• TABLE OF CONTENTS
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 Front Cover
 Acknowledgement
 Abstract
 Introduction and literature...
 Bahadur efficiencies of general...
 The combination of binomial...
 Applications and future resear...
 Bibliography
 Biographical sketch
 Back Cover














Group Title: comparison of methods for combining tests of significance
Title: A comparison of methods for combining tests of significance
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 Material Information
Title: A comparison of methods for combining tests of significance
Physical Description: vii, 122 leaves : ill. ; 28 cm.
Language: English
Creator: Louv, William C., 1952-
Publication Date: 1979
Copyright Date: 1979
 Subjects
Subject: Statistical hypothesis testing   ( lcsh )
Statistics thesis Ph. D
Dissertations, Academic -- Statistics -- UF
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non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by William C. Louv.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 118-121.
General Note: Typescript.
General Note: Vita.
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Table of Contents
    Front Cover
        Page i
        Page ii
    Acknowledgement
        Page iii
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
    Introduction and literature review
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
    Bahadur efficiencies of general combination methods
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    The combination of binomial experiments
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
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        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
    Applications and future research
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
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        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
    Bibliography
        Page 118
        Page 119
        Page 120
        Page 121
    Biographical sketch
        Page 122
        Page 123
        Page 124
    Back Cover
        Page 125
Full Text







A COMPARISON OF METHODS FOR COMBINING
TESTS OF SIGNIFICANCE













BY

WILLIAM C. LOUV


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



UNIVERSITY OF FLORIDA


1979




















Digitized by the Internet Archive
in 2009 witfn unaing from
University of Florida, George A. Smathers


Libraries


http://www.archive.org/details/comparisonofmeth00louv















ACKNOWLEDGMENTS


I am indebted to Dr. Ramon C. Littell for his guidance and

encouragement, without which this dissertation would not have been

completed. I also wish to thank Dr. John G. Saw for his careful

proofreading and many helpful suggestions. The assistance of Dr.

Dennis D. Wackerly throughout my course of graduate study is greatly

appreciated.

My special thanks go to Dr. William Mendenhall who gave me the

opportunity to come to the University of Florida and who encouraged

me to pursue the degree of Doctor of Philosophy.











TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . . . . .. . . . . . iii

ABSTRACT . . . . . . . .. . . . . . vi

CHAPTER

I INTRODUCTION AND LITERATURE REVIEW . . . . . . 1

1.1 Statement of the Combination Problem . . . . 1
1.2 Non-Parametric Combination Methods . . . . . 2
1.3 A Comparison of Non-Parametric Methods . . . . 5
1.4 Parametric Combination Methods . . . . . . 8
1.5 Weighted Methods of Combination . . . . .. 11
1.6 The Combination of Dependent Tests . . . . .. 12
1.7 The Combination of Tests Based on Discrete Data . 13
1.8 A Preview of Chapters II, III, and IV . . . .. 18

II BAHADUR EFFICIENCIES OF GENERAL COMBINATION METHODS . . 19

2.1 The Notion of Bahadur Efficiency . . . . .. 19

2.2 The Exact Slopes for TA) and T() . . . . . 21
2.3 Further Results on Bahadur Efficiencies . . .. 26
2.4 Optimality of T(F) in the Discrete Data Case . .. 28

III THE COMBINATION OF BINOMIAL EXPERIMENTS . . . . .. 32

3.1 Introduction . . . . . . . . . . 32
3.2 Parametric Combination Methods . . . . . .. 33
3.3 Exact Slopes of Parametric Methods . . . . .. 37
3.4 Approximate Slopes of Parametric Methods . . .. 44
3.5 Powers of Combination Methods . . . . . .. 54
3.6 A Synthesis of Comparisons . . . . . . .. 57
(F)
3.7 Approximation of the Null Distributions of T ,
(LR) (ALR)79
T T.. . . . . . . .... . 79

IV APPLICATIONS AND FUTURE RESEARCH . . . . . ... 96

4.1 Introduction . . . . . . . . . . 96
4.2 Estimation: Confidence Regions Based on
Non-parametric Combination Methods . . . . .. 96







TABLE OF CONTENTS (Continued)


CHAPTER IV (Continued) Page

4.3 The Combination of 2 x2 Tables . . . . .. 110
4.4 Testing for the Heterogeneity of Variances ..... 113
4.5 Testing for the Difference of Means with
Incomplete Data . . . . . . . ... .115
4.6 Asymptotic Efficiencies for k-.. . . . ... 116

BIBLIOGRAPHY . . . . . . . . ... .... .. .118

BIOGRAPHICAL SKETCH . . . . . . . . ... . . 122






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


A COMPARISON OF METHODS FOR COMBINING
TESTS OF SIGNIFICANCE

By

William C. Louv

August 1979

Chairman: Ramon C. Littell
Major Department: Statistics

Given test statistics X( ),...,X(k) for testing the null

hypotheses H1,...,Hk, respectively, the combining problem is to select

a function of X ,...,(k) to be used as an overall test of the hypoth-

esis H = H n H2 n ... n Hk. Functions based on the probability integral

transformation, that is, the significance levels attained by X(1),...,X(k)

form a class of non-parametric combining methods. These methods are com-

pared in a general setting with respect to Bahadur asymptotic relative

efficiency. It is concluded that Fisher's omnibus method is at least as

efficient as all other methods whether X(),...(k) arise from contin-

uous or discrete distributions.

Given a specific parametric setting, it may be possible to

improve upon the non-parametric methods. The problem of combining binom-

ial experiments is studied in detail. Parametric methods analogous to

the sum of chi's procedure and the Cochran-Mantel-Haenszel procedure as

well as the likelihood ratio test and an approximate likelihood ratio

test are compared to Fisher's method. Comparisons are made with respect

to Bahadur efficiency and with respect to exact power. The power

vi






comparisons take the form of plots of contours of equal power. If

prior information concerning the nature of the unknown binomial success

probabilities is unavailable, Fisher's method is recommended. Other

methods are preferred when specific assumptions can be made concerning

the success probabilities. For instance, the Cochran-Mantel-Haenszel

procedure is optimal when the success probabilities have a common value.

Fisher's statistic has a chi-square distribution with 2k degrees

of freedom when X(1),...,X(k) are continuous. In the discrete case,

however, the exact distribution of Fisher's statistic is difficult to

obtain. Several approximate methods are compared and Lancaster's mean

chi-square approximation is recommended.

The combining problem is also approached from the standpoint

of estimation. Non-parametric methods are inverted to form k-dimensional

confidence regions. Several examples for k=2 are graphically displayed.














CHAPTER I


INTRODUCTION AND LITERATURE REVIEW



1.1 Statement of the Combination Problem


The problem of combining tests of significance has been studied

by several writers over the past fifty years. The problem is: Given

test statistics X(),...,X(k) for testing null hypotheses H1,...,Hk,

respectively, to select a function of X(,...,X(k) to be used as the

combined test of the hypothesis H = H1 n H2 n ... n k. In most of the
(i)
work cited, the X are assumed to be mutually independent, and, except

where stated otherwise, that is true in this paper.

Some practical situations in which an experimenter may wish to

combine tests are:

i. The data from k separate experiments, each conducted to
(1) (k)
test the same H, yield the respective test statistics X ,...,X

It is desired to pool the information from the separate experiments to

form a combined test of H. It would be desirable to pool the informa-

tion by combining the X(i) if (a) only the X(i), instead of the raw

data, are available, if (b) the information from the ith experiment is

sufficiently contained in X or if (c) a theoretically optimal test

based on all the data is intractible.
th (i)
ii. The i of k experiments yields X to test a hypothesis

H., i = l,...,k, and a researcher wishes to simultaneously test

1








the truth of l ,..., 'k. Considerations (a), (b), and (c) in the preced-

ing paragraph again lead to the desirability of combining X) as a test

of H = H1 n ... n Hk.

iii. A simultaneous test of H = H1 n ... n Hk is desired, and the

data from a single experiment yield X( ) ,X) as tests of H, ...,Hk,

respectively. Combining the X(i) can provide a test of H.

In Section 1.2 several non-parametric methods of combination

are introduced. A literature review of comparisons of these procedures

is given in Section 1.3. The remainder of this chapter is primarily

a literature review of more specific aspects of the combination problem.

We make some minor extensions which are identified as such.



1.2 Non-parametric Combination Methods


(i)
Suppose that H. is rejected for large values of X Define
1
L(i) = 1 F.(X(i), where F. is the cumulative distribution function

of X under H.. If X ) is a continuous random variable, then L)
1
is uniformly distributed on (0,1) under H.. Many of the well-known
1
methods of combination may be expressed in terms of the L Such

methods considered here are:

(1) T(F) -2QenL) (Omnibus method, Fisher [13])

(2) T(N) =_Z-1-(L(i)) (Normal transform, Liptak [26])

(3) (m) = -min (i) (Minimum significance level, Tippett [42])

(4) T = -max L (Maximum significance level, Wilkinson [44])

(5) T() = 21Ln(l L(i) (Pearson [361)

(6) T(A) -EL(i) (Edgington [12]).









As the statistics are defined here, H is rejected when large values are

observed. Figure 1 (page 4) shows the rejection regions for the sta-

tistics defined above when k = 2.

In the continuous case, the null distributions of these statis-

tics are easily obtained. They are all based upon the fact that the

L are uniformly distributed under H. It is easily established that

this is true. The cumulative distribution function for L(i) is

P{L < } = P{1 F(x) < }

= 1 P{F(x) < 1 I }
-l
= 1 P{x < F -(1 )}

= 1 F{F-(1 ()}

= 1 (1 9) = 9.

That T(N) has a normal distribution with mean 0 and variance k

follows trivially. The statistics T) and T are seen to be based

on the order statistics of a uniform random variable on (0,1) and

therefore distributed according to beta distributions.

(P) (F)
That T and T(F) are distributed as chi-squares on 2k degrees

of freedom is established as follows. The probability density function

(i)
of L is


fL(k) = I(0,1)(P)

Let S = -2ZnL. Then
-S/2 dL -1 -S/2
L' dS 2

It follows that
dL 1 -S/2
f (S) = f (S) = e I (2 (S).


Edgington's statistic, T(A), is a sum of uniform random variables.

As shown by Edgington, significance levels can be established for






L(2) L(-2)











T(F) L) (N

L(2) L(2)









() L
L









(2)
T(m) L 1(M)


0)2)









T(D
T (P) T(A)
Figure 1. Rejection Regions in Terms of the Signific
for k=2.


ance Levels


)








values of TA on the basis of the following equation [12]:

k k k k k
(A) t k (t-l) k (t-2) k (t-3) k (t-S)
P{T >-t} =-- ( ) k ( ) ( ) k .+ (S) (1.1i)
k! 1 k! 2 k! 3 k! S k!

where S is the smallest integer greater than t.



1.3 A Comparison of Non-parametric Methods


The general non-parametric methods of combination are rules

prescribing that H should be rejected for certain values of

(L (), L(2),..., L(k)). Several basic theoretical results for non-

parametric methods of combination are due to Birnbaum [7]. Some of

these results are summarized in the following paragraphs.

(i)
Under H., L is distributed uniformly on (0,1) in the con-
1
(i)
tinuous case. When H. is not true, L is distributed according to
1
a non-increasing density function on (0,1), say gi(L(i)), if X(i)

has a distribution belonging to the exponential family. Some overall

alternative hypotheses that may be considered are:

(i)
H : One or more of the L 's have non-uniform densities
gA
gi (L(i)).

HB: All of the L(i)'s have the same non-uniform density g(L).

HC: One of the L(i) s has a non-uniform density g (L()).

H is the appropriate alternative hypothesis in most cases where prior

knowledge of the alternative densities g (L(i)) is unavailable [7].

The following condition is satisfied by all of the methods

introduced in Section 1.2.

Condition 1: If H is rejected for any given set of L 's,then

it will also be rejected for all sets of L(i)*'s such that L (i) L)

for each i [7].








