82 11 -0.60
-0.79 -0.95 12 -1.15 10 6.89 0.53 0.03 0.06 0.06
-0.13 0.96 12 3.13
0.99 0.80 12 2.88
0.38 -0.87 12 2.15
-0.86 0.60 12 1.86
0.07 0.16 13 2.31 11 6.54 0.61 0.02 0.03 0.03
-0.49 -0.13 13 1.58
0.89 -0.95 13 0.04
0.19 0.36 13 2.80
0.96 -0.80 13 -0.27
-0.89 0.18 14 0.35 12 7.00 0.57 0.02 0.03 0.03
-0.98 1.00 14 4.43
0.85 -0.18 14 -0.20
-0.21 -0.75 14 1.30
0.14 -0.89 14 0.37
0.89 0.96 15 3.46 13 6.38 0.59 0.00 0.01 0.01
0.79 0.64 15 1.39
-0.25 0.96 15 6.05
-0.41 -0.84 15 0.55
0.06 -0.33 15 2.75
-0.09 -0.01 16 2.39 14 6.24 0.61 0.00 0.00 0.00
0.64 -0.37 16 1.60
-0.03 -0.20 16 1.00
0.92 -0.94 16 -0.62
0.79 0.96 16 3.63
0.94 0.55 17 1.98 15 6.46 0.59 0.00 0.00 0.00
0.19 -0.74 17 1.28
-0.92 0.99 17 4.23
-0.55 0.20 17 2.46
-0.95 0.64 17 2.87
-0.93 0.05 18 0.73 16 6.89 0.56 0.00 0.00 0.00
-0.56 0.78 18 3.61
0.95 -0.95 18 -1.87
0.98 -0.97 18 0.01
-0.49 -0.85 18 -0.09
0.90 0.96 19 3.12 17 6.60 0.56 0.00 0.00 0.00
0.93 0.50 19 2.21
0.05 0.75 19 4.00
-0.61 0.42 19 2.20










Table 4-2. Continluue


21 Z2 boky iteration maximum trace o; o, I r*
-0.90 0.98 19 2.90
-0.92 -1.00 20 -2.18 18 6.51 0.54 0.00 0.00 0.00
0.32 -0.35 20 1.56
-0.79 -0.95 20 -0.64
-0.60 -0.28 20 0.80
-0.07 -0.04 20 0.34
0.31 -0.43 21 1.05 19 6.00 0.55 0.00 0.00 0.00
0.15 -0.21 21 3.00
0.99 0.92 21 3.58
-0.88 0.48 21 2.72
-0.53 0.96 21 4.31









CHAPTER 5
COMPARISON OF DESIGNS FOR CORRELATED RESPONSE MODELS WITH AN
UNKNOWN DISPERSION MATRIX

5.1 Introduction

Various authors discussed the problem of obtaining exact or approximate tests for

comparing two independent regression models; see, for example, Ali and Silver (1985)

and Conerly and Mansfield (1988). The problem of comparing correlated response

models was discussed by Zellner (1962). This procedure requires that an estimate of

the variance-covariance matrix, C, he used in place of E. To circumvent the problem

of an unknown C, Smith and Choi (1982) developed an exact method to compare two

correlated response models without having to estimate E. K~huri (1986) introduced a

general procedure involving exact multivariate tests for the equality of parameters from

several correlated response models with an unknown E. This procedure assumed the

response models to be of the same form and to contain the same set of control variables.

This chapter deals with the comparison of designs for correlated response models with

an unknown variance-covariance matrix, E. The novelty of our approach lies in applying

the quantile dispersion graphs in an investigation of the power of several multivariate

tests concerning such models. Our proposed approach is based on considering quantiles

of a certain criterion function (namely, power of each of three multivariate tests) on

concentric surfaces within a particular region of the so-called alternative .space. Power

comparisons of the four multivariate tests, namely, Roy's largest root, Wilks' likelihood

ratio, Hotelling-Lawley's trace, and Pillai's: trace wer-e conlsider-ed h sever-al authour-s

(see Pillai and Jayachandran (1967), Roy et al. (1971, Ch. 5), Seher (1984, Section

8.6.2)). The dependence of these quantiles on the unknown values of the variances and

correlations obtained from the variance-covariance matrix, E is depicted by plotting the

so-called quantile li~spersion Il g/t -, (QDGs) of the criterion function. These plots provide

a clear assessment of the magnitude of the power value associated with a given design. A

numerical example is presented to illustrate the proposed methodology.










5.2 Correlated Response Models

Consider a system of r correlated response variables, Yl, Y2, *, Yr, each of which

depends on the same p control variables denoted by xx~, ..2 ** Let us assume that the

relationship between yi (i = 1, 2, ..., r) and xl, xa2, ,p can be represented by a linear

model of the form

ye=#i Skk i i = ,2 ,(5-1)

where pio and the Pik S are unknown parameters and as is a random error corresponding

to the ith response (i = 1, 2, .., r). Assuming that the above r models can have different

design matrices and each design consists of a experimental runs, these models can be

rewritten as



yi = Pio1, + XiPi + Esi = ,,., (5-2)

where yi is an a x 1 vector of observations, 1, is an a x 1 vector of ones, Pi

(Pil, Pi2, Pip)', Xi denotes the n x p design matrix of rank p, and Ei is a vector of

random errors corresponding to the n observations from the ith response (i = 1, 2, .. ., r).

In matrix notation, the above models can be written as



Y = 1,0'o + X B + E: (5-3)

where
Y =(yl:y : y2 W

O'o = (Plo, P20, Pr0

X =(X1 : X2 : : X,) (5-4)

B = diag(P,,2) r,.. ,



The above matrices are of orders n x r, 1 x r, n x (pr), (pr) x r and n x r, respectively.

diag implies that the matrix B is block diagonal. It is assumed that the rows of E: are










independent random vectors from the multivariate normal distribution NV(0, E). The
unknown variance-covariance matrix E is of order r x r and rank r.

Let us consider the following hypothesis Ho : plo P~o020 = Pr; P1 2 P

= O against H, : Pio d Po for some i / 1 = 1, 2, .. ., r, or m, / P for some

m / u = 1, 2, .. ., r. This is known as the hypothesis of concurrence.

5.3 Development of a Multivariate Test

Consider the multiresponse model (5-3). Let p denote the rank of X. Note that since

each Xi(i = 1, 2, ..., r) in X is of rank p, we have p < p < pr. We shall assume that

r I n p. Let C be the r x (r 1) matrix of rank r 1,


(5-5)


The null hypothesis Ho can be expressed as


Ho : obC = O', WBC = 0,


where W is a matrix of order p x (pr) of the form


W = (I, : I, : : I,)


(5-6)


where I, is the identity matrix of the order p x p. Multiplying model (5-3) on the right by

C, we get


(5-7)


YC = Zr + EC









where Z = (1, : X), and r is the matrix


r = oCe (5-8)
BC

The hypothesis Ho can then be expressed as Ho : Gr = 0 where


G = 1 0',", (5-9)


and 0', and 02 arT ZeoO VeCtoTS of orders 1 x (pr) and p x 1 respectively. Note that G is of

full row rank equal to p+ 1.

The development of a multivariate test for testing Ho depends on Roy's union-

intersection principle. This is illustrated as follows (see K~huri (1986) for details): Let a

=(al, a2, tr-1)' be an arbitrary nonzero vector of order (r 1) x 1. The multivariate
model given in (5-7) can be reduced to a univariate model by the transformations

y, = YCa, (a = ra, and E, = eCa. Then,

ya = Ze, + E, (5-10)

Note that E, N 1(0, o-f,), where of = a'C'ECa. The multivariate hypothesis stated

earlier also reduces to the univariate hypothesis


Ho (a) : Ge, = 0 (5-11)

Clearly, Ho is true if and only if Ho(a) is true for all a / 0. But, for each a / 0, the

hypothesis Ho(a) is a general linear hypothesis associated with the univariate model

(5-10). The hypothesis Ho(a) can, therefore, be rejected for large values of the statistic

y' Z(Z'Z) -G'[G(Z'Z)- G'] -1G(Z'Z)- Z'y,
R(a) = (5-12)
y's [I, Z(Z'Z)- Z'] y,

where (Z'Z)- is a generalized inverse of Z'Z. This statistic is invariant to the choice of

the generalized inverse (see Searle (1971, Section 5.5)). Since y, = YCa, then R(a) can









be rewritten as


R(a) = (a'Sha)/(a'Sea) (5-13)

where

Sh = C'Y'Z(Z'Z)- G'[G(Z'Z)- G'] G(Z'Z)- Z'YC (5-14)

and

Se C'Y' [1,L Z(Z'Z)-Z'] YC (5 -15)

The matrices Sh and Se are called, respectively, the matrix: due to the 1 i,;.-HI,~ .: and the

matrix: due to the error. The matrix Sh is positive semidefinite of rank = mintr 1, p+ 1),

and Se, under the condition n p 1 > r 1, is positive definite with probability 1,

and has the central Wishart distribution W(n p 1, C'EC) with n p 1 degrees of

freedom. Since G in (5-9) is of full row rank equal to p + 1, Sh is independent of Se and

has the noncentral Wishart distribution W(p + 1, C'EC, 02) with p + 1 degrees of freedom

and noncentrality parameter matrix given by (see Seber (1984, p. 414))


0 = (C'EC)- T'G'[G(Z'Z)-G']- GT 5-6


Roy's largest root test statistic is given by (see K~huri (1986, p. 350)) emax [ShS;'], which

denotes the largest eigenvalue of the matrix ShS; Other multivariate test statistics for

testing Ho include the following:

Wilks' likelihood ratio: A = [det(Se)]/[det(Se + Sh)], (small values of A lead to

rejection of Ho)

Hotelling-Lawley's trace: U = tr(ShS;'), where large values of U are significant, and

Pillai's trace: V = tr [Sh(Sh + Se)-l], where large values of V are significant.

As a special case, let us consider two independent regression models,


yi = X $4 + Ei, i = 1, 2, (5-17)









where yi is an as x 1 vector of observations on the response variable for the ith model,

Xi is a known matrix of order as x F and rank 7, and pi is a fx 1 vector of unknown

parameters (i = 1, 2). It is assumed that the esi's are normally distributed random vectors

with E(E,) = 0, E(EE() = eflmi for i = 1, 2, and E(E1E'2) = 0. A reasonable hypothesis

of interest is Ho : p, = 2, With the alternative H, : Si 02P. The likelihood ratio test

for Ho, when aeye? is known, is based on the test statistic (see for example, Ali and Silver

(1985, Eq. 2.2))

n. 2p (0 -0)'e X )l-' + "2X2 )~-']-'(( P )
F = x7 ~ (5-18)


where n. = ni + n2) 1, = (X Xi)- X y and s2 = (yi Xi0 )'(yi XiP1) for

i = 1, 2. Under Ho, F has an exact F distribution with y and n 2# degrees of freedom.

Under the alternative hypothesis, F is distributed as a noncentral F with F and n 2#

degrees of freedom and noncentrality parameter (see for example, Conerly and Mansfield

(1988, p. 815))


\=(0 0> [2)' X' X1)-l + a2 fX2Xo)-l ]- (P1 P2) (5-19)


Note that, in this case, knowledge of both a~ and a,2 is needed for the computation of the

power for specified alternative values of P, P2. This situation is a generalization of the

classical Behrens-Fisher problem of testing equality of two means when the population

variances are different.

5.4 Design Criteria

The power of each of the multivariate tests Roy's largest root, Wilks' likelihood

ratio, Hotelling-Lawley's trace, and Pillai's: trace canl be used as: a designl crit~erionr. As:

pointed out by Pillai (1977, p. 24), the non-null distribution of the Roy's largest root has

only been derived in finite or infinite series forms, involving zonal polynomials, whose

general terms are difficult to compute. In general, exact power functions of the four

multivariate tests are extremely difficult to obtain in closed-form expressions. Since the









power of Roy's largest root test is below those of the other three tests (see Pillai and

.I li- Ill i li di a (1967) and Muirhead (1982, Section 10.6.5)), we consider the power of the

other three tests as design criteria. It should be noted that power comparisons of the four

multivariate tests by Pillai and Jayachandran (1967), Lee (1971) and Roy et al. (1971, Ch.

5) indicated that no single test was uniformly better than any of the other tests in terms

of power.

We now present .I-i-uspind ilc formula that provide adequate approximations for the

power of three multivariate tests (see, for example, K~huri (1986)). The null and the

alternative hypotheses can be written as Ho : Gr = 0 and He : Gr = a where a / 0.
5.4.1 Wilks' likelihood ratio

The power of Wilks' likelihood ratio test, using -6logA as a test statistics in place of

A, under the alternative hypothesis H, and at a level of significance a~ is approximately

given by (see, for example, K~huri (1986, formula 3.3))


Pr (- blog A > rw (a~) |H,) =P(y(a) w()


46 (r + p+ 1)aPr( 26) T )
4fi (5-20)




where 6 = n p 1 (r p 1)/2, A = [det(Se)]/[det(Se + Sh)], Tw(a) denotes the upper

100.. .; point of -5 logA under the null hypothesis Ho, I = (r 1)(p + 1), a6 = tr(D2),

a2 tr7) ~2) iS the noncentrality parameter matrix defined in (5 16) and 7)(a6) denotes
the noncentral chi-squared variate with f degrees of freedom and noncentrality parameter









5.4.2 Hotelling-Lawley's trace

The power of Hotelling-Lawley's trace under the alternative hypothesis H, and at a
level of significance a~ is approximately given by (see, for example, K~huri (1986, formula

3.4))



P~r((n, p 1) U > TH (a) |H,) =Pr (X (a) > 7H 0)

4(n p-1)

+ 2(p 1) [A (p ) I(r )]Pr ~,(7 I +2 Fl H (N))

+ [(P + 1)(r 1(p + r + ) (p+r + 2)$~ + ]Pr(X +4~I TH 0L))

+ [2(p + r + 1)$ 21T]Pr(X y}+6( 1 Fl TH ))


(5-21)
wnhere U= rSizS. ) is Hotelling-Lawley's trace, TrH(a~) denotes the upper 100. .'. point of
the statistic (n p 1)U, f, a6, and a4 are the same as defined in Section 5.4.1.
5.4.3 Pillai's trace

The power of Pillai's trace under the alternative hypothesis H, and at a level of

significance a~ is approximately given by (see, for example, K~huri (1986, formula 3.5))



Pr((n; p 1)V >7p(a)|H,) =Pr(X:(A;) > 7p(a))

4(n p-1)
+ [2r(p + 1)(r ) 2(p) + 1)6IPr(X +2~T) Fl TP 0)j


+ [(p + 1)(1r -1(p + (r + ) 2rp $P( + F P )

ar(2fi(pl + r ) rPr(yf+ 1 P

(5-22)









where V = tr[Sh(Sh + Se)- ], TP(a~) denotes the upper 100. .'. point of the statistic

(n p 1) V, f aI, and a2 are the same as defined in Section 5.4.1.

5.5 Quantile Dispersion Graphs

The three power functions will be used to compare designs for the correlated

response models. The power depends on the design (through the matrix Z), r (number

of responses), the degrees of freedom of Sh and Se, a~ (level of significance), and 02

(the noncentrality parameter matrix) only through its eigenvalues. Recall that the

noncentrality parameter matrix, 02, as defined in (5-16) depends on the design, E and

Gr = a. To address the problem of unknown E, we consider simultaneous confidence

intervals on the elements of E. In addition, the specification of the so-called alternative

space for a will be needed. In order to study and compare the performances of two

designs, D1 and D2, throughout the alternative space, we consider surfaces of several

concentric hyperspheres of radius -r which are located within the alternative space. The

surface of a sphere of radius -r is denoted by T,. Please refer to Section 3.3.1 for an

illustration of the use of the quantile dispersion graphs approach.

5.5.1 Simultaneous Confidence Intervals on the elements of E

In order to obtain simultaneous confidence intervals on the elements of E, we adopt

the following procedure given in Seber (1984, Section 3.5.7). Let oyj denote the (i, j)th
element of E. Define the r x r matrix Q as Q = '-(y y)(yf )', hr

y = y? andU yi (yil, Yi2) ***, ir)l 1S the ith row of Y in (5-3). It can be easily

show tha L < U,' for all nonzero 1, if and only if L < 7min,, < 7max, < Ul where

7min and 7max are the minimum and maximum eigenvalues, respectively, of QE- The

readers are referred to Seber (1984, Section 3.5.7) in order to have a better understanding

about the choice of the values of L and U.

Cl wIg~-!! L and U such that P[7min > L] = c0/2 and P[7max < U] = ca/2, we

get 1 a~ = P[ l'E i The distribution of 7ms, and 7max are tabulated

in Hanumara and Thompson (1968), but since the exact values are difficult to obtain










they computed approximate values of L and U. As -II_0-r-- -1. by Clemm et al. (1973,

Section 3) we choose L and U such that L = 1/U and then use the tables given in their

paper. Setting I equal to (1,0,...,0), (0,1,...,0), (1,1,...,0) and so on, we obtain the following

intervals for the variances and covariances, namely, aln, a22,***, arr and a12, al137---.



"' < ail < for i = 1, 2, ..., r and

2 + (qii qjj)(' ) < 2aij < 2 +j (qii + qjj)( -)fri
where q 4 = (i, j)th element of Q. These are simultaneous confidence intervals with a joint

coverage probability '> 1 a~.

5.5.2 An Alternative Space for a = Gr

Let 6 = vec(A), where vec(.) is a matrix operator (see Searle (1982, pp. 332-333))

which stacks the columns of the matrix, a one under the other to form a single column.

This technique was used by Valeroso and K~huri (1999, p. 163). Let q be equal to the

product of the number of columns and rows in the matrix a. In order to construct an

alternative space for 6 = (61, ..., 6,)', which is a subset of sRV, one ca select 6 points

that fall on the surface of a hypersphere of radius -r. Let T, denote the surface of such

a hypersphere. The points on T, depend on only q 1 independent variables. Using,

for example, the spherical coordinates associated with 61, ..., by, we obtain the following

equations (see Section 3.3.3)


61 = r COS ~1

62 = d i 1 COS 2~



6,1= 7 sin I1 sin 2~ sin q-3 i q-2~ COS q-_1

6, = -r sin I1 sin 2~ sin q-3 si q-2~ sin q-1


where 01, ..., 4,_1 are angles chosen independently such that 0 < 44 < xr for i =

1, 2, ..., q 2, and 0 < ~,_1 < 2xr. In order to generate points on T,, ~1,..., 4,_1 are chosen










randomly from independent uniform distributions, namely, U(0, xr) for fi, ..., #q-2 and

U(0, 2xr) for 4,_1. We can obtain several alternative values for 6 by varying the value of -r.

5.6 Example

To illustrate our proposed methodology let us consider an example of a repeated

measures design given in K~uehl (2000, p. 514). It is based upon an experiment conducted

to determine the effects of X1 = soil compaction and X2 = SOil moisture on the soil

microbes activity. Treated soil samples are placed in airtight containers and incubated

under conditions conducive to microbial activity. The microbe activity in each soil sample

was measured as the percent increase in the production of CO2 above atmospheric

levels. The design used to generate the data was a 3 x 3 factorial with three levels of

soil compaction and three levels of soil moisture. For each treatment, two replicate soil

container units were prepared and CO2 eVOlution/kg soil/d w- was recorded on three

successive d we~ giving the response values yl, y2 and y3. Table 5-1 gives the data for each

soil container unit and Table 5-2 shows the coded variables (obtained by subtracting the

mean, and then dividing by the standard deviation) of Design 1 and 2, which represent

the initial design and a randomly generated design within the region of experimentation,

respectively (also, see Fig 5-1). The QDGs help in comparing these two designs based

on the power of the three multivariate tests. The program used to generate the QDGs is

written using the R language. High values of the power are obviously desirable. All the

three figures 5-2, 5-3 and 5-4 show that the maximum quantiles of D1 are above those of

D2, foT Tr = 2. For -r = 0.1 and 1, maximum quantiles of D1 are either above or same as

those of D2. Hence, D1 is preferred over D2-










Table 5-1. Design 1 original variables along with responses
)I )(2 91 2 3
1.10 0.10 2.70 0.34 0.11
1.10 0.10 2.90 1.57 1.25
1.10 0.20 5.20 5.04 3.70
1.10 0.20 3.60 3.92 2.69
1.10 0.24 4.00 3.47 3.47
1.10 0.24 4.10 3.47 2.46
1.40 0.10 2.60 1.12 0.90
1.40 0.10 2.20 0.78 0.34
1.40 0.20 4.30 3.36 3.02
1.40 0.20 3.90 2.91 2.35
1.40 0.24 1.90 3.02 2.58
1.40 0.24 3.00 3.81 2.69
1.60 0.10 2.00 0.67 0.22
1.60 0.10 3.00 0.78 0.22
1.60 0.20 3.80 2.80 2.02
1.60 0.20 2.60 3.14 2.46
1.60 0.24 1.30 2.69 2.46
1.60 0.24 0.50 0.34 0.00



























Table 5-2. Design 1 and 2 Coded Variables
Design 1 Design 2
.I' x- 2 1 -
-1.08 -1.14 -0.34 -0.36
-1.08 -1.14 0.91 0.8:3
-1.08 0.29 0.50 -0.1:3
-1.08 0.29 -0.32 0.00
-1.08 0.86 0.76 0.64
-1.08 0.86 -0.87 -0.78
0.12 -1.14 0.6:3 -0.54
0.12 -1.14 0.00 0.74
0.12 0.29 0.4:3 0.26
0.12 0.29 0.17 -0.67
0.12 0.86 0.10 -0.39
0.12 0.86 0.28 0.78
0.92 -1.14 -0.17 -0.82
0.92 -1.14 0.59 0.55
0.92 0.29 -0.80 -0.1:3
0.92 0.29 0.1:3 0.54
0.92 0.86 -0.69 -0.92
0.92 0.86 -0.36 0.45





















































Ln



a




a


N a
X a

Ln
4

a




~-


I C


1 5 1O -0.5 0.0


xl


0.5 1.0 1.5


D


N O
2~ D

an
4





u,


IC


1.5 -1.0 -0.5 0.0 O 5 1 O 1.5


x1


Design 1


Design 2


Figure 5-1. Plot of the design points




































radius -0.1








01 0 06D
PP




re us-







0 0 0 2 04 6 0 8 1.0




radlus=2







0 002040 .
atP



Fiue52 D~ ftepwr ucinfrD n 2(aedo ik'LkeiodRto

































radius=0.1

















PP

ra iu =









0 0 0.2 0.4 0 6 0 1.0




P P

















88




































radius -0.1






M
0 040 E

PP










01 0
rP




radlus=2









PP


Figure 5-.Qr ftepwrfnto o D adD bsdo ilisTae










CHAPTER 6
SIT1ll\ARY AND FITTIRE RESEARCH

In Cl'I I'lter 2, we demonstrated the application of QDGs as a powerful graphical

tool for comparing designs for response surface models with random block effects. The

design dependence problem was circumvented by the consideration of the dispersion of the

quantiles of the scaled prediction variance over a certain parameter space associated with

the unknown parameter, rl. The methodology discussed in this chapter can he adapted

to compare split plot designs using a linear mixed model similar to the one considered by

Goos and Vandebrock (2003) and Liang et al. (2006).

In Cl'I I'lter 3, we applied QDGs to compare designs for response surface models with

random block effects based on the power of a statistical test. Crucial to this approach

is the specification of the parameter space and the alternative space. As a sequel to this

work, we would like to consider the problem of comparing two independent regression

models with heteroscedastic error variances. This situation is a generalization of the

classical Behrens-Fisher problem of testing equality of two means when the population

variances are different.

