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 Permanent Link:
 http://ufdc.ufl.edu/AA00062827/00001
Material Information
 Title:
 Quantal response assays by inverse regression
 Alternate title:
 Inverse regression, Quantal response assays by
 Alternate title:
 Regression, Quantal response assays by inverse
 Creator:
 Dietrich, Frank H
 Publication Date:
 1975
 Language:
 English
 Physical Description:
 viii, 90 leaves : ; 28cm.
Subjects
 Subjects / Keywords:
 Statistics thesis Ph. D
Dissertations, Academic  Statistics  UF
 Genre:
 Academic theses. ( lcgft )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Includes bibliographical references (leaves 8889).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Frank Hain Dietrich II.
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 University of Florida
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Full Text 
QUANTAL RESPONSE ASSAYS BY
INVERSE REGRESSION
By
FRANK HAIN DIETRICH Il
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
1975
To my Mother and Father
for their love and faithful support
ACKNOWLEDGMENTS
I wish to express my deepest thanks to Dr. J. J. Shuster for his expert and helpful guidance in this effort.
I also wish to thank Dr. J. T. McClave for many helpful discussions and comments.
Finally, I wish to thank Mrs. Nancy McDavid for the outstanding job of transforming the rough draft I gave her into this typing masterpiece.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ......... ............... .. ii
LIST OF TABLES vi ... . .. .
ABSTRAQT ... . .... .. vii
CHAPTER
I STATEMENT OF THE PROBLEM . 1
1.0 Preamble. ........ ...... 1
1.1 Introduction 1
1.2 HistoryPrevious Methods of Analyzing Quantal Response Curves .
1.3 HistoryInverse Regression 8
1.4 Summary of Results. ........ ......13
II INVERSE REGRESSION OF QUANTAL RESPONSE
ASSAYS: ASYMPTOTIC THEORY . 15
2.0 Preamble...... .. .. .. .. . 15.
2. 1. Introduction. ......... .. 15
2.2 Parametric Model and Estimators 16
2.3 Asymptotic Theory....... .......... 18
2.4 Summary.... ........ .. .37
III APPLICATION TO THE ANGLE TRANSFORMATIONS 38
3.0 Preamble. .. ...... ........38
3.1 Introduction. .. ...... ......38
3.2 Estimation of LD(50) .. .. .......47
3.3 Estimation of Relative Potency. 72
3.4 Test for Parallelism.. .... ....... 76
3.5 Summary ......... .. ...... ..80
IV NUMERICAL APPLICATIONS .. ... ..... .. 82
4.0 Preamble .. .. ......
4.1 Exact Coverage Probability (95% *8
Nominal Confidence Interval) .. .. ...82
i v
TABLE OF CONTENTS (continued)
CHAPTER Page
IV 4.2 Estimation of Relative Potency by
(cont.) Various Linear Techniques 85
4. 3 Summary 87
BIBLIOGRAPHY 88
BIOGRAPHICAL SKETCH 90
v
LIST OF TABLES
Table Page
3.1 Notation Chart 73
4.1 Exact Coverage Probability 84
4.2 Estimation of Relative Potency 86
v i
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
QUANTAL RESPONSE ASSAYS BY INVERSE REGRESSION
By
Frank Hain Dietrich II
August, 1975
Chairman: Dr. Jonathan J. Shuster Major Department: Statistics
Numerous methods are available to analyze quantal response assays. Some of the more popular methods of analysis are discussed. The general aspects of inverse regression are also discussed.
A general inverse regression procedure for
estimating dose response curves in quantal response assays is presented. Asymptotic distributional properties are developed. Procedures to form (1cc)100% nominal confidence intervals for quantities of interest are given. Methods of testing hypotheses of interest are also developed.
The particular method of applying inverse regression to quantal response assays by use of the angle transformation is presented. The special case is given where, after application of the transformation, the dose response curve is linear. The inverse method has a decided advantage over the more classical methods in
vi i
this case, both in flexibility and in ease of application. The procedure will be shown to be fully efficient in the asymptotic sense.