It can be shown that the best test of H versus any particular

alternative in HA must satisfy Condition 1. It seems reasonable,

therefore, that any method not satisfying Condition 1 can be elim-

inated from consideration [7].

In the present context, Condition 1 does little to restrict

the class of methods from which to choose. In fact, "for each non-

parametric method of combination satisfying Condition 1, we can find

some alternative H represented by non-increasing functions

gl(L(1) ,...,g k(L(k) against which that method of combination gives

a best test of H" [7].

It should be noted that direct comparison of general combining

methods with respect to power is difficult in typical contexts. The

precise distributions of the gi(L(i)) under the alternative hypothesis

are intractible except in very special cases.
(i)
When the X have distributions belonging to the one-parameter

exponential family, the overall null hypothesis can be written H:
(1) = (1),.. (k) (k)
S= 0 0 Rejection of H is based upon
0 0
(X(1),...,X(k) It is reasonable to reject the use of inadmissible

tests. A test is inadmissible if there exists another test which is

at least as powerful for all alternatives and more powerful for at

least one alternative. Birnbaum proves that a necessary condition for

the admissibility of a test is convexity of the acceptance region in

the (X(1),...,X(k)) hyperplane. For X(i) with distributions in the

exponential family, T) and T( do not have convex acceptance regions

and are therefore inadmissible [7].

Although Birnbaum does not consider Edgington's method, we see

that it is clear that T(A) must also be inadmissible. For instance,









for k=2, consider the points (O,c), (c,O), and (c/2,c/2) in the

(L(1),L(2)) plane which fall on the boundary of the acceptance region

T(A) > c. The points in the (X(1),X(2) plane corresponding to (O,c)

and (c,O) would fall on the axes at o (and -"). The point correspond-

ing to (c/2,c/2) certainly falls interior to the boundaries described

by the points corresponding to (c,0) and (O,c). The acceptance region

can not, therefore, be convex and hence T is inadmissible. This

argument is virtually the same as that used by Birnbaum to establish

(PM (M)
the inadmissibility of T and T

For a given inadmissible test it is not known how to find a

particular test which dominates. Birnbaum, however, argues that the

choice of which test to use should be restricted to admissible tests.

The choice of a test from the class of admissible tests is then contin-

gent upon which test has more power against alternatives of interest [7].

In summary of Birnbaum's observations, since T( and T do

not in general form convex acceptance regions in the (X(1),...,X(k))

hyperplane, they are not in general admissible and can be eliminated as

viable methods. We can extend Birnbaum's reasoning to reach the same

conclusion about T By inspecting the acceptance regions formed by

the various methods, Birnbaum also observes than T) is more sensitive

(F)
to HC (departure from H by exactly one parameter) that T(F). The test
(F)
T ( however, has better overall sensitivity to HA [7].

Littell and Folks have carried out comparisons of general non-

parametric methods with respect to exact Bahadur asymptotic relative

efficiency. A detailed account of the notion of Bahadur efficiencies

is deferred to Section 2.1.








In their first investigation [26], Littell and Folks compare

T(F) T(N), T and T(m). The actual values of the efficiencies are

(F)
given in Section 2.3. The authors show that T(F) is superior to the

other three procedures according to this criterion. They also observe

that the relative efficiency of T(m) is consistent with Birnbaum's

observation that T performs well versus HC.

(F)
Further, Littell and Folks show that T with some restric-

tions on the parameter space, is optimal among all tests based on the

X as long as the X) are optimal. This result is extended in

(F)
a subsequent paper [28] by showing that T(F) is at least as efficient

as any other combination procedure. The only condition necessary for

this extension is equivalent to Birnbaum's Condition 1. A formal state-

ment of this result is given in Section 2.3.



1.4 Parametric Combination Methods


(F)
The evidence thus far points strongly to T(F) as the choice

among general non-parametric combination procedures when prior knowledge

of the alternative space is unavailable. When the distributions of the

X(i) belong to some parametric family, or when the alternative param-

(F)
eter space can be characterized, it may be possible that T(F) and the

other general non-parametric methods can be improved upon. A summary

of such investigations follows.

Oosterhoff [33j considers the combination of k normally dis-

tributed random variables with known variances, and unknown means

P1l. 2,... k. The null hypothesis tested is H; P = )2 = ... = k 0

versus the alternative HA :i > 0, with strict inequality for at least
A i









one i. lie observed that many combination problems reduce to this

situation asymptotically. The difference in power between a particular

test and the optimal test for a given (p 2" ,...,i k) is called the short-

coming. Oosterhoff proves that the shortcomings of T(F) and the maximum

likelihood test go to zero for all (l' .. k) as the overall signif-

icance level tends to zero. The maximum shortcoming of the likelihood

(F)
ratio test is shown to be smaller than the maximum shortcoming of T

Oosterhoff derives a most stringent Bayes test with respect to

a least favorable prior. According to numerical comparisons (again

with respect to shortcomings), the most stringent test performs sim-

ilarly to the likelihood ratio test. The likelihood ratio test is much

easier to implement than the most stringent test and is therefore pre-

ferable. Fisher's statistic, T(F) is seen to be slightly more powerful

than the likelihood ratio test when the means are similar; the opposite

is true when the means are dissimilar. A simple summing of the normal

variates performs better than all other methods when the means are very

similar [33].

Koziol and Perlman [20] study the combination of chi-square
(i) 2
variates X ~ X (6.). The hypothesis test considered is H:
Vi 1
6 = .. = = 0 vs HA: 6.i 0 (strict inequality for at least one i)

where the 6. are non-centrality parameters. The V. correspond to the
1 1
respective degrees of freedom. An earlier Monte Carlo study by

Bhartacharya [6] also addressed this problem and compared the statistics

T(F), T(m), and EX(i) Bhattacharya concluded that X(i) and TF) were

almost equally powerful and that both of these methods clearly dominated

T(m). Koziol and Perlman endeavor to establish the power of T( and









EXi) in some absolute sense. To do this, they compare T(F) and EX(

to Bayes procedures since Bayes procedures are admissible and have

good power in an absolute sense [20].

(i)
When the v. are equal, EX is Bayes with respect to priors

giving high probability to (01,...,9k) central to the parameter space

(Type B alternatives). The test Eexp{ X ) is Bayes with respect

to priors which assign high probability to the extremes of the param-

eter space (Type C alternatives). For unequal v.'s the Bayes tests
i
have slightly altered forms. The Bayes procedures are compared to

T(F) T(m) and T(N) for k=2 for various values of (1,' 2) via numerical

tabulations and via the calculation of power contours.

(i)
The statistic T is seen to have better power than the other

tests for Type C alternatives but performs rather poorly in other situa-

tions. The Bayes test performs comparably to T for Type C alterna-

(m)
tives and is much more sensitive to Type B alternatives than T .The

(N)
statistic, T is relatively powerful over only a small region at the

(N)
center of the parameter space. The statistic, T is seen to be dom-

inated by some other procedure for each value of k investigated. The
(F) (i) (F)
statistics, T and EX are good overall procedures, with T more

sensitive to Type C alternatives and EX more sensitive to Type B

alternatives. For v E 2, T(F) is more sensitive to Type B alternatives

than EXi) is to Type C alternatives and T(F) is therefore recommended.

The opposite is true for v = 1. These observations were supported for

k>2 through Monte Carlo simulations.

Koziol and Perlman also consider the maximum shortcomings of

the tests. In the context of no prior information, they show that T(F)









minimizes the maximum shortcomings for vi. 2 while X minimizes the

maximum shortcoming for v. =1. An additional statistic can be consid-

ered when vi = 1. It is T X) = Z(X(i) M the sum of chi's procedures.

For k=2, T is powerful only for a small region in the center of the

parameter space. For large k, the performance of T(X) becomes progres-

sively worse. It can be said that T(X) performs similarly to T(N)



1.5 Weighted Methods of Combination


Good [14 suggests a weighted version of Fisher's statistic,
(G) (i)
T( = -i .nL He showed that, if the Ai are all different, signif-

icance probabilities can be found by the relationship


(C k
P{T( > x) = E A exp(-x/A )
r=l

where

k-1
A (r
r 1 r 2 r r-l r r+l r k

Zelen [45] illustrates the use of T(G) in the analysis of

incomplete block designs. In such designs, it is often possible to

perform two independent analyses of the data. The usual analysis

(intrablock analysis) depends only on comparisons within blocks. The

second analysis (interblock analysis) makes use of the block totals

only. Zelen defines independent F-ratios corresponding to the two

types of analysis. The attained significance level corresponding to

the interblock analysis is weighted according to the interblock effi-

ciency which is a function of the estimated block and error variances.









A similar example is given by Pape [34 Pape extends Zelen's

method to the more general context of a multi-way completely random

design.

Koziol and Perlman [20] also considered weighted methods for

the problem of combining independent chi-squares. They conclude that

when prior information about the non-centrality parameters is available,

increased power can be achieved at the appropriate alternative by a

(i)
weighted version of the sum test, Eb.X if v. > 2 for all i and by
1 1
(G)
the weighted Fisher statistic, T when v i 2 for all i.



1.6 The Combination of Dependent Tests


The combinations considered up to this point have been based

on mutually independent L) arising from mutually independent statis-

tics, X As previously indicated, in such cases, the functions of

the L) which comprise the general methods have nulldistributions which

are easily obtained. When the Xi (and thus the L ) are not inde-

pendent the null distributions are not tractible in typical cases.

Brown [9] considers a particular example of the problem of

combining dependent statistics. The statistics to be combined are

assumed to have a joint multivariate normal distribution with known

covariance matrix f and unknown mean vector (ul ,2 2,..., k)'. The hypoth-

esis test of interest is H: p. = versus H: p > p. (strict in-
1 10 1 i- 10
equality for at least one i). A likelihood ratio test can be derived

[31], but obtaining significance values from this approach is difficult.

Brown bases his solution on T The null distribution of T(

is not chi-square on 2k degrees of freedom in this case. The mean of









(F)
T is 2k as in the independent case. The variance has covariance

terms which Brown approximates. The approximation is expressed as

a function of the correlations between the normal variates. These

first two moments are equated to the first two moments of a gamma dis-

tribution. The resultant gamma distribution is used to obtain approx-

imate significance levels.



1.7 The Combination of Tests Based on Discrete Data


As noted in previous sections, the literature tends to support
(F)
T as a non-parametric combining method in the general, continuous

data framework. Those authors who have addressed the problem of com-
(F)
bining discrete statistics have utilized T(F) assuming that the opti-

mality properties established in the continuous case are applicable.

The problem then becomes one of determining significance

(F)
probabilities since T(F) is no longer distributed as a chi-square on

2k degrees of freedom. We describe the problem as follows. Suppose
(i)* (i)*
L derives from a discontinuous statistic, X and that a and b

are possible values of L(i)*, < a < b 1, such that a < L
is impossible. For a < E < b, P{L (i) } = P{L a} = a.

If L(i) derives from a continuous statistic, X(i), then P{L (i) =

(i)* (i)
Since a < &, L is stochastically larger than L It follows that

Fisher's statistic is stochastically smaller in the discrete case than

(F)
in the continuous case. The ultimate result is that if T(F) is com-

pared to a cli-square distribution with 2k degrees of freedom when the

data are discrete, the null hypothesis will be rejected with too low

a probability.









When k, the number of statistics to be combined, is small and

(nl,n2 ...,nk), the numbers of attainable levels of the discrete sta-

tistics are small, the exact distribution of T(F) can be determined.

Wallis [43] gives algorithms to generate null distributions when all

of the X(i) are discrete and when one X( is discrete. Generating

null distributions via Wallis' algorithms becomes intractible very

quickly as k and the number of attainable levels of the X) increase.

The generation of complete null distributions is even beyond the capa-

bility of usual computer storage limitations in experiments of modest

size. The significance level attained by a particular value of T(F)

can be obtained for virtually any situation with a computer, however.

A transformation of T(F) which can be referred to standard tables is

indicated.