In C'!s Ilter 4, we showed how to generate D-optimal designs sequentially for mixed

response surface models by augmenting an initial design with additional points. The

Equivalence Theorem provides a powerful tool for constructing optimal designs for

response surface model with a random block effect. The response surface model with a

random block effect was treated as a multivariate model with the responses in a block

forming an m-variate response (nz is the size of each block). After deriving the moment

matrix, the multivariate version of the Equivalence theorem was applied. Price's algorithm

was used to optimize the objective function. It should be noted that the proposed

methodology requires the assumption of equal block sizes. The experimental error variance

in a response surface model with a block effect has traditionally been assumed to be

constant. However, in many experimental situations, this variance may not he the same










for the different blocks that make up the associated design. We would like to extend the

proposed methodology to study the choice of designs for such models with heterogeneous

error variances.

In C'!s Ilter 5, we applied the QDGs to compare designs for correlated response

surface models with an unknown dispersion matrix. A niultiresponse experiment, by

definition, is one which involves a number of responses measured for each setting of a

group of control variables. In many cases, the subdivision of experimental units into

blocks necessitates the addition of block effects to the nmultirepsonse surface model. The

modeling of a niultiresponse experiment with random blocks was considered by Valeroso

and K~huri (1999). We hope to extend our proposed methodology to address the problem

of comparison and sequential generation of designs for niultiresponse models with random

block effects.

A part of my current research work at GlaxoSnxithE~line involves designs for

niulticentre trials. 1\ost clinical trials require a large number of patients, which cannot he

provided by a single centre, and to ensure that the trial is completed in a reasonable time,

patients are treated at more than one centre. In the pharmaceutical industry niulticentre

trials are used as evidence in submissions to regulatory agencies seeking approval. Our

perspective is that of statisticians providing advice on the design and analysis of such

trials within the pharmaceutical industry. Fedorov and Jones (2005) demonstrated the

fitting of fixed and random effects models to data collected in niulticentre trials and

indicated preference for the randon1-effects model. As observed by them, a randon1-effects

model takes into account the within-centre and between-centre variation. It also provides

a basis for sample size and power calculations involving the number of centres and the

number of patients. For the case of repeated measures data, that is, when multiple

responses are collected front each patient, it is of interest to determine the optimal number

of centers and patients.













APPENDIX A

R CODE USED IN CHAPTER 2



# Program to generate the Quantile dispersion graphs for 2

Designs .



library(MASS); #Since we need to compute inverse, ginv() is in

MASS source("bdiag.txt") ;



Calculating the confidence interval for eta --- from bisection

method --- SAS program



etalow <- 0.0763243 ; etaup <- 0.9667678 ;



# Read in the design matrix for the two designs. Read in the

values of X1 and X2 from the dataset to create the design matrix X



X1 <- matrix(scan("QDGdata.txt" n =118+4), 118, 4, byrow=TRUE) ;

xlx1 <- X1[,1]+X1[,1]; x2x2 <- X1[,2]+X1[,2]; x1x2

9<-X1[C,1]+X1[,2] ; W.D1 <- cbind(1,X1[,1] ,X1[,2] ,x1x1,x2x2,x1x2) ; n

<- length(X1[,1]);

rm(xlx1);

rm(x2x2);

rm(xlx2);

#Generate second design within the rectangular region xl <-

runif (108,-1,2) ; x2 <- runif (108,-1,1) ; design.pts.arbitrary <-

cbind(x1,x2);

write(t(design.pts. arbitrary),f ile="design_pts_arbitrary .txt" ,ncolumns=2 ,sep="\t") ;

W.D2 <- cbind(1,x1,x2,x1+x1,x2+x2,x1+x2); n2 <- length(W.D2[,1]);



# Quantile plots are to be generated for different values of

lambda lambda is # a constraint on the space of X1, X2. The

values of lambda can be between 0.5 and 1



lambda.min <- 0.5; lambda.max <- 1.0; lambda.interval <-lambda.max

-lambda.min; no.1ambda <- 5; # Calculating only for 5 values of

lambda lambda.skip <- lambda. interval/no.1ambda; lambda <-

lambda.min; lambda <- lambda + lambda.skip; check.1ambda.flag <-

0; # Just a flag to create a final concatenated dataset



# Start loop for different values of lambda













fftftftftftftftftftftftftftftftftftftftf


# Defining the experimental region


while(lambda <= lambda.max){

eta <- max(O,etalow); # eta cannot be negative. Hence start from O, if estimate is negative

check.eta.flag <- 0;

no.eta <- 2000;

eta.interval <- (etaup-eta)/no.eta;

while(eta <= etaup){

A12 <- diag(rep(1,12)) + etao(matrix(1,12,12));

All <- diag(rep(1,11)) + etao(matrix(1,11,11));

A9 <- diag(rep(1,9)) + etao(matrix(1,9,9));

A10 <- diag(rep(1,10)) + etao(matrix(1,10,10));

mats <- list(A12,A9,A9,A12,A9,A9,A9,A9,A11,A9,A119) # Creating the list of elements

Al <- bdiag(mats) ;

mats <- list(A9,A9,A9,A9,A9,A9,A9,A9,A9,A9,A9,A9) # Creating the list of elements

A2 <- bdiag(mats) ;

Var.tau.hat.D1 = ginv(t(W.D1) X+X ginv(A1) X+X W.D1); # Calculate the var-cov matrix of tau hat

Var.tau.hat.D2 = ginv(t(W.D2) X+X ginv(A2) X+X W.D2); # Calculate the var-cov matrix of tau hat

rm(A12);

rm(A9);

rm(All);

rm(A10);

rm(mats);


x0 <- 1;

min.x1.D1 <- min(W.D1[,2]);

max.x1.D1 <- max(W.D1[,2]);

min.x2.D1 <- min(W.D1[,3]);

max.x2.D1 <- max(W.D1[,3]);

min.x1.D2 <- min(W.D2[,2]);

max.x1.D2 <- max(W.D2[,2]);

min.x2.D2 <- min(W.D2[,3]);

max.x2.D2 <- max(W.D2[,3]);

al <- max(min.x1.D1,min. x1.D2) ;

bl <- min(max.x1.D1,max.x1.D2);

a2 <- max(min. x2.D1,min. x2. D2) ;

b2 <- min(max.x2.D1,max.x2.D2);

max.x1.val <- bl- (-lambda)+(bi

min.x1.val <- al + (1-lambda)+(bi

max.x2.val <- b2 (1-lambda)+(b2


al); # Calculating the range for different lambda

al);

a2);
















KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK


min.x2.val <- a2 + (1-lambda)+(b2-a2);


# Generate the expt region -- note that the points should be on the boundary



no.x <- 500;

temp. expt. x1.min <- matrix(O,no. x,2);

temp. expt. x1.max <- matrix(O,no. x,2);

temp. expt. x2. min <- matrix(O,no. x,2);

temp. expt. x2. max <- matrix(O,no. x,2);

temp. expt .x1 .minC, 1] <- rep(min. x val ,no .x) ;

temp. expt .x1 .maxC, 1] <- rep(max.x1.val,no. x) ;

temp. expt .x2 .min[C,2] <- rep(min. x2. val ,no .x) ;

temp. expt .x2.max[C,2] <- rep(max.x2.val,no. x) ;

x1. skip <- (max. x1.val-min. x1.val) /no. x;

x2.skip <- (max.x2.val-min.x2.val)/no.x;

for(i in 1:no.x){

temp.expt.x1.min~i,2] <- min.x2.val + iox2.skip;

temp.expt.x1.max~i,2] <- min.x2.val + iox2.skip;

temp.expt.x2.min~i,1] <- min.x1.val + i~x1.skip;

temp.expt.x2.max~i,1] <- min.x1.val + i~x1.skip;



expt. region. pts <- rbind(temp. expt. x1.min, temp. expt. x1.max, temp. expt. x2. min, temp. expt. x2. max) ;

rm(temp. expt. x1.min) ;

rm(temp. expt. x1.max) ;

rm(temp. expt. x2. min) ;

rm(temp. expt. x2. max) ;

no.pts <- length(expt.region.pts[,1]);



flag <- 0; # just a flag to create the vector of prediction variance



# For the 2 Designs calculate the prediction variance of y for

different combination of x1,x2



for(i in 1:no.pts){

xl <- expt.region.pts~i,1];

x2 <- expt.region.pts~i,2];

x1x2 <- x1+x2;

xlx1 <- x1+x1;

x2x2 <- x2+x2;

gx <- cbind(x0,x1,x2,xlx1,x2x2,x1x2);

if (flag==0){Var. y.hat .D1 <- (gx X+X Var .tau.hat.D1 X+X t(gx))+n














Var.y.hat.D2 <- (gx X+X Var.tau.hat.D2 X+X t(gx))+n2

flag <- 1}else{Varyhat.D1 <- (gx X+X Var.tau.hat.D1 X+X t(gx))+n

Var. y. hat .D1 <- rbind(Var. y. hat. D1, Varyhat. D1)

Varyhat.D2 <- (gx X+X Var.tau.hat.D2 X+X t(gx))+n2

Var.y.hat.D2 <- rbind(Var. y.hat.D2,Varyhat .D2)};


};###################


# Calculate the quantiles for Design 1 and 2. Note: quantile

function produces a list, hence we use unlist to create arrays.


qi.1
qi.D2<-quantile(Var .y.hat.D2,probs=c(. 1,1,5,10,15,25,30, 35,40,45,50,55,60 ,65,70,75,80, 85,90,95,100)/100) ;


if(check. eta. flag==0){q.D1 <- array(unlist(qi.D1) ,c(1,1ength(qi.D1)) )

q. D2 <- array(unlist(qi. D2), c(1,1length(qi. D2)))

eta.1ist <- eta

check. eta. flag <- 1}else~q2 .D1<- array(unlist(qi.D1) ,c(1,1ength(qi.D1)) )

q.D1 <- rbind(q.D1,q2.D1)

q2 .D2<- array(unlist(qi.D2) ,c(1,1length(qi.D2)))

q.D2 <- rbind(q.D2,q2.D2)

eta.1ist <- rbind(eta.1ist,eta)};

eta <- eta + eta.interval;

} # end of eta loop noquantile.pts <- length(qi.D1l);

quantile.1list<-array(c( .1,1,5,10,15,25,30 ,35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100) ,dim=c(noquantile .pts,1));



# Saving the min and the max quantiles for each design



min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

for(i in 1:noquantile.pts){min.quantile.D1[i] <- min(q.D1[,i])

max.quantile.D1[i] <- max(q.D1[,i])};

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

for(i in 1:noquantile.pts){min.quantile.D2[i] <- min(q.D2[,i])

max.quantile.D2[i] <- max(q.D2[,i])};

if (check.1lambda. flag==0){plot .D1 <- cbind(lambda,quantile .1ist,min. quantile.D1,max .quantile.D1)

plot. D2 <- cbind(lambda, quantile.1ist ,min. quantile. D2, max. quantile. D2)

check.1ambda.flag <- 1

}else~plotl.D1 <-cbind(lambda,quantile .1ist,min. quantile.D1,max. quantile.D1)

plot.D1 <- rbind(plot.D1,plotl.D1)

plotl.D2 <- cbind(lambda,quantile .1ist,min.quantile .D2,max. quantile .D2)

plot.D2 <- rbind(plot.D2,plotl.D2)
















lambda <- lambda + lambda.skip; } # End if for lambda



# # Saving the values to a permanent data set in a tab

delimited .txt file where # first column is for lambda, second

column is for the quantiles, third column is the min # value of

the quantile for the design, 4th col is for the max quantile. # #

Saved in 2 files for design 1 and design 2 #


write##plotD1),fie="pltD1.tt",nclumns=, sep"\t")

write(t(plot.D2),f ile="plot_Dl. txt",ncolumns=4, sep="\t") ;




# To read data from the text file---------- PLOT #

rows <- 21+5; # We have 21 quantiles, 5 lambda

plot .D1 <- matrix(scan( "plot_D1.txt", n =rows*4), rows, 4, byrow=TRUE) ;

plot .D2 <- matrix(scan( "plot_D5 .txt", n =rows*4), rows, 4, byrow=TRUE) ;

noquantile.pts <.- 21;



# Read in the min and max for the X and Y axis



min.quantile <.- min(min(plot.D1[,3] ),min(plot.D2[C,3]))

max.quantile <.- max(max(plot.D1[,4] ),max(plot.D2[C,4]))

ymin <.- min.quantile 0.01

ymax <.- max.quantile + 2



# # Read in min and max quantiles for each lambda for designs 1

and 2, and create plots



Ior(k in 1:5){

if (k == 1) {

pdf (f ile="qDG_designiandD5diff 1.pdf ")}

else~if (k == 2){

pdf (f ile="qDG_designiandD5diff 2 .pdf ")}

else~if (k == 3){

pdf (f ile=" qDG_designiaandD5diff 3.pdf )}

else~if (k == 4){

pdf (f ile="qDG_designiandD5diff4.pdf ")}

else{

pdf (f ile=" qDG_des igniandD5dif f5 .pdf )

















row <;- 21 + (k-1) + 1;

lambda.val <- plot.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1) ; # To

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.D2[21 + (k-1) +

max.quantile.D2[i] <- plot.D2[21 + (k-1) + :

min.quantile.D1[i] <- plot.D1[21 + (k-1) +

max.qiuantile.D1[i] <- plot.D1[21 + (k-1) +

qiuantile.1ist~i] <- plot.D1[21 + (k-1) + i,:


create points at 0, 0.1, 0.2..


i,3]

i,4]

i,3]

i,4]

2]/100;


f or (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1[C2i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



min. quantile <- min(min(min. quantile. D1), min(min. quantile. D2))

max. quantile <- max(max(max. quantile. D1), max(max.qunantile. D2))

ymin <- min.quantile 0.01 # ymax <- max.quantile + 2.00

plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p",

ylab, = "Quantiles of Scaled Prediction Variance" ,main=substitute(lambda == lambdaval,

list (lambdaval=1ambda. val)) ,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

points(quantile .points,min.qunantile .points .D2,pch=17, col="red") ;

points(quantile .points,max.qunantile .points .D2,pch=17, col="red") ;

points(quantile .points,min.qunantile .points.D1,pch=16, col="blue") ;

points(quantile .points,max.qunantile .points.D1,pch=16, col="blue") ;














legend( "topleft ",c("Arbitrary Design in all blocks","Design 1"),1ty=c(1,2),pch = c(17,16),cex=0.7);

dev.off();

}; #dev.off(); pdf(f ile="qDG_designiandD5diff .pdf "); # Saving plot

to a pdf file #pdf (f ile="qDG_designiandD5dif I_samepage .pdf ");

#par(mfrow=c(2,3)); # multiple plots in a page for(k in 1:5){

row <;- 21 + (k-1) + 1;

lambda.val <- plot.D1[row,1];

min. quantile. D1 <;- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <;- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <;- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <;- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <;- matrix(O,11,1); # To create points at 0, 0.1, 0.2..

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.D2[21 + (k-1) + i,3]

max.quantile.D2[i] <- plot.D2[21 + (k-1) + i,4]

min.quantile.D1[i] <- plot.D1[21 + (k-1) + i,3]

max.quantile.D1[i] <- plot.D1[21 + (k-1) + i,4]

quantile.1ist~i] <- plot.D1[21 + (k-1) + i,2]/100;



f or (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1[C2+i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2+i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



# min. quantile <- min(min(min. quantile. D1) ,min(min. quantile. D2))

# max. quantile <- max(max(max. quantile. D1) ,max(max.qunantile. D2))

# ymin <- min.quantile 0.01 # ymax <- max.quantile + 1.00

plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p",

ylab, = "Quantiles of Scaled Prediction Variance",

main=substitute(lambda == lambdaval,1list(lambdaval=1ambda. val)) ,type="l",1Ity=1, col="red") ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

points(quantile .points,min.qunantile .points .D2,pch=17, col="red") ;














points(quantile .points,max.qu~antile .points .D2,pch=17, col="red") ;

points(quantile .points,min.qu~antile .points.D1,pch=16, col="blue") ;

points(quantile .points,max.qu~antile .points.D1,pch=16, col="blue") ;

legend( "topleft ",c("Arbitrary Design in all blocks","Design 1"),1ty=c(1,2),pch = c(17,16),cex=0.7);

}; dev.off();













APPENDIX B

R CODE USED IN CHAPTER 3



# Generate QDG for two designs note that the design criterion

now is the non-centrality parameter and the # power function.



# Read in the values of X1 and X2 from the dataset to create the

design matrix X # for the first design. # We can compute the MSE

for this design. # # NOTE : We use the MSE for this design

throughout



#1ibrary(nlme); library(MASS); #Since we need to compute inverse,

ginv() is in MASS source("bdiag.txt"); X1 <-

matrix(scan( "powerdata-D1. txt", n =60+4), 60, 4,byrow=TRUE) ; xl

<- X1[,1]; x2 <- X1[,2]; xlx1 <- X1[,1]+X1[,1]; x2x2

<-X1[,2]+X1[,2]; x1x2 <=- X1[,1]+X1[,2];



# Define the test matrix C



tC <- array(c(0,0,0,1,0 ,0,0,0,0,0,1,0) ,dimn=c(6,2)) ; C <- t(tC) ;

c .df <-length(C[, 1]);



# Read in the design matrix for the two designs



X.D1 <- cbind(1,x1,x2,xlx1 ,x2x2,x1x2) ; n.D1 <- length(X.D1[,1]) ;



X1 <- matrix(scan( "D3data. txt", n =60+3), 60, 3, byrow=TRUE) ;

X.D2

<-cbind(lX C1,C1,X11]X[2,X 1[,1]+X1[,1],X1[,]X[2,1,]X[2)

n.D2 <- length(X.D2[,1]);



df.D1 <- n.D1 5 1; df.D2 <- n.D2 5 1 ; # degrees of

freedom = n-p-1



# Drop necessary matrices



rm(X1); rm(x1); rm(x2); rm(xlx1); rm(x2x2); rm(xlx2);



# To vary the plot over different ranges of r



r.max <- 0.03; r.min <- 0.01; r.interval <- r.max r.min;














r.no <-4; r.skip <- r.i~nterval/r.no;

r.values <- 0.01; for (i in 1:r.no){

r.incre <(- r.min + i~r.skip;

r.values <- rbind(r.values,r.incre);

}; no.of .r <- length(r.values); r.f lag <- 0; # Just a flag to set

for all r values alpha <- 0.05;



# start loop for different values of r



Ior(i in 1:no.of .r){



# Within each r, to generate quantiles Ior the eta



r <- r.values~i];

check.eta.f lag <- 0;

sigma.values <- read.table("sigma_values.txt" header = FALSE, sep = "");

no.sigma <- length(sigma.values[C,1])



# Start eta loop



f or(sig in 1:no.sigma){

gamm~a <- sigma.values Csig,2] ;

epsi <- sigma.values~sig,1] ;

eta <- sigma.values~sig,2]/sigma.values~sig,1];

V12 <- diag(rep(1,12)) + etao(matrix(1,12,12));

V16 <- diag(rep(1,16)) + etao(matrix(1,16,16));

V15 <- diag(rep(1,15)) + etao(matrix(1,15,15));

mats1 <- list(V16,V16 ,V16,V12) ; # Creating the list of elements

V.D1 <- bdiag(mats1);

mats2 <-- list(V15,V15,V15,V15);

V.D2 <- bdiag(mats2);

rm(V12);

rm(V16);

rm(V15);

rm(mats1) ;

rm(mats2) ;



# Define inv[C(X'V-XC'] /sigma2 f or the two designs. Next is to generate

# random numbers Ior delta and obtain the dataset



CVX.D1 <- ginv(C X+X ginv(t(X.D1) X+X ginv(V.D1) X+X X.D1) X+X t(C))/epsi;














CVX.D2 <- ginv(C X+X ginv(t(X.D2) X+X ginv(V.D2) X+X X.D2) X+X t(C))/epsi;


delta.flag <- 0;

delta.no <- 1000;

for(i in 1:delta.no){

psi <- (180/pi)+runif (1,0,2+pi)

deltal <- r + cos(psi)

delta2 <- r + sin(psi)

delta <- rbind(deltal,delta2)

noncentr.par.D1 <- t(delta) X+X CVX.D1 X+X delta;

noncentr.par.D2 <- t(delta) X+X CVX.D2 X+X delta;

Ialpa.D1 <- qf (alpha,c.df ,df .D1,lower.tail = FALSE,log.p = FALSE);

Ialpa.D2 (<- qf (alpha,c.df ,df .D2,lower.tail = FALSE,log.p = FALSE);

powr.D1 (<- pf (f alpa.D1,c.df ,df .D1,ncp=noncentr.par.D1lwrti = FALSE,log.p = FALSE);

powr.D2 (<- pf (f alpa.D2,c.df ,df .D2,ncp=noncentr.par.D2lwrti = FALSE,log.p = FALSE);

z.D1 <- powr.D1/(1-powr.D1);

z.D2 <- powr.D2/(1-powr.D2);

if (delta.f lag==0){noncentral.D1 <- noncentr.par.D1

noncentral.D2 <- noncentr.par.D2

power.D1 <- powr.D1

power.D2 <- powr.D2

Zplot.D1 <- z.D1

Zplot.D2 <- z.D2

delta.f lag <- 1}else~noncentral .D1 <- rbind(noncentral.D1 ,noncentr .par.D1)

noncentral.D2 <- rbind(noncentral.D2 ,noncentr. par. D2)

power.D1 <- rbind(power.D1 ,powr.D1)

power.D2 <- rbind(power.D2,powr.D2)

Zplot.D1 <- rbind(Zplot.D1,z.D1)

Zplot.D2 <- rbind(Zplot.D2,z.D2)}


};###################


# Generate the quantiles Ior each eta Ior power and noncentrality parameter


ncp.D<-uatie#nncntalD1prbs#(.,15,0,5,5,3,3,4,4,5,55606570758085,0,5,00/10)

ncpl .D2<-quantile(noncentral .D2,probs=c(.1,1,5, 10,15,25,30,35,40 ,45,50,55,60 ,65,70,75,80, 85,90,95,100)/100) ;

npol .D1<-quantile(powenr.D1.D,probs=c(.1,1,5, 10,15,25,30,35,40 ,45,50,55,60 ,65,70,75,80, 85,90,95,100)/100); ;

powl .D2<-quantile(power .D2,probs=c( .1,1,5,10,15,25,30 ,35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100)/100) ;

zol .D1<-quantile(Zplote.D1,probs=c( .1,1,5,10,15,25,30 ,35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100)/100) ;

zl .D2<-quantile(Zplot .D2,probs=c( .1,1,5,10,15,25,30, 35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100)/100) ;


if (check. eta. flag==0){ncp.D1 <- array(unlist(ncpl .D1),c(1,1length(ncpl.D1)))

ncp.D2 <- array(unlist(ncpl.D2),c(1,1ength(ncpl.D2))














pow. D1 <- array(unlist(powl. D1), c(1,1length(powl. D1)))

pow.D2 <- array(unlist(powl.D2),c(1,1ength(powl.D2))

powtrans.D1 <- array(unlist(zl.D1) ,c(1,1ength(zl.D1)) )

powtrans .D2 <- array(unlist(zl.D2) ,c(1,1length(zl.D2)))

eta.1ist <- eta

check.eta.flag <- 1}else~ncp2.D1<- array(unlist(ncpl.D1) ,c(1,1ength(ncpl.D1)))

ncp.D1 <- rbind(ncp.D1,ncp2.D1)

ncp2.D2<- array(unlist(ncpl.D2),c(1,1ength(ncpl.D2))

ncp.D2 <- rbind(ncp.D2,ncp2.D2)

pow2.D1<- array(unlist(powl.D1),c(1,1ength(powl.D1))

pow.D1 <- rbind(pow.D1,pow2.D1)

pow2.D2<- array(unlist(powl.D2),c(1,1ength(powl.D2))

pow.D2 <- rbind(pow.D2,pow2.D2)

powtrans2 .D1<- array(unlist(zl.D1) ,c(1,1ength(zl.D1)) )

powtrans. D1 <- rbind(powtrans. D1, powtrans2. D1)

powtrans2 .D2<- array(unlist(zl.D2) ,c(1,1length(zl.D2)))

powtrans. D2 <- rbind(powtrans. D2, powtrans2. D2)

eta.1ist <- rbind(eta.1ist,eta)};