Numerical examples are presented to demonstrate
the applicability of inverse regression to quantal response assays. The numerical examples deal with linear response curves since classical methods of analysis are only applicable in this case. Inverse regression may also be used when the response curves are nonlinear.
v i i i
CHAPTER I
STATEMENT OF THE PROBLEM
1.0 Preamble
In Chapter I the general problems of quantal
response assay and inverse regression are presented. In Section 1.1 we will introduce the classical quantal response problem along with different objectives of a quantal response assay. Section 1.2 gives a history of frequently used methods of analyzing quantal response curves. In Section 1.3 we will discuss the general topic of inverse regression. Inherent in this discussion is a comparison with classical regression. In Section 1.4 we will give a summary of Chapter I along with the results obtained in the remaining chapters.
1.1 Introduction
In the classical quantal response problem subjects (plants, insects, patients, etc.) are subjected to a stimulus (fungicide, insecticide, physical therapy, etc.), and an all or nothing response is recorded. Although it would usually be desirable to measure the response quantitatively, it is often only possible to measure a
2
response as occurring or not occurring. It is this type of response we will be interested in analyzing.
Trhe stimulus is often referred to as a dose, and the dose is administered at different levels. Generally, we independently sample n.i subjects at dose, d.,
1 = 1, 2, k. For each dose, di we are interested in the true fraction of positive responses, pi. Thus, for each dose level the number of positive responses observed in a sample of n.i subjects is a binomial random variable with probability of success equal to pi. For each dose, d., we calculate the observed fraction of positive responses,, ^Pi, the maximum likelihood estimator of pi. A quantal response curve is then fit. The response curve is basically found by fitting the fraction of positive responses observed against dose. Usually both the fraction of responses and the doses are transformed before the curve is actual ly fit. This type,of analysis is often used to assess the potency of drugs of all types when it is either impractical or impossible to determine the potency by chemical analys is.
The actual objective of a quantal response assay
may be the solution of one of a number of related problems. An objective of many assays is to estimate LD(lO0p), the true dose at which 100p% of the subjects have a positive response. In particular, LD(50), called the median lethal
3~J
dose, is often of prime interest. One reason for this is that it is used in an attempt to classify dIrugs as to their effectiveness. At one time it was attempted to classify drugs by a minimal lethal dose or a maximal YjK
lethal dose. The minimal lethal dose would b'e the smallest dose at which a positive response is attained for at least one subject. The maximal lethal dose would be the smallest dose at which all the subjects would exhibit a positive response. Needless to say, it would be very difficult to estimate these quantities. For a fixed number of subjects LD(50) can be estimated more accurately than a minimal lethal dose or a maximal lethal dose. Thus, LD(50) is now often used to attempt to measure the effectiveness, or potency of a drug. There are however instances, such as toxicological problems, where doses producing 100% response are of more interest than LD(50).
If two or more drugs are to be compared, it is
often done in terms of the relative potency, the ratio of equally effective doses. Even if a new drug is to be compared to a standard, the tolerance of the population may change, and both drugs must be experimented with at. the same perio d of time. Thus, an estimate of relative potency is obtained~ rather than measure the performance of the new drug singly and measure its effectiveness in relation to the standard as an absolute effect. Relative ,potency is a
valuable measure only if it is foundI that the quantal response curves are parallel, Thus, the ratio would be the same at all equally effective doses. The relative potency is therefore usually measured as the ratio of median effective doses.
Whenever two or more drugs are under consideration in a particular problem, it is desired to know if a mixture of the drugs might be more effective than applying the drugs individually. In ge neral, the joint action of a mixture of drugs can be classified in three categories. The three categories as given by Bliss [1] are independent
joint action, similar joint action, and synergistic action.
If drugs have independent joint action, they act independently and have different modes of action., The drugs. may or may not be correlated in terms of the susceptibility of one component as compared to anoth er. The potency of the mixture can be predicted from the fitted 'ki curve for each drug alone and the correlation in suscepti ~Y; bility to the drugs. The potency of the mixture can be
*Drugs are classified as having similar joint action if they produce similar~ efects so that one component can be substituted at a constant proportion for the other. Variations in individual susceptibility to the drugs are
/5
completely correlated or parallel. The potency of a mixture is predictable from the relative proportions of the individual components.
The last classification is synergistic action. The
potency of the mixture cannot be assessed from a knowledge of the individual potencies. 'It must be based upon a study of their combined potency when used in different propor. tions. If the potency of the mixture is greater than that expected by studying the mixtures singly, the drugs are said to synergize. One drug antagonizes another if the mixture has a smaller potency than expected.
We have now stated the basic problems of interest in a quanta] response assay. The next section will deal with a history of methods for analyzing quanta] response curves.