A method suggested by Pearson [37] involves the addition, by

a separate random experiment, of a continuous variable to the original

discrete variable and thus yielding a continuous variable. Suppose X(i)

can take on values 0, 1, 2,..., n. with probabilities p0 p ,..., p
ni 1 0 1 ni
Let P() = E p Note that P(i) J x x ni ni-1 1 0

If the null hypothesis is rejected for large values of X(i), then the

P j = 0,1,2,...,n are the observable significance levels for the

.th
i test; i.e., the observable values of the random variable

L(i) = FX(i) 1) under the null hypothesis. Denote by U(i)

i = 1,2,...,k, mutually independent uniform random variables on (0,1).

Pearson's statistic is defined as

L(i)(i),(i)) = L(i)(i) U) P{X)}.
P









We now establish that Lp i (X(),U i) is uniformly distributed

on (0,1) if and only if X (i) and U() are independent and Ui) is uni-

formly distributed on (0,1). Omitting the superscripts for convenience,

define the random variable (X,U) by Lp = L(X) U P{X} where 0 < U < 1.

It follows that

P{X = x, U u} = P{L(x) u P(x) < Lp < L(x)}

=uP
x
= P{X = x} P{U < u}.

The statistic, -2EnL (, thus has an exact chi-square distri-
p
bution with 2k degrees of freedom. Exact significance levels can be

determined for any combination problem. The concept of randomization

to obtain statistics with exact distributions has been debated by

statisticians. That a decision may depend on an extraneous source of

variation seems to violate some common sense principle. Pearson [37]

argues, however, that his randomization scheme is no more an extran-

eous source of variation than is the a priori random assignment of

treatments to experimental units.

Lancaster [21] considers a pair of approximations to T(F)

Although Lancaster does not consider Pearson's method, the statistics

he introduces can be expressed in terms of the L(i) [19]:
P
i. Mean chi-square (X2):
m

E(-22nL(i) = (-2nL ) du
p p


= 2 {L i)(X)nL (X)


-L )(X+l)nL )(X+l) }/P(X). (1.3)








'2
ii. Median chi-square (X ):
m

Median (-2knL ) = -2Zn{L ( (X) + L(i)X+l)} if L (X+l) # 0
p 2

= 2 2 nL( X) if L(i)(X+) = 0.

2 2
The expectation of X is 2. The variance of X is slightly less
m m
than 4. The median chi-square is introduced because of its ease of

calculation. With the ready availability of pocket calculators with

in functions, this justification no longer seems valid. The expecta-
'2 '2
tion of X is less than 2. The alternate definition of X for when
m m
(i)
L (X+1) = 0 is intended to reduce the bias (without increasing the

difficulty of calculations) [21].

David and Johnson [10] undertake a theoretical investigation of

'2
the distribution of X They prove that as n, the number of attainable
m
levels of X(i) (and hence L (i)), increases without limit, the moments of
'2
X converge to those of a chi-square distribution on 2 degrees of
m

freedom. We obtain a similar result for X by adapting David and John-
m
'2
son's proof for X (superscripts are omitted for convenience). From
m
2
the definition of X given in equation (1.3), it follows that
m
2 I b1
lim E(X2)b = lim E( (-2gnL )du)b
no m p
P(xi)- 0
(j =1,2,...,n.)
1
Then,
2nn
lim E(X b = lim E P(x.)[ (-2LnL )du]
m j=l J P

b n l b
= (-2) lim E P(x.)[ n(L(x.) -uP(x.))du]
j=l J 0 j J


= (-2) lim EP(x.)[ {(L(x) -uP(x) -1)


lim E(X2) b 1 (L(x.) uP(x.)-l)2 + ...}du]b
m (2 +









1 2 1 3
(Note: kn(a) = (a-1) -(a-1) + -(a-) ...)
2 3

Upon performing the integration,


lim E(X2) = (-2)b lim P(x.)[(L(x.) -P(x.) )
m 3J 2 j

22 1
+ -([L(x.) 1] 2 P(x.)(L(x.) 1)

1 2 b
3 J
-3 [P(xj +
1b
Since all of the terms in the expansion of [ n(L(x.) uP(x))du]
0J
are multiplied by P(x.) it is evident that u(P(x.)) gives rise only to

second-order terms in P(x.) which can be ignored in the limit. The

limit thus reduces to

n
b b
(-2) lim E P(x.)[ knL(x.)du]
n- 0 i=l

= (-2)b lim EP(x.)[ZnL(x.)]
b b

= (-2)b [nL(x)]b dL(x).
0

Letting Y = ZnL(x) yields


b-Y b b
(-2)b(l)b e- Y dY = 2bb!
0

which gives the bth moment of a chi-square distribution with 2 degrees

of freedom.

The convergence of moments does not in general imply convergence

in distribution. However, if there is at most one distribution function

F such that lim X dF = X dF, then F F in distribution [8].
n n n
Since the chi-square distribution is uniquely defined by its moments,

2 2
it follows that X X in distribution.
m 2










1.8 A Preview of Chapters II, III, and IV


In Section 2.1, the notion of Bahadur asymptotic relative

efficiency is introduced. In Section 2.2, we derive the Bahadur exact

(A) (P)
slopes for T and T The results due to Littell and Folks men-

tioned in Section 1.3 are summarized in detail in Section 2.3. In

Section 2.4, we extend the optimality property for T(F) given by Littell

and Folks to the discrete data case.

Chapter III deals with a particular combination problem: the

combination of mutually independent binomial experiments. Fisher's

(F)
method, T is compared to several methods which are based directly

on the X) (rather than the L ). Comparisons are made via both

approximate and exact slopes in Sections 3.3 and 3.4. The tests are

also compared by exact power studies. These results are given in

Section 3.5. A summary of the tests' relative performances follows.

Recommendations as to the appropriate use of the methods are given.

Section 1.6 describes some proposed approximations to the null

(F)
distribution of T In Section 1.7, these methods are shown to be

less reliable than might be expected. Alternative approaches are also

o'valuated.

In Section 4.1, the combination problem is approached from the

standpoint of estimation. Confidence regions based upon T(F) and T(

are derived. The remainder of Chapter IV introduces future research

problems which are related to the general combination problem.















CHAPTER II


BAHADUR EFFICIENCIES OF GENERAL COMBINATION METHODS



2.1 The Notion of Bahadur Efficiency


Due to the intractibility of exact distribution theory, it is

often advantageous to consider an asymptotic comparison of two compet-

ing test statistics. In typical cases, the significance level attained

by each test statistic will converge to zero at an exponential rate as

the sample size increases without bound. The idea of Bahadur asymp-

totic relative efficiency is to compare the rates at which the attained

significance levels converge to zero when the null hypothesis is not

true. The test statistic which yields the faster rate of convergence

is deemed superior. A more detailed definition follows.

Denote by (Y1,Y2,...) an infinite sequence of independent

observations of a random variable Y, whose probability distribution P0

depends ona parameter 6 E0.

Let H be the null hypothesis H: 6 E0 and let A be the alterna-

tive A: 6 E0 00. For n = 1,2,..., let X be a real valued test
0 n
statistic which depends only on the first n observations Y1,..., n'

Assume that the probability distribution of X is the same for all

6 0 0. Define the significance level attained by X by
n
Li =1-F (X ) where F (x) =P IX < x}. Let {X }=X X ,....,X ,...}
n n n n 0 n n 1 2' n








denote an infinite sequence of test statistics. In typical cases, there

exists a positive valued function c(6), called the exact slope of {X },

such that for 6e 0-O.

(-2/n) Zn L c(6)
n
with probability one [0]. That is,

-2
P{lim n L = c(O)} = 1.
n n

If {x(n)} and {X(2) have exact slopes cl(0) and c2(0), respectively
n n 1 2
then the ratio 12(0) = c (0)/c2(0) is the exact Bahadur efficiency of

{X(1)} relative to {X(2).
n n
An alternative interpretation of 12(0) is that it gives the

limiting ratio of sample sizes required by the two test statistics to

attain equally small significance levels. That is, if for E >0, N(i)()

(i)
is the smallest sample size such that L < for all sample sizes
n
n N (i), i = 1,2, then as C tends to zero [3]

(2)
lim N= ().
N (1E) 12

The following theorem due to Bahadur [3] gives a method for

calculating exact slopes. A proof is given by I. R. Savage [40].


Theorem 1. Suppose {T } is a sequence of test statistics which
n
the following two properties:

1. There exists a function b(6), O T //n-b(O) with probability one [0].
n
2. There exists a function f(t), O continuous in some open set containing the range of
b(O) such that for each t in the open set
-1
i n[l-F (/n t)]-f(t), where F (t) = 1-P (T n t).
n n n o n
Then the exact slope of {T } is c(O) = 2f(b(9)).
n


satisfies










2.2 The Exact Slopes for T() and T(P)




In terms of the above discussion the general combination problem


can be defined as follows. There are k sequences {X (1),...,{X(k) of
n1 nk


statistics for testing H: 0 E00. For all sample sizes nl,...,n k the


statistics are independently distributed. Let Li) be the level attained
1
(ii
by X i = 1,2,...,k.
n.



n.
1

slope c.(0); that is


-2 (i)
ZnL --c.(O) as n. c
n. n. 1i
1i
with probability one [0]. Assume also that the sample sizes nl,...,nk

n.
satisfy n + ... + n = nk and lim = A, i = 1,...,k. Then
n +o

S+ ... = k and
1 k

-2 (i)
ZnL c.( () as n m
n n. i i
1

with probability one [9]. As defined here, n can be thought of as the


average sample size of the k tests.



Two general combining methods introduced in Section 1.2 are


T (A) Edgington's additive method, and T Pearson's method. Deriva-


tions of the exact slopes of T(A) and T(P) follow.








k
(A) -2 (i)
Proposition 1. Let T = n Z L The exact slope of
n /n i=l n
(A)
T is k min (c.(e)).

(A)
Proof: This proof requires the above definition of T although
n
T(A) = L(i) is a more obvious definition and is the form given in
n ni
Section 1.2 where the non-parametric methods are introduced. Nothing

is lost, however, since equivalent statistics yield identical exact

slopes. Proposition 1 is proved by using Theorem 1 (Bahadur-Savage).

The first step is to establish b(O) of Part 1 of Theorem 1. To accom-

plish this, first suppose that

Acl(6) = min {(.c.()}.
1 i 1 1


It follows that for all > 0, there exists

n >N,


N=N(E) such that for


with probability




with probability





with probability


-2 (1) -2 (i)
Qn L __- knL
n n n ni

one [6]. Then, for n >N,

L(1) (i)
n1 ni '

one [6]. It follows that
k
(1) < L(i) n1 i=l ni n1

one [6]. Thus, for n -N,


i=1,2,...,k,




i = 1,2,...,k,


for n N,


-2n L(1) -2 n i) -2 n kL1)
n n n n. n nk
n n n n. n n
1 1


with probability one [0]. It follows that as n tends to infinity










-2 (1) -2 (i) -2 (1)
lim n L 2 lim Zn EL lim n kL
n n n n. n n

with probability one [0]. Hence,


( >-2 n
Alcl(6) > n in


n. 1 1
i
1


with probability one [0] since, as n tends to infinity,


-2 (1)
lim -2 n kL(1)
n nl


= lim {-2 Zn L()
n nl


- k n k} = lim --nL(1)
n n nl


with probability one [0]. Thus,

T(A)
n -- c (0)
r- 11


with probability one [0].