} # end eta loop

noquantile.pts <- 21;

quantile.1list<-array(c(.1,1 ,5,10,15,25,30,35 ,40,45,50,55,60,65 ,70,75,80,85,90,95, 100) ,dim=c(noquantile .pts,1));



# Saving the min and the max quantiles for each design

min. quant ile. ncp.D1 <- matrix(O,noquant ile.pts,1) ;

max. quant ile. ncp.D1 <- matrix(O,noquant ile.pts,1) ;

min. quant ile. ncp.D2 <- matrix(O,noquant ile.pts,1) ;

max. quant ile. ncp.D2 <- matrix(O,noquant ile.pts,1) ;

min. quant ile. pow.D1 <- matrix(O,noquant ile.pts,1) ;

max. quant ile. pow.D1 <- matrix(O,noquant ile.pts,1) ;

min. quant ile. pow.D2 <- matrix(O,noquant ile.pts,1) ;

max. quant ile. pow.D2 <- matrix(O,noquant ile.pts,1) ;

min. quantile .powtrans .D1 <- matrix(O,noquantile .pts,1);

max. quantile .powtrans .D1 <- matrix(O,noquantile .pts,1);

min. quantile .powtrans .D2 <- matrix(O,noquantile .pts,1);

max. quantile .powtrans .D2 <- matrix(O,noquantile .pts,1);

for(i in 1:noquantile .pts){min.quantile .ncp.D1[i] <- min(ncp.D1[C,i])

max .quantile .ncp.D1[i] <- max(ncp.D1C, i])

min.quantile .pow.D1[i] <- min(pow.D1C, i])

max .quantile .pow.D1[i] <- max(pow.D1C, i])

min. quantile. powtrans. D1[i] <- min(powtrans. D1[, i])

max. quantile. powtrans. D1[i] <- max(powtrans. D1[, i])














min.quantile.ncp.D2[i] <- min(ncp.D2[,i])

max.quantile.ncp.D2[i] <- max(ncp.D2[,i])

min.quantile.pow.D2[i] <- min(pow.D2[,i])

max.quantile.pow.D2[i] <- max(pow.D2[,i])

min. quantile.powtrans.D2 [i] <- min(powtrans.D2 [,i i)

max.quantile.powtrans.D2[i] <- max(powtrans.D2[,i] )};

if(r. flag==0){plot .ncp.D1 <- cbind(r,quantile.1list ,min.quantile.ncp .D1,max.quantile .ncp.D1)

plot. ncp.D2 <- cbind(r,quant ile.1ist,min. quantile. ncp. D2,max. quantile. ncp. D2)

plot .pow.D1 <- cbind(r,quantile.1list ,min. quantile .pow.D1,max. quantile .pow.D1)

plot. pow.D2 <- cbind(r,quant ile.1ist,min. quantile. pow. D2,max. quantile. pow. D2)

plot. powtrans. D1 <- cbind(r, quantile.1list,min. quant ile.powtrans. D1,max. quantile.powtrans. D1)

plot. powtrans. D2 <- cbind(r, quantile.1list,min. quant ile.powtrans. D2,max. quantile.powtrans. D2)

r.flag <- 1

}else~plotl.ncp.D1 <-cbind(r,quantile.1list ,min. quantile .ncp.D1,max. quantile .ncp.D1)

plot.ncp.D1 <- rbind(plot.ncp.D1,plotl.ncp.D1)

plotl.ncp.D2 <-cbind(r,quantile.1list,min.quantile .ncp.D2,max. quantile.ncp.D2)

plot.ncp.D2 <- rbind(plot.ncp.D2,plotl.ncp.D2)

plotl.pow.D1 <-cbind(r,quantile.1list ,min. quantile .pow.D1,max. quantile .pow.D1)

plot.pow.D1 <- rbind(plot.pow.D1,plotl.pow.D1)

plotl.pow.D2 <-cbind(r,quantile.1list,min.quantile .pow.D2,max. quantile.pow.D2)

plot.pow.D2 <- rbind(plot.pow.D2,plotl.pow.D2)

plotl.powtrans .D1 <-cbind(r,quantile.1list ,min.quantile .powtrans.D1,max .quantile .powtrans.D1)

plot .powtrans .D1 <- rbind(plot .powtrans.D1,plotl .powtrans.D1)

plotl.powtrans.D2 <-cbind(r,quantile.1list,min.quantile .powtrans.D2,max .quantile .powtrans.D2)

plot .powtrans .D2 <- rbind(plot .powtrans .D2,plotl.powtrans .D2)}

} # end loop for r



# Save the matrices in a tab delimited file



write(t(plot .ncp.D1),f ile="plot_noncentralityD1 .txt" ,ncolumns=4,sep="\t" );

write(t(plot .ncp.D2),f ile="plot_noncentralityD2 .txt" ,ncolumns=4,sep="\t" );

write(t(plot .pow.D1),f ile="plot_powerDl. txt" ,ncolumns=4,sep="\t") ;

write(t(plot. pow. D2), file="plot_powerD2. txt ", ncolumns=4, sep=" \t ");

write(t(plot .powtrans.D1),f ile="plot_powertransD1 .txt" ,ncolumns=4, sep="\t") ;

write(t(plot .powtrans .D2),f ile="plot_powertransD2 .txt" ,ncolumns=4, sep="\t") ;



# Once the data has been saved, the above part of the program need

not be run. # The next part is to get the plots from the data



r.no <- 5;

noquantile.pts <- 21;
































KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK


min.quantile <- min(min(plot.D1[,3] ),min(plot.D2[C,3]))

max.quantile <- max(max(plot.D1[,4] ),max(plot.D2[C,4]))

ymin <- min.quantile

ymax <- max.quantile



# # Read in min and max quantiles for each lambda for designs 1

and 2, and create plots


rows <- noquantile.pts~r.no; # We have 21 quantiles, 5 r

plot. D1 <- matrix(scan( "plot_noncentral ityD1. txt ", n =rows*4), rows, 4, byrow=TRUE) ;

plot. D2 <- matrix(scan( "plot_noncentral ityD2. txt ", n =rows*4), rows, 4, byrow=TRUE) ;

#png(file=" qDGncp_rotorth. png") ; # Saving plot to a pdf file

png(f ile="PowerqDG_D3_ncpsamepage .png") ; # Saving plot to a png

file #pdf (f ile="PowerqDG_D3_ncpsamepage_dif range .pdf ") ; # Saving

plot to a png file

#pdf (f ile= "PowerqDGncpdif fpage_rot orth_dif frange .pdf );

par(mfrow=c(2,3)); # multiple plots in a page


# Read in the min and max for the X and Y axis


for(k in 1:r.no){

row <- 21 + (k-1) + 1;

r.val <- plot.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1) ; # To

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.D2[21 + (k-1) +

max.quantile.D2[i] <- plot.D2[21 + (k-1) + :

min.quantile.D1[i] <- plot.D1[21 + (k-1) +

max.quantile.D1[i] <- plot.D1[21 + (k-1) +

quantile.1ist~i] <- plot.D1[21 + (k-1) + i,:



f or (i in 1: 11) {


create points at 0, 0.1, 0.2..


i,3]

i,4]

i,3]

i,4]

2]/100;














min. quantile. points. D1[i] <- min. quantile. D1[C2i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p",

ylab = "Quantiles of ncp" ,main=substitute(radius == lambdaval,

list(lambdaval=r .val)) ,type="l",1Ity=1, col="red") ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

points(quantile .points,min. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,max. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,min. quantile .points.D1,pch=16, col="blue") ;

points(quantile .points,max. quantile .points.D1,pch=16, col="blue") ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch = c(17,16), cex=0.7) ;

}; dev.off();

min. quantile <- min(min(min. quantile.D1) ,min(min. quantile. D2))

max. quantile <- max(max(max .quantile.D1) ,max(max. quantile.D2))

ymin <- min.quantile 0.0001

ymax <- max.quantile

if ((k == 4)) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 400000}

else{

if (k == 2) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 100000



else{

if (k == 3) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 200000



else{

ymin <- min.quantile 0.0001

ymax <- max.quantile + 5000














fftftftftftftftftftftftftftftftftftftftf


r.no <- 5;

noquantile.pts <- 21;

rows <- noquantile.pts~r.no; # We have 21 quantiles, 20 r

plot.D1 <- matrix(scan("plot_powertransD1.txt", n =rows*4), rows, 4, byrow=TRUE) ;

plot .D2 <- matrix(scan( "plot_powertransD2. txt", n =rows*4), rows, 4, byrow=TRUE) ;

#png(file=" qDGpowertrans_rotorth .png") ; # Saving plot to a png

file png(file=" qDG_D3_powertranssamepage .png") ; # Saving plot to a

png file #pdf (f ile=" qDG_D3_powertranssamepage_dif frange.pdf ") ; #

Saving plot to a png file

#png(f ile=" qDGpowertransdif fpage_rotorth_dif frange .png" ); # Saving

plot to a png file par(mfrow=c(2,3)); # multiple plots in a page


min.quantile <- min(min(plot.D1[,3] ),min(plot.D2[C,3]))

max.quantile <- max(max(plot.D1[,4] ),max(plot.D2[C,4]))

ymin <- min.quantile

ymax <- max.quantile



# Read in min and max quantiles for each r for designs 1 and 2,

and create plots



for(k in 1:r.no){

row <- 21 + (k-1) + 1;

r.val <- plot.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1); # To create points at 0, 0.1, 0.2..

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.D2[21 + (k-1) + i,3]

max.quantile.D2[i] <- plot.D2[21 + (k-1) + i,4]


# PLOT FOR POWER transformation


# Read in the min and max for the X and Y axis














min.quantile.D1[i] <- plot.D1[21 + (k-1) + i,3]

max.quantile.D1[i] <- plot.D1[21 + (k-1) + i,4]

quantile.1ist~i] <- plot.D1[21 + (k-1) + i,2]/100;



f or (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1[C2i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p",

ylab = "Quantiles of transformed power" ,main=substitute(radius == lambdaval,

list(lambdaval=r .val)) ,type="l",1Ity=1, col="red") ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

points(quantile .points,min. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,max. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,min. quantile .points.D1,pch=16, col="blue") ;

points(quantile .points,max. quantile .points.D1,pch=16, col="blue") ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch = c(17,16), cex=0.7) ;

}; dev.off();

min. quantile <- min(min(min. quantile.D1) ,min(min. quantile. D2))

max. quantile <- max(max(max .quantile.D1) ,max(max. quantile.D2))

ymin <- min.quantile 0.001

ymax <- max.quantile

if ((k == 4)) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 0.0003}

else{

if (k == 2) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 0.2



else{

if (k == 3) {

ymin <- min.quantile 0.0001

ymax <- max.quantile + 0.01



else{














ymin <- min.quantile 0.0001

ymax <- max.quantile + 0.3









# PLOT FOR POWER



r.no <- 5;

noquantile.pts <- 21;

rows <- noquantile.pts~r.no; # We have 21 quantiles, 20 r

plot .D1 <- matrix(scan( "plot_powerD1. txt", n =rows*4), rows, 4, byrow=TRUE) ;

plot .D2 <- matrix(scan( "plot_powerD2. txt", n =rows*4), rows, 4, byrow=TRUE) ;

#pdf (f ile=" QDGpower_orth .pdf ") ; # Saving plot to a png file

png(file=" qDGpower_D3_samepage .png") ; # Saving plot to a png file

#png(file=" qDGpowersamepage_diffrnertrhpng"); # Saving plot

to a png file #pdf (f ile=" qDGpower_D3_samepage_dif frange.pdf ") ;

#Saving plot to a png file par(mfrow=c(2,3)); # multiple plots in

a page



# Read in the min and max for the X and Y axis



min.quantile <- min(min(plot.D1[,3] ),min(plot.D2[C,3]))

max.quantile <- max(max(plot.D1[,4] ),max(plot.D2[C,4]))

ymin <- min.quantile

ymax <- max.quantile + 0.24



# Read in min and max quantiles for each r for designs 1 and 2,and

create plots



for(k in 1:r.no){

row <- 21 + (k-1) + 1;

r.val <- plot.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1); # To create points at 0, 0.1, 0.2..

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;
































Ior (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1[C2i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p",

ylab = "Quantiles of power" ,main=substitute(radius == lambdaval,

list(lambdaval=r .val)) ,type="l",1Ity=1, col="red") ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1,col="red") ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2,col="blue") ;

points(quantile .points,min. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,max. quantile .points .D2,pch=17, col="red") ;

points(quantile .points,min. quantile .points.D1,pch=16, col="blue") ;

points(quantile .points,max. quantile .points.D1,pch=16, col="blue") ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch = c(17,16), cex=0.7) ;

}; dev.off();


min. quantile <- min(min(min. quantile.D1) ,min(min. quantile. D2))

max. quantile <- max(max(max .quantile.D1) ,max(max. quantile.D2))

ymin <- min.quantile 0.001

ymax <- max.quantile + 0.21

if (k==4) {

ymax <- max.quantile + 0.01

}else{

if (k==5) {

ymax <- max.quantile + 0.0002

}else{


quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.D2[21 + (k-1) + i,3]

max.quantile.D2[i] <- plot.D2[21 + (k-1) + i,4]

min.quantile.D1[i] <- plot.D1[21 + (k-1) + i,3]

max.quantile.D1[i] <- plot.D1[21 + (k-1) + i,4]

quantile.1ist~i] <- plot.D1[21 + (k-1) + i,2]/100;


# Read in the min and max for the X and Y axis















ymax <- max.quantile + 0.23



















# Program to generate a G-optimal design


library(nlme); library(MASS); #Since we need to compute inverse,

ginv() is in MASS library(lasso2); # to calculate the trace



# Specify the initial values for the fixed effects and variance

components # parameters



# Variance Components sigma.sq.gamma.hat <- 9 ; MSE <- 1 ; # Fixed

Effects bO <- 1 ; bl <- 2 ; b2 <- 2 ; b11 <- 1 ; b22 <- 1 ; bl2 <-

1 ; fe <- rbind(b0,bi,b2,bl1,b22,bl2);



# Specify the number of blocks and number of points in each block

to start with. # NOTE : The number of points per block remains same

through out.



no.blocks <- 2; no.points.block <- 5;



# Specify/Calculate some more miscellaneous parameters



eta.hat <- sigma.sq.gamma.hat/MSE; eta.star <-

sigma.sq.gamma.hat/(MSE+sigma.sq.gamma.hat; sd.gamma <-

sqrt(sigma.sq.gamma.hat) ; sd.error <- sqrt(MSE); fp <- 0 ; # Trace

- just specify an arbitrary value iteration.no<- 0; run.time <-

date() ;



# Specify the starting design points # # NOTE : The total number

of inputs should be equal to number of blocks times # number of

points per block (as specified above) #



x1.design.pts <- matrix(c(-0.7, -0.2, 0, 0.5, 0.2, -0.6, -0.7, 0,

0.8, 0.2), nrow=no.points.block,ncol=no.blocks); x2.design.pts <-

matrix(c(-0.8, -0.9, 0, 0.8, 0.9, 0.5, 0.7, 0, -0.7, -0.9),

nrow=no.points.block,ncol=no.blocks);



# Generate the response values based on initial input (parameters


Call required libraries


APPENDIX C

R CODE USED IN CHAPTER 4
















KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK


# Save the values for eta hat for the very first time we fit a

model based on a standard dataset



fp.block.MSE.gamma <-

cbind(iteration.no ,no .blocks,fp,MSE,sigma, sq.gamma.hat, eta.hat,run.time)

write(t (fp.block.MSE. gamma),f ile="Values2 .txt" ,ncolumns=7, sep="\t") ;



#ul.unique <- matrix(O,no.points.block,1) ; #u2.unique

<-matrix(O,no. points.block,1) ; #ul.unique <- x1.design.pts[C,1] ;

#u2.unique <- x2.design.pts[,1]; #n.i <- 12; # no of times the

unique point is repeated ul.unique <- x1.design.pts[,1]; u2.unique

<- x2.design.pts[,1]; unique.flag <- 0; for(i in 1:no.blocks-1){

uniq.chk <- 5;

for(j in (i+1):no.blocks){

diff.x1 <- abs(x1.design.pts[,j] x1.design.pts[,i]);

diff .x2 <- abs(x2.design.pts[,j] x2.design.pts[,i]);

if ((sum(diff I.xl) == 0)&&(sum(diff .x2)==0)){

uniq.chk <- 1





if (uniq. chk == 5) {

if (unique.f lag ==0){

ul.unique <- x1.design.pts[,i];

u2.unique <- x2.design.pts[,i];

unique.flag <- 1;

}else{

ul.unique <- cbind(ul.unique,x1. design.pts[C,i]) ;


and design points)


for (i in 1:no.blocks){

xl <- x1.design.pts[,i]

x2 <- x2.design.pts[,i]

y.append <- cbind(1,x1,x2,x1+x1,x2+x2,x1+x2) X+X fe ;

y. append <- y. append +rnorm(no .points.block,0, sd. gamma) + rnorm(no .points .block,0,sd. error) ;

if (i == 1) {

X <- cbind(x1, x2, i, y.append)

}else{

X <- rbind(X, cbind(x1, x2, i, y.append))
































































































no.uniq <- length(n. i) ; u <- matrix(O,no .points.block,2) ; M.design


u2.unique <- cbind(u2.unique,x2. design.pts[C,i]) ;





} # end Ior i in 1 ul.unique <-

cbind(ul.unique,x1 .design.pts [,no.blocks]) ; u2.unique <-

cbind(u2.unique,x2.design.pts [,no.blocks]); no.unique <-

length(ul.unique[1,]) ; n.i <- matrix(0,no.unique,1) ; Ior(i in

1:no.unique) {

f or(j in 1:no.blocks){

if ((sum(abs(ul .unique[C,i]-x1.design.pts[C,j])) == 0) & (sum(abs(u2.unique[C,i] -x2.design.pts[C,j])) == 0))









# Iteration number Starting is 1, now can specify the maximum

number of iteration required



iteration.no <- 1; #max.iteration.no <- 81; # Always specify one

more than the desired number of max so if you want 10

iterations, specify 11 #while(iteration.no < max.iteration.no){

fp.tol <- 2; # Specifying an arbitrary start value while(fp.tol >

0.01){



# Re calculate the value of eta and continue from here



xl <- X[,1]; x2 <- X[,2]; xlx1 <- x1+x1; x2x2 <- x2+x2; x1x2 <-

x1+x2; y <- X[,4]; block <- X[,3]; qDGseq <-

data.frame(cbind(x1,x2,x1x1,x2x2,x1x2,blok); fm <- Ime(y ~ xl

+ x2 + x1+x1+x2xx2 +x, 2 data=qDGseq, random = ~1 block) ; V

<- array(VarCorr(fm) ,c(2,2)); # Get the variance components fe <-

array(f ixed .ef fects(fm) ,c (6 ,1) ); sd .error <- as .numeric (V[C2 ,2] );

sd.gamma <- as.numeric(V[1,2]); var.error <- as.numeric(V[2,1]);

MSE <- var.error; sigma.sq.gamma.hat <- as.numeric(V[1,1]);

eta.hat <- sigma.sq.gamma.hat/MSE;



V <- diag(rep(1,no .points.block)) + eta.hat +

matrix(1,no .points .block,no .points .block) ;



Compute the moment matrix














<- matrix(O,6,6); for(i in 1:no.uniq){

u[,1] <- ul.unique[C,i]

u[,2] <- u2.unique[C,i]

G.prime <- cbind(1~C~l,C,21],u[,2],u[,1]+u[,21],u[]u,]u,]u,]

M.design <- M.design + n.i~i]+(t(G.prime)%+Xginv(V)%+XG.prime)

}; M.design <- (1/no.blocks) + M.design;



# Check for non-singularity



det.design <- det(M.design) ; if (det.design == 0){

error <- "Design matrix is singular";

error


}####################


# Create the initial pot set of N points


N <- 50 ; # number of points in the initial pot x1.expt <-

matrix(O, no. points. block, N) ; x2. expt

<-matrix(O,no.points.block,N); fval <- matrix(O,N,1); x1.min <-

-1; x1.max <- 1; x2.min <- -1; x2.max <- 1; for(i in 1:N){

x1.expt[C,i] <- runif(no .points.block,x1 .min,x1.max)

x2.expt[C,i] <- runif(no.points. block,x2.min,x2. max)

gp <-cbind(1,x1.expt [,i] ,x2.expt [,i i,x1.expt [,i i]x1.expt [, i] ,x2.expt [,i]+*x2.expt [,i i,x1.expt [,i i]x2.expt [, i])

fval~i] <- tr(ginv(V)%+X(gp%+Xginv(M. design)%+Xt(gp))) ;

} tol <- 100; while(tol > 0.0001){



# Select 3 distinct points from the stored points # p.flag is to

select the 3 distinct points



p.flag <- 1; while(p.flag == 1){

pos1 <- round(runif (1,1,N))

pos2 <- pos1

while(pos2 == pos1){

pos2 <- round(runif (1,1,N))



pos3 <- pos2

while((pos3 == pos1) I (pos3 == pos2)){

pos3 <- round(runif (1,1,N))





# Calculate the centroid













fftftftftftftftftftftftftftftftftftftftf


G <- matrix(O, no. points. block,2) ;

G [,1] <- 0.5 + (x1.expt[C,posl]+x1.expt[C,pos2])

G [,2] <- 0.5 + (x2.expt[C,posl]+x2.expt[C,pos2])

P <- 2 + G cbind(x1.expt[C,pos3] ,x2.expt[C,pos3])

p.flag <- 0 # Check if P within bounds

for(i in 1:no.points.block){

if((P~i,1] < x1.min) || (P~i,1] > x1.max) || (P~i,2] < x2.min) || (P~i,2] > x2.max)){

p.flag <- 1





} # end while for P flag gp

<-cbind(lP,1,P[,IP,1],P[,2],P[,21]+[1,[2+,]P,]P,]

fp <- tr(ginv(V)%+X(gp%+Xginv(M.design)%+Xt(gp))



# Get the minimum value of function in the N points converse of

Price # Price algo reduces the maximum in the set to the minimum

...... hence finds maximum and # replaces by new I-value. We

want to maximize the function. Hence, we get the minimum # and

replace it by new function value if it is greater ... i.e.

increasing our scope of minimum # towards the maximum



min.f <- min(fval) min.pos <- which.min(fval) # Get the position

for the minimum value if(fp > min.f){ # If the mi

f val [min.pos] <- f p

x1.expt[,min.pos] <- P[,1]

x2.expt[,min.pos] <- P[,2]

} max.f <- max(fval) min.f <- min(fval) tol <- abs(max.f min.f)

} #end Price algo run.time <- date();



# Generate response for the new set based on previous estimates



y. append <- matrix(O,no .points.block, 1);

y.append <- cbind(1,P[C,1] ,P[,2] ,PC,1]+PC,1] ,P[,2]+PC,2] ,P[C,1]+P[,2]) X+X fe;

y .append <- y. append +rnorm(no.points .block,0,sd.gamma) +

rnorm(no.points.block,0,sd.error); no.blocks <- no.blocks + 1; #

increment the number of blocks fp.block.MSE.gamma <-

rbind(fp.block .MSE .gamma,cbind(iteration.no ,no .blocks,fp,MSE, sigma. sq.gamma.hat, eta.hat,run. time))

iteration.no <- iteration.no + 1; # increment the number of

iteration X <- rbind(X,cbind(P[C,1] ,P[,2] ,no.blocks,y. append)) ; #

Append to the old dataset ul.unique <- cbind(ul.unique,P[, 1]);














u2.unique <- cbind(u2.unique,P[,2]); n.i <- rbind(n.i,1); n <-

length(n. i) ; write(t(X), file="X_augmented2. txt ", ncolumns=4,

sep="\t ");

write(t (fp.block.MSE. gamma),f ile="Values2 .txt" ,ncolumns=7, sep="\t") ;

write(t(ul.unique) ,file="ul_unique2 .txt",ncolumns=n, sep="\t") ;

write(t(u2.unique) ,file="u2_unique2 .txt",ncolumns=n, sep="\t") ;

fp.tol <- abs(fp 6.000000000); } # For merging purpose

merge.block <- X[,3] X.merge <- cbind(X,merge.block) merge.block

<- fp.block.MSE .gamma[,2] fp.block.MSE .gamma.merge <-

cbind(fp.block.MSE.gamma,merge.block)