1.2 HistoryPrevious Methods of Analyzing.>j
Quantal Response Curves
Although numerous methods have been proposed 'for;4
analyzing quanta] response curves, the most frequntly / <~
used method is probit analysis. A thorough discussion of probit analysis is given by Finney [1
, In the classical quanta]response problem we indepenently sample ni subjects at dose, d obti P,
the fraction of positive responses, i = 1, 2, k.
In order to use.probit analys.is n analyzing quantal response curves, the profit of ^., Z., is found by the following transformation,
Z. e1/2x dx, i = 1, 2, ,k.(.2.1)
Once the probits have been determined, a linear response curve is fit against log dose by iterative weighted least' squares.
A procedure similar to probit analysis was suggested by Knudsen and Curtis [5]. Rather than use the probit transformation given in equation (.1.2.1), Knudsen and Curtis suggest the use of the angle transformation
Z.=Arcsine('.k (12)
1 1i I =, 2, l22
where Z, is recordedin degrees. Once the angle transformation has been performed, a linear response curie is fit against log dose by ordinary least squares if the sample sizes are approximately equal, and by weight[ least squares otherwise. For all practical purposes th1e angles transform nation is a linear function of the profit transformati on.
Moore and Zeigler [ 7 ] discuss the use of nonlinear regression methods for analyzing quantal response
7
curves. They demonstrate that any methods based on maximum likelihood estimation of appropriate parameters may be formulated as nonlinear regression problems. It should be noted that both probit analysis and the angle trans <
formation are based on maximum likelihood principles, and thus fall in this category. Moore and Zeigler conclude that a reasonably general least squares computer program could replace several specialized quantal response analysis programs.
It has also been pointed out by Nelder [8 ] that there is an important class of estimation problems which leads to a form of solution which is closely analogous to linear rather than nonlinear regression. Basically, the condition which must be satisfied to be in this class ofestimation problems is that the first derivative of the likelihood can be put in a form where p, the true fraction of positive responses, is a linear function of the unknown~ parameters. Again, probit analysis and use of the ang/ tr ansformation fall into this class of problems. Thus a wellconstructed linear regression program could be adapted to cope with this type of problem. Although quanital res ponse assays usually involve discrete distributions,
Nelder also shows that the' same iterative linear regression procedure can be used on a class of nonlinear models which involve continuous rather than discrete distribut i s .
1.3 HistoryInverse Regression
Krutchkoff [6] discusses the general problem of
inverse regressionand in particular as it applies to the problem of calibrating an instrument. He uses the example of calibrating a pressure gauge. To calibrate the gauge, one subjects it to two or more controlled pressures, and notes the gauge markings. From these data, the calibration parameters are estimated,.and the gauge is calibrated, Unknown pressures are' then estimated by reading the calibrated markings.
If x represents the controlled variable, and y represents the measured variable, then the relationship between x and y can be expressed by the usual linear model,
y + .x + C. (1 .3 1,
The classical approach to calibration using model (1.3.1) with k values of x, .and independent identically distributed errors with zero mean, uses the usual leas squares estimates of c't and eThese estimates are found by
k
nA i ...........
k I (1 .3. 2)
Z (x.iY2
a n d
where J i
X k~l and y ( .3 4
The least squares line is,then represented by : ;i
A IN
y ".: ( 1 3. 5
a nd the LcalIibration. equa tion i s
.
X= ya (1,3.3)
Thus, frmagauge, reading of Y h lsia
estimate, Xc for the pressure is
Yaa
i1 (1 .3i.7)
Thelesn squae inverse teresnte byracmoe
(:1.3.1)..is rewritten as
X = y + (1.3.8)
where e y. = a 6 =:1/., and F::. ... ,
Again, the usual leatti sq esaimateis of y andt are
found by
k
k= (1 .3.9)
and
y x 6 y .(1.3.10)
The least squares line is now one and the same as the calibration equation and is expressed by
x = (1.3.11)
Thus, using inverse regression, for a reading of
Y of the gauge, the inverse regression estimate, of the
pressure is
A, = A Y (1.3. 12
The estimates given by equations (1.3.6) and
(1.3.11) are not generally the same. It is therefore o interest to judge which estimate is better by the use of certain criteria. Krutchko'ff uses the criterion~ of mean square error to judge the relative effectiveness of the estimates.