The choice of A1 c(0) as the minimum was


arbitrary. Hence,


(A)
n
J^


+ min A.c.(9)
i 1


giving b(9) of Part 1 of Theorem 1. Now, as n tends to infinity,


lim -- n[l F (vn t)]
n n

-1 -2 (i) t
= lim -r- n P{-2 Zn L(i) /n t
n /n n.


n n. 2
1

= lim -r n [exp ( k)/k!)].
n 2


The last equality if true since t is positive and therefore

-nk is less than one.
exp ( ) is less than one.










lim -I n[l F nn t]
n n

tk 1 tk
lim [ + 1 Yn k!] = -
2 n 2


This gives f(t) of Part 2 of Theorem 1. Thus, from Theorem 1,


C (0) = 2f(b(O)) = kmin A.c.(8).
A 1 1


Proposition 2. Let T(P = n[-E n(l L )]. The exact
n F- n.
n ni

slope of T is C (6)=k min A.c.(9).
n P 1i


The form of Pearson's statistic given in this proposition is

equivalent to the form given in Section 1.2. The proof of Proposition 2

entails use of the Bahadur-Savage Theorem (Theorem 1). The derivation

of b(0) of Part 1 of the theorem parallels the derivation of b(9) in

Proposition 1. In order to establish f(t) for part two of the theorem,

a result due to Killeen, Hettmansperger, and Sievers [18] is required.

They show that under broad conditions,


1 n f (V'n t) I- n P{X Fn t} = o(l) (2.1)
n n n n


as n->- where f (t) is the density function of X .
n n


Proof of Proposition 2. By a first-order Taylor series approximation,


-E n(l L (i) = E L + E I| L)
n. I. n n.
1 1 1
(i)
where -+0 as L -0 for all i. It follows that as n tends to infinity,
n n.
I










T
lim -- = lim -2 kn[-EZ n(l L(i)]
An n ni

= lim -2 kn[ L(i)]
n n.
1

= min X c (O).
i i


The last step is shown in Proposition 1. Now, to derive f(t) of


Part 2 of Theorem 1 via equation (2.1), note that the U( =1-Li)
n. n.
1 i

are mutually independent uniform random variables on (0,1). Letting


V = -n i it follows that
n. n.
1 1

-v
f (v) = e < <
(i)
n.
1

Letting W =E V = -EZn U it follows directly that
n n. n.
1 1

1f () k-l -w
f (w) = (k w e
n



since V is a gamma with parameters (11) Now, letting


1

f (y) = exp{-ky e } (2.2)
n A

It follows that, as n tends to infinity


-1 (P)
lim -1 n P{T /n t}
n n

-1 2
= lim -- n P{-- Y > n t}
n /n n










= lim -i pn P{Y > 1t
n n 2

-1 ut
= lim -n n fy (-) (2.3)
n Y 2
n

by the result given in equation (2.1). Substituting (2.2) into (2.3),

we find that

-1 1 nt -nt
lim n[- exp{-k(-) exp( )}]
n 2 2

S-1 -knt -nt
=lim [ exp( -)]
n 2 2

kt 1 -nt tk
= lim [k + exp( --)] = t
2 n 2 2

Applying the result of Theorem 1,


C (6) = 2f(b(O)) = k min X.c.().
i



2.3 Further Results on Bahadur Efficiencies


Bahadur exact slopes are derived in Littell and Folks [26] for

other general combining methods. Their results are summarized below.

Test Exact Slope

(F)
T c F(6) = X Z.c.(6)


T(N) c, (0) E c (0)) 2
N k i i







(A) (P) [Y(i
i
(m)
T c. (0) = max ).c.(0)
m 1i 1

Thus T and T have the same exact slope as T The relationship

among these quantities is displayed in Figure 2.











X2 C2


=(i +V-2)2XIOC


2C2= 2X ci


XI c = X2C2


TI'


Iy


Note: I and T-I
IT and Y :

UI and _F:


CM
CM < Cm < CF
Cm

Figure 2. Relative Sizes of Exact Slopes for k=2.










The optimality property of T(F) established by Littell and

Folks [28] is mentioned in Section 1.3. The detailed result is given

here as a theorem.


Theorem 2. Let T be any function of T ,...,T(k) which is
n n1 nk

(i) ,
non-decreasing ineach of the T ', that is, t1 < t',...,tk tk
1

implies T (t ,...,t ) < T (tl ....t). Then the exact slope c(6) of


T satisfies c(O) < Z A.c.(6).
n 1 1


The non-decreasing condition is equivalent to Birnbaum's

Condition 1. (See Section 1.3.) The condition is not very restric-

tive; it is satisfied by any method thus far introduced and by virtually

all other reasonable statistics.



(F)
2.4 Optimality of TF) in the Discrete Data Case


Littell and Folks established that the exact slope for T(F)

is c F() = E A c .(). This derivation is contingent on the fact that
F 1 1
(F)
T has a chi-square distribution on 2k degrees of freedom. This is

true only when the X(i) are distributed according to continuous distri-

butions. A proof that the exact slope of T(F) is Z A.c.(9) when the X(i)
1 1
are discrete follows.


Suppose X(i) can take on values 0, 1, 2,...,n. with probabil-
n. 1
1
nn
1
ities pO, p ...,p Let P i= p Note that
0 1 n. j x
iJ x=j
1 x=j

(i) < i) < < (i) n. n.-l 1 0
1 1









(i) (i)
for large values of X ), then the P. j = 0,1,2,...,n. are the
n. j
1
th
observable significance levels for the i test; that is, the observ-


(i) (i)
able values of random variable L = 1 F(X 1) under the null
n. n.
1i

hypothesis. Assume that an exact slope exists for all tests; that is,


assume that there exist functions c l(), c2(0),...,c k() such that


-2 (i)
-- nL c.(0) as n -
n. n. I i
1 1

with probability one [6] for i = 1,2,...,k.


(d) (i) i
Proposition 3. Let T = (-2 EZ n L )2. If
II n
1

lim -- n[l- F (d)n t)] exists then the exact slope of T(d) is
n n n
n o
k
cd(0) = Z c (0).
i=l

Proof: This proof utilizes the Bahadur-Savage Theorem (Theorem 1).

To establish the first part of Theorem 1, observe that

(1) (k)
(d) 2 (nL) 2nnL) ;
T n_ _
n n n




(A cl(a) + ... + AkCk(0))k as n + om


with probability one [0]. Consistent with the notation of Theorem 1,

denote this limiting quantity bd(0).

Now, to establish f(t) of Part 2 of Theorem 1, choose

Z.c (0,1), i = 1,2,...,k. For each i, there exists a j such that

1 [P(i) (i)
i j Pj-1]









(i)
Now, since L is a discrete random variable,
n.
i

P{(i) .) = P{L (i) n i n. j 1
1 1
Thus

P{L() e .)} P{u( ) M .}
n. i i
1
(ii
where the U are mutually independent uniform random variables on

(0,1). It follows that

P{-21nL(i) >_ } < P{-2n U(i) > .},
n. 1 1
1
and hence that

k k
P{ Z -2nL) >z"} P{ E -2n U(i) "}.
i=l i i=l
Thus,

-1 Mi)
-1 n P{[E 2nL (i)] t} n P{[ -2Zn U (i) > n t}. (2.4)
n n. n
1

The quantity Z = [Z 2Rn U (i) is distributed as the square root of
n
a chi-square with 2k degrees of freedom. It follows that the density

of Z is
n 2
f (Z 1 2k-1 -Z2/2
fZ (Z) k- Z e
n 2 F(k)

Thus, from the result given in (2.1), the limit of the right-hand side

of (2.4) can be written as

-1
lim -4n f (Wn t)
n Z

Sn 2k-1 -1 2 n
= lim n [ (/n t)2 exp( (/n t) )]
n- 21 F(k)

-1 1 2
=lim -- [(2k-1) kn(n t) - nt]
n 2
nco

1 2
t
2










Hence, it follows from (2.4) that

1 2
f (t) > t ,

since it is assumed that the limit of the left-hand side exists.

Applying the result of Theorem 1,

Cd(e) ZiA.c.i ().
d 11

By Theorem 2,


Cd(X) 11().

Hence,

cd(9) = ic. (e).


That is, the exact slope of T(F) is EX.c.(6) regardless of whether the
11
X are continuous or discrete. The condition imposed in Proposition 3


that lim -_. n[lF (d) (/n t)] exist is not very restrictive. It is
n n
n -co
satisfied in most typical cases and in every example considered in this

dissertation.














CHAPTER III


THE COMBINATION OF BINOMIAL EXPERIMENTS



3.1 Introduction


Chapter III deals with the combination of binomial experiments.

That is, suppose k binomial experiments are performed. Let nl,n2,... ,nk

(1) (2) (k)
and X X ...,X he the sizes and the observed numbers of suc-
n1 n2 nk

cesses, respectively, for the experiments. Denote the unknown success

probabilities as pl,p2,...,pk. Suppose one wishes to test the overall

null hypothesis H: pl = P10' p2 = P20' ". = PkO versus the alter-

native hypothesis HA : p > p10' .. k PkO (with strict inequality

for at least one p.). The problem, then, is to choose the best function

of (X ... X (k) for this hypothesis test.
n I nk
1 k.

(F)
The results of Chapters I and II support T(F) as a non-

parametric method with good overall power when there is no prior infor-

mation concerning the unknown parameters. The method based on the

minimum significance level, T is sensitive to situations where

exactly one of the individual hypotheses is rejected. That Is, T) is

powerful versus the alternative IC of Section 1.3.

The investigations of Koziol and Perlman [20] and Oosterhoff [33]

show that the general non-parametric combining methods can be improved


32









on for certain parametric combining problems. It follows that there

may be combination methods based directly on (X ... ) that are
n nk
1 k
superior to Fisher's omnibus procedure.

Chapter III is a detailed comparison of T(F) and several

parametric combination methods.



3.2 Parametric Combination Methods


As stated in Section 1.3, no method of combination is most

powerful versus all possible alternative hypotheses. There are, however,

certain restricted alternative hypotheses against which most powerful

tests do exist.

th
Let the likelihood function for the i binomial experiment

be denoted by

n. (i) n.-X(
L(p) = ( i)pX (1- p.) (3.1)
X

According to the Neyman-Pearson Lemma, if a most powerful test of the

null hypothesis H: pi = Pi, all i, versus the alternative hypothesis

HA: pi > io (with strict inequality for at least one i) exists, it is

to reject H if

k L(p )
iO
71 < C.
i=l L(

Upon substituting (3.1) and taking logs, an equivalent form of the test

is to reject H if

EX() M n{Pi(l Pi )/Pi (l-Pi)} > C. (3.2)

It follows that rejecting H when

SXi) > C (3.3)

is most powerful if p.(l p i)/p i(1-p.) is constant on i.
1 i iO 1









The problem of combining 2 x2 contingency tables is closely

related to the problem being considered. The purpose of each 2x2 table

can be interpreted as testing for the equality of success probabilities

between a standard treatment and an experimental treatment. The overall

null hypothesis is that the experimental and standard success probabil-

ities are equal in all experiments. The overall alternative hypothesis

is that the experimental success probability is superior in at least

one experiment.

Cochran [LO] and Mantel and Haenszel [29] suggest the statistic

E widi//Ew.p.qi (3.4)

where


S= nilni2/(nil + ni2) d = Pil Pi2

for combining 2x2 tables. Mantel and Pasternak [34] have discussed

this statistic in the context of combining binomial experiments. Each

individual binomial experiment is similar to an experiment resulting

in a 2x2 table, two cells of which are empty because the control

success probability is considered known and need not be estimated from

(CHM)
the data. The statistic defined by (3.4) will be denoted by T(

It can easily be shown that the test T(crH) > c is equivalent to the

test E X(i) > C, thus T(CMH) is the most powerful test when

p.(l p )/pi(1 p.) is constant on i.

In many practical combination problems with binomial experiments,

Pi =1/2 for all i. The null hypothesis is then H: p. =1/2, i=,2,...,k

and the general alternative hypothesis is HA: pi > 1/2 (strict inequal-

ity for at least one i). This is the hypothesis testing problem under









consideration throughout the remainder of Chapter III. For pi0 =1/2,

all i, T( ) is uniformly most powerful for testing H: pi0=1/2, all i,

versus HB: pi = p2 = p3 = ... = k > 1/2.