X. output<-merge(X. merge, fp. block. MSE. gamma. merge, by="merge. block") ;

write(t(X. output),f ile="X_0uput.txt" ,ncolumns=12, sep="\t") ;
























































































# rho = rank of X # download lasso2, to use tr () to compute the

rank rho <- qr(X)$rank


APPENDIX D

R CODE USED IN CHAPTER 5



Multivariate Power QDG code based on Kuehl example # n = 18, p =

5, r= 3



library(MASS); #Since we need to compute inverse, ginv() is in

MASS library(lasso2); source("bdiag.txt") ; # read in the data

K. data <- matrix(scan( "Kuehl-Data. txt "), 18,12, byrow=TRUE) ; #set

up the dataset from for Design 1 (D1) xl <-K.data[,2]; x2 <-

K.data[,4]; D2x1 <- K.data[,8]; D2x2 <- K.data[,9]; # r=3

responses y1 <- K.data[,5]; y2 <- K.data[,6]; y3 <- K.data[,7];

#This is for Design 1 X1 <- cbind(x1, x2, x1+x1, x2+x2, x1+x2); X

<- cbind(X1, X1, X1); # Since r=3 responses n <- 18; # Number of

observations p <- 5; # Number of fixed effects parameters r <- 3;

# Number of responses pr <- por; # Setting up C matrix tC <-

array(1,dim=c(1,r-1)); C <- rbind(tC,diag(-1,r-1,r-1)); # Setting

up W matrix Ip <- diag(1,p,p); W <- cbind(Ip, Ip, Ip); dim(W) #

Setting up Y matrix Y = cbind(y1, y2, y3); dim(Y) # Setting up Z

matrix Onen <- array(1,dim=c(n,1)) ; Z <- cbind(Onen, X); dim(Z) #

Setting up G matrix G <- rbind(cbind(1,array(0,dim=c(1,pr))),

cbind(array(0,dim=c(p,1)),W)); dim(G) # Setting up Sh matrix

ZpZ <- giny (t(Z) X+X Z) ;

GpG <- t(G) X+X ginv( G X+X ZpZ X+X t(G) ) X+X G;

Sh <- (t(C) X+X t(Y)) X+X (Z X+X ZpZ) X+X GpG X+X ZpZ X+X t(Z) X+X Y X+X C;

dim(Sh) # Setting up Se matrix

Se <- (t(C) X+X t(Y)) X+X ( diag(1,18,18) ( Z X+X ZpZ X+X t(Z) ) ) X+X Y X+X C;

dim(Se)



# Multivariate test statistics Wilk's, Hotteling-Lawley, Pillai



Wilks.LR.Lambda <- det(Se)/det(Se+Sh) ;

Hotteling.trace.U <- tr(Sh%+Xginv(Se));

Pillai.trace.V <- tr(Sh%+X(ginv(Sh+Se)));

multivariate .report <- rbind(cbind("Test", "Wilks

LR", "Hotell ing-Lawley's trace", "Pillai's

trace") ,cbind( "Value" ,Wilks.LR. Lambda,Hotteling. trace.U,Pillai. trace. V))

write(t (multivariate. report),f ile="Multivariate_test_statistics .txt" ,ncolumns=4, sep="\t") ;
















# Define some fixed parameters for the power functions delta <- n

- rho 1-((r-p-1)/2); Lambda <- det(Se)/de~t(Se+Sh) ; I <-

(r-1)+(p+1); W.critical <- 17.07973; # Critical value for Wilks

Lambda U.critical <- 34.2882; # Critical value for

Hotelling-Lawley's trace V.critical <- 12; # Critical value for

Pillai's trace



X1.D2 <- cbind(D2x1, D2x2, D2x1+D2x1, D2x2+D2x2, D2x1+D2x2); X.D2

<- cbind(X1.D2, X1.D2, X1.D2); Z.D2 <- cbind(Onen, X.D2);

ZpZ.D2 <- giny (t(Z.D2) X+X Z.D2);



# Note that in order to calculate power, # need sigmal & sigma2

(defined above see Wilks' Power section) # which is dependent on

the noncentality parameter matrix # For each Design and each

combination of (s11, s22, s33, s12, s13, s23), # calculate

quantiles of power for p = 0(0.10)1 on a hypersphere for a #

specified value of a radius (choose 3 values of radius) # Obtain

the max & min quantiles [out of several quantiles calculated # for

different combinations of (s11, s22, s33, s12, s13, s23)] # Note

max & min are taken over the parameter space # To decrease

runtime, we can choose 3 ^ {6} = 729 combinations only



# Calculate the confidence interval for the elements of Sigma

matrix



Y.star <- rbind(t(y1),t(y2) ,t(y3)) ; Y.star.bar <-

rowMeans(Y.star); Y.star.minusbar <- Y.star Y.star.bar; q <-

array(0, dim=c(3,3)); for(i in 1:n){

Q <- Q + Y star.minusbarC, i] X+Xt(Y. star .minusbar[C, i])





# Based on the q matrix, we can get the confidence interval for

each element of the Sigma matrix. Before we start the loop, # we

need to define 2 more parameters U and L (based on Seber 1984) U

<- 35.35; L <- 1/U ;



# Specify the number of values to consider for each sigma no.sigma

<- 3; #Set upper and lower bound for the sigma.11, sigma.22,

sigma.33 sigma.ii.upper <- array(0,dim=c(r,1)); sigma.ii.lower <-

array(0,dim=c(r,1)); step.ii <- array(0,dim=c(r,1)); for (i in













1:r) {

sigma. ii.upper~i] <- q~i,i]/L

sigma. ii. lower~i] <- Q~i,i]/U

step. ii Ci] <- (sigma. ii.upper Ci] sigma. ii.10wer Ci])/(no.sigma-1)

} # set the upper bound for sigma.12, sigma.13, sigma.23

sigma. 12.upper <- ((q[1,2]/L) + (q[1, 1]+q[2,2])+(1/L 1/U))/2;

sigma. 13.upper <- ((q[1,3]/L) + (q[1, 1]+q[3,3])+(1/L 1/U))/2;

sigma.23.upper <- ((q[2,3]/L) + (q[2,2]+q[3,3] )+(1/L 1/U))/2 ; #

The lower bound for sigma.12, sigma.13, sigma.23 are assigned #as

the starting value for those parameters. sigma.12.1 <- ((Q[1,2]/U)

+ (q[1,1]+q[2,2])+(1/U 1/L))/2 ; sigma.13.1 <- ((Q[1,3]/U) +

(q[1,1]+q[3,3] )+(1/U 1/L))/2 ; sigma.23.1 <- ((q[2,3]/U) +

(q[2,2]+q[3,3])+(1/U 1/L))/2 ; #Calculate the step value for the

c.i. for each sigma step.12 <- (sigma.12.upper -

sigma. 12.1)/(no.sigma-1) ; step.13 <- (sigma.13.upper -

sigma. 13.1)/(no.sigma-1) ; step.23 <- (sigma.23.upper-

sigma.23.1l)/(no.sigma-1) ; # start the loops to create the Sigma

matrix sigma.11 <- sigma.ii.10wer[1]; sigma.22 <-

sigma. ii. lower[C2]; sigma.33 <- sigma. ii. lowerC3] ;

sigma. iteration.count <- O;



no.tau <- 3; no.delta <- 100; # Specify how many random delta to

generate to get the quantiles for each Sigma # Specify the

different values for tau tau.mat <- c(0.1,1,2); # start the loop

for tau for(tau.i in 1:no.tau){ tau <- tau.mat~tau.i]

for(sigma.11.i in 1:no.sigma){

sigma.12 <- sigma.12.1;

for(sigma.12.i in 1:no.sigma){

sigma.13 <- sigma.13.1;

for(sigma. 13.i in 1:no.sigma){

sigma.22 <- sigma. ii.lowerC2] ;

for(sigma.22.i in 1:no.sigma){

sigma.23 <- sigma.23.1;

for(sigma.23.i in 1:no.sigma){

sigma.33 <- sigma. ii. lowerC3] ;

for(sigma.33.i in 1:no.sigma){

Sigma <- matrix(c(sigma.11i,sigma. 12,sigma.13,sigma. 12,sigma.22, sigma.23, sigma. 13, sigma. 23,sigma.33), nrow=r, ncol=r )

sigma.iteration.count <- sigma.iteration.count + 1

sigma.iteration.temp <- cbind(sigma.11.i, sigma.12.i, sigma.13.i, sigma.22.i, sigma.23.i,sigma.33.i,sigma.iteration.cou

sigma.11l,sigma. 12,sigma. 13, sigma.22, sigma.23,sigma.33)

if (sigma. iteration.count == 1){













sigma. iteration <- sigma. iteration.temp

}else{

sigma .it erat ion <- rbind(s igma. it erat ion, sigma .iterat ion .temp)

}~f~~f~~f~~~~f~~f~~f~


# For each Sigma, compute the noncentrality parameter Omega

# Omega is dependent the alternative space Delta.

# Delta = G + Gamma and is of dimension (p+1) + (r-1) (for the example 6+2)

# Following method suggested in the paper generate a vector of length (p+1) + (r-1) = 12



# Generate multiple delta and calculate the power to get the quantiles


sigmal2 .D1D2 <- array(0, dim=c (no. delta,5))
Wilks.i.D1 <- O

Hotelling.i.D1 <- O

Pillai.i.D1 <- O

Wilks.i.D2 <- O

Hotelling.i.D2 <- O

Pillai.i.D2 <- O

for(delta.i in 1:no.delta){



delta.q <- array(0,dim=c(q, 1))

psi <- array(0,dim=c(q-1,1))

for(i in 1:q-2){

psi~i] <- runif (1,0,pi)


psi~q-1] <- runif (1,0,2+pi)

delta.q[1] <- tau +cos(psi[1])

delta.q[2] <- tau +sin(psi[1])

delta.q[3] <- tau +sin(psi[1])

delta.q[4] <- tau +sin(psi[1])

delta.q[5] <- tau +sin(psi[1])

delta.q[6] <- tau +sin(psi[1])

delta.q[7] <- tau +sin(psi[1])

delta.q[8] <- tau +sin(psi[1])

cos(psi[8])

delta.q[9] <- tau +sin(psi[1])

sin(psi[8]) + cos(psi[9])


+ cos(psi[2])

+ sin(psi[2])

+ sin(psi[2])

+ sin(psi[2])

+ sin(psi[2])

+ sin(psi[2])

+ sin(psi[2])


+ cos(psi[3])

+ sin(psi[3])

+ sin(psi[3])

+ sin(psi[3])

+ sin(psi[3])

+ sin(psi[3])


+ cos(psi[4])

+ sin(psi[4]) + cos(psi[5])

+ sin(psi[4]) + sin(psi[5]) + cos(psi[6])

+ sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + cos(psi[7])

+ sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + sin(psi[7]) +


+ sin(psi[2]) + sin(psi[3]) + sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + sin(psi[7]) +


delta.q[10] <- tau +sin(psi[1]) + sin(psi[2]) + sin(psi[3]) + sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + sin(psi[7]) +

sin(psi[8]) + sin(psi[9]) + cos(psi[10])

delta.q[11] <- tau +sin(psi[1]) + sin(psi[2]) + sin(psi[3]) + sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + sin(psi[7]) +














sin(psi[8]) + sin(psi[9]) + sin(psi[10]) + cos(psi[11])

delta.q[12] <- tau +sin(psi[1]) + sin(psi[2]) + sin(psi[3]) + sin(psi[4]) + sin(psi[5]) + sin(psi[6]) + sin(psi[7]) +

sin(psi[8]) + sin(psi[9]) + sin(psi[10]) + sin(psi[11])

Delta <- array(delta.q,dim=c((p+1),(r-1)))



# Define the noncentrality parameter

Omega.D1 <- ginv(t(C) X+X Sigma %+X C) %+X t(D~elta) X+X ginv(G X+X ZpZ X+X t(G)) X+X Delta

Omega.D2 <- ginv(t(C) X+X Sigma %+X C) %+X t(D~elta) X+X ginv(G X+X ZpZ.D2 X+X t(G)) X+X Delta


# Calculate the design criterion

sigmal.D1 <- tr(0mega.D1)

sigma2.D1 <- tr(0mega.D1 X+X Omega.D1)

sigmal.D2 <- tr(0mega.D2)

sigma2.D2 <- tr(0mega.D2 X+X Omega.D2)

#sigmal2.D1D2[delta.i,1] <- delta.i

#sigmal2.D1D2[delta.i,2] <- sigmal.D1l

#sigmal2.D1D2[delta.i,3] <- sigma2.D1

#sigmal2.D1D2[Cdelta. i,4] <- sigmal.D2

#sigmal2.D1D2[Cdelta. i,5] <- sigma2.D2


KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK


# Calculate Wilks' likelihood ratio Test Statistic value


(W) & Power (Power.W)


if((sigmal.D1 > 0) && (sigmal.D2 > 0)){

P.W.D1 <- pchisq(W.critical, i, ncp=sigmal .D1,lower.tail = FALSE,log.p = FALSE:

((r+p+1)+sigmal .D1+pchisq(W. critical, f+2, ncp=sigmal. D1, log.p =

FALSE) ((r+p+1)+sigmal.D1 sigma2.D1)+ pchisq(W.critical, f+4,

ncp=sigmal.D1,log.p = FALSE) sigma2.D1+pchisq(W.critical, f+6,

ncp=sigmal.D1,log.p = FALSE))

P.W.D2 <- pchisq(W.critical, i, ncp=sigmal .D2,lower.tail = FALSE,log.p = FALSE:

((r+p+1)+sigmal .D2+pchisq(W. critical, f+2, ncp=sigmal. D2, log.p =

FALSE) ((r+p+1)+sigmal.D2 sigma2.D2)+ pchisq(W.critical, f+4,

ncp=sigmal.D2,log.p = FALSE) sigma2.D2+pchisq(W.critical, f+6,

ncp=sigmal.D2,log.p = FALSE))

if((P.W.D1 >= 0) && (P.W.D1 <= 1) ){

if ((Wilks.i.D1 == 0)){

Power .W.D1 <- cbind(delta. i,P.W.D1,sigmal.D1 ,t(delta. q))

Wilks.i.D1 <- Wilks.i.D1 + 1

}else{

Power .W.D1 <- rbind(Power .W.D1, cbind(delta. i,P.W.D1,sigmal.D1 ,t(delta. q)))


) (1/(4+delta))+


) (1/(4+delta))+














if((P.W.D2 >= 0) && (P.W.D2 <= 1) ){

if ((Wilks.i.D2 == 0)){

Power.W.D2 <- P.W.D2

Wilks.i.D2 <- Wilks.i.D2 + 1

}else{

Power .W.D2 <- rbind(Power .W.D2,P.W.D2)





rm(P.W.D1)

rm(P.W.D2)



# Calculate Hotelling-Lawley's trace Test Statistic value (H) & Power (Power.H)



H.D1.ti <- pchisq(U.critical, i, ncp=sigmal.D1,lower.tail = FALSE,log.p = FALSE)

H.D1.t2 <- (1/(4+(n-rho-1)))+( (p+1)+(r-1)+(p-r+1) +pchisq(U. critical, i, ncp=sigmal.D1, log.p = FALSE)+2+(p+1)+

(sigmal.D1 -(p+1)+(r-1))+pchisq(U.critical, f+2,

ncp=sigmal.D1,log.p = FALSE) + ((p+1)+(r-1)+(p+r+1) 2+(2+p + r

+2)+ sigmal. D1 +sigma2. D1)+pchisq(U. critical, f+4,

ncp=sigmal.D1l,log.p = FALSE) + (2+(p+r+1)+sigmal.D1-

2+sigma2.D1)+ pchisq(U.critical, f+6, ncp=sigmal.D1l,log.p = FALSE)

+ sigma2.D1+pchisq(U.critical, f+8, ncp=sigmal.D1l,log.p = FALSE))

P.H.D1 <- H.D1.ti H.D1.t2

H.D2.ti <- pchisq(U.critical, i, ncp=sigmal.D2,lower.tail = FALSE,log.p = FALSE)

H.D2.t2 <- (1/(4+(n-rho-1)))+( (p+1)+(r-1)+(p-r+1) +pchisq(U. critical, i, ncp=sigmal.D2, log.p = FALSE) + 2+(p+1)+

(sigmal.D2 -(p+1)+(r-1))+pchisq(U.critical, f+2,

ncp=sigmal.D2,log.p = FALSE) + ((p+1)+(r-1)+(p+r+1) 2+(2+p + r

+2)+ sigmal. D2 +sigma2. D2) +pchisq(U. critical, f+4,

ncp=sigmal.D2,log.p = FALSE) + (2+(p+r+1)+sigmal.D2 -

2+sigma2.D2)+ pchisq(U.critical, f+6, ncp=sigmal.D2,log.p = FALSE)

+ sigma2.D2+pchisq(U.critical, f+8, ncp=sigmal.D2,log.p = FALSE))

P.H.D2 <- H.D2.ti H.D2.t2

if((P.H.D1 >= 0) && (P.H.D1 <= 1)){

if ((Hotelling.i.D1 == 0)){

Power.H.D1 <- P.H.D1

Hotelling.i.D1 <- Hotelling.i.D1 + 1

}else{

Power. H.D1 <- rbind(Power. H.D1,P. H.D1)





if((P.H.D2 >= 0) && (P.H.D2 <= 1)){

if ((Hotelling.i.D2 == 0)){














Power.H.D2 <- P.H.D2

Hotelling.i.D2 <- Hotelling.i.D2 + 1

}else{

Power. H. D2 <- rbind(Power. H. D2, P.H. D2)





rm(P.H.D1)

rm(P.H.D2)

rm(H.D1.t1)

rm(H.D1.t2)

rm(H.D2.t1)

rm(H.D2.t2)



# Calculate Pillai's trace Test Statistic value (V) & Power (Power.P)



P.P.D1 <- pchisq(V.critical, i, ncp=sigmal.D1,lower.tail = FALSE,log.p = FALSE) (1/(4+(n-rho-1)))+

((p+1)+(r-1)+(p-r+1)+sigmal.D1+pchisq(V.crtcl i,

ncp=sigmal.D1,log.p = FALSE) + (2+r+(p+1)+(r-1) +

2+(p+1)+sigmal.D1)+ pchisq(V.critical, f+2, ncp=sigmal.D1,log.p =

FALSE) + (-(p+1)+(r-1)+(p-r+1) + 2+rosigmal.D1 +sigma2.D1)+

pchisq(V.critical, f+4, ncp=sigmal.D1,log.p = FALSE)-

2+(p+r+1)+sigmal .D1+pchisq(V. critical, f+6, ncp=sigmal.D1, log.p =

FALSE) sigma2 .D1+pchisq(V. critical, f+8, ncp=sigmal.D1,log.p =

FALSE))

P.P.D2 <- pchisq(V.critical, i, ncp=sigmal.D2,lower.tail = FALSE,log.p = FALSE) (1/(4+(n-rho-1)))+((p+1)+(r-1:

sigmal .D2+pchisq(V.critical i, ncp=sigmal.D2,log.p = FALSE) +

(2+r+(p+1)+(r-1) + 2+(p+1)+sigmal.D2)+ pchisq(V.critical, f+2,

ncp=sigmal.D2,log.p = FALSE) + (-(p+1)+(r-1)+(p-r+1) +

2+rosigmal.D2 +sigma2.D2)+ pchisq(V.critical, f+4,

ncp=sigmal.D2,log.p = FALSE)-

2+(p+r+1)+sigmal. D2+pchisq(V. critical, f+6, ncp=sigmal. D2, log.p =

FALSE) sigma2 .D2+pchisq(V. critical, f+8, ncp=sigmal.D2,log.p =

FALSE))

if((P.P.D1 >= 0) && (P.P.D1 <= 1)){

if ((Pillai.i.D1 == 0)){

Power.P.D1 <- P.P.D1

Pillai.i.D1 <- Pillai.i.D1 + 1

}else{

Power. P. D1 <- rbind(Power. P. D1, P. P.D1)














if((P.P.D2 >= 0) && (P.P.D2 <= 1)){

if ((Pillai.i.D2 == 0)){

Power.P.D2 <- P.P.D2

Pillai.i.D2 <- Pillai.i.D2 + 1

}else{

Power. P. D2 <- rbind(Power. P. D2, P. P.D2)





rm(P.P.D1)

rm(P.P.D2)

} # end if loop for checking sigmal is positive for both D1 and D2

} # end loop for delta.q



# For a Sigma value, get the quantiles for each of the function

Wilks. q.t.D1 <- quantile(Power .W.D1[,2],probs=c(. 1,1,5,10,15,25,30, 35,40,45,50,55,60 ,65,70,75,80, 85,90,95,100)/100) ;

Hotelling. q. t.D1 <- quantile(Power. H.D1,probs=c(.1,1, 5,10,15,25,30,35,40 ,45,50,55,60,65 ,70,75,80,85,90,95, 100)/100) ;

Pillai. q. t.D1 <- quantile(Power. P.D1,probs=c(.1,1, 5,10,15,25,30,35,40 ,45,50,55,60,65,70 ,75,80,85,90,95, 100)/100) ;

Wilks. q.t.D2 <- quantile(Power .W.D2,probs=c( .1,1,5,10,15,25,30, 35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100)/100) ;

Hotelling. q. t.D2 <- quantile(Power. H.D2,probs=c(.1,1, 5,10,15,25,30,35,40 ,45,50,55,60,65 ,70,75,80,85,90,95, 100)/100) ;

Pillai. q. t.D2 <- quantile(Power. P.D2,probs=c(.1,1,5, 10,15,25,30,35,40 ,45,50,55,60,65,70 ,75,80,85,90,95, 100)/100) ;



# Save the quantiles

Wilks. q.D1 <- array(unlist(Wilks. q. t.D1), c(1,1ength(Wilks. q. t.D1)))

Hotelling.q.D1 <- array(unlist(Hotelling~q ~t.D1)c(,egt(Hel~ling~qt D))

Pillai.q.D1 <- array(unlist(Pillai.q.t.D1),c(1,1ength(Pila~~.1)

Wilks. q.D2 <- array(unlist(Wilks. q. t.D2), c(1,1ength(Wilks. q. t.D2)))

Hotelling.q.D2 <- array(unlist(Hotelling~q ~t.D2)c(,egt(Hel~ling~qt D2)

Pillai.q.D2 <- array(unlist(Pillai.q.t.D2),c(1,1ength(Pila~~.2)

if (sigma. iteration.count == 1){

Wilks. quantile. Sigma.D1 <- Wilks. q.D1

Hotelling. quantile. Sigma. D1 <- Hotelling. q. D1

Pillai. quant ile. Sigma. D1 <- Pillai. q. D1

Wilks. quantile. Sigma.D2 <- Wilks. q.D2

Hotelling.quantile. Sigma.D2 <- Hotelling.q.D2

Pillai. quant ile. Sigma. D2 <- Pillai. q.D2

}else{

Wilks. quantile .Sigma.D1 <- rbind(Wilks. quantile. Sigma.D1,Wilks. q.D1)