Krutchkoff concludes on the basis of a Monte
Carlo study (in which values of 01<.001 were replaced by +.001 as appropriate) that the mean square error of the inverse estimate is uniformly less than that of the classical estimate. The Monte Carlo study involved different values of a and different variances, different designs, and normal as well as nonnormal error distributions. Thus, on the basis of mean square error, it appears that the inverse estimate is more desirable than the classical estimate.
Williams [11] points out that under the assumption of normally distributed errors the classical estimate has undefined expectation and infinite variance, and hence infinite mean square error. Under the same assumption the inverse estimate has finite mean square error. Thus, Williams concludes that the inverse estimate is better
than the classical estimate from the mean square error point of view.
Williams goes on to point out, however, that this
conclusion is not very satisfying. He reaches this conclusion because all that was shown is that the mean square error of the inverse estimate is less than infinity. He questions using mean square error at all as a criterion for comparing the two estimators.
It is also of interest to note that Williams shows~ that there is no unbiased estimator with fjiite variance.
In somewhat the same spirit as Williams, Halperin [4] notes that a random drawing from any distribution with finite variance would provide a better estimate than the classical estimate inthe mean 'square error sense.
Rather than dwelling on the mean square error argument, Halperin considers the criterion of relative "closeness" of two estimators, and 2 to X. Here closeness" is in the Pitman sense. That is, X is a closer estimate of X than 22 if, for all X,
P[I 1X
Halperin shows that the inverse estimate is a closer estimator than the classical estimate for all values in a closed interval of X.~ This interval depends on quantities such~ as 5, a, x, and the sample size. It turns out that if Ipi is large, where p = o/a, Y is well determined, or the values of the independent variable ar widely disperse the estimates are indistinguishable.
Saw [10] shows that for any distribution on the
errors, the slope of the inverse regression line is always of the same sign, but greater modulus, than the slope of the classical line. Thus, at X = x, the inverse estimate is closer to X than the classical estimate with probability o ne .
Saw goes on to point out that any line through< (y, x) with slope of the same sign, but greater modulus than the classical regression line,will perform better (as an estimate of X) than will the classical estimate within some neighborhood of x.
A similar statement can be made in reference to the inverse regression line. Thus, there exists no best way to estimate X uniformly over an interval of X. This being the case, Saw concludes the specific use of inverse calibration is unappealing.
1.4 Summary of Results
In Chapter I we have presented the general problem of quantal response assay. We have also discussed the general method of inverse regression. We have chosen to apply inverse regression to quantal response assays for a number of reasons. ..
In quantal response assays we usually seek solutions of
F(dose) = p (1.4.1) '
and relationships among such solutions, for two or more drugs. The following criticisms >are
1) the< least squars~ process minimizes the residual sum of squares in the transforme d probability scale (vertical), while estimates of the solutions of (1.4.1<) have errors measured in the logcfose scale (horizont'al).
2) Serious problems occur in estimating solutions~ to (1.4.1) when we model
F (Probability of Response) = Z Br (1ooer( 42 for ir I r.og?.dose r (1.4,2)
The main problem is that the solutions to (1.4.1) may not exist or may not be unique when they do exist. Thus, the classical approach is pretty well limited to situations where a linear relationship exists (thatis to situations when m=l in (1.4.2)).
In Chapter II we will develop the general theory
necessary to apply inverse regression to find solutions to (1.4.2) when intl. The solutions will minimize residual sum of squares in the logdose scale.
In Chapter III we will use the angle transformation with inverse regression to develop a particular method of analysis.
Chapter IV will give~ some numerical applications of the methods developed in the preceding chapters. B actual application of our results it is seen that inverse regression offers the mos elementary computations as compared to other methods.
CHAPTER II
INVERSE REGRESSION OF QUANTAL RESPONSE
.ASSAYS: ASYMPTOTIC THEORY
2.0 Preamble
In Section 2.1 we will introduce the basic reasons for studying the asymptotic theory. Section 2.2 will deal with developing a parametric model along with estimation of population parameters. In Section 2.3 we will develop the asymptotic distribution of the estimators. 'We will also develop methods of forming confidence intervals and testing hypotheses of interest. Section 2.4 will be a summary of the results.
2.1 Introduction
As previously stated, the classical quantal response assay consists of independently sampling n. subjects at dose, di and obtaining the' f racti onof. positive responses, Pi1 i = 1, 2, k. If only one drug is of interest in the assay, it is often of interest to estimate LDcl00p), the true dose of which Op percent of the subjects exhibit a positive response. In particular, LD(50) is a quantity often estimated.