For the hypothesis test just described, T( ) can be written

k k
(i) 1 1 ).
S(X n.)/( E n.) (3.5)
i=l 2 i 4 1
i=l j=1

This variate is asymptotically standard normal. It is of note that

this form is standardized by a pooled estimate of the standard devia-

tion. An alternative statistic can be formed by standardizing each

X( yielding

(X) =1 {(x(i) 1 1 )A
T {(X -n)/(- n)
2 i 4 i
which also has an asymptotic standard normal distribution. The statistic,
(X)
T is analogous to the sum of chi's procedure which has been recommended

for combining 2x2 tables. The statistic T(X) is not in general equiva-

lent to T(CmlI); in fact, the test T(X) > c is equivalent to the test

k (i) (x) (CMI)
E n X > c. When the n. are all equal, T and T are equivalent.
i=l
(i) (i)
Weighted sums of the X say S(g) = Z gi X form a general

class of statistics. Oosterhoff considers this class and makes the fol-

lowing observations concerning their relationship with the individual

sample sizes [32]. It follows from (3.2) that if kn(p./l-p.) = a gi

then the most powerful test of H versus H is E g. X( > c.
1
Let pi = + C It follows that
i 2 1









n(p./l p.) = kn(l + 2L./l 2t.)


(2e.)3 (2c.)5
= 2{2 + + ...}
1 3! 5!

= 4 + (E2)
i O+0 i
1

This implies that for alternatives close to the null hypothesis, H,

S(g) is most powerful if Ei = e gi; that is, if the deviations from the

null values of the pi are proportional to the respective g.. The sum

of chi'sprocedure, T is a special case where g. (n.) It fol-

(x)
lows that the alternatives against which T) is powerful is strongly

related to the sample sizes, n,n2,....,nk.

The weighted sum, S(g), may be a viable statistic if prior

information concerning the p. is available. Under the null hypothesis,

S(g) is a linear combination of binomial random variables, each with

success probability 1/2. The null distribution of S(g) will therefore

be asymptotically normal. The proper normalization of S(g) is analogous

to that of CmllH) given in (3.5).

A well-known generalization of the likelihood ratio test is to

reject the null hypothesis for large values of -2 kn{ sup L(O,X)/sup L(O,X)}.
c60 ~ 6
It is easily shown that for the hypothesis test being considered, the

likelihood ratio statistic is


(LR) k (i) (i) (i) X(i) X(i) 1
_i= 1 2 n. i n. n. 2
i=l1 1 1
where


1 if -- 1/2
.(i n

i (i)
0 if < 1/2.
i










Under broad conditions, which are satisfied in this instance, the
(LR)
statistic T(LR) has an asymptotic chi-square distribution with degrees

of freedom.

Suppose z., i = 1,2,...,k are normal random variables with

means U. and variance 1. The likelihood ratio test for H: P. = 0,
1 1

i = l,...,k versis H1: 1i i 0 (with strict inequality for at least one i)

is to reject H for large values of

k 2
E z. I{z. 0}. (3.6)
i=l

For the binomial problem, an "approximate likelihood ratio" test is then

to reject for large values of
k
(ALR) (i) 1 2 1 (i) 1
T =E (X n) /- n. Ix > n.}
2 4 1 -2 i
i=l

1 1 1
since (X n )/( n. ) is asymptotically a standard normal random var-

iable under H. The exact null distribution of (3.6) is easily derived.

Critical values are tabled in Oosterhoff. When p = 1/2, the normal

approximation to the binomial is considered satisfactory for even

fairly small sample sizes. It follows that the exact null distribution

of (3.6) should serve as an adequate approximation to the null distribu-

tion of T(R)



3.3 Exact Slopes of Parnmetric Methods


In this section, the exact slopes of r(F) T(CI) and T(LR)

are compared. We have not been successful in deriving the exact slope

for T(). A more complete comparison of methods is given in Section 3.4

with respect to approximate slopes.










Suppose X) is a binomial random variable based on n. observa-
n 1
tions with unknown success probability p.. Consider testing the single

null hypothesis H: pi = 1/2 versus the single alternative hypothesis

H : p. > 1/2.


(i 1 (i) (i)
Proposition 4. Let T = X The exact slope of T is
1n. n n.
i n. i i
1

c.(9) = 2{pi n 2pi + (1 pi) n 2(1 pi)}.


Theorem 1 is used to prove Proposition 4. There are several

means by which the function f(t) of Part 2 of Theorem 1 can be obtained.

Perhaps the most straightforward way is by using Chernoff's Theorem [1].

Bahadur, in fact, suggests that the derivation of f(t) provides a good

exercise in the application of Chernoff's Theorem.

Theorem 3. (Chernoff's Theorem). Let y be a real valued random

ty
variable. Let (t) = E(e ) be the moment generating function of y.

Then, 0 < #(t) < for each t and P(0) = 1. Let P = inf{((t): t > 0}.

Let yl, Y2,..., denote a sequence of independent replicates of y and

for n = 1,2,..., let Pn = P{yi + ... + yn 01. Then


1
in P Zn P as n -c .
n n


Proof of Proposition 4. For Part 1 of Theorem 1,


,(i) X(i)
T X
1 i
-+ Pi
n i
1

with probability one [6] giving b(6). For the binomial problem

0 = (pl, 2 ... ,Pk)










Now, as n Lends to infinity,


lim -1
n.
lim
n.
1


= lim -
n.
1


= lim -
n.
1


= lim -
n.
1


Zn(l -F (n-. a))



Zn P{X()/n. > A7 a}
n. i i



(i)
n. 1
1


Zn P{X(
n.
i


- n.a > 0}.
1


The random variable


(i)
X
n.
1


(i)
X = (y
n. 1
1


- n.a can be expressed as



- a) + (y2 a) + ... (n a)
2 n


where the y. are independent replicates of a Bernoulli random variable

y with parameter 1/2. Therefore, P(t) of Chernoff's Theorem is



(t) = e (l + et).
y -a 2
1


It follows that


1 -at
The quantity e
T


1 -at t
P = inf{- e (l + e ):
2

(1 + et) is minimized for


a
t = n -
1-a


Thus,


1 -a[Zn a/l-aj [[Zn a/1-a]
S = 2e (1+ eJ /l-)



1
Zn P = -a Zn a/1-a + Zn (1 + a/1-a).
2


(3.7)


t > }.


and









Hence,

-1 (i) 1
li -- n P{X(i) > n.a} =a An(a/1-a) n( (l +a/1-a)) (3.8)
n. n. 2
1 1

giving f(a) of Part 2 of Theorem 1. Thus,


c.(0) = 2{pi kn p./l-Pi -n -(1 + p./l-p.)}
1i i 2 i i

= 2{pi n 2pi + (1 p.) An 2(1 p.)}



Following the notation of Section 2.2, suppose k binomial

experiments are to be combined and the sample sizes nl,n2,..., nk

satisfy nl + n2 + ... + nk = nk and


n.
lim = X., i = l,...,k.
n 1


Then 1 + ... + k = k and
1 k

-2 (i)
n L .c. (0) as n m.
n n. i i
1

According to Proposition 3, c F() = Z X c (6) in both the continuous and
F i i
discrete case if lim -- n[l F (/To)] exists. The existence of this
n n
limit for a single binomial experiment is shown in (3.8) of the proof

of Proposition 4. Therefore, for the binomial combination problem, the

exact slope for Fisher's method, T(F) is

k
c ,() = X Ai{pi Zn 2pi + (1 p.i)n 2(1 pi)}.
i=l


A property of likelihood ratio test statistics is that they

achieve the maximum possible exact slope [2]. Theorem 2 states that

the exact slope for the combination problem is bounded above by

Si.c.(O). Proposition 3 shows that T(F) achieves this. If follows
11I





41



(F)
that T and the maximum likelihood test have the same exact slope;

that is,

cF(0) = CLR(6).


This relationship is true regardless of whether the data are discrete

or continuous.

(CMH) 1
Let T( -C) X). This form of the Cochran-Mantel-
n -n k ni

Haenszel statistic is equivalent to those previously given in (3.3)

and (3.5).


Proposition 5. The exact slope of T is
n

c (0) = 2k{p kn 2p + (1-p) Zn 2(1-p)} where p = E X.p..
CMH k ii

Proof. To get b(O) of Part 1 of Theorem 1,

S(CmH)
-n = X i) /nk
n n
nn.




1
X(i)
1 n n
k n, n



k i i


with probability one [a]. Now, for Part 2 of Theorem 1, as n tends to

infinity,


lim -1 kn[l F(JI a)]
n


= lim r- in P{-- E i) ,n a}
n n k i

-1 (i)
= lim k in P{E X n k a} = f(a). (3.9)
nk n.
i








(i)
Under the null hypothesis, Z X is a binomial random variable based
n.
I
on nl + ... + nk = nk trials with success probability 1/2. The quantity

(3.9) is the same as the quantity (3.7) except that n. has been replaced
1
by nk. Theorem 3 can be directly applied to line (3.9), yielding

-1
lim -_- n [1 F(/n a)]
n

= k{a en -a n 1(1 + a/l-a)} = f(a)
1-a 2

and therefore

cCMH(0) = 2f(b(O))


= 2k{p ln(p/l-p) ln(-(l + pl-p)}


= 2k{p ln 2p + (l-p) ln 2(l-p)}.

A comparison of T(CME) relative to T(F) and F(LR) with respect

to exact slopes is given in the next section.

Derivation of the exact slope of the sum of chi's procedure,
(X)
T has not been accomplished. An incomplete approach to the problem

follows.

Let T(X) = Z n X ). To derive b(O) of Part 1 of Theorem 1,



T n n. n.
T(X) n.
n 1n s
,n n n


X p. as n m
i i

with probability one [8]. Now, as n tends to infinity,
-1
lim Ln P{1 F(Vn a)}


-1 P(n1 (1) nk (k)
= lim- p{ I ( + ... ) n a).
n ( nl nk









The left-hand side of the above probability statement is a weighted sum

of independent binomial random variables based on varying sample sizes

nl,n2,...,nk each with success probability 1/2. The moment generating

function of this random variable is therefore




n n



= -+ e +... e+ e
2 2

(3.10)

From the form of the moment generating function given in (3.10), it is

apparent that the random variable in question can be regarded as a sum

n independent identically distributed variates each with moment generat-

ing function



1 1 t)nl/n k 1 nk /n
+ Te I... + e


Then, since as n tends to infinity

n-1 n(1) ^(k)
lim P-1 x ( +. ... + X na
n n n nk


n n M
= lim -1{ Z X a 0},
n n. n
j=1 i

#(t) of Theorem 3 is


-at 1 1 1 1 1 k
(t) = e [( + e ) ... ( + e )]
2 2 2 2

and P = inf{f(t): t a 0}

The quantity p has not been found.










3.4 Approximate Slopes of Parametric Methods


Exact slopes are defined in Section 2.1. In Section 3.3 some

comparisons among methods are made with respect to exact slopes and

corresponding efficiencies. Bahadur also defines a quantity called the

approximate slope [3]. Suppose that X has an asymptotic null distri-

bution F; that is,

lim F (x) = F(x)
n
n- oo

for all x. For each n, let


L = 1 F(x)
n n

be the approximate level attained. (Consistent with Bahadur's notation,

the superscript a stands for approximate.) If there exists a c(a)(

such that

-2 Zn L(a) + (a)
n n
(a)
with probability one [0], then c ()() is called the approximate slope

of (X }.
n

If ca) () is the approximate slope of a sequence {x ),
1 n(
i = 1,2, then cl (a)/c2 (a ) is known as the approximate asymptotic


efficiency of {x()} relative to {x(2)
n n

A result similar to Theorem 1 is given by Bahadur [3] for the

calculation of approximate slopes. Suppose that there exists a

function b(0), 0 < b(O) < co, such that

T
n
-- b(6)









with probability one [0]. Suppose that for some a, 0 < a < m, the

limiting null distribution F satisfies

1 2
Zn[l F(t)] ~ at as t c
2
(a) [b() 2
Then the approximate slope is c (() = a[b(6)]. This result is

applicable with a = 1 for statistics with asymptotic standard normal

distributions [3]. This result can be shown directly by applying the

result of Killeen et al. given in (2.1).