Hotelling.quantile. Sigma.D1 <- rbind(Hotelling .quantile. Sigma.D1,Hotelling .q.D1)

Pillai .quantile. Sigma.D1 <- rbind(Pillai .quantile. Sigma.D1,Pillai. q.D1)



Wilks. quantile. Sigma. D2 <- rbind(Wilks. quantile. Sigma. D2, Wilks. q. D2)













Hotelling.quantile. Sigma.D2 <- rbind(Hotelling .quantile. Sigma.D2,Hotelling .q.D2)

Pillai .quantile. Sigma.D2 <- rbind(Pillai .quantile. Sigma.D2,Pillai. q.D2)



sigma.33 <- sigma.33 + step.ii[3]

} # end loop for sigma.33

sigma.23 <- sigma.23 + step.23

} # end loop for sigma.23

sigma.22 <- sigma.22 + step.ii[2]

} # end loop for sigma.22

sigma.13 <- sigma.13 + step.13

} # end loop for sigma.13

sigma.12 <- sigma.12 + step.12

} # end loop for sigma.12

sigma.11 <- sigma.11 + step.ii[1]

} # end loope for sigma.11



# After getting quantiles for calculate the maximum and minimum

for each quantile noquantile.pts <- 21;

quantile.1list<-array(c( .1,1,5,10,15,25,30 ,35,40,45,50,55,60, 65,70,75,80,85,90 ,95,100) ,dim=c(noquantile .pts,1));

Wilks.min.q.D1 <- array(0,dim=c (noquantile.pts,1)) Wilks.max. q.D1

<- array(0,dim=c(noquantile.pts,1)) Hotelling.min.q.D1 <-

array(0,dim=c(noquantile.pts, 1)) Hotelling.max.q.D1 <-

array(0,dim=c(noquantile.pts, 1)) Pillai.min.q.D1 <-

array(0,dim=c(noquantile.pts, 1)) Pillai.max.q.D1 <-

array(0, dim=c (noquantile. pts,1)) Wilks. min. q. D2 <-

array(0, dim=c (noquantile. pts,1)) Wilks. max. q. D2 <-

array(0,dim=c(noquantile.pts, 1)) Hotelling.min.q.D2 <-

array(0,dim=c(noquantile.pts, 1)) Hotelling.max.q.D2 <-

array(0,dim=c(noquantile.pts, 1)) Pillai.min.q.D2 <-

array(0,dim=c(noquantile.pts, 1)) Pillai.max.q.D2 <-

array(0,dim=c(noquantile.pts,1)) for(i in 1:noquantile.pts){

Wilks .min. q.D1[i] <- min(Wilks. quantile. Sigma.D1C, i])

Wilks .max. q.D1[i] <- max(Wilks. quantile. Sigma.D1C, i])

Hotelling.min.q.D1[i] <- min(Hotelling.quantile.Sigma.D1[C,i])

Hotelling.max.q.D1[i] <- max(Hotelling.quantile.Sigma.D1[C,i])

Pillai. min. q. D1[i] <- min(Pillai. quantile. Sigma. D1C, i])

Pillai.max. q.D1[i] <- max(Pillai. quantile. Sigma.D1C, i])

Wilks.min.q.D2[Ci] <- min(Wilks.quantile. Sigma.D2[C,i])

Wilks.max.q.D2[Ci] <- max(Wilks.quantile. Sigma.D2[C,i])

Hotelling.min.q.D2[Ci] <- min(Hotelling. quantile.Sigma.D2[C,i])

Hotelling.max.q.D2[Ci] <- max(Hotelling. quantile.Sigma.D2[C,i])














Pillai.min.q.D2[Ci] <- min(Pillai. quantile.Sigma.D2[C,i] )

Pillai.max.q.D2[Ci] <- max(Pillai. quantile.Sigma.D2[C,i] )


};###################


# For plotting purpose, save the data for each tau if(tau.i ==

1){Wilks.plot.D1 <-

cbind(tau,quantile.1list ,Wilks .min. q.D1,Wilks.max. q.D1)

Hotelling.plot .D1 <- cbind(tau,quantile.1list ,Hotelling.min. q.D1,Hotelling.max. q.D1)

Pillai.plot .D1 <- cbind(tau,quantile.1list ,Pillai .min. q.D1,Pillai.max .q.D1)

Wilks .plot .D2 <- cbind(tau,quantile.1list,Wilks.min. q.D2,Wilks .max.q.D2)

Hotelling.plot .D2 <- cbind(tau,quantile.1list ,Hotelling.min. q.D2,Hotelling.max. q.D2)

Pillai.plot .D2 <- cbind(tau,quantile.1list ,Pillai .min. q.D2,Pillai.max .q.D2)

}else{Wilks.plot.D1 <-

rbind(Wilks .plot.D1, cbind(tau,quantile.1list ,Wilks .min. q.D1,Wilks.max. q.D1))

Hotelling.plot .D1 <- rbind(Hotelling .plot.D1, cbind(tau,quantile.1list ,Hotelling.min.q.D1 ,Hotelling.max. q.D1))

Pillai.plot .D1 <- rbind(Pillai .plot.D1, cbind(tau,quantile.1list ,Pillai.min. q.D1,Pillai.max. q.D1))

Wilks .plot .D2 <- rbind(Wilks .plot .D2, cbind(tau,quantile.1list,Wilks.min. q.D2,Wilks .max.q.D2))

Hotelling.plot .D2 <- rbind(Hotelling .plot .D2, cbind(tau,quantile.1list ,Hotelling.min.q.D2 ,Hotelling.max. q.D2))

Pillai.plot .D2 <- rbind(Pillai .plot.D2, cbind(tau,quantile.1list ,Pillai.min. q.D2,Pillai.max. q.D2))}

} #end loop for tau

write(t(Wilks.plot .D1),f ile="plot_Wilks_D1. txt" ,ncolumns=4,sep="\t") ;

write(t(Hotelling .plot.D1),f ile="plot_Hotelling_D1 .txt" ,ncolumns=4,sep="\t" );

write(t(Pillai .plot.D1),f ile="plot_Pillai_D1 .txt" ,ncolumns=4, sep="\t") ;

write(t(Wilks.plot .D2),f ile="plot_Wilks_D2. txt" ,ncolumns=4,sep="\t") ;

write(t(Hotelling .plot .D2),f ile="plot_Hotelling_D2 .txt" ,ncolumns=4,sep="\t" );

write(t(Pillai .plot .D2),f ile="plot_Pillai_D2 .txt" ,ncolumns=4, sep="\t") ;

write(t(sigma. iteration),f ile="sigma_iteration.txt" ,ncolumns=13 ,sep="\t") ;



# PLOT FOR POWER # no.tau <- 3; noquantile.pts <- 21; rows

<- noquantile.ptsono.tau; plot.Wilks.D1 <-

matrix(scan( "plot_Wilks_D1. txt", n =rows*4), rows, 4,

byrow=TRUE) ; plot .Wilks .D2 <- matrix(scan("plot_Wilks_D2 .txt", n =

rows*4), rows, 4, byrow=TRUE) ; png(file="Wilks_power .png") ; #

Saving plot to a png file par(mfrow=c(3,1)); # multiple plots in a

page



# Read in the min and max for the X and Y axis



min.quantile <- min(min(plot.Wilks. D1[,3] ),min(plot.Wilks.D2[C,3]))

max. quantile <- max(max(plot .Wilks.D1[,4]) ,max(plot .Wilks.D2[,4]))

ymin <- min.quantile


















# # Read in min and max quantiles for each r for designs 1 and 2,

and create plots


ymax <- max.quantile + 0.01 #ymin <- 0; # ymax <-1;


for(k in 1:no.tau){

row <- 21 + (k-1) + 1;

r.val <- plot.Wilks.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1) ; # To

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

# to show 11 points not 21 quantile points

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.Wilks.D2[21 + (k-

max.quantile.D2[i] <- plot.Wilks.D2[21 + (k-

min.quantile.D1[i] <- plot.Wilks.D1[21 + (k-

max.quantile.D1[i] <- plot.Wilks.D1[21 + (k-

quantile.1ist~i] <- plot.Wilks.D1[21 + (k-1:


create points at 0, 0.1, 0.2..


-1)

-1)

-1)

-1)

) +


+ i,3]

+ i,4]

+ i,3]

+ i,4]

i,2]/100;


Ior (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1C2+i-

max. quantile. points. D1[i] <- max. quantile. D1C2+i-

min.quantile. points.D2[Ci] <- min.quantile.D2C2+i-

max.quantile. points.D2[Ci] <- max.quantile.D2C2+i-

quantile.points~i] <- quantile.1ist[C2+i-1]


min. quantile <- min(min(min. quantile. D1), min(min. quantile. D2))

max. quantile <- max(max(max. quantile. D1), max (max. quantile. D2))

ymin <- min.quantile 0.001 ymax <- max.quantile + 0.002

plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax), xlab="p", ylab = "Quantiles of power (Wilks)",

main=substitute(radius ==

l ambdaval,1ist( lambdaval=r. val)), type="l",1Ity=1) ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1) ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2) ;














lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2) ;

points(quantile .points,min. quantile .points .D2,pch=17) ;

points(quantile .points,max. quantile .points .D2,pch=17) ;

points(quantile .points,min. quantile .points.D1,pch=16) ;

points(quantile .points,max. quantile .points.D1,pch=16) ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch= c(17,16), cex=0.7) ;

}; dev.off ();


no.tau <- 3;

noquantile.pts <- 21;

rows <- noquantile.ptsono .tau;

plot. Hotelling.D1 <- matrix(scan( "plot_Hotelling_D1 .txt", n =rows*4), rows, 4, byrow=TRUE) ;

plot. Hotelling.D2 <- matrix(scan( "plot_Hotelling_D2 .txt", n =rows*4), rows, 4, byrow=TRUE) ;

#pdf (f ile=" qDGpower_orth .pdf ") ; # Saving plot to a png file

#png(f ile=" qDGpower_D3_samepage .png") ; # Saving plot to a png file

png(f ile="Hotelling_power .png") ; # Saving plot to a png file

par(mf row=c(3,1)); # multiple plots in a page


min.quantile <- min(min(plot. Hotelling.D1[C,3]) ,min(plot .Hotelling.D2[C,3]))

max.quantile <- max(max(plot. Hotelling.D1[C,4]) ,max(plot .Hotelling.D2[C,4]))

ymin <- min.quantile

ymax <- max.quantile + 0.01



# Read in min and max quantiles Ior each r for designs 1 and 2,

and create plots



Ior(k in 1:no.tau){

row <- 21 + (k-1) + 1;

r.val <- plot.Wilks.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1); # To create points at 0, 0.1, 0.2..

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;


# PLOT FOR POWER #


# Read in the min and max f or the X and Y axis






















































l ambdaval,1ist( lambdaval=r. val)), type="l",1Ity=1) ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1) ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2) ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2) ;

points(quantile .points,min. quantile .points .D2,pch=17) ;

points(quantile .points,max. quantile .points .D2,pch=17) ;

points(quantile .points,min. quantile .points.D1,pch=16) ;

points(quantile .points,max. quantile .points.D1,pch=16) ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch = c(17,16), cex=0.7) ;

}; dev.off();


quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.Hotelling.D2[21 + (k-:

max.quantile.D2[i] <- plot.Hotelling.D2[21 + (k-:

min.quantile.D1[i] <- plot.Hotelling.D1[21 + (k-:

max.quantile.D1[i] <- plot.Hotelling.D1[21 + (k-:

quantile.1ist Ci] <- plot .Hotelling.D1[21 + (k-1)



f or (i in 1: 11) {

min. quantile. points. D1[i] <- min. quantile. D1C2+i-

max. quantile. points. D1[i] <- max. quantile. D1C2+i-

min.quantile. points.D2[Ci] <- min.quantile.D2C2+i-

max.quantile. points.D2[Ci] <- max.quantile.D2C2+i-

quantile.points~i] <- quantile.1ist[C2+i-1]


+ i,3]

+ i,4]

+ i,3]

+ i,4]

i,2]/100;


min. quantile <- min(min(min. quantile. D1), min(min. quantile. D2))

max. quantile <- max(max(max. quantile. D1), max (max. quantile. D2))

ymin <- min.quantile 0.001 ymax <- max.quantile + 0.002

plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(l

main=substitute(radius ==


ymin,ymax), xlab="p", ylab = "Quantiles of power (Hotelling)",


# # PLOT FOR POWER #

no.tau <- 3;

noquantile.pts <- 21;

rows <- noquantile.ptsono .tau;

plot. Pillai .D1 <- matrix(scan( "plot_Pillai_D1 .txt", n =rows*4), rows, 4, byrow=TRUE) ;

plot. Pillai .D2 <- matrix(scan( "plot_Pillai_D2 .txt", n =rows*4), rows, 4, byrow=TRUE) ;

#pdf (f ile=" QDGpower_orth .pdf ") ; # Saving plot to a png file

#png(f ile=" qDGpower_D3_samepage .png") ; # Saving plot to a png file

png(f ile="Pillai_power .png") ; # Saving plot to a png file

par(mf row=c(3,1)); # multiple plots in a page













fftftftftftftftftftftftftftftftftftftftf


min.quantile <- min(min(plot. Pillai.D1[C,3] ),min(plot .Pillai.D2[C,3]))

max.quantile <- max(max(plot. Pillai.D1[C,4] ),max(plot .Pillai.D2[C,4]))

ymin <- min.quantile

ymax <- max.quantile + 0.01



# Read in min and max quantiles for each r for designs 1 and 2,

and create plots


# Read in the min and max for the X and Y axis


for(k in 1:no.tau){

row <- 21 + (k-1) + 1;

r.val <- plot.Wilks.D1[row,1];

min. quantile. D1 <- matrix(O, noquantile. pts,1) ;

max. quantile. D1 <- matrix(O, noquantile. pts,1) ;

min. quantile. D2 <- matrix(O, noquantile. pts,1) ;

max. quantile. D2 <- matrix(O, noquantile. pts,1) ;

min.quantile.points .D1 <- matrix(O,11,1) ; # To

max. quantile. points. D1 <- matrix(O,11,1) ;

min. quantile. points. D2 <- matrix(O,11,1) ;

max. quantile. points. D2 <- matrix(O,11,1) ;

quantile .points <- matrix(0,11,1) ;

quantile.1ist <- matrix(0,noquantile .pts,1) ;

for(i in 1:noquantile.pts){

min.quantile.D2[i] <- plot.Pillai.D2[21+ (1

max.quantile.D2[i] <- plot.Pillai.D2[21+ (1

min.quantile.D1[i] <- plot.Pillai.D1[21+ (1

max.quantile.D1[i] <- plot.Pillai.D1[21+ (1

quantile.1ist~i] <- plot.Pillai.D1[21 + (k-:


create points at 0, 0.1, 0.2..


k-1)

k-1)

k-1)

k-1)

1) +


+ i,3]

+ i,4]

+ i,3]

+ i,4]

i,2]/100;


for(i in 1:11){

min. quantile. points. D1[i] <- min. quantile. D1[C2i-1]

max. quantile. points. D1[i] <- max. quantile. D1[C2i-1]

min.quantile. points.D2[Ci] <- min.quantile.D2[C2+i-1]

max.quantile. points.D2[Ci] <- max.quantile.D2[C2+i-1]

quantile.points~i] <- quantile.1ist[C2+i-1]



min. quantile <- min(min(min. quantile. D1), min(min. quantile. D2))

max. quantile <- max(max(max. quantile. D1), max (max. quantile. D2))

ymin <- min.quantile 0.001 ymax <- max.quantile + 0.002














plot(quantile.1ist ,max.quantile.D2, xlim = c(O,1), ylim = c(ymin,ymax),

xlab="p", ylab = "Quantiles of power (Pillai)" ,main=substitute(radius =

l ambdaval,1ist(l ambdaval=r. val)), type="l",1Ity=1) ;

lines(quantile.1list ,min. quantile .D2,type="l" ,1ty=1) ;

lines(quantile.1list ,max. quantile.D1 ,type="l" ,1ty=2) ;

lines(quantile.1list ,min. quantile.D1 ,type="l" ,1ty=2) ;

points(quantile .points,min. quantile .points .D2,pch=17) ;

points(quantile .points,max. quantile .points .D2,pch=17) ;

points(quantile .points,min. quantile .points.D1,pch=16) ;

points(quantile .points,max. quantile .points.D1,pch=16) ;

legend( "topleft ", c("D2","D1"),1Ity=c(1,2),pch = c(17,16), cex=0.7) ;

}; dev.off();


























Define the level of significance --- want to construct a 95% c.i.



alpha = 0.05/2; beta= 1- alpha; prec = 0.0001;


Read in values to matrices from the dataset



USE dat.qDGeg; READ ALL VAR {xl} INTO xl; READ ALL VAR {x2} INTO

x2; READ ALL VAR {Block} INTO block ; READ ALL VAR {y} INTO Y; n =

NROW(x1); /+ Number of observations/ rows in the matrix X +/


X =J(n,1,1) Ilxl lx21 lxl#xil lx2#x21 lx1#x2; Z = DESIGN(block); +CALL

EIGEN(eigvalX,eigvecx,X'+X); +CALL EIGEN(eigvalZ,eigvecx,Z'+Z);

+PRINT eigvalX eigvalZ (Z'+Z);


C = Z'+(I(n)-X+GINV(X'+X)+X')+Z; q = Z'+(I(n)-X+GINV(X'+X)+X')+Y;

print C;


Get the Eigenvalues and Eigen vectors for the matrix C eigvall =

vector storing the eigenvalues eigvec = matrix storing the eigen

vectors


Correcting for round-off errors on eigenvalues


KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK


eigval2 = EIGVAL(X'+X); eigvalX = eigval2>.0001; PRINT eigval2,


APPENDIX E

SAS CODE: TO OBTAIN CONFIDENCE INTERVAL FOR r7
DM LOG 'CLEAR'; DM OUT 'CLEAR'; LIBNAME dat

'C: \Sourish\qDG\qDGO63006\Data' ; DATA dat.qDGeg; INFILE

"C: \Sourish\qDG\qDGO63006\at\qDdata\QDtatatx; INPUT xl x2 block y;

RUN; PROC IML;


Design matrx X and Z for the fixed and random effects


Define the matrix C and Q


CALL EIGEN(eigvall,eigvec,C); PRINT eigvall;


eigvalC = eigvall>.0001; print eigvalC;


Calculate the eigenvalues for the matrix X'X













(X'+X);



Calculate the rank and degrees of freedom



r = SUM(eigvalC); pstar = SUM(eigvalX); I = n r pstar; PRINT r

pstar f ;



Calculate the F values



F_upper = FINV(beta,r,f ); F_10wer = FINV(alpha,r,f );



Construct a diagonal matrix with the positive eigenvalues



D = DIAG(eigvall(11:r,1)); PRINT D;



Extract the min and max eigenvalues



delta_min = MIN(eigvall(ll:r,1)) ; delta_max =

MAX(eigvall(11:r, 1)) ; PRINT delta_min delta_max;



Get the corresponding eigenvectors -- for matrix C



V = eigvec(1,1:rl); PRINT V;



T = INV(D##0.5)+V'+Q ; SSE = Y'+Y Y'+X+GINV(X'+X)+X'+Y T'+T; k

= f/(roSSE) ; L_min = k+T'+INV(D) +T/F_upper 1/delta_min ; L_max =

L_min + 1/delta_min 1/delta_max ; U_max = k+T'+INV(D)+T/F_10wer

- 1/delta_max ; U_min = U_max + 1/delta_max 1/delta_min ; PRINT

SSE k L_min L_max U_max U_min;



Computing the exact upper bound



ac =MAX(0,U_min) ; bc =U_max; precnt = INT(LOG((1l/delta_min -

1/delta_max) /prec)/LOG(2)+1) ; PRINT precnt; IF ((T'+T~f)/(roSSE) <

F_10wer) THEN en = 0; ELSE DO cnt = 1 TO precnt;

cn = (ac + bc)/2;

IF (k+T'+INV(I(r)+cn*D)+T F_10wer > 0) THEN ac = en;

ELSE bc = en;

END; ustr = en;



Compute exact lower bound





129















ac =MAX(0,L_min) ; bc =L_max; IF ((T'+T~f)/(r+SSE) < F_upper)

THEN dn = 0; ELSE DO cnt = 1 TO precnt;

dn =(ac +bc)/2;

IF (k+T'+INV(I(r)+dn*D)+T F_upper > 0) THEN ac = dn;

ELSE bc = dn;

END; 1str = dn; PRINT ustr 1str; QUIT;









APPENDIX F
PRICE'S CONTROLLED RANDOM SEARCH PROCEDURE

Let us consider the following mathematical problem


Minimize g(0) subject to 8 E 8


where 8 is a k-dimensional vector and 8 is a bounded set within a k-dimensional rectangle

of finite volume in sRk

The controlled random search (CRS) algorithm (Price (1977)) stores function values

(gl,..., g,) at p points which are distributed randomly throughout 8. Let 9max and

gmin denote the largest and smallest function values in the storage. Let Omax and 8mi,

denote the corresponding points where the largest and smallest values are attained. At

each iteration, a new point 8* is obtained by reflecting a randomly selected point from

the storage through the centroid of k other points selected from the storage without

replacement. If g(8*) < 9max then 8* replaces Omax in the storage. The process is stopped

when gmax and gms, are close together. The steps of algorithm are as follows:

1. Build a storage of size p by selecting points 01, .. ., 0, from 8 and storing the

corresponding function values, namely g (81), .. ., g(8,).

2. At each iteration:

(a) Generate a new point 8* E 8 as

0* <- 2 x 0 O'.


where 8' is chosen randomly from 81, .. ., 8, and 8 is the centroid of k points

chosen without replacement from r, O.

(b) Check to see if 0* E 8. Otherwise, repeat generation of 8*.

(c) If g(0*) < gmax then replace 9max with g(0*), replace Omax with 0*, and obtain

new values for gmax and gmin.










(d) If gmax -gms, is less than a user specified small value or the number of iterations

has reached above a certain limit, then stop the process.

3. Return 9min as the minimum value and Oms, as the location of the minimum.










REFERENCES


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Ali, 31. A., Torn, A., and Viitanen, S. (1997). A numerical comparison of some modified
controlled random search algorithms. Journal of Global Op~timization, 11::377-285.

Atkins, J. E. and Cheng, C. S. (1999). Optimal regression designs in the presence of
random block effects. Journal of Statistical Planning and Inference, 77::321-3:35.

Atkinson, A. C. (1972). Planning experiments to detect inadequate regression models.
Biometrikes, 59:275-29:3.

Atkinson, A. C. and Doney, A. N. (1989). The construction of exact d-optimum
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76:515-526.

Atkinson, A. C. and Doney, A. N. (1992). Optimum Ex~lperimentol Designs. Oxford
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BIOGRAPHICAL SKETCH

Sourish Saha was born in Calcutta, India, on June 18, 1978. He earned his bachelor's

degree in statistics from Presidency College, Calcutta, in 2000 and master's degree in

statistics from Indian Institute of Technology, K~anpur, in 2002. He joined the Department

of Statistics at the University of Florida in 2002. Besides serving as a teaching assistant

to nine different statistics courses, he was instructor of two statistics courses STA :3024

and STA 4:322. He worked as a research assistant at the Department of Clinical Health

& Psychology and as a statistical consultant at the Department of Psychology. In the

summer of 200:3 and 2004, he interned at US Department of Health and Human Services.

He started working for GlaxoSmithE~line as a senior statistician in August, 2007. He

expects to receive his Ph.D in December, 2007.