15
If more than one drug is involved in the assay, other aspects of the analysis may be of interest. It is often desirable to compare LD(50) values in terms of relative potency, the ratio of the true LD(50) values. If the assay involves drug mixtures, it is of interest to know if one drug synergizes or antagonizes the other.
In order to use inverse regression to analyze a
quantal response assay, a model for the problem is necessary. We will develop a parametric model for the classical quantal response assay. Once the model has been formulated, estimators of population parameters will be developed. Since confidence intervals for, or test hypotheses about, population quantities are of interest, the asymptotic distribution of the estimators will be studied. The results of the asymptotic theory will be stated in terms of linear combinations of the estimators. From this, confidence intervals and tests of hypotheses of interest will follow.
2.2 Parametric Model and Estimators
We will now develop a para metric model to express the relationship between the observed fraction of positive responses at different doses and the corresponding true fraction. In order to do this certain matrices and their relationships will be defined.
~~< 17
Let M be a k x r matrix with r <.k. M will1 beof rank r and will usually consist of two different types of elements,. M will contain elements which are functions of the true fraction of positive response. M may also contain dummy variables. A k x r matrix Y will be of a form similar to M. If M contains a dummy variable in position m iY n will contain the same element in position y. The remaining elements of Yn wil be the maximum keli hood estimates of the corresponding elements of M. Thus, rather than containing functions of the true fraction of positive responses, as M does, Y n will contain the corresponding functions of observed fraction of positive responses. For a k x 1 (transformed) dose vector, X, we hypothesize the following relationship:
X M (2.2.1)
where : is an r x 1 vector of parameters. Let En be a k x r matrix such that {e: } the rows of pendent random vectors. If we let n be a linear function of the n, = 12,. k w, we will assume that
n1/2 (e i) L >Nr(01 V as n (2.2.2)
nr
where Nr represents an rvarlate normal randoI variable, and V. is a continuous matrix function of M.
With all matrices defined as above, we propose
the following model for the relationship between the ob~servecd fraction of positive' responses and the true fraction of positive responses:
Yn = M+ En* (2.2.3)
Multiplying equation (2.2.3) on the right by g
Y n M + En .(2. 2.4)
Using the relationship given in (2.2.1) we see that (2.2.4) can be rewritten as
Yn =x +E n (2.2.5)
or equivalently as
.= .. E n P (2.2.6)
Thus usi ng the unweighted least squares estimator of we obtain
'"4(2 2.7)
2.3 Asym'pto:tic Theory "
Now that a parametric model has bee developed with the estimators of these parameters, we will obtain the limiting distribution of the quantity
19
Tn n 112 ( (2.3. 1)
n n
where X is an r x 1 specified vector.
Before we actually find the asymptotic distribution of Tn, we will first introduce some lemmas needed in later proofs. The first three lemmas may be found in Rao [ 9].
Lemma 2.3.1
Let {X n Y, n = 1, 2, be a sequence of pairs of random variables. Then
Xn YnI > 0 Y i > Y X > Y, (2.3.2)
that is, the limiting distribution of X exists and is the same as that of Y.
Lemma 2.3.2
Let {Xn5 Yn}, n = 1, 2, be a sequence of
pairs of random variables. Then:
(a) X L > X, Y n > 0 = X nY n P > 0. (2.3.3)
n nnn
(b) Xn > n2> C = X +Yn X+c (2. 3.4)
fiIf
=> X Y n L > cX (2.3.5)
=. X. .n n > X/c, if cf0.(2.3.6)
Lemma 2.3.3
Let g be a continuous function. Then;
(a) X L > X = g(X ) L > g(X). (2.3.7)
(b) Xn P > X g(X n) > g(X). (2.3.8)
(c) Xn Y p> 0, Y L> Y g(X )g(Yn) > 0.(2.3.9) Lemma 2.3.4
Let g be a continuous matrix valued function of Yn a matrix. Then
Y > M g(Y ) p > g(M). (2.3.10)
Proof
Since g is a continuous matrix valued function of Y we can let E>0 be arbitrary and let 6 >0 be such that
Y MI < 6 = Ig(Y) g(M)I
1 > P[jg(Y ) g(M) I
 n
>P [IYI n Mu < 6 1 > 1 as n (2.3.12)
since Y p > M.
n
21
Since s is arbitrary, g(Y ) P >g(M). This completes the
proof.