The approximate slope, c(a)(0), and the exact slope, c(6),

of a sequence of test statistics are guaranteed to be in agreement only

for alternative hypotheses close to the null hypothesis. Otherwise,

they may result in very different quantities. One notable exception is

the likelihood ratio statistic. When the asymptotic null distribution

is taken to be the chi-square distribution from the well-known -2 Zn

(likelihood ratio statistic) approximation, the approximate slope of the

likelihood ratio statistic is the same as the exact slope. The approx-

imate slope is based upon the asymptotic distribution of the statistic.

Equivalent test statistics may have different asymptotic null distribu-

tions giving rise to different approximate slopes. This apparent short-

coming does not exist with exact slopes.

In typical applied situations, the significance levels attained

by T(C0H) and T(X) will be ascertained by appealing to their asymptotic

(LR)
normal distributions. Similarly, T(R) will be compared to the appropri-

ate chi-square distribution and approximate levels for T(ALR) will be

obtained from the asymptotic distribution given in Section 1.2. Approx-

imate slopes based upon these asymptotic distributions would therefore

seem to afford a more appropriate comparison of the methods. In other









words, it is appealing to consider the null distribution that will be

used to obtain significance levels in practice when comparing the

statistics. The only statistic which will not usually be compared to



(CMHI)
asymptotic distributions is perhaps T .(CMH) The null distribution of

T is binomial based on nI + ... + nk trials with success probabil-

ity 1/2. However, even with the availability of extensive binomial

tables, T(CMH) will often be standardized as in (3.5) and compared to

standard normal tables since the normal approximation to the binomial

when p = 1/2 is satisfactory even for fairly small sample sizes.

The asymptotic null distribution of T(F) in the discrete case

is easily shown to be chi-square with 2k degrees of freedom. This is

(F)
also the exact distribution of T(F) in the continuous case. It follows

that the approximate slope in the discrete case is the same as the

exact slope in the continuous case. In summary,


(a) (a)( k
CLR () =c =c R() =C ( =2 E X.{p kn 2p +(l-p.i)n 2(l-pi)}.
LR F LR F ill i i 1 1
i=l
(Cuf) (x)
In order to derive the approximate slopes for T and T

consider the linear combination Zn X of which T and T are
1 n.
(i)
special cases. The variate X has an asymptotic normal distribution
ni
1
1 1
with mean n. and variance n under the null hypothesis. It follows
2 4 i
directly that

(U) 1 ++l / 2u+l
T (Zn.Xni n2 n )/ vn
n i 2 i 2 i

is asymptotically standard normal.

Proposition 6. The approximate slope of T) is
n
c+(a) +1 2 2c+1+
c(a)() = [A (2p -l)] /i
cti i










Proof. First, to get b(O),


1 C+1
-Zni )
2 i


1 f-2c+l
21/ Eni


1
(E(-)
nn


(i)
i


cE+1
n
1 i
E )
2 x
n


2"+1
2cA+l
1 i
2/ 2a
v n


X(i)




1 /( 2a+1
2 ni


Ac 1 ac+l
( (.iPi) -iz
i 2 i

1 / Z2A+
2 i


with probability one [0].


(Ca)
Now, since T is asymptotically standard
n


normal,


(a)
co


= [b(O)]2


-a+l 1 E+1 2 1 2a+l
i i 2 i 4 i


= [Zi (2Pi


2 2a+1
- 1)] /ZA.
i


Letting a = 0 yields the approximate slope of'lT(CI)


C(a() = [EiX(2p 1)]2/k
CHH 1 i

Letting c = -2 yields the approximate slope of T ,
(a) 2
c(a() = [E(2p 1)]2/k.
X 1 i


(")
T
n

11


I aQ(i)
- (EnX M
i n.
v n 1


as n CO









By inspection of the above approximate slopes, it is apparent
(CMIH)
that T is more efficient when the p. are proportional to the A.
1 1
(X)
(the relative sample sizes) and T() is more efficient when the p. are

inversely related to the X.. The boundary of the parameter space where

(a) (a)
c (6) = c (6) is not p = p = ... = k' however. The statistic

T(CIMH) is more efficient than T(X) in more than half of the parameter

(a) (CHH) (x)
space. As a further comparison, e (T T ), the approximate

efficiency of T(C with respect to T(X), can be integrated over the

parameter space. The result is greater than one which again supports
(CMH)
use of T It should be noted, however, that when the pi are pro-

portional to the A. both tests have high efficiencies relative to when
1
(X)
p. are inversely related to the A.. Therefore T() is more efficient in

a region of the parameter space where both tests have relatively low

efficiency. This is a good property for T(X)

An "approximate" likelihood ratio test is introduced in

Section 3.2. A statistic which is equivalent to the form given in

Section 3.2 is
k n
(ALR) [(i) 1 2 i (i)
T = (E [(X n) / IX n
i=l i 4 n 2

Proposition 7. The approximate slope of TALR) is
n

c (o) = eA.(2p.-)1).

Proof. To find b(0) of Part 1 of Theorem 1,

(i)
,(ALRO) X
Tn 1 (i 1 2 Ix(i) 1
-4- --( )n.}}
S n ni 2 2 1


{4E A.(p --1)2 as n co
i i 2









with probability one [0]. Thus,


b(O) = {4EXA(p --1)2 = {ZA.(2p- l)2
I i 2 1 1

To find f(t) of Part 2 of Theorem 1, the asymptotic null distribution

of T(ALR) is required. According to Oosterhoff [33],
n
k
2 -k k 2
P{z.z I{z. >0} s} = 2 Z (.)P[x2 s)
1 1 j=l J
j=1
where z. is a standard normal random variable. Since, under the null
1
hypothesis

(X 2- n.)/ / ./4 z.
n. 2 1 1 1
1

in distribution, it follows that

(ALR) (ALR) 2 2 -k k 2 2
P{T (ALR) >s} = P{(T (ALR) s 2}-2 -k( k )P{X 2s (3.11)
n n j j
as n-, oo for all s. It follows that the associated density function is

a linear combination of chi-square densities. The result of Killeen et al.

can be applied to verify that

1 2
kn[l F(t)] ~ as t -*
2
(ALR)
where F is the asymptotic null distribution of T Hence,
n

c (0) = [b(0)] = Ei(2i- 1)2.
ALR i 1

Before proceeding to a further comparison of approximate slopes, the

slopes are summarized in the following listing.










Approximate Slope


Fishers (T(F)
Fisher's (T )


2EA (pi Zn 2p. + (l-pi) en 2(l-pi)}
i 1 1 1 1


Likelihood Ratio (T(LR) 2EA {pi n 2p + (1-p )n 2(l-p.)}


"Approximate Likelihood Ratio"(T(ALR) Zi(2i 1)2


Sum of Chi's (T(X) [ 1 (2p 1)]2


(CM) 1 2
Cochran-Mantel-Haenzel (T ) -[Ei(2p 1)]



Letting A = 5A (2i 1), it is easy to see that cALR() (
i i ALR

(ak 2 1 2 (a) (a)
c (0O) since Z A 2 [A.] It is also true that c ( c c().
X i=l1 k 1 ALR CM
i=1

Let B. = (2p. 1). It can easily be shown that
1 1

2 1 2 1 2
EX Bi [A.B] = E XX.(B. B.)2
ii k 1 1 k i ] i
i
(ALR) (CMH) (X)
given that EX. = k. Therefore T dominates both T and T
1
with respect to approximate slopes.

Approximate efficiencies of T(C) and T(ALR) with respect to

T( (and equivalently T(LR ) for A1 = 2 = 1 are given for several points

in the parameter space in Table 1. In this case of equal sample sizes,

T() is equivalent to T(C Table 2 gives efficiencies of T(CN), T(x)

and T(ALR) with respect to T (and equivalently T ( ) for A = 1/3,

A2 = 5/3. The values of A1 and A2 imply that the second test is based

on five times as many observations as the first test. When the exact

(CMH) k
null distribution of T (binomial with parameters Z n. and 1/2) is
i=1


Test









to be used to determine significance levels it is more appropriate to

employ the exact slope, Cc (0), rather than the approximate slope,

(a) (TMH) (F)
cC(a ). The exact efficiencies of T relative to T (and

equivalently T(LR)) are given in Tables 1 and 2 in parentheses.

The efficiencies listed in Tables 1 and 2 support several

previously made observations:

1. The statistic T(ALR) dominates T(CM) and T(X) with respect

to approximate slopes (efficiencies).

2. The test based on T( ) dominates the test based on T(X)

when the success parameters are proportional to the sample sizes. The

test based on T() is more efficient in the reverse case. The test
(X)
based on T is more efficient in a region of relatively low effi-

ciencies for both T() and T(CH)

3. Exact and approximate slopes are not, in general, equivalent.

They are in close agreement for parameters close to the null hypothesis.

4. All of the tabled efficiencies are at most one. This is

expected from the optimality properties of T(F) and ILR) given by Theorem 2

and Proposition 3. A value of one is achieved only for the exact effi-

ciency of T(C when pi = p2. This is consistent with the fact that

T(CH) is the most powerful test (and the likelihood ratio test) when

P = P2'















Table 1

Efficiencies of T(CP TX and T(L


Relative to


(LR) (F)
T( or Equivalently, to T (F) 1 = 1




.5 .6 .7 .8 .9 1.0


(0.497)
0.497
0.993

(0.489)
0.486
0.972

(0.474)
0.467
0.934

(0.447)
0.435
0.869

(0.377)
0.361
0.721


(0.497)
0.497
0.993

(1.000)
0.993
0.993

(0.892)
0.879
0.976

(0.773)
0.752
0.939

(0.674)
0.644
0.876

(0.540)
0.505
0.729


(0.489)
0.486
0.972

(0.892)
0.879
0.976

(1.000)
0.972
0.972

(0.951)
0.909
0.945

(0.856)
0.799
0.888

(0.697)
0.632
0.748


(0.474)
0.467
0.934

(0.773)
0.752
0.939

(0.951)
0.909
0.945

(1.000)
0.934
0.934

(0.964)
0.871
0.892

(0.831)
0.722
0.768


(0.447)
0.435
0.869

(0.674)
0.644
0.876

(0.856)
0.799
0.888

(0.964)
0.874
0.892

(1.000)
0.869
0.869

(0.932)
0.763
0.773


(0.377)
0.361
0.721

(0.540)
0.505
0.729

(0.697)
0.632
0.748

(0.831)
0.722
0.768

(0.932)
0.763
0.773

(1.000)
0.721
0.721


1.0












Table 2

Efficiencies of T(CMH)and (ALR) to T(LR)
Efficiencies of T T and T Relative to T


or Equivalently,


to T(F), X = 1/3, X = 5/3
toT 1 2


.9 1.0


0.166
(0.165)
0.497
0.993

0.162
(0.162)
0.486
0.972

0.156
(0.156)
0.467
0.934

0.145
(0.145)
0.435
0.869

0.120
(0.121)
0.361
0.721


0.828
(0.832)
0.497
0.993

0.993
(1.000)
0.867
0.993

0.893
(0.901)
0.981
0.984

0.737
(0.736)
0.934
0.954

0.576
(0.585)
0.830
0.896

0.420
(0.428)
0.660
0.756


0.810
(0.826)
0.486
0.972

0.935
(0.957)
0.694
0.973

0.972
(1.000)
0.848
0.972

0.932
(0.964)
0.924
0.960

0.838
(0.872)
0.921
0.924

0.673
(0.751)
0.812
0.815


0.778
(0.814)
0.467
0.934

0.867
(0.914)
0.604
0.935

0.921
(0.978)
0.925
0.937

0.934
(1.000)
0.815
0.934

0.904
(0.976)
0.861
0.916

0.805
(0.878)
0.827
0.845


0.725
(0.791)
0.435
0.869

0.790
(0.872)
0.531
0.871

0.839
(0.938)
0.623
0.874

0.867
(0.982)
0.702
0.896

0.869
(1.000)
0.759
0.869

0.823
(0.962)
0.767
0.829


0.601
(0.703)
0.361
0.721

0.646
(0.771)
0.426
0.723

0.685
(0.836)
0.490
0.727

0.714
(0.897)
0.550
0.732

0.731
(0.952)
0.601
0.736

0.723
(1.000)
0.629
0.721


1.0









3.4 Powers of Combination Methods


In the previous two sections, competing methods were compared

with respect to asymptotic efficiencies. Asymptotic efficiencies com-

pare sequences of test statistics in some sense as the sample sizes tend

to infinity. Such comparisons may or may not be applicable to situa-

tions when small sample sizes are encountered. Therefore, the methods

of combinations are compared in this section with respect to exact power.