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Iamgratefultomyadvisor,Dr.AndreI.Khuri,forhisexcellentsupervisionduringmydoctoralresearch.Thisdissertationwouldnothavebeenpossiblewithouthisguidanceandtremendouspatience.Iwouldalsoliketothankallmycommitteemembers:Dr.MalayGhosh,Dr.RonaldRandles,Dr.TrevorParkandDr.MuraliRao.Specialthanksgotomywife,SoumitaLahiri. 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 1.1LiteratureReview ................................ 10 1.2DissertationObjectives ............................. 11 2COMPARISONOFDESIGNSONTHEBASISOFTHEPREDICTIONVARIANCECRITERIONUSINGQUANTILEDISPERSIONGRAPHS ........... 13 2.1Introduction ................................... 13 2.2ModelandNotation .............................. 14 2.3QuantileDispersionGraphs .......................... 16 2.4CondenceIntervalfor 17 2.5Example ..................................... 22 2.6ABriefReviewofSplitPlotDesigns ..................... 24 3COMPARISONOFDESIGNSONTHEBASISOFTHEPOWERCRITERIONUSINGQUANTILEDISPERSIONGRAPHS ................... 36 3.1Introduction ................................... 36 3.2DevelopmentofDesignCriteria ........................ 37 3.3DesignComparisonsusingQDGs ....................... 38 3.3.1QuantileDispersionGraphs ...................... 38 3.3.2AConservativeCondenceRegionfor(2;) ............. 40 3.3.3AnAlternativeSpacefor 40 3.4ANumericalExample ............................. 41 4SEQUENTIALGENERATIONOFD-OPTIMALDESIGNS ........... 56 4.1Introduction ................................... 56 4.2ModelandNotation .............................. 56 4.3Methodology .................................. 57 4.4EquivalenceTheorem .............................. 60 4.5SequentialGenerationofD-optimaldesigns ................. 61 4.6ANumericalExample ............................. 62 4

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................. 68 5.1Introduction ................................... 68 5.2CorrelatedResponseModels .......................... 69 5.3DevelopmentofaMultivariateTest ...................... 70 5.4DesignCriteria ................................. 73 5.4.1Wilks'likelihoodratio .......................... 74 5.4.2Hotelling-Lawley'strace ........................ 75 5.4.3Pillai'strace ............................... 75 5.5QuantileDispersionGraphs .......................... 76 5.5.1SimultaneousCondenceIntervalsontheelementsof 76 5.5.2AnAlternativeSpacefor=G 77 5.6Example ..................................... 78 6SUMMARYANDFUTURERESEARCH ..................... 85 APPENDIX ARCODEUSEDINCHAPTER2 .......................... 87 BRCODEUSEDINCHAPTER3 .......................... 95 CRCODEUSEDINCHAPTER4 .......................... 107 DRCODEUSEDINCHAPTER5 .......................... 113 ESASCODE:TOOBTAINCONFIDENCEINTERVALFOR 128 FPRICE'SCONTROLLEDRANDOMSEARCHPROCEDURE ......... 131 REFERENCES ....................................... 133 BIOGRAPHICALSKETCH ................................ 139 5

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Table page 2-1Designsettingsandresponsevalues(shearstrengthinpsi) ............ 27 2-2DesignsettingsforD1,D2,D3,D4andD5 28 3-1Theexperimentaldesignalongwiththeresponsevalues .............. 45 3-2DesignsettingsforD1,D2andD3 47 3-3Numberofpointsineachdesign ........................... 48 4-1Initialdesignalongwithresponsevalues ...................... 64 4-2Augmenteddesignalongwithupdatedestimates .................. 65 5-1Design1-originalvariablesalongwithresponses ................. 79 5-2Design1and2-CodedVariables .......................... 80 6

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Figure page 2-1PlotsofthedesignpointsforD1,D2,D3,D4,andD5 29 2-2QDGsforD1andD2 30 2-3QDGsforD1andD3 31 2-4QDGsforD1andD4 32 2-5QDGsforD1andD5 33 2-6QDGsforD3andD4 34 2-7QDGsforD4andD5 35 3-1Condenceregionfor(2;) ............................. 48 3-2Plotsofthedesignpoints(inBlock1)fordesignsD1,D2,andD3 49 3-3QDGsofthenoncentralityparameter(ncp)forD1andD2 50 3-4QDGsofthepowerfunctionforD1andD2 51 3-5QDGsofthenoncentralityparameter(ncp)forD1andD3 52 3-6QDGsofthepowerfunctionforD1andD3 53 3-7QDGsofthenoncentralityparameter(ncp)forD2andD3 54 3-8QDGsofthepowerfunctionforD2andD3 55 4-1InitialDesign ..................................... 64 5-1Plotofthedesignpoints ............................... 81 5-2QDGsofthepowerfunctionforD1andD2(basedonWilks'LikelihoodRatio) 82 5-3QDGsofthepowerfunctionforD1andD2(basedonHotelling-Lawley'sTrace) 83 5-4QDGsofthepowerfunctionforD1andD2(basedonPillai'sTrace) ....... 84 7

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Responsesurfacemethodology(RSM)isasetoftoolsthatincludessettingupaseriesofexperimentsthatproducereliablemeasurementsoftheresponseofinterest,ttingandevaluatingagivenmodel,anddeterminingthesettingsofthefactorsthatyieldoptimumvalueofthepredictedresponse.IntheareaofRSM,oneofthemainconsiderationsisthechoiceofanexperimentaldesign.Extensivestudieshavebeenundertakeninthedesignareawithregardtoresponsesurfacemodels,thesameisnottruewithregardtosuchmodelsinthepresenceofarandomblockeectinthettedmodel.Designsforthelattertypedependoncertainunknownparametersconcerningthemodel'svariancecomponents.Hence,theconstructionofsuchdesignsrequiressomepriorknowledgeoftheunknownparameters.Thedesigndependenceproblemformixedresponsesurfacemodelsisaddressedbyapplyingquantiledispersiongraphs(QDGs),whichisapowerfulgraphicaltoolforcomparingdesignsforsuchmodels.ThegenerationofD-optimaldesignssequentiallyformixedresponsesurfacemodelsisalsodiscussed.TheuseofQDGstocomparedesignsforcorrelatedresponsesurfacemodelswithanunknowndispersionmatrixispresentedfollowedbysummaryandpossibleareasforfurtherresearch. 8

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Thetoolsrequiredfortheadequateselectionofadesignandthesubsequentttingandevaluationofthehypothesizedmodel,usingthedatageneratedbythedesign,havebeendevelopedinanareaofexperimentaldesignknownasresponsesurfacemethodology(RSM).Theformaldevelopmentofthisareastartedwiththeworkof BoxandWilson ( 1951 ).ManybooksandpapersdiscussingRSMhavebeenpublishedsincetheappearanceofthatseminalpaper.Thearticlesby HillandHunter ( 1966 ), Myersetal. ( 1989 )and MeadandPike ( 1975 )provideabroadreviewofRSM.Thebooksby BoxandDraper ( 2007 ), MyersandMontgomery ( 1995 ),and KhuriandCornell ( 1996 )giveacomprehensivecoverageofthevarioustechniquesusedinRSM.Oneoftheobjectivesofresponsesurfacemethodologyistheselectionofaresponsesurfacedesignaccordingtoacertainoptimalitycriterion.Thedesignselectioninvolvesthespecicationofthesettingsofagroupofinputorcontrolvariablesthatcanbeusedasexperimentalrunsinagivenexperiment.Thechoiceofadesignplaysanimportantroleintheprecisionofestimatingtheunknownparametersofagivenmodel. Responsesurfacemodelsarewidelyusedtodescribeanempiricalrelationshipbetweentheresponseandasetofcontrolvariables.Theexperimentaltrialsperformedinaresponsesurfaceinvestigationareoftenconductedunderhomogeneousconditions.However,therearesituationswherethisassumptionisinvalid.Heterogeneityofexperimentalconditionscanbecausedbyextraneoussourcesofvariation.Insuchsituations,theexperimentaltrialsshouldbeperformedingroups,orblocks,withineachofwhichhomogeneousconditionscanbemaintained.Althoughextensivestudieshavebeenundertakeninthedesignareawithregardtoresponsesurfacemodels,thesamecannotbesaidwithregardtosuchmodelsinthepresenceofarandomblockeectinthettedmodel.Designsforthelattertypedependoncertainunknownparametersconcerningthemodel'svariancecomponents.Thustheconstructionofsuchdesignsrequiressomeprior 9

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AtkinsonandDonev ( 1989 )and TrincaandGilmour ( 2000 )). CookandNachtsheim ( 1989 )proposedanexchangealgorithmforthecomputationofD-optimaldesignsinthepresenceofaxedblockeect.Thechoiceofdesignwhentheblockeectisrandomwasdiscussedby Cheng ( 1995 ), AtkinsandCheng ( 1999 ),and GoosandVandebroek ( 2001a )whoconsideredD-optimalresponsesurfacedesigns.Theobservationsobtainedwithinablockwereassumedtobecorrelated,whereasobservationsindierentblockswereconsidereduncorrelated. Themostcommonalgorithmsforconstructingexactoptimaldesignsareexchange-typealgorithms.TheystartwithaninitialN-pointdesign,afterwhichthepointsareaddedordeletedeithersequentiallyorsimultaneouslyinordertoimprovethevalueofagivendesigncriterion.Someofthewell-knownexamplesofexchange-typealgorithmsarethoseof Federov ( 1972 ),theDETMAXalgorithm( Mitchell ( 1974 )),themodiedFedorovalgorithm( CookandNachtsheim ( 1989 )),theKL-exchange(whereKandLdenotethekthandlthpointsrespectively)andtheBLKL(whereBLstandsforblock,andKandLdenotethekthandlthpointsinablockrespectively)algorithms( AtkinsonandDonev ( 1989 ), AtkinsonandDonev ( 1992 )),andthecoordinateexchangealgorithm( MeyerandNachtsheim ( 1995 )).Themainadvantageofthesealgorithmsisthatdierentdesignproblemsarereducedtoacommonoptimizationstructure.Whetherornotthedesignproblemsinvolvequantitativeorqualitativefactorsormixturevariables,theoptimaldesignisfoundbyselectingdesignpointsfromalistofcandidatepoints.Thealgorithms 10

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AtkinsonandDonev ( 1992 ). GoosandVandebroek ( 2001a )presentedapoint-exchangealgorithmtoconstructD-optimaldesignsunderrandomblockeects.Theiralgorithmisbasedonadirectsearchmethod,soitonlyprovideslocallyoptimalsolutions.Inthegenericalgorithmdescribedintheirpaper,morethanonestartingdesignwasusedinordertoincreasethechancesofattainingagloballyoptimalsolution.Theinitialdesignwasimprovedbysequentiallyaddingcandidatepointswiththelargestpredictionvariance.Optimaldesigns,ingeneral,dependontheextenttowhichobservationswithinablockarecorrelated.Theyshowedthatoptimaldesignsdonotdependuponcorrelationsinanyofthethreesituations:(i)blocksizegreaterthanthenumberofmodelparameters,(ii)minimumsupportdesign(orsaturateddesignwherethenumberofdistinctdesignpointsisequaltothenumberofmodelparameters),(iii)orthogonally-blockedrst-orderdesigns.Theyalsoshowedthatwhentheblocksizeishomogeneous,orthogonally-blockeddesignswillbebetterthandesignsthatarenotblockedorthogonallywithrespecttoanygeneralizedoptimalitycriterion.However,theauthorspointedoutthatorthogonalblockingmaynotyieldoptimaldesignswhentheblocksizesareheterogeneous.Whentheblockeectsareassumedtobexed, GoosandVandebroek ( 2001a )showedthatorthogonalblockingisindeedanoptimalstrategyforhomogeneousaswellasheterogeneousblocksizes. Cheng ( 1995 )and AtkinsandCheng ( 1999 )resortedtotheapproximatetheoryofoptimaldesignstodealwiththespecialfeatureofwithin-blockcorrelation. 1.Todemonstratetechniquesforcomparisonofdesignsforresponsesurfacemodelswitharandomblockeectbasedontwocriteria,namely,thepredictionvarianceandthepowerofastatisticaltest. 11

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3.Toproposemethodologyforcomparisonofdesignsforcorrelatedresponsesurfacemodelswithanunknowndispersionmatrix. 12

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( 1992 )discussedtheanalysisofaresponsesurfacemodelinthepresenceofarandomblockeect,butdidnotconsiderthedesignaspect. Giovannitti-JensenandMyers ( 1989 )introducedtheso-calledvariancedispersiongraphs(VDGs)toassesstheoverallpredictioncapabilityofaresponsesurfacedesigninsidearegionofinterest,R.TheVDGsaretwo-dimensionalplotsofthemaximumandminimumvaluesofthepredictionvarianceonthesurfacesofseveralconcentricspheresinsideR.Inaneorttoprovidemoreinformationconcerningthepredictionvarianceonsuchspheres, Khurietal. ( 1996 )proposedtheuseofthequantileplotsofthepredictionvarianceonthesurfacesofthespheres.Suchplotsdescribetheentiredistributionofthepredictionvarianceandtherebygivemorecompleteassessmentoftheeectofthedesignonthepredictionvariance. Forresponsesurfacemodelswitharandomblockeect,thepredictionvariancedepends,unfortunately,onanunknownparameter,namely,theratiooftwounknownvariancecomponents,aswillbeseenlater.Consequently,theassessmentofthedesigneectonthepredictionvariancedependsonthevalueofthisunknownratio.Thisdependenceproblemisacommonfeatureofdesignsforvariancecomponentestimation,andfordesignsfornonlinearmodels,includinggeneralizedlinearmodels.Thereareseveralapproachestodealwiththisdependenceproblemingeneral.Theseapproachesincludetheuseof (i)locallyoptimaldesignsbyspecifyinginitialvaluesoftheunknownparameters. (ii)thesequentialmethodwhichallowstheexperimentertoobtainupdatedestimatesoftheunknownparametersinsuccessivestagesfollowingtheuseoftheinitialvaluesintherststage. (iii)theBayesianmethodologywhichrequiresthespecicationofsomepriordistributionsontheunknownparameters. 13

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Khuri ( 1997 )forthecomparisonofdesignsforarandomtwo-waymodel(seealso LeeandKhuri ( 1999 2000 )). ( 1992 )consideredthefollowingmodeloforderd(1)inkinputvariablesx1;x2;:::;xk.Theexperimentalrunsintheassociateddesignarearrangedinbblocksofsizesn1;n2;:::;nb: whereyisavectorofn=(Pbi=1ni)observationsontheresponse,0and=(1;2;:::;p)0areunknownparametersassociatedwiththepolynomialportionofthemodel,=(1;2;:::;b)0,whereldenotestheeectofthelthblock(l=1;2;:::;b),andisarandomerrorvector.Here,XandZareknownmatricesofordersnpandnbandrankspandb,respectively.TherowsofXconsistofthevaluesoff0(xu),avectorwhoseelementsarepowersandcrossproductsofpowersofx1;x2;:::;xkofdegreedevaluatedatxu,thevectorofdesignsettingsattheuthexperimentalrun(u=1;2;:::;n).ThematrixZisoftheform: NotethatthecolumnsofZsumto1n.TherandomvectorisassumedtobedistributedasN(0;2Ib)independentlyof,whichisassumedtofollowthenormaldistributionN(0;2In). Letusrewritethemodel( 2{1 )as 14

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where If=2=2isknown,thenthebestlinearunbiasedestimator(BLUE)ofisthegeneralizedleastsquaresestimator^givenby ^=(W0A1W)1W0A1y: Itsvariance-covariancematrixis Let^y(x)=g0(x)^denotethepredictedresponseatapointxintheexperimentalregionR,whereg0(x)=[1;f0(x)].Thepredictionvariance,Var[^y(x)],isoftheform Wedenethescaledpredictionvarianceas 2Var[^y(x)]=n[g0(x)(W0A1W)1g(x)]: Itistobenotedthatthedivisionofthepredictionvarianceby2makesthisquantityscalefree,andthemultiplicationbynputsthepredictionvarianceintermsoftheweighteddesignmomentmatrix,W0A1W 15

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MyersandMontgomery ( 1995 ,p.295)and KhuriandCornell ( 1996 ,p.209)). ( 1997 )introducedagraphicaltechniquetocomparedesignsforanalysisofvarianceestimatesofvariancecomponents.Thesegraphsdescribethedispersioninthequantilevaluesofaparticulardesigncriterionfunctionoveracertainparameterspaceassociatedwiththeunknownparameter(orparameters).Weshallusethescaledpredictionvariancegivenin( 2{10 )asadesigncriterionforcomparingdesignsformodel( 2{1 ).Thesegraphsoeramorecomprehensiveassessmentofcompetingdesignsthanwhatisoeredwhencomparingdesignsusingasingle-numberoptimalitycriteria.Unlikelocallyoptimaldesigns,thisgraphicaltechniquetakesintoconsiderationthedesigndependenceproblem. NotethatVar[^y(x)]dependsonx(throughg(x)),D(thedesignunderconsideration)throughthematrixW,and.LetusthereforedenotenVar[^y(x)]=2byD(x;),wherenisthetotalnumberofexperimentalruns.IfdesignDisusedtotthemodel,thepredictioncapabilityofDcanbeassessedbyconsideringthequantilesofthedistributionofD(x;)overconcentricsurfaceswithintheexperimentalregionRobtainedbyshrinkingtheboundaryofRusingashrinkagefactor.Toaddresstheproblemofunknown,weconsiderseveralvaluesofthatbelongtoaparameterspaceC.WechooseCtobea(1)100%condenceintervalon.Suchacondenceintervalcanbeconstructedusingresponsedataobtainedunderaninitialdesign.Amethodby HarvilleandFenech ( 1985 )isusedtoconstructsuchacondenceinterval,aswillbeshowninSection2.4. InordertostudytheperformanceofadesignDthroughoutR,weconsiderseveralconcentricsurfacesdenotedbyRlocatedwithinR.ThesesurfacesareobtainedbyreducingtheboundaryofRusingashrinkagefactor.ThepredictioncapabilityofDcanthenbeevaluatedbyconsideringthedistributionofD(x;),asdeterminedbyitsquantilesonR,foragiveninC.SmallvaluesofD(x;)areobviouslydesirable. 16

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ByplottingQmaxD(p;)andQminD(p;)againstpforeachDandweobtaintheso-calledquantiledispersiongraphs(QDGs).Suchplotscanbeconstructedfortwogivendesigns,D1andD2,andforeachofseveralvaluesof.TheplotsprovideacomprehensiveassessmentofthepredictioncapabilitiesofD1andD2throughouttheregionR.Clearly,D1ispreferredoverD2iftheQDGsforD1showsmallervaluesofQmaxD1(p;)andQminD1(p;)thanthoseforD2.Furthermore,theclosenessofQmaxD(p;)toQminD(p;)foragivendesignindicatesrobustness,orlackofsensitivity,thatisinducedbythedesignofthescaledpredictionvariancetochangesintheparametervaluesovertheparameterspaceC. ( 1985 )derivedacondenceintervalforavarianceratioforanunbalancedmixedlinearmodel.Wenowbrieyoutlinetheirprocedureforconstructingsuchanintervalfor.Letusconsiderthegeneralmodel( 2{3 ).Dener=rank(W;Z)rank(W),p=rank(W),andf=nrank(W;Z).From( 2{5 )itfollowsthat 17

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HarvilleandFenech ( 1985 ,Eq.2.3)).AninterestingpropertyofthematrixMisthatrank(M)=r(see MarsagliaandStyan ( 1974 )).AlsonotethatS=R(j)=R(;)R()andS=R(),whereR(j)denotesthereductioninthesumofsquaresduetottingafter,R(;)isthetotalregressionsumofsquaresobtainedfromboththexedandrandomeectsinthemodel,R()istheregressionsumofsquaresduetottingamodelwithjustthexedeects,andSSEistheresidualsumofsquaresformodel( 2{3 ).ApplyingFormula(79)of Searle ( 1971 ,p.445),wehave:E[S=r]=2+1 18

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Recallthatr=rank(M).Let1;:::;rdenotethenonzeroeigenvaluesofMandletD=diag(1;:::;r)with0<1:::r.DeneNasabrmatrixsuchthatN0N=IrandMN=ND,thatis,thecolumnsofNareorthonormaleigenvectorsofMcorrespondingtotheeigenvalues1;:::;r.Treatinglikeasavectorofunknownparameters, HarvilleandFenech ( 1985 ,p.140)concludedthefollowing:(i)anecessaryandsucientconditionforalinearfunctiong0tobeestimableistheexistenceofavectorhsuchthatg0=h0M,and(ii)thereexistsanr1(wherer=rank(M))vectoroflinearlyindependentestimablefunctionsof.Onesuchr1vectorist=(t1;:::;tr)0where Since~isasolutiontothelinearequationM~=q,t=(t1;:::;tr)0canbeestimatedby~t=(~t1;:::;~tr)0,where ~t=D1=2N0~=D1=2N0M~=D1=2N0q; whichwillbeusedtoobtainapivotalquantityforconstructingacondenceintervalfor.NotingthatPWW=W,PW=P0W=P2W,itcanbeshownthatthedistributionofqisamultivariatenormalwithE(q)=0Var(q)=2Z0(IPW)(I+ZZ0)(IPW)Z=2(M+M2);

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r)Pri=1~t2i=(1+i) Itcanbeshownthat~t0(I+D)1~t=2=~t0[Var(~t)]1~t2r,andhenceG(;y)F(r;f).NotethatalthoughthematrixNisnotunique,Var(~t)=2(I+D)isinvarianttothechoiceofN.AsthedistributionofG(;y)doesnotdependonanyunknownparametersexcept,itcanbeusedasapivotalquantityforconstructingacondenceintervalfor. Burdicketal. ( 1986 )usedtheso-calledbisectionmethodtocomputetheaforementionedexactinterval.Theexacttwo-sided(1)100%condenceintervalonisgivenby[l;u],wherel;uare,respectively,therootsofthefollowingequations: whereProb[Fr;fF=2;r;f]==2.NotethatG(;y)canberewrittenas r)Pri=1~t2i(1i+)1=i ThefunctionG(;y)isconvexandmonotonicallynonincreasingwithrespecttofor0<1andG(0;y)=fPri=1~t2i=(rSSE).Ifwereplace(1i+)1with(11+)1and(1r+)1wegetthefunctionsG1(;y)andGr(;y),respectively,suchthatG1(;y)G(;y)Gr(;y).Thisistruebecause0<1:::r.Thebisection 20

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2{19 )),whichareobtainedbyequatingG1(;y)andGr(;y)withF1=2;r;fandF=2;r;f,respectivelyandsolvingfor.Thusbyreplacingwithl1;lr;u1;urwegetthefollowingequations: Theaboveequationsyieldthefollowingbounds Forexample,G1(l1;y)=F1=2;r;fimpliesthat(seeEq.( 2{17 ))(f r)Pri=1~t2i(11+l1)1=i r)Pri=1~t2i=i 2{16 ).Insteadofsolvingthetwonon-linearequations,thebisectionmethodusestheselowerandupperboundstocomputelandu.Ademonstrationofthesecalculationsisgivenin BurdickandGraybill ( 1992 ,AppendixB). 21