Lemma 2.3.5
Let E = e11 e12 e Ir (2.3.13)
e21 e22 e2r
e1k ek2 kr
be a k x r matrix of random variables such that the asymptotic distribution of. nl /2e the ith row vector of nl/2E has variance covariance matrix Vi i = 1, 2, .. k. Assume that ACov(n/2e ., n 112e ,.) = 0, Vii', where ACov
13 1'j
is the covariance of the asymptotic distribution. Let
a' = (al a2, a (2.3.14)
b = (b1, b2, br (2.3.15)
C' = (c1, c2, ck) (2.3.16)
and
d' = (d1, d 2 r)(2 3 17)
be vectors of constants.
Then,
AVarrnl/'2a ,~ n1/2c' Edl = a2, (2.3.18)
where a2 k 2 121
Cy F [a b V b + c d "Vd A 2 a c b 4 d] (2.3. 19)j .
and AVar refers to the variance of the asymptotic clistribut ion.
Proof
1/2 1/2 k r
n a Eb n Ei Z a b e. (2.3.20)
i=1 j=1 1
and
1/2 1/2 k r
n c'Ed =n z c d e. (2. 3.21
 i=1 j] ~ 1
Thus,
2 Fk r12
a = AVar E Z (a b. c id )n e
i=1 j=i i 32l~;
= r c dab 2 cd)AVar(n 1/ 2e.
+ Z(a b..c d )(a.,b.,Ci,d.,)ACOV(n 1/2 e n1/2 e~j)
H 1 3 13 1 i~
23
where
H={(i j jIi 1j': 0(ij Ii ')ij 1i1,2,. .k; j j' l=12,...rl.(2 .3 .22) Since ACov(n 1/2e..j, n 1/2 e .i) 0= oVitv, equation (2.3.22) can be written as
Z rZ (a ib.j c d.i) AVar (n "e.i
i=1 j=1 13 131
+ (a ib.c id .)(a ib1.Ic d.,)ACov(n 1/ 'e.i .f1/2 e. 1)
k 2 2
where
Hl=f(i,j,jl): (i,j)f(i,j'), i=1,2,...,k; j,j'=i,2,,..,r}. (2A. 22 This completes the proof.
We are now ready to derive the asymptotic distribution of Tn' as given in (2.3.1). Theorem 2.3.1
Under the conditions specified in section 2.2,
T n N(Q,0~2) as n +, (2.3.23)
24~
where
2 k 2
E {a b v b+c d kV d 2 a c bkVd} (2.3.24) W
wi th
a' (a, a k) =X .{I M(M .M) 1~ Wl (2..3.25~)
b =(MM e (2.3.26)
C, (c1, c k ~ (M.M)1 M, (2.3.27)
and
d =(MIM)"1 M' x *(2.3.28) Proof
From equation (2.2.1) we see that
= (M'M) Mk, X..( 3 29
Using equations (2.2.7) and (2.3.29) we obtain
k (.n9L = Z"{(Y'nyn))1Yx (M' M)1 'x n~~ n
From equation (2.2.3) we observe that
=k M' + E' 231
and
Yly =WMM + M'E(, + Ek"M + EE .( 2
n n n n n n( .2
Substituting (2.3.31) and (2.3.32) into (2.3.30) yields
V(O_)=VI'(M'M+ME +E'M+E ME
n nl n n) M IM+ )x+ZV(MM) E' x.(2.3.33)
By making use of the identity
(U + V)1 = (I + U V) Ul (2.3.34)
and letting
U =M'iI, V W'E + P'M + E'E (2.3.35)
n n n n
(2.3.33) can be written as Z,(^ .)=vK[+(MMyl 1(M'E +E'M+E'EM) (WE~
nn n n En M') n
+ (I'M) 1 E'x. (2.3.36)
(I + V' (I V) I V2 (2.3.37){~
implies that
2
(I V) =(I + V) (I + V)~ V2 (2.3.38)~
Recalling that n112e (i) L> N(0, V as n
n
and letting V be as defined in (2.3.35), we observe that
n '12 6 V >0 6 >0 (2. 3. 39)
Thus, from Lemma 2.3.3
n I 6V2 Pa> 0, I6> 0. (2. 3. 40)
Combining the results of (2.3.30), (2.3.39), and (2.3.40) and applying Lemma 2.3.2, we obtain
(I V) (I + V) + 0(n1 (2.3.41)
or
(I + V). (I V) + 0(n 1(2.3 .42)
Using the relationship given in (2.3.42) and ing Lemma 2.3.2, equation (2.3.36) can be written as
27
(MIM) 1 MIX Z,(MM) MIX
+ Z'(M'M) E I X (M' M) E I X n n
,(MM) (M'En+EnM+EnE n)(M'M) MIX '(MM) 1 (M'E +E M+EnE Enx
n n n)(M.M)V (M'M) E I X + 0(n 1 ( .3.43)
Since any matrices involving EnE n are of order n 1, application of Lemma 2.3.2 reduces equation .(2.3.43) to V Z'(M") M'En(M'M) I MIX
Z'(M'M) En'M(M'M)'M'x
+X'(M'M).l P x + O(nn
V (M' M) 1 E.[,M(MM)'M.]x
TI(MIM)l
WE (MM) MIX +0)(n
n
X,[IM(M.M) 1 M']En(MM)ZI(MIM) 1 M'En (M I.M)lM'x + O(n 1
28
c'E nd + O(n 1 2... .