As mentioned previously, exact power studies are often intractible.

For the test statistics considered here, power functions are not obtain-

able in any simple form which would allow direct comparisons between

competing methods. However, through the use of the computer, it is

possible to plot contours of equal power in the parameter space. From

such plots, the relative powers of the competing methods can be surmised.

The first step in obtaining the power contours is the generation

of the null distributions for each of the five statistics: T(LR) T(F)

T(C T(X), and T(ALR). Size a = .05 acceptance regions for each of

the statistics for varying sample sizes are shown in Figures 3 8.

Acceptance regions for tests with equal sample sizes

(nl = n2 = 10, 15, 20, 30) appear in Figures 3 6. The statistics
T(LR) a (ALR)
T(LR) and T(ALR) define very similar, but not identical tests for

n1 = n2 = 10, 15, 20, 30. They define exactly the same a = .05 accep-

tance regions in all four cases, and will therefore yield identical

power contours. Fisher's statistic, T(F) defines a test similar to T(LR)

and T(ALR) for n = n2 = 10, 15; in fact, T(F) defines the same a = .05

acceptance region for those sample sizes. The major difference between









(F) (LR) (ALR)
T and the two likelihood ratio statistics, T and T( is that

(F)
T (F) has many more attainable levels. For sample sizes nI =n2 =20, 30,

(F) (LR) (ALR)
T defines different a = .05 acceptance regions than T and T

The statistics T( ) and T(X) are equivalent for nI = n2.

Figures 7 8 portray acceptance regions for cases of unequal

sample sizes (nl = 10, n2 = 20 and nI = 10, n2 = 50). The difference

between T and T(C) is apparent for the case of unequal sample sizes.

The statistics T(LR) and T(ALR) define different a = .05 acceptance

(F) (LR) (ALR)
regions. In both figures, it is seen that T), T and TLR) define

similar regions.

In Section 1.3 it was stated that Birnbaum [7] has shown that

combination procedures must produce convex acceptance regions in the

(1) (2) (k)
(X X .., X ) hyperplane in order to be admissible. Each of

the acceptance regions in Figures 3 8 appear to satisfy this convex-

ity condition.

The acceptance regions given in Figures 3 8 are not exact

a = .05 size regions. They are the nominal acceptance regions which are

the closest to size a = .05. In order to make a fair comparison among

the powers of the competing methods, all of the acceptance regions must

be of exactly the same size. This can be accomplished by admitting

(1) (2)
certain values of (X X ) to the acceptance region with probabil-

ities between zero and one. A more precise definition of this procedure

follows. Suppose


P{T(i)
n.
1


P{T(i
n.
i


St } = .05 a,



< t } = .05 + b,
u




56



and T() does not take on any values between t and t Then all T(
S1. u n.
1 1

such that T = t are included in the acceptance region with probabil-
n. L
1
(i)
ity one and T = t is included in the acceptance region with prob-
n. u
1

ability a/a+b.


th(i
The power of the i test is one minus the probability that T
n.
1

fails in the acceptance region. More precisely, define the power of
th
the i test to be


Hi(p1P2) =1- [p{T) < ti(plP2)} + (a/a+b) P"T = tu (p'p)]
i 1


nI (1)
= 1 -[ ( ) )p
(1) (2) (i) M ( tx
(x ,x ): T t
i


n -x n2 (2)
(I-P) ( )p2 (1-
X(2)


(I \


(2)
n -x
P2)


n (1) n -x n (2) n
+ (a/a+b)E E ( 1)P (1-P) (2)2 (l-p
x(1) 1 (2)(2(i) x x
(x ,x ):T n.
ni U

For each test statistic, power is calculated for 2500 values of


(pp 2) in the alternative parameter space. This was accomplished with


a FORTRAN computer program. A data set consisting of these calculated


powers is then passed into SAS [4,16]. The plotting capabilities of


SAS are then exploited to portray contours of equal power in the


(p1P'2) plane.


2-x
2










Figures 9 14 are .90 power contours corresponding to the

acceptance regions of Figures 3- 8, respectively. The Cochran-Mantel-

Haenszel procedure, T is most powerful in the center of the

parameter space; that is, when pl and p2 are nearly equal. This is

expected since T(CM) is uniformly most powerful when pl = P2 for any

choice of sample sizes. The statistic T(C) is clearly inferior

to T(LR), TLR), and T( in the extremes of the parameter space, that

is, when pl and p2 are quite different. Further, the deficiency of

T(CMH) compared to the other methods when pl and p2 are different is

larger than the deficiency of the other methods when pi = p2. From

Figures 9 12 it can also be seen that the central wedge of the param-

eter space where T( ) is more powerful shrinks as the sample sizes
(F)
increase. Fisher's statistic, T and the likelihood ratio statis-
(LR) (ALR)
tics, TLR and T have similar power. Fisher's method gives

slightly more power in the central region of the parameter space while

T(LR) and T(ALR) are slightly more powerful when pl and p2 ave very

different.
(F) (LR)
For unequal sample sizes (Figures 13, 14), T and T yield

power contours too similar to be separated on the drawings. The approx-

imate likelihood ratio test, T(ALR), has almost the same power as T(LR)
(F)
and T(F), having slightly more power when the experiment based on more

observations has the larger p., and slightly less power in the reverse

case. The sum of chi's procedure, T(X) is not equivalent to T(CM) when
(CMII)
n # n2. The power contours are very different with T being more

powerful when the larger experiment matches with a large p i The

statistic T() is more powerful in the opposite case.





58



Figures 15 20 are .60 power contours concomitant with the .90

power contours in Figures 9 14. The comparison of competing methods

may be more appropriately made for low powers. When all of the powers

of the tests are high it is probably unimportant which test is used.

The patterns observed in the .90 power contours are virtually the same

in the .60 power contours, however. No additional information is appar-

ent except that the patterns are consistent over a wide range of powers.






















T(F), T(LR) T(ALR)
T(CMH)

10-------


L



6- -- ---







L


2



0 I
0 2 4 6 8 10 )


Figure 3. Acceptance Regions for n = n2 = 10.




60






X(2)
S(F) T(LR) T(ALR)
14 L- '
14 - (CMH)
IT


12 .--.
I


10- -


8- -














0 12 14X
6 '1








0 ,
0 2 r 4 6 8i0 12 14


Figure 4. Acceptance Regions for n1 = n2 = 15.


































T(F)
T (LR) T(ALR)

......... T(CMH)


i. ...
r.51
rT.-l

I
: I
''"'



""


Figure 5. Acceptance Regions for n, = n2 = 20.


X (2)


20

S8

16

14


S 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20


( I)
X


-

-




62








X(2)
30 -. --...... .-- I.. ....
.... T (F)
28 "--. T (LR), T(ALR)
26 ..T(CMH)
2 6 ..., ......... T CM )

24- :-

22-

20 ------------------- -

18



14 i ....

12,

10-
10 :--:

8-

6-
6 L-:-

4-

2-


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30


Figure 6. Acceptance Regions for n = n2 = 30.




63

X(2)
20 .
(F) (LR)
S(F T(LR) (virtually the same)
S(ALR)
T (CMH)
|3 .......... .. .... ....



6T......... .
16



.. ....... .. .

1 ........







I0



s
8-




6-




2 4 6 8 10


Acceptance Regions for n, = 10, n2 = 20.


Figure 7.




64




X (2)

46 -
46T(F)

44 oooo T(LR)
ST(ALR)

42 ........ T(CMH)
4 2 -. T(X)
. T
40

38

36

34 -
324 .... i



30.- o
0o
28 -

26o ......
26 001 ---
0


24- !0
o

22 ...
: I 0

20-
II
lo -
I s I

2 4 6 8 10


Acceptance Regions for nl = 10, n2 = 50.


Figure 8.










T_(F) T(LR) T(ALR)
T-- (CMH)


.6 .7 .8 .9 1.0 = n2 = 10.
Figure 9. .90 Power Contours for n = n2 = 10.


P
1.0











S(F), T(LR),T(ALR)
T- (CMH)


.90 Power Contours for n = n2 = 15.


P
2
1.0




.9-




.8-




.7-




.6-


.5 .6 .7 .8 .9


-1
1.0 I


Figure 10.









P2
2


T(F)
__ T(LR) T(ALR)
......... T(CMH)


-1.0
1.0 1


.5 .6 .7 .8 .9
Figure 11. .90 Power Contours for nl = n2 = 20.




68



P
p2
1.0- T(F)

T(LR) T(ALR)
........... T(CMH)

.9-




.8-




.7-




.6-




.5- I
.5 .6 .7 .8 .9 I.0

Figure 12. .90 Power Contours for n = n= 30.
1 2









P2
1.0




.9-





.8-




.7-




.6-


N


K


.5 .6 .7 .8 .9

Figure 13. .90 Power Contours for n1 = 10, n2 = 20.


-1.
1.0


T(--- F) T(LR)(virtually the same)
T(ALR)
......... T(CMH)
_._T(X)


69


































PI








P
1.0


N


.5 .6 .7 .8 .9

Figure 14. .90 Power Contours for n1 = 10, n2 = 50.


T(F) ,T(LR)(virtually the so
__ T(ALR)
........T(CMH)
_._T(X)


70





me)
























I
1.0-
1.0 1











- T(F) T(LR), T(ALR)
-__ T(CMH)


.60 Power Contours for nl = n2 = 10.


P
2
1.0




.9




.8-




.7




.6-


.5 .6 .7 .8 .9


1.0P


Figure 15.











-T(F(F) ,T(LR), T(ALR)
---T(CMH)


.60 Power Contours for n1 = n2 = 15.


P0
1.0


.5 .6 .7 .8 .9


-1 p
1.0


Figure 16.









P
2
1.0- _F)

--- T(LR) T(ALR)
......... T(CMH)

.9-




.8




.7




.6




.57-
.5 .6 .7 .8 .9 1.0 I
Figure 17. .60 Power Contours for n = n2 = 20.










T(F)
--- T(LR) T(ALR)
......... T(CMH)


P
2
I.0




.9




.8-




.7-




.6-




.5


-. P
1.0


.5 .6 .7 .8 .9

Figure 18. .60 Power Contours for n1 = n2 = 30.











---T(F) ,T(LR)(virtually the same)
__ T(ALR)
......... T(CMH)
__ WT(X)


.8-
\
\




.7-








.5 .6 .7 .8 .9 1.0 1
Figure 19. .60 Power Contours for n = 10, n2 = 20.


P
1.0
I.0








P2
2.0
.0O


.5 .6 .7 .8 .9

Figure 20. .60 Power Contours for n1 = 10, n2 = 50.


o P
1.0 I


- T(F) ,T(LR)(virtually the some)
-_T(ALR)
......... T(CMH)
_._T(X)


-\










3.6 A Synthesis of Comparisons


When detailed prior knowledge of the unknown parameters is

(F)
unavailable the class of competing methods can be restricted to T
(LR) (ALR) (x)
T T and T(. These methods are compared with respect to

various criteria in previous sections. In this section, the results

of these comparisons are synthesized to make recommendations concern-

ing the optimum choice of method for various situations.