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Khuri ( 1992 ).Theexampleinvestigatedtheeectsoftwofactors,temperature(x1)andcuringtime(x2),ontheshearstrengthofthebondingofgalvanizedsteelbarswithacertainadhesive.Thegalvanizedsteelbarswereselectedoveraperiodof12days.Eachbatchofsteelbarswasselectedatrandomonagivendate.Hence,the12datesweretreatedasrandomblocks.Threelevelswerechosenforeachofthetwofactorsaccordingtoa32factorialdesign.Thesamedesignwasusedoneachofthese12dates,exceptthatoncertaindays,replicationsweretakenatthedesigncenterinordertotestforlackoftofthettedmodel.Werefertothis32factorialdesignasD1.Acompletesecond-degreemodelwasttedtothedata.Theactualdataset,consistingofn=118observationsin12blocksisreproducedinTable 2-1 .Notethatx1andx2denotethecodedvaluesoftemperatureandtime,respectively,namely,x1=temperature400 25x2=time35 5 Thesettingsofx1andx2fromthe32designare1;0;2forx1,and1;0;1forx2. WecompareD1withfourothersecond-orderdesigns.Thedesignswerechosensothatasecond-degreemodelcanbetted.TheyincludethedesignsD2,D3,D4,andD5.TherstoneissimilartoD1withtheonlydierencebeingthatthecodedsettingsofx1areequallyspacedhavingthevalues-1,0.5,2.DesignsD3andD4aremodiedversionsoftheso-calleduniformshelldesigns(see Doehlert ( 1970 )and DoehlertandKlee ( 1972 )).Theoriginaluniformshelldesignsconsistofpointsthatareuniformlyspacedonconcentricsphericalshells.Intwo-dimensionalspaces,thesedesignsareobtainedfromthepointsofadoublesimplexwithacenterpoint.TheproposedmodieddesignpointsforD3andD4resemblethepointsofatwo-dimensionaluniformshelldesign,buttheyarenot 22

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2-2 andthecorrespondingdesignpointsareshowninFigure 2-1 Theexperimentalregion,R,isrectangularwith1x12,1x21.Foreachdesign,weconsiderthedistributionofD(x;)oneachofseveralconcentricrectangles,R,whichareobtainedbyareductionoftheboundaryofRusingashrinkagefactor,0:5<1.ThusRisdeterminedbytheinequalitiesai+(1)(biai)xibi(1)(biai);i=1;2; 2-1 ,weobtainthe95%condenceinterval,C,namely(l;u)=(0.0763243,0.9667678).ForeachdesignandaselectedvalueofinC,quantilesofthedistributionofD(x;)areobtainedforx2R,whereisoneofseveralvaluesofchosenfromtheinterval(0:5;1].ThenumberofpointschosenoneachRwas2000consistingof500oneachside.Thequantilesarecalculatedforp=0(0:05)1.TheprocedureisrepeatedforothervaluesofinC.ThenQmaxD(p;)andQminD(p;)arecalculatedusingtheformulasin( 2{11 ).TheRsoftwarewasusedinconductingthenumericalinvestigationandobtainingtheactualplots. TheQDGsforthecomparisonofdesignD1withD2,D3,D4,andD5aregiveninFigures 2-2 2-3 2-4 and 2-5 ,respectively.TheQDGsforthecomparisonofD3withD4,andD4withD5arepresentedinFigures 2-6 and 2-7 ,respectively.FromFigure 2-2 ,itcanbeseenthatD2performsslightlybetterthanD1for=0.6,0.7and0.8.ThedierenceinpredictioncapabilityisdistinctlyevidentinfavorofD2for=0.9and1.Figures 2-3 and 2-4 suggestthatthesemiuniformshelldesigns,D3andD4,performbetterthanD1

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2-5 )clearlydepictbetterpredictioncapabilitywithdesignD1.Figures 2-6 and 2-7 indicatethatdesignD4isbetterthanD3andD5.ItshouldbenotedthatchangingthecondencecoecientforthecondenceintervalCfrom0.95to0.90or0.99didnotchangemuchthepatternoftheQDGs.ThisindicatesthattheQDGsarenottoosensitivetochangesintheboundsofthecondenceintervalon. TrincaandGilmour ( 2001 )). MathewandSinha ( 1992 )derivedoptimumtestsinunbalancedsplit-plotdesignsundermixedandrandommodels. Robinsonetal. ( 2004 )consideredtheanalysisofsplit-plotexperimentswithnon-normalresponses.Completerandomizationformanyindustrialandagriculturalexperimentsisnotpracticalduetotimeorcostconstraints,orexistenceofsomehard-tochangefactors.Insituationslikethese,restrictionsonrandomizationleadtosplit-plotdesigns,allowingcertainfactorlevelstoberandomlyappliedtothewholeplotunits,whiletheremainingfactorlevelsrandomlyappliedtothesubplotunits.Theconceptofsplit-plotting,bi-randomization,ortwo-stagerandomizationisheavilyusedinindustrialexperimentation. Huangetal. ( 1998 )and BinghamandSitter ( 1999 2001 )havederivedminimumaberrationtwo-levelfractionalfactorialsplit-plotdesignsthatareusedforscreeningexperiments. GoosandVandebroek ( 2001b )consideredanexchangealgorithmforconstructingD-optimalsplit-plotdesigns.Aproperclassicalstatisticalanalysisrequirestheuseofgeneralizedleastsquaresestimationandinferenceproceduresand,hence,theestimationofthevariancecomponentsin 24

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GoosandVandebroek ( 2003 )demonstratedtheuseofanecientalgorithmtocomputeD-optimalsplit-plotdesignsassumingaxednumberofwholeplotsandpre-speciedwholeplotsizes.Thetwo-levelfactorialandfractionalfactorialdesignswereshowntobeD-optimalforestimatingrst-orderresponsesurfacemodelsforspecicnumbersandsizesofwholeplots. Goosetal. ( 2006 )discussedthepracticalinferencefromindustrialsplit-plotdesigns.Theanalysisofthedatainvolvedmixedmodelusingthegeneralizedleastsquares(GLS)estimation.Theauthorscomparedtheanalysiswiththeoneinvolvingordinaryleastsquares(OLS)whichtreatstheexperimenttobecompletelyrandomized.Itwasrecommendedthatsplit-plotexperimentsinvolvingextremelylownumbersofwholeplotsshouldbeavoided.Whentheobservationsarecorrelatedonlytoasmallextent,itwasfoundthattheredoesnotexistmuchdierencebetweentheanalysisperformedusingOLSandtheonecarriedoutbyGLS. Thecommonassumptioninasplit-plotexperimentisthatthesameerrorvarianceexitsinallsubplottreatments. Curnow ( 1957 )dealtwithtestsofsignicanceforthedeparturefromequalityofvariancesfordierentsubplottreatments,andalso,extended Pitman ( 1939 )methodtoconstructcondencelimitsforthevarianceratio.Thedesignandanalysisofsplit-plotdesignsformixtureexperimentswereconsideredby Kowalskietal. ( 2002 )(seealso Kowalski ( 1999 )and Cornell ( 1988 )).Aclassicexampleofasplitplotdesignisgivenin Montgomery ( 2005 ,Section14-4).Heconsideredanexperimentdonebyapapermanufacturerwhoisinterestedinstudyingtheeectsofthreedierentpulppreparationmethodsandfourdierentcookingtemperaturesonthetensilestrengthofthepaper.However,thelinearmodelforthesplitplotdesigngivenin Montgomery ( 2005 ,Eq.(14-15))isdierentfromthelinearmixedmodelusedby GoosandVandebroek ( 2003 )and Liangetal. ( 2006 ).Thelatterauthorsconsideredthefollowingmodel 25

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Liangetal. ( 2006 )usedthree-dimensionalvariancedispersiongraphstocomparecompetingsplit-plotdesignsbasedonthepredictionvariancecriterion.Itcanbeshownthatthepredictionvariancedependsontheunknownvariancecomponents.Inordertoapplythetechniqueofquantiledispersiongraphsapproach,theparameterspacefortheunknownvariancecomponentsneedstobeconstructed.Themethodologydiscussedinthischaptercanbeadaptedtocomparesplitplotdesigns. 26

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Designsettingsandresponsevalues(shearstrengthinpsi) Block(Batch) Temp.Time123456x1x2(July11)(July16)(July20)(Aug.7)(Aug.8)(Aug.14) -1-11226107511721213128211420-11898179018041961194016992-1214218432061218420951935-10147211211506160615721608002010,1882217522792450,2355229123741915,21062420,224020235222742168229821472413-111491169117071882174118460120782513239225312366239221253125882617260924312408 Block(Batch) Temp.Time789101112x1x2(Aug.20)(Aug.22)(Sep.11)(Sep.24)(Oct.3)(Oct.10) -1-11281130510911281130512070-11833177415881992201117422-1211621331913221321921995-1015021580134316911584148600247123932205,226821422052,203223392103219020243024402093220822012216-111645168815821692174417510123922413239224882392239021251726042477260125882572 Note:Theoriginaldesignsettingsofx1correspondingto-1,0,2are375F,400F,and450F,respectively;thoseforx2correspondingto-1,0,1are30,35,and40sec.,respectively.(Source: Khuri ( 1992 ,Table1)) 27

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DesignsettingsforD1,D2,D3,D4andD5 a:D1istheoriginaldesignshowninTable1. b:D2isthesameasD1,butthesettingsofx1are-1,0.5,2. c:D3isthesemi-uniformshell1design. d:D4isthesemi-uniformshell2design. e.D5istherandomlygenerateddesign. 28

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PlotsofthedesignpointsforD1,D2,D3,D4,andD5

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QDGsforD1andD2

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QDGsforD1andD3

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QDGsforD1andD4

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QDGsforD1andD5

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QDGsforD3andD4

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QDGsforD4andD5

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BoxandDraper ( 1975 )listed14propertiesthatadesignshouldhave(seealso, KhuriandCornell ( 1996 ,pp.53,273)).Oneofthesepropertiesstatesthatadesignshouldbeabletodetectmodelinadequacy.Byaproperchoiceofdesign,itispossibletomaximizethepowerofthelackofttesttodetectdeparturefromthettedmodel.Forexample,multiresponsedesignsbasedonthepowercriterionaregivenin WijesinhaandKhuri ( 1987a ).Theseauthorsintroducedtwodesigncriteriatoimprovethepowerofthemultivariatelackofttestforalinearmultiresponsemodel(seealsothereferencescitedtherein).Thedevelopmentofthistestwasdiscussedin Khuri ( 1985 ).Inthecaseofasingleresponse, Atkinson ( 1972 ), AtkinsonandFedorov ( 1975 ),and JonesandMitchell ( 1978 )deriveddesigncriteriatoenhancethedetectionofmodelinadequacy. Aswillbeseenlaterinthecaseofaresponsesurfacemodelwitharandomblockeect,thepowerfunctiondependsontheunknownvariancecomponentsofthemodel.Theconstructionofdesignsforsuchamodelwouldthereforerequiresomepriorknowledgeoftheseunknownparameters.Thepowerofageneralizedleastsquares(GLS)F-testisusedtoassesstheeciencyassociatedwithagivendesign. RaoandWang ( 1995 )studiedseveralpropertiesofthepowerofthistestforregressionmodelswithanestederrorstructure.Sincethepower,inthecaseofasingleresponsemodel,isamonotoneincreasingfunctionofthenoncentralityparameter,thelatterquantityisalsousedasadesigncriterionforcomparingdesigns.Foragivendesign,quantilesofthesetwocriteria 36

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where=Z+.Supposethatitisofinteresttotestthehypothesis,H0:Ld=0againstthealternativehypothesis,Ha:Ld=,whereLisaknown[q(p+1)]matrixofrankq(
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RaoandWang ( 1995 )consideredthepowerofasimilartestforaregressionmodelwithanestederrorstructure. 3{2 )areusedasdesigncriteriaforcomparingdesignsforaresponsesurfacemodelwitharandomblockeect.Thenoncentralityparameterandthepoweraredependentontwoparameters,namely(2;),whichareunknown.Toaddresstheproblemofunknown(2;),weconsiderseveraloftheirvaluesinsideacertainparameterspacefor(2;)denotedbyS.InadditiontotheparameterspaceS,wealsoneedtospecifyanalternativespacefor,whichisasubsetofRq.RecallthatisthevaluespeciedinthealternativehypothesisHainSection3.2.ThealternativespaceisneededbecauseanyconsiderationofthepoweroftheF-testin( 3{2 )requiresthespecicationofvaluesof.Inordertostudyandcomparetheperformancesoftwodesigns,D1andD2,throughoutthealternativespace,weconsidersurfacesofseveralconcentrichyperspheresofradiuswhicharelocatedwithinthealternativespace.ThesurfaceofasphereofradiusisdenotedbyT.Thenoncentralityparameterin( 3{4 )dependson,D(thedesignunderconsiderationthroughthematrixW),and(2;).LetusthereforedenotethevalueofthenoncentralityparameterbyD(;2;).IfdesignDisusedtotthemodel,thecapabilityofDtoproducehighvaluesofD(;2;)canbeassessedbyconsideringthedistributionofD(;2;),determinedintermsofitsquantilesonthesurfaceT,fordierentvaluesof. 38

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(3{6) whereQD(p;2;;)denotesthepthquantileofD(;2;)for2T. LargevaluesofD(;2;)aredesirable.ByplottingQmaxD(p;)andQminD(p;)againstpweobtainthequantiledispersiongraphs(QDGs)forthenoncentralityparameter.Suchplotscanbeconstructedfortwogivendesigns,D1andD2,andforeachofseveralvaluesof.DesignD1ispreferredoverdesignD2,iftheQDGsforD1depicthighervaluesofQmaxD1(p;)andQminD1(p;)thanthoseforD2.Furthermore,theclosenessofQmaxD(p;)toQminD(p;)foragivendesignDindicatesrobustness,orlackofsensitivity,ofthenoncentralityparametertotheparametervaluesinthesetS. AswasmentionedinSection3.2,thepowerofFGLS()isamonotonicallyincreasingfunctionofthenoncentralityparameter.Hence,asasecondcriterionforcomparingdesigns,wecanconsidertheactualpowervalueitself.ForagivendesignD,highpowervaluesaredesirableforseveralvaluesof.Thepowerfunctiondependson,whichdependson2and.Asaresult,theuseofthepowerfunctionasadesigncriterionalsosuersfromthedesigndependenceproblem.TheQDGapproachcanthenbeusedtocircumventthisdependenceproblembyreplacingin( 3{6 )quantilesofbythoseofthepowerfunctionin( 3{5 ). ItshouldbeemphasizedagainthattheparameterspaceSisneededinordertoassessthedependenceofthechosencriterionfunction(thenoncentralityparameterorthepowerfunctionoftheF-testin( 3{2 ))onthevaluesoftheunknownvariancecomponents,2and2(or2and=2 39

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A(1)100%condenceintervalon=2=2,namely,l<
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Khurietal. ( 1996 ))1=cos12=sin1cos2...q1=sin1sin2sinq3sinq2cosq1q=sin1sin2sinq3sinq2sinq1 Byvaryingthevalueof,wecanobtainseveralvaluesofwithinthealternativespace. KhuriandCornell ( 1996 ,p.328)concerningtheeectsofstorageconditionsonthequalityofacertainvarietyofapples(seeTable 3-1 ).Theappleswereharvestedfromfourdierentorchardswhichwereselectedatrandom.Thus,`orchard'canbeconsideredasarandomblockeect.ThecontrolvariableswereX1=numberofweeksinstorageafterharvestandX2=storagetemperature(oC)afterharvest.Theresponsevariableofinterestwasy=amountofextractablejuice(ml/100g).A44factorialdesignwasusedinorchards1,2and3,anda34factorialdesignwasusedinorchard4.Thecodedvaluesofthecontrolvariableswereobtainedasfollows:xi=(Xi 41

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3-1 ,weusethecodedvaluesx1=X12:4 1:083x2=X211:0 9:772: 3-2 .ThistableincludesthesettingsofX1andX2inallthefourblocks.WecompareD1withtwootherdesigns,D2andD3,chosensuchthatasecond-degreemodelcanbettedineachblock.DesignD2isasemicentralcompositedesign(SCCD)consistingofacomplete22factorialdesignhavingtwosettingscodedas-1and+1foreachfactor,7centerpoints(n0=7),and4axialpoints.ThecenterpointswerechosensothatthetotalnumberofdesignpointsinD1andD2issame(seeTable 3-3 ).Thisdesignislabeledassemicentralbecausetheaxialpointsarenotsymmetricwithrespecttotheorigin,arequirementforacentralcompositedesign(see,forexample, KhuriandCornell ( 1996 ,Section4.5.3)).Thesamedesignwasusedinallfourorchards.DesignD3containsarbitrarilygenerateddesignpointsintheexperimentalregionfordesignD1,thatis,R=f(x1;x2):1:29x11:48;1:13x21:43g.Thesamepointswereusedinallfourorchards.Table 3-2 showsthecodeddesignsettingsofD1,D2andD3.ThenumberofpointsconsideredineachdesignisgiveninTable 3-3 andthecorrespondingdesignpoints(inBlock1)areshowninFigure 3-2 RecallthatthehypothesisofinterestisingeneralH0:Ld=0versusHa:Ld=,whereLisaknown[q(p+1)]matrixofrankq(
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Here,Lisa26matrixofrank2.Thepurposeofthishypothesisistodeterminewhetherornotacompletesecond-degreemodelisneededtorepresentthemeanresponse.Inthiscase,q=2.Therefore,Tisacircleofradiuscenteredat(0,0).Onethousand'saregeneratedonthiscircleusingthemethoddescribedinSection3.3.3.Wehaveselected5dierentvaluesofthatareequallyspacedbetween0.01and1.Theyare0.01,0.2575,0.505,0.7525and1.Forhighervaluesof,thepowerincreasedtovaluescloseto1forallthedesignsunderconsiderationandthequantileplotsofthedesignswerenotdistinguishable.Foreachdesignandaxedvalueof(2;)selectedfromtheparameterspaceS,thequantilesofthenoncentralityparameterandcorrespondingpowervaluesforp=0(0:05)1areobtainedonTforaspeciedvalueof.Theprocedureisrepeatedforothervaluesof(2;)inS,whichisconstructedbyusingthemethoddescribedinSection3.3.2.UsingthedatasetinTable 3-1 ,weobtainedthefollowing95%condenceintervals,namely,(a;b)=(1:797;3:935)for2,and(l;u)=(0:0;3:205)for.Atrst,100equally-spacedvaluesof2arechosenfromtheinterval(a,b).Foreachsuchselectedvaluesof2,100equally-spacedvaluesofarethenchosenfromtheinterval(l;u).Hence,thetotalnumberofvaluesof(2;)choseninsideSis10,000.Finally,aftercalculatingQmaxD(p;)andQminD(p;)using( 3{6 ),theQDGplotsforthenoncentralityparameterwereobtained.Theprocesswasrepeatedusingothervaluesof.SimilarQDGplotswereobtainedforthepowerfunction.TheRsoftwarewasusedforconductingthenumericalinvestigationandobtainingtheactualplots. 43

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3-3 and 3-4 .Highvaluesofthenoncentralityparameterandpowerareobviouslydesirable.BothgraphsshowthatthemaximumquantilesofD1areabovethoseofD2,especiallyforlarge.Hence,D1ispreferredoverD2.TheQDGsforthecomparisonofdesignsD1andD3(seeFigures 3-5 and 3-6 )indicatethatD1performsfarbetterthanD3,whichisexpectedsinceD3consistsofarbitrarilygenerateddesignpoints.Finally,theQDGsforthecomparisonofdesignsD2andD3(seeFigures 3-7 and 3-8 )showthatD2performsbetterthanD3.Thus,inconclusion,amongthe3designs,bestpowervaluesfortestingH0in( 3{9 )canbeachievedbyusingD1,followedbyD2,andthenD3. 44

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OriginalcontrolvariablesCodedcontrolvariablesResponse OrchardX1X2x1x2y 45

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OriginalcontrolvariablesCodedcontrolvariablesResponse OrchardX1X2x1x2y Source: KhuriandCornell ( 1996 ,p.328) 46

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DesignsettingsforD1,D2andD3 Note:ThedesignsettingsinBlocks2,3,and4arethesameasthoseinBlock1,exceptfordesignD1withregardtoBlock4.D1istheoriginaldesign.D2isthesemicentralcompositedesign.D3istherandomlygenerateddesign. 47

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Numberofpointsineachdesign DesignBlock1Block2Block3Block4Total Figure3-1. Condenceregionfor(2;) 48

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Plotsofthedesignpoints(inBlock1)fordesignsD1,D2,andD3

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QDGsofthenoncentralityparameter(ncp)forD1andD2

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QDGsofthepowerfunctionforD1andD2

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QDGsofthenoncentralityparameter(ncp)forD1andD3

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QDGsofthepowerfunctionforD1andD3

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QDGsofthenoncentralityparameter(ncp)forD2andD3

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QDGsofthepowerfunctionforD2andD3

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Khuri ( 1992 )discussedtheanalysisofthexedpolynomialtermsandtherandomblockeectsinsuchmodels.Inthischapter,thesamemodelisconsideredundertheassumptionthatnointeractionsexistbetweentheblocksandthemodel'sxedpolynomialeects. KieferandWolfowitz ( 1960 )showedthatD-optimalandG-optimaldesignsforclassicallinearmodelsareidenticalaccordingtotheirEquivalenceTheorem.Themultivariateversionofthistheoremisgivenin Federov ( 1972 ,p.212)and Kiefer ( 1974 ,Section5). Federov ( 1972 )introducedanalgorithmfortheconstructionofaD-optimalmultiresponsedesignusingasequentialprocedure.However,hisalgorithmrequiresthatthevariance-covariancematrixforthemultiresponsevectorbeknown.BasedonFedorov'swork, WijesinhaandKhuri ( 1987b )proposedasequentialalgorithmtogenerateaD-optimalmultiresponsedesignusinganestimateofthatvariance-covariancematrix.Inthischapter,theresponsesurfacemodelwitharandomblockeectistreatedasamodelforamultivariateresponseasin AtkinsandCheng ( 1999 ).ThisenablesustoapplythemultivariateversionoftheEquivalenceTheorem. whereyisavectorofnobservationsontheresponse,0and=(1;2;:::;p)0areunknownparametersassociatedwiththepolynomialportionofthemodel,= 56

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ItisassumedthatisdistributedasN(0;2Ib)independentlyof,whichisassumedtofollowtheN(0;2In)distribution. ( 1995 )and AtkinsandCheng ( 1999 )resortedtotheapproximatetheoryofoptimaldesignstodealwiththespecialfeatureofwithin-blockcorrelation.Inthissectionweshowhowtheapproachof AtkinsandCheng ( 1999 )canbeadoptedtoderiveoptimaldesignsinthepresenceofarandomblockeect.Letusrewrite( 4{1 )as where=(0;0)0andW=[1n:X],wheren=mb.LetusalsopartitionWasW=0BBBBBBB@W1W2...Wb1CCCCCCCA=0BBBBBBB@1m:X11m:X2......1m:Xb1CCCCCCCA Eachsetofmpointsincanberepresentedasapointuinm=f(x01;x02;:::;x0m)0:xi2;i=1;2;:::;mg.Also,eachpointinmcanberegardedasablock.Thereis, 57

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Var(yi)=2Im+2Jm=2V(4{5) whereV=Im+Jmand=2=2.Notethat0<<1.Dening=2=(2+2)==(+1)makesliebetween0and1.Theparameterisoftenreferredtoastheintra-classcorrelationcoecient.Hence,thevariance-covariancematrixofy,thevectorof 58

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whereA=IbV.Ifisknown,thenthebestlinearunbiasedestimator(BLUE)ofinmodel( 4{3 )isthegeneralizedleastsquaresestimator^givenby ^=(W0A1W)1W0A1y; anditsvariance-covariancematrixis where=2A. Notethatinpracticalsituationsisunknown.Anestimateof,canbeobtainedbyreplacingtheunknownvariancecomponents2and2bytheircorrespondingestimates,suchasANOVA,mainlyHenderson'sMethodIII,maximumlikelihood(ML),orrestrictedmaximumlikelihood(REML)estimates.Therstmethodmayyieldnegativeestimates,butdoesnotneedthenormalityassumptionoranyotherdistributionalassumption.Thelattertwoestimatesarenonnegativeandhavedesirablepropertiessuchasconsistencyandasymptoticnormality.However,theyneedanunderlyingprobabilitydistributionforthedata,usuallyconsideredtobethenormaldistribution.TheseestimatescanbeeasilyobtainedbyusingPROCMIXEDin SAS ( 2000 )orthelmefunctioninR( PinheiroandBates ( 2000 )). Atanypointu2m,thepredictedresponsevectoris ^y(u)=G0(u)^ Thepredictionvariance-covariancematrixis (4{10) 59