where a', b, c', and d are given in (2.3.25) (23.26), (2.3.27), and (2.3.28) respectively.
Thus ,
n/L7g 1 n) n a 'E nb + n 2c'E d + (1/ (2.3.45
From Lemma 2.3.1, we observe that both sides of equation (2.3.45) have the same limiting distribution. Thus, from the asymptotic properties of Eand the application of Lemma 2.3.3 and Lemma 2.3.5, we observe that
Tn T L N(0,c2 (2.3.46)
where u2 is as given in (2.3.24). 4;4
This completes the proof.
In Theorem 2.3.1 the asymptotic variance component, 02, was given in terms of M. Elements of M involve he true fraction of positive responses. Since the truefraction of positive responses i s unknown in a practical situations, we will wish to estimate them and obtain a consistent estimator of a2.
29.
'7 7. ; .. 2
Corollary 2.3.1
By substituting Y for M in equation (2.3.24) including V terms, we obtain 92, a consistent estimator
2
ofa and hence
A1 L
a n Tn > N(O, 1). (2.3. 47)
Proof
Chebyshev's Inequality states that for any random variable X with mean, p, and variance, a2
P(I X 1 ) < 1 X>0. (2.3.478)
XA
For a binomial random variable, p, the maximum likelihood estimator,of p, has mean, p, and variance, Pq[ Thus, using Chebyshev's Inequality we observe that
nn
pPp DO) (2.3.49
The refore
SlimP( pp > < (235)
A
Thus', p converges to p in probability.
=!. ; .. .:i ; ;'' 1 = !{ : .i./ 'J ;
Recalling that Y is identical to M except tha
*where M contains functions of p, Y n contains the same functions of p. Thus, from *Lemma 2.3.3 we observe that each element of Y n converges in probability to the correspon'ding element of M. That is,
Yn I> M. (2.3.51)
By application of Lemma 2.3.4, we observe that
2P
Y n > a, (2.3.52),
where ao2 is found by substituting Y forM in equation
n n
(2.3.24). Since
T L>NO 2 (2.3.3
Lemma 2.3.2 justifies that
0n Tn N(O, 1) (2.3....
This completes the proof.
Now that the asymptotic distribution of the
estimators has been developed, we will give nominal
(1a)100% confidence interval for i We will give a
confidence interval of this form because many of the
31
estimation problems of interest can be phrased in terms of linear combinations of the parameters. Corollary 2.3.2
eC' nn 1 Zc/ 2 (2.3. 55)
C~ ~~ ~z/ ni ~ r :.
forms a nominal (la)lO0% confidence interval for V2
where z/ is such that
P[Z > z X/ a/2 (2.3. 56).
when Z is the standard normal random variable. Proof
Corollary 2.3.1 implies that
1/2 "1
n a~ [n EV n > N(O, 1), (2.3.57)
Thus, as n ,
S < (2.3. 58)
Therefore in the asymptotic sense
+n Z an (2. 3. 59)
" ~ a /2.. ... n'S t. . .)7.... <:,: .:
forms a nominal (100)l00% confidence interval for VThis completes the proof.