For the comparisons in the previous sections, the null hypothesis

considered is H : p = 2 = ... = = 1/2. The most general alternative

hypothesis considered is HA: p. i 1/2 (strict inequality for at least

one i). In some situations, it is reasonable to assume that the success

probability is consistent from experiment to experiment. In such cases

the alternative hypothesis of interest is HB: Pk = 2 = "' = k > 1/2.

A third alternative hypothesis of possible interest is H : p. > 1/2
Cj
(exactly one j). This alternative is appropriate if the researcher

believes that at most one p. will be greater than 1/2. The hypotheses

HA and H are probably the more frequently encountered alternatives in

practical situations.

The following recommendations are based on evidence presented

thus far in this dissertation:

(i)
1. The minimum significance level, T has good power versus

the HC alternative. It performs poorly, however, versus other alterna-

tives.

2. The Cochran-Mantel-Haenszel statistic, T( forms the

uniformly most powerful test against HB. Its use is therefore indicated
Bi










whenever it can be assumed that the pi are not very different. The
(CM)
statistic T performs relatively poorly versus alternatives in the

extremes of the parameter space (Type B alternatives).

(F)
3. Fisher's combination, T(F), is not, in general, the most

powerful test versus a particular simple alternative hypothesis. Its

power, however, is never much less than that of the optimum test.

Fisher's method gives good coverage to the entire parameter space and

its use is therefore indicated whenever specification of the alternative

hypothesis cannot be made more precisely than HA.

4. There seems to be no compelling reasons to recommend the

(x)
use of the sum of chi's procedure, T unless it is known, a priori,

that the pi are inversely related to the sample sizes of the individual

binomial experiments.
(LR)
5. The likelihood ratio statistic, T and the approximate

likelihood ratio statistic, T(ALR), define tests very similar to T(

They obtain approximately the same powers throughout the parameter space.

Choosing among these three statistics then depends upon which yields

significance levels with the greatest ease and accuracy. This problem

is addressed in Section 3.7.










3.7 Approximation of the Null Distributions

(F) (JR) (ALR)
of T T T




In Section 1.6 the problem of obtaining significance levels for
(F)
Fisher's statistic, T(F), when the data are discrete is discussed.
2 '2
Lancaster's transformations, X and X are introduced. It is estab-
m m
2 '2
lished in Section 1.6 that X and X both converge to chi-squares with
m m

2k degrees of freedom. Although Lancaster's approach can be expected

to yield good approximate levels for large sample sizes, the degree

of accuracy has not been established for small or moderate sample sizes.

Some indication of the accuracy of significance levels obtained from

2 '2
X and X is given by observing the mean and variance of these variates.
m m

Table 3 (page 79) lists the means and variances for n = 1,2,...,20 for
2 '2
X and X when applied to one experiment. Since the altered form of
m m

T(F) will be compared to a chi-square distribution with 2k degrees of
2 '2
freedom, it is desirous that the mean and variance of X and X are as
m m

close as possible to the mean and variance of the chi-square distribution

with degrees of freedom, which are 2 and 4, respectively. For n 3,

the mean and variance of X2 are closer to 2 and 4, respectively, than
m

the mean and variance of X2. This suggests that X should, in general,
m m
'2
be a more accurate approximation than X
m

In Section 3.2, the likelihood ratio statistic, T(LR), and the

approximate likelihood ratio statistic, T(AL are introduced. The
(LR)
necessary regularity conditions can be shown to be satisfied for T(

so that the statistic can be deemed asymptotically a chi-square with

k degrees of freedom. As previously stated, the null distribution of














Table 3

'2 2
Mean and Variance of Lancaster's X and X
m m


Median chi-square (X 2)
m


Mean

1.9808
1.9531
1.9394
1.9362

1.9385
1.9430
1.9481

1.9530
1.9574

1.9612
1.9645
1.9674

1.9698
1.9719
1.9738
1.9754
1.9768
1.9781

1.9793
1.9803


Mean chi-square (X 2)
m


Mean
2.000


Variance
1.9753
2.8587

3.2223
3.3754

3.4514

3.5014
3.5428

3.5804
3.6151

3.6465

3.6748

3.6998
3.7218
3.7412
3.7582
3.7733
3.7866
3.7986
3.8092
3.8188


Variance
1.9218
2.8036
3.2419

3.4760

3.6101
3.6923

3.7462
3.7836
3.8110

3.8320

3.8486
3.8621

3.8733
3.8828
3.8909
3.8980
3.9042
3.9097
3.9145
3.9189









T(ALR) can be approximated with a distribution derived by Oosterhoff.

Significance levels are then determined by the relationship

k
(ALR) -k k 2
P{T ( c} = 2 E ( ) P{X. c)
j=l J

(F) (LR) (ALR)
The null density functions of TF), T and TALR) are

plotted in Figures 21 22 for k=2, nI =n2= 6. These plots give an

indication of the difficulty of approximating the respective null

density functions. More extreme (larger) values of the statistics do

not always occur with smaller probabilities. This fact gives the jagged

appearances for the density functions. This lack of smoothness caused

difficulty in approximating a discrete density with a continuous one.

The remainder of this section contains numerical comparisons

of the above-mentioned approximations. The goal is to choose the approx-

imation which yields significance levels closest to the exact levels

of the respective statistic.

Tables 4 and 5 correspond to the density functions pictured
(F)
in Figures 21-22. Table 4 lists the possible events as ordered by TF
2 '2
Lancaster's approximate statistics, X and X are calculated for each
m m
2 '2
event. Although it is not generally true, X and X maintain the same
m m

ordering of events as the uncorrected T(F). Significance levels obtained
2 '2
by comparing X and X to a chi-square distribution with four degrees
m m

of freedom are then compared to exact levels. The inaccuracies of these

approximations are then reflected in the columns labeled percentage error.
(LR)
Table 5 gives an evaluation of the approximations given by T and

T(ALR) These statistics define equivalent tests, but yield different

approximations to the exact densities.










The percentage errors given in Tables 4 and 5 tend to favor
(LR) (ALR)
Lancaster's approximations over the approximations to T and T

Both X and X2 yield generally conservative results in this partic-
m m
2
ular case. The mean chi-square, X is somewhat more accurate than
m
'2
the median chi-square X
m

All of the approximations can be expected to improve as the

sample sizes increase. To indicate the behavior of the contending

approximations for increasing sample sizes, nominal c = .05 and a = .01

values for each statistic are given in Tables 6-8 for n =n2 = 3,4,5,...

The data in Tables 4-8 indicate that Lancaster's approxima-
(LR) (ALR)
tions clearly dominate the approximations to T and T The
2 '2 (F)
optimal choice then becomes either the X or the X correction to T
m m
2 '2
Table 6 gives no clear indication as to whether X or X
m m

yields a better approximation. Both statistics give large errors for

small sample sizes. Both statistics yield errors less than 12% for

both a = .05 and a = .01 levels for n16.

The superiority of the mean chi-square, X over the median
'2
chi-square, X becomes clear for k= 3. Table 9 gives the nominal
m
2 '2
a = .05 and a = .01 values for X and X for k=3, n = n2 = n
m m 2 3
2
2,3,4...,10. The mean chi-square, X is more accurate in all cases
but two ( = .01, n and = .05, n5)
but two (0 = .01, n = 8 and 0 = .05, n =5).
















































































I I I I I I


83





0






CO






(D


II



O:
II



0
t4-'
oo






0
C!

c

40



r-4
0O .,4
C:




(D 4
tO



















-0


I- rO RJ -I 0 0 O -- Io o I I oJ
- - - o QQ oQ Q QQ QQ













(LR)
T(ALR)
(equivalent for T(LR) T(ALR)=O)


O 3 6 9 12 15


(LR) (ALR)
Figure 22. Density Functions of T and T


for n, = n2 = 6.


.40


30







.20-







.0-










Table 4


(F)
Lancaster's Approximations to T for k=2,


Event


6,6

6,5


6,4


6,3


5,5


6,2


6,1


6,0


n = n =6
1 2


2 '2
X X
m m

20.6355

15.8629
16.0951

13.2872
13.3847

11.7041
11.7376

11.0904
11.5546

10.8316
10.8393

10.4468
10.4477

10.3335
10.3335
8.5146
8.8442

6.9315
7.1972

6.0590
6.2988

5.9389
6.1338

5.6743
5.9072

5.5609
5.7930

4.3558
4.4867

3.4833
3.5884

3.0985
3.1968

2.9852
3.0826


Approximate
Level


.000374

.003209
.002894

.009954
.009541

.01969
.01941

.02557
.02099

.02852
.02843

.03354
.03353
.03517
.03517

.07445
.06511

.1396
.1258

.1948
.1779
.2038
.1894

.2248
.2062

.2344
.2152

.3600
.3441

.4804
.4646

.5415
.5254

.5603
.5441


Exact
Level


.0002441

.003174


.01050


.02026


.02905


.03638


.03931


.03979


.08374


.1423


.1863


.2412


.2588


.2617


.4082


.5181


.5620


.5693


4,4


5,0


Percent
Error


53.2

1.1
-8.8

-5.2
-9.1

-2.8
-4.2

-21.0
-27.7

-21.6
-21.9

-14.7
-14.7

-11.6
-11.6

-11.1
-22.2

-1.9
-11.6
4.6
-4.5
-15.5
-21.5

-13.1
-20.3

-10.4
-17.7

-11.8
-15.7

-7.3
-10.3

-3.6
-6.5

-1.6
-4.4









Table 4 (Continued)


2 '2
Event X, X Approximate Exact Percent
m m Level Level Error

2.7726 .5966 -10.6
2.8397 .5850 .60 -12.3

2 1.9001 .7541 -7.3
1.9414 .7465 .85 -8.2
,1 1.5154 .8239 -5.5
1.5498 .8178 .-6.2
0 1.4020 .8438 -4.3
1.4356 .8380 8818 -5.0

22 1.0276 .9056 -3.3
1.0431 .9032 -3.6
.6429 .9582 -2.3
2,1 .9807
.6514 .9572 -2.4

.5295 .9706 -1.8
2,0 .9880
.5372 .9698 -1.8

.2582 .9924 -0.4
.2598 .9923 8 -0.5
.1448 .9975 -0.2
0 .1456 .9975 999-0.2
.03142 .9999 0.0
0 .03142 .9999 1.00.0














Table 5

(LR) (ALR)
Approximations to T and T
for k=2, n1 =n2 =6


Event


(6,6)


(6,5)


(6,4)

(6,3),(6,2)
(6,1),(6,0)

(5,5)


(5,4)

(5,3), (5,2)
(5,1), (5,0)

(4,4)

(4,3),(4,2)
(4,1),(4,0)

Remainder


(LR) (ALR)
T T


16.44
12.00
11.23
8.667
8.977
6.667

8.318
6.000
5.822
5.333
3.591
3.333


2.911
2.667
1.359
1.333
.6790
.6667
0
0


Approximate
Level
.002
.000886
.0036
.004901
.0111
.01383
.0156
.0196
.0544
.02783
.1660
.08117


.2333
.1171
.5069
.2525
.7119
.3862
1.0000
1.0000


Exact
Level

.0002441


.003209


.0150


.0310


.0398


.0837


.2068


.2617


.5693


1.0000


Percent
Error
0.0
-63.7
12.5
52.7
5.7
31.7
-49.7
-36.8
36.7
-30.1
98.3
-3.0


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-43.4
93.7
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The statistics T(F) T(LR), and T(ALR) define very similar

tests. In Section 3.6 it is concluded, therefore, that the choice

among these three statistics should depend upon which affords the best

approximation to its null distribution. The evidence of this section
2 (F)
indicates that Lancaster's mean chi-square (X ) approximation to T(F)
m

is the best choice. Even this approximation yields large errors for

small sample sizes. Tables 10 and 11 give nominal a =.05 and a = .01

levels for HLL an equivalent and more convenient form of T for

k = 2 and k=3. The exact significance levels are also given.




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