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(4{11) Letsdenotethenumberofdistinctvaluesofu1;u2;:::;ub.Also,letnidenotethenumberofreplicationsattheithofsuchdistinctpoints(i=1;2;:::;s).Then,Psi=1ni=b.Forthediscretedesignmeasurebthatassignstheweightni=btoui(assumingu1;u2;:::;usaredistinct),theinformationmatrixperblock,1 (4{12) Hence,foranygivendesignmeasure(u)onm,theinformationmatrixis (4{13) KieferandWolfowitz ( 1960 )forthesingle-responsecase. 60

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( 1972 )describedanalgorithmfortheconstructionofaD-optimalmultiresponsedesignassumingthatthevariance-covariancematrixforthemultiresponsevectorisknown.Tocircumventtheproblemofunknownvariance-covariancematrix, WijesinhaandKhuri ( 1987b )proposedasequentialalgorithmusinganestimateofthatvariance-covariancematrix.ThefollowingtheoremisbaseduponthemultivariateversionoftheEquivalenceTheoremgivenin WijesinhaandKhuri ( 1987b ). Thefollowingassertionsareequivalent: (i)ThedesignmeasureisD-optimal,thatis,jM(;V)j=sup2HjM(;V)j (ii)isG-optimal,thatis,itminimizesmaxu2mtr[V1G0(u)M1(;V)G(u)] withrespecttoall2H. (iii)maxu2mtr[V1G0(u)M1(;V)G(u)]=p+1 wherep+1isthenumberofxedunknownparametersinmodel( 4{1 ). (i)Aninitialdesignb0consistingofb0blockseachofsizemischosensuchthatM(b0;V0)isnonsingular,whereV0=Im+0Jm,0=initialguessof.Thismeasureisdenedbyassigningtheweight1=b0toeachpointinb0. 61

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Theweight1 (iii)Repeatsteps(i)and(ii)severaltimesuntilforthe(l+1)staddedblockmaxu2mtr[V1lG0(u)M1(bl;Vl)G(u)](p+1)< 4-1 .Twocontrolvariables,x1andx2,areconsidered,andtheexperimentalregionis=f(x1;x2):1x11;1x21g.Inordertotacompletesecond-degreemodelwitharandomblockeectweneedn>p+b,wheren=mbisthetotalnumberofexperimentalruns,pisthenumberofxedeects(excludingtheintercept)andbisthenumberofblocks.Hereb=b0=2,m=5.Inordertogeneratetheresponsevalues,initialguessesforthevaluesofthexedunknownparametersandvariancecomponentsareneeded.Theseinitialvaluesareusedtogeneratetheresponsevaluesonly,butarenotusedinthesequentialgenerationoftheD-optimaldesign.Since2;2areunknown,weobtaintheirestimates,namelyrestrictedmaximumlikelihood(REML)estimatesusingthelmefunctionintheRsoftwarefromthedatageneratedbytheinitialdesign,thenupdatetheseestimatesasweproceed. Themaximizationofthetracefunctionwithrespecttou25ateveryiterationwasperformedusingacomputerprogramwrittenintheRsoftware( VenablesandRipley 62

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2002 ))andisbasedonthecontrolledrandomsearch(CRS)procedureof Price ( 1977 )(also,seeAppendixF).CRSisadirectsearchtechniqueandiswellsuitedforobjectivefunctionsthatarenotdierentiable.Thesequentialprocedureiscomputationallyintensive.Withoutaproperglobaloptimizationalgorithm,thecomputationcanbeveryslowandsometimesmayfailtoconverge. Conlon ( 1985 )wroteasimilarprograminFORTRANimplementingPrice'salgorithmforfunctionminimization.Morerecently, Alietal. ( 1997 )providedcomparisonsofmodiedcontrolsearchalgorithms. WehaveusedthealgorithmdescribedinSection4.5tocomputeanoptimaldesignsequentiallystartingwithaninitialdesignpointanddierentvaluesoftheunknownparameters.Thenumberofiterationsrequiredfortheconvergenceofouralgorithmcorrespondingto3initialvaluesof,namely,0.10,0.50,and0.90(with2heldxedas1)are87,19and100respectively. WijesinhaandKhuri ( 1987b ,Section4.1)madeaconjectureregardingthechoiceofinitialdesign.Theyobservedthatthenumberofiterationsrequiredtosatisfythestoppingcriterionisgreatlyreducedbytheinclusionoftheboundarypointsoftheexperimentalregionasinitialdesignpoints.Whenweincludedboundarypointsasinitialdesignpoints,theprocedureattainedconvergenceatafasterrate.Inthischapter,wearereportingtheresultsfor=0:50(i.e.2=2=1).Theinitialvaluesofthexedunknownparametersofasecond-degreemodelwerechosenasfollows:0=1,1=2,2=2,11=1,22=1,and12=1.TheinitialdesignpointstogetherwiththeaugmenteddesignpointsformanapproximateD-optimaldesign.InTable 4-2 ,theaugmenteddesignpointsalongwithupdatedestimatesof2,2,andareprovided.Thevalueofwaschosenas0.001andconvergencewasattainedatthe19thiteration. 63

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Initialdesignalongwithresponsevalues Figure4-1. InitialDesign 64

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AliandSilver ( 1985 )and ConerlyandManseld ( 1988 ).Theproblemofcomparingcorrelatedresponsemodelswasdiscussedby Zellner ( 1962 ).Thisprocedurerequiresthatanestimateofthevariance-covariancematrix,,beusedinplaceof.Tocircumventtheproblemofanunknown, SmithandChoi ( 1982 )developedanexactmethodtocomparetwocorrelatedresponsemodelswithouthavingtoestimate. Khuri ( 1986 )introducedageneralprocedureinvolvingexactmultivariatetestsfortheequalityofparametersfromseveralcorrelatedresponsemodelswithanunknown.Thisprocedureassumedtheresponsemodelstobeofthesameformandtocontainthesamesetofcontrolvariables. Thischapterdealswiththecomparisonofdesignsforcorrelatedresponsemodelswithanunknownvariance-covariancematrix,.Thenoveltyofourapproachliesinapplyingthequantiledispersiongraphsinaninvestigationofthepowerofseveralmultivariatetestsconcerningsuchmodels.Ourproposedapproachisbasedonconsideringquantilesofacertaincriterionfunction(namely,powerofeachofthreemultivariatetests)onconcentricsurfaceswithinaparticularregionoftheso-calledalternativespace.Powercomparisonsofthefourmultivariatetests,namely,Roy'slargestroot,Wilks'likelihoodratio,Hotelling-Lawley'strace,andPillai'stracewereconsideredbyseveralauthors(see PillaiandJayachandran ( 1967 ), Royetal. ( 1971 ,Ch.5), Seber ( 1984 ,Section8.6.2)).Thedependenceofthesequantilesontheunknownvaluesofthevariancesandcorrelationsobtainedfromthevariance-covariancematrix,isdepictedbyplottingtheso-calledquantiledispersiongraphs(QDGs)ofthecriterionfunction.Theseplotsprovideaclearassessmentofthemagnitudeofthepowervalueassociatedwithagivendesign.Anumericalexampleispresentedtoillustratetheproposedmethodology. 68

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wherei0andtheik'sareunknownparametersandiisarandomerrorcorrespondingtotheithresponse(i=1,2,:::,r).Assumingthattheabovermodelscanhavedierentdesignmatricesandeachdesignconsistsofnexperimentalruns,thesemodelscanberewrittenas whereyiisann1vectorofobservations,1nisann1vectorofones,i=(i1;i2;:::;ip)0,Xidenotesthenpdesignmatrixofrankp,andiisavectorofrandomerrorscorrespondingtothenobservationsfromtheithresponse(i=1,2,:::,r).Inmatrixnotation,theabovemodelscanbewrittenas where Theabovematricesareofordersnr,1r,n(pr),(pr)randnr,respectively.diagimpliesthatthematrixBisblockdiagonal.Itisassumedthattherowsofare 69

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LetusconsiderthefollowinghypothesisH0:10=20==r0;1=2==ragainstHa:i06=l0forsomei6=l=1;2;:::;r;orm6=uforsomem6=u=1;2;:::;r.Thisisknownasthehypothesisofconcurrence. 5{3 ).LetdenotetherankofX.NotethatsinceeachXi(i=1,2,:::,r)inXisofrankp,wehaveppr.Weshallassumethatrn.LetCbether(r1)matrixofrankr1, ThenullhypothesisH0canbeexpressedasH0:00C=00;WBC=0; whereIpistheidentitymatrixoftheorderpp.Multiplyingmodel( 5{3 )ontherightbyC,weget 70

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ThehypothesisH0canthenbeexpressedasH0:G=0where and001and02arezerovectorsoforders1(pr)andp1respectively.NotethatGisoffullrowrankequaltop+1. ThedevelopmentofamultivariatetestfortestingH0dependsonRoy'sunion-intersectionprinciple.Thisisillustratedasfollows(see Khuri ( 1986 )fordetails):Leta=(a1;a2;:::;ar1)0beanarbitrarynonzerovectoroforder(r1)1.Themultivariatemodelgivenin( 5{7 )canbereducedtoaunivariatemodelbythetransformationsya=YCa,a=a,anda=Ca.Then, NotethataN(0;2aIn),where2a=a0C0Ca.Themultivariatehypothesisstatedearlieralsoreducestotheunivariatehypothesis Clearly,H0istrueifandonlyifH0(a)istrueforalla6=0.But,foreacha6=0,thehypothesisH0(a)isagenerallinearhypothesisassociatedwiththeunivariatemodel( 5{10 ).ThehypothesisH0(a)can,therefore,berejectedforlargevaluesofthestatistic where(Z0Z)isageneralizedinverseofZ0Z.Thisstatisticisinvarianttothechoiceofthegeneralizedinverse(see Searle ( 1971 ,Section5.5)).Sinceya=YCa,thenR(a)can 71

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where and ThematricesShandSearecalled,respectively,thematrixduetothehypothesisandthematrixduetotheerror.ThematrixShispositivesemideniteofrank=min(r1;p+1),andSe,undertheconditionn1r1,ispositivedenitewithprobability1,andhasthecentralWishartdistributionW(n1;C0C)withn1degreesoffreedom.SinceGin( 5{9 )isoffullrowrankequaltop+1,ShisindependentofSeandhasthenoncentralWishartdistributionW(p+1;C0C;)withp+1degreesoffreedomandnoncentralityparametermatrixgivenby(see Seber ( 1984 ,p.414)) Roy'slargestrootteststatisticisgivenby(see Khuri ( 1986 ,p.350))emax[ShS1e],whichdenotesthelargesteigenvalueofthematrixShS1e.OthermultivariateteststatisticsfortestingH0includethefollowing: Wilks'likelihoodratio:=[det(Se)]=[det(Se+Sh)],(smallvaluesofleadtorejectionofH0) Hotelling-Lawley'strace:U=tr(ShS1e),wherelargevaluesofUaresignicant,and Pillai'strace:V=tr[Sh(Sh+Se)1],wherelargevaluesofVaresignicant. Asaspecialcase,letusconsidertwoindependentregressionmodels, 72

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AliandSilver ( 1985 ,Eq.2.2)) (n1ep)s21=21+(n2ep)s22=22(5{18) wheren:=n1+n2,^f1=(fX0ifXi)1fX0iyi,ands2i=1 ConerlyandManseld ( 1988 ,p.815)) ~=(f1f2)0[21(fX01fX1)1+22(fX02fX2)1]1(f1f2)(5{19) Notethat,inthiscase,knowledgeofboth21and22isneededforthecomputationofthepowerforspeciedalternativevaluesoff1f2.ThissituationisageneralizationoftheclassicalBehrens-Fisherproblemoftestingequalityoftwomeanswhenthepopulationvariancesaredierent. Pillai ( 1977 ,p.24),thenon-nulldistributionoftheRoy'slargestroothasonlybeenderivedinniteorinniteseriesforms,involvingzonalpolynomials,whosegeneraltermsarediculttocompute.Ingeneral,exactpowerfunctionsofthefourmultivariatetestsareextremelydiculttoobtaininclosed-formexpressions.Sincethe 73

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PillaiandJayachandran ( 1967 )and Muirhead ( 1982 ,Section10.6.5)),weconsiderthepoweroftheotherthreetestsasdesigncriteria.Itshouldbenotedthatpowercomparisonsofthefourmultivariatetestsby PillaiandJayachandran ( 1967 ), Lee ( 1971 )and Royetal. ( 1971 ,Ch.5)indicatedthatnosingletestwasuniformlybetterthananyoftheothertestsintermsofpower. Wenowpresentasymptoticformulathatprovideadequateapproximationsforthepowerofthreemultivariatetests(see,forexample, Khuri ( 1986 )).ThenullandthealternativehypothesescanbewrittenasH0:G=0andHa:G=where6=0. Khuri ( 1986 ,formula3.3)) 4f(r+p+1)e1Pr(2f+2(f1)W())[(r+p+1)e1e2]Pr(2f+4(e1)W())e2Pr(2f+6(e1)W())g(5{20) where=n1(rp1)=2,=[det(Se)]=[det(Se+Sh)],W()denotestheupper100%pointoflogunderthenullhypothesisH0,f=(r1)(p+1),e1=tr(),e2=tr(2),isthenoncentralityparametermatrixdenedin( 5{16 )and2f(e1)denotesthenoncentralchi-squaredvariatewithfdegreesoffreedomandnoncentralityparametere1. 74

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Khuri ( 1986 ,formula3.4)) 4(n1)f(p+1)(r1)(pr+1)Pr(2f(e1)H())+2(p+1)[e1(p+1)(r1)]Pr(2f+2(e1)H())+[(p+1)(r1)(p+r+1)2(2p+r+2)e1+e2]Pr(2f+4(e1)H())+[2(p+r+1)e12e2]Pr(2f+6(e1)H())+e2Pr(2f+8(e1)H())g(5{21) whereU=tr(ShS1e)isHotelling-Lawley'strace,H()denotestheupper100%pointofthestatistic(n1)U,f,e1,ande2arethesameasdenedinSection5.4.1. Khuri ( 1986 ,formula3.5)) 4(n1)f(p+1)(r1)(pr+1)Pr(2f(e1)P())+[2r(p+1)(r1)+2(p+1)e1]Pr(2f+2(e1)P())+[(p+1)(r1)(p+r+1)+2re1+e2]Pr(2f+4(e1)P())2(p+r+1)e1Pr(2f+6(e1)P())e2Pr(2f+8(e1)P())g(5{22) 75

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5{16 )dependsonthedesign,andG=.Toaddresstheproblemofunknown,weconsidersimultaneouscondenceintervalsontheelementsof.Inaddition,thespecicationoftheso-calledalternativespaceforwillbeneeded.Inordertostudyandcomparetheperformancesoftwodesigns,D1andD2,throughoutthealternativespace,weconsidersurfacesofseveralconcentrichyperspheresofradiuswhicharelocatedwithinthealternativespace.ThesurfaceofasphereofradiusisdenotedbyT.PleaserefertoSection3.3.1foranillustrationoftheuseofthequantiledispersiongraphsapproach. Seber ( 1984 ,Section3.5.7).Letijdenotethe(i;j)thelementof.DenetherrmatrixQasQ=Pni=1(yi 5{3 ).ItcanbeeasilyshownthatLl0Ql l0lU,forallnonzerol,ifandonlyifLminmaxU,whereminandmaxaretheminimumandmaximumeigenvalues,respectively,ofQ1.Thereadersarereferredto Seber ( 1984 ,Section3.5.7)inordertohaveabetterunderstandingaboutthechoiceofthevaluesofLandU. ChoosingLandUsuchthatP[minL]==2andP[maxU]==2,weget1=P[l0Ql HanumaraandThompson ( 1968 ),butsincetheexactvaluesarediculttoobtain 76

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Clemmetal. ( 1973 ,Section3)wechooseLandUsuchthatL=1=Uandthenusethetablesgivenintheirpaper.Settinglequalto(1,0,...,0),(0,1,...,0),(1,1,...,0)andsoon,weobtainthefollowingintervalsforthevariancesandcovariances,namely,11,22,.....,rrand12,13,.....,(r1);r: 2qij whereqij=(i;j)thelementofQ.Thesearesimultaneouscondenceintervalswithajointcoverageprobability1. Searle ( 1982 ,pp.332-333))whichstacksthecolumnsofthematrix,oneundertheothertoformasinglecolumn.Thistechniquewasusedby ValerosoandKhuri ( 1999 ,p.163).Letqbeequaltotheproductofthenumberofcolumnsandrowsinthematrix.Inordertoconstructanalternativespacefor=(1;:::;q)0,whichisasubsetof
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Kuehl ( 2000 ,p.514).ItisbaseduponanexperimentconductedtodeterminetheeectsofX1=soilcompactionandX2=soilmoistureonthesoilmicrobesactivity.Treatedsoilsamplesareplacedinairtightcontainersandincubatedunderconditionsconducivetomicrobialactivity.ThemicrobeactivityineachsoilsamplewasmeasuredasthepercentincreaseintheproductionofCO2aboveatmosphericlevels.Thedesignusedtogeneratethedatawasa33factorialwiththreelevelsofsoilcompactionandthreelevelsofsoilmoisture.Foreachtreatment,tworeplicatesoilcontainerunitswerepreparedandCO2evolution/kgsoil/daywasrecordedonthreesuccessivedaysgivingtheresponsevaluesy1,y2andy3.Table 5-1 givesthedataforeachsoilcontainerunitandTable 5-2 showsthecodedvariables(obtainedbysubtractingthemean,andthendividingbythestandarddeviation)ofDesign1and2,whichrepresenttheinitialdesignandarandomlygenerateddesignwithintheregionofexperimentation,respectively(also,seeFig 5-1 ).TheQDGshelpincomparingthesetwodesignsbasedonthepowerofthethreemultivariatetests.TheprogramusedtogeneratetheQDGsiswrittenusingtheRlanguage.Highvaluesofthepowerareobviouslydesirable.Allthethreegures 5-2 5-3 and 5-4 showthatthemaximumquantilesofD1areabovethoseofD2,for=2.For=0.1and1,maximumquantilesofD1areeitheraboveorsameasthoseofD2.Hence,D1ispreferredoverD2. 78

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Design1-originalvariablesalongwithresponses 79

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Design1and2-CodedVariablesDesign1Design2 80

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Plotofthedesignpoints 81

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QDGsofthepowerfunctionforD1andD2(basedonWilks'LikelihoodRatio) 82

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QDGsofthepowerfunctionforD1andD2(basedonHotelling-Lawley'sTrace) 83

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QDGsofthepowerfunctionforD1andD2(basedonPillai'sTrace) 84

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InChapter2,wedemonstratedtheapplicationofQDGsasapowerfulgraphicaltoolforcomparingdesignsforresponsesurfacemodelswithrandomblockeects.Thedesigndependenceproblemwascircumventedbytheconsiderationofthedispersionofthequantilesofthescaledpredictionvarianceoveracertainparameterspaceassociatedwiththeunknownparameter,.Themethodologydiscussedinthischaptercanbeadaptedtocomparesplitplotdesignsusingalinearmixedmodelsimilartotheoneconsideredby GoosandVandebroek ( 2003 )and Liangetal. ( 2006 ). InChapter3,weappliedQDGstocomparedesignsforresponsesurfacemodelswithrandomblockeectsbasedonthepowerofastatisticaltest.Crucialtothisapproachisthespecicationoftheparameterspaceandthealternativespace.Asasequeltothiswork,wewouldliketoconsidertheproblemofcomparingtwoindependentregressionmodelswithheteroscedasticerrorvariances.ThissituationisageneralizationoftheclassicalBehrens-Fisherproblemoftestingequalityoftwomeanswhenthepopulationvariancesaredierent. InChapter4,weshowedhowtogenerateD-optimaldesignssequentiallyformixedresponsesurfacemodelsbyaugmentinganinitialdesignwithadditionalpoints.TheEquivalenceTheoremprovidesapowerfultoolforconstructingoptimaldesignsforresponsesurfacemodelwitharandomblockeect.Theresponsesurfacemodelwitharandomblockeectwastreatedasamultivariatemodelwiththeresponsesinablockforminganm-variateresponse(misthesizeofeachblock).Afterderivingthemomentmatrix,themultivariateversionoftheEquivalencetheoremwasapplied.Price'salgorithmwasusedtooptimizetheobjectivefunction.Itshouldbenotedthattheproposedmethodologyrequirestheassumptionofequalblocksizes.Theexperimentalerrorvarianceinaresponsesurfacemodelwithablockeecthastraditionallybeenassumedtobeconstant.However,inmanyexperimentalsituations,thisvariancemaynotbethesame 85

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InChapter5,weappliedtheQDGstocomparedesignsforcorrelatedresponsesurfacemodelswithanunknowndispersionmatrix.Amultiresponseexperiment,bydenition,isonewhichinvolvesanumberofresponsesmeasuredforeachsettingofagroupofcontrolvariables.Inmanycases,thesubdivisionofexperimentalunitsintoblocksnecessitatestheadditionofblockeectstothemultirepsonsesurfacemodel.Themodelingofamultiresponseexperimentwithrandomblockswasconsideredby ValerosoandKhuri ( 1999 ).Wehopetoextendourproposedmethodologytoaddresstheproblemofcomparisonandsequentialgenerationofdesignsformultiresponsemodelswithrandomblockeects. ApartofmycurrentresearchworkatGlaxoSmithKlineinvolvesdesignsformulticentretrials.Mostclinicaltrialsrequirealargenumberofpatients,whichcannotbeprovidedbyasinglecentre,andtoensurethatthetrialiscompletedinareasonabletime,patientsaretreatedatmorethanonecentre.Inthepharmaceuticalindustrymulticentretrialsareusedasevidenceinsubmissionstoregulatoryagenciesseekingapproval.Ourperspectiveisthatofstatisticiansprovidingadviceonthedesignandanalysisofsuchtrialswithinthepharmaceuticalindustry. FedorovandJones ( 2005 )demonstratedthettingofxedandrandomeectsmodelstodatacollectedinmulticentretrialsandindicatedpreferencefortherandom-eectsmodel.Asobservedbythem,arandom-eectsmodeltakesintoaccountthewithin-centreandbetween-centrevariation.Italsoprovidesabasisforsamplesizeandpowercalculationsinvolvingthenumberofcentresandthenumberofpatients.Forthecaseofrepeatedmeasuresdata,thatis,whenmultipleresponsesarecollectedfromeachpatient,itisofinteresttodeterminetheoptimalnumberofcentersandpatients. 86

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LetusconsiderthefollowingmathematicalproblemMinimizeg()subjectto2 whereisak-dimensionalvectorandisaboundedsetwithinak-dimensionalrectangleofnitevolumein
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Ifgmaxgminislessthanauserspeciedsmallvalueorthenumberofiterationshasreachedaboveacertainlimit,thenstoptheprocess. 3. Returngminastheminimumvalueandminasthelocationoftheminimum. 132

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SourishSahawasborninCalcutta,India,onJune18,1978.Heearnedhisbachelor'sdegreeinstatisticsfromPresidencyCollege,Calcutta,in2000andmaster'sdegreeinstatisticsfromIndianInstituteofTechnology,Kanpur,in2002.HejoinedtheDepartmentofStatisticsattheUniversityofFloridain2002.Besidesservingasateachingassistanttoninedierentstatisticscourses,hewasinstructoroftwostatisticscourses-STA3024andSTA4322.HeworkedasaresearchassistantattheDepartmentofClinicalHealth&PsychologyandasastatisticalconsultantattheDepartmentofPsychology.Inthesummerof2003and2004,heinternedatUSDepartmentofHealthandHumanServices.HestartedworkingforGlaxoSmithKlineasaseniorstatisticianinAugust,2007.HeexpectstoreceivehisPh.DinDecember,2007. 139