Although estimation is often of prime importance, it may also be of interest to test hypotheses of the general, form
H 0 A =0 ,(2.3.60)
where A is a q x r matrix with rank q (1 < q < r).
We will now develop a test statistic appropriate
for this general hypothesis. In order to achieve this end, we will consider the asymptotic distribution of n in terms of a multivariate normal framework. We will first
give a definition and two lemmas from Rao [9 1 ..
Definition 2.3.1
A pdimensional random variable U, that is, a
random variable U taking values in E (Euclidean space of pdimension) is said to have a pvariate normal distribution; N if and only if every linear function of U has a univariate normal distribution.
Lemma 2.3.6
If U has a pvariate normal distribution, then the
joint distribution q linear functions of U is N Let
U have mean vector, p, and dispersion matrix, Z. if
Y =CU, where C is (q x p), represents thie q linear
tions, then Y has mean vector, C}j, and dis version matrix
Lemma 2.3.7
Let U be pvariate normal with mean vector, and
dispersion matrix, .. Then the necessary and sufficient condition that
Q U R (U (2. 3.61)
has a chisquared distribution with k degrees of freedom is
E(R E R' R) E =0 (2. .62)
in which case
k = trace (RE). (2.3.63)
Theorem 2.3.2
n 1L > Nr ( E) (2.3. 3 .64)
where
 f" "
34
Q.. = 1 /2(coefficient of 4. 1j (2. 3. 65)
in (2.3.24), and
==2(coefficient of Z )(2.3.66)
2
in (2.3.24). n is similarly obtained by using as
n n
defined in Corollary 2.3.1. 0i is a consistent estimator of Q.
Proof
In Theorem 2.3.1 we proved that every linear
1/2 A
combination of n (8 8) is asymptotically univariate
n
1/2
normal. Thus, since n ( ) is an rdimensional random variable, by Definition 2.3.1 in the asymptotic sense n/2 ) has an rvariate normal distribution.
The mean vector and dispersion matrix follow directly from Theorem 2.3.1. It should be noted that the elements of Q are defined as they are because in (2.3.24),
4 4
coefficient Z. = 2ACov(', 3) (2. 3. 67)
1 n n
where is the ith element of the vector and
n n
2 Ai
coefficient = AVar(o ) (2.3.68)
i". n.
It again follows from Lemma 2.3.3 that each eleenrt of converges in probability to the corresponding element'' of Q. Thus,
P
Q n,~. (2,3.69) and is thus a consistent estimator.
This completes the proof. Corollary 2.3.3
To test the general hypothesis
A 0 3 : 0, (2.3.70)
the test statistic is ~ A)1AI L 2 '3
n(A n)''(A n An) > X q
Proof < ~ "
From Lemma 2.3.6 and Theorem 2.3.2,
1/2 L J ~<
n A n> N q(0, A ~2A')
if A =0. Thus,, from Lemma 2.3.7, if A g 0
36
n(A )'(AA Q1(A ) 2
n e nA''( *'" Xq # 444
since in the notation used in Lemma 2.3.7
n A A (2.3.72)
and
R = = (A Q A') (2.3.73)
Thus,
Z(R Z R' R) Z = (Z ) = 0, (2.3.74) and
trace(RE) = trace(E1 ) = trace (I) = q. (2.3.75) Application of Lemma 2.3.2 implies that if A = 0,
n(A An )'(A A) in) X2 (2.3.76)
This completes the proof.
24 Sumr
We have now developed the general asymptotic
theory necessary to solve problems of interest in classical quanta] response assays. In Chapter III we will apply these general results to the particular use of the angle transformation. In particular we will discuss estimation of L(lOOp) and relative potency. We wiN also discuss testing the hypothesis of parallelism. ,<
iN/
:r :);; i;2 ,;
: .. ?' ) J>
i 4~<
?' : 74
" =. 4t 447 ". j
CH P E II AP ICA INT H AL
TRANSFORMATION
3. Preamble
In Sec io 3.1 we wil dicus te r tio al
behid chosin theangl trnsfomatin toappl th
me ho of invrs rersso Wew llas j sif h
pro ose moel InA Se t o w ild sus h si mation of LD ) Wew l nld eslsfrtecs
3.1 ICHAPTERtion
3. Preamble 1
behind choin th n trnfrato oaplh
mhethdorvre regesio. We will emlywigtdlaso jqustify the propsedmode. I Secion32w ildscs h si
;Q
fit the model. Appealing to the notation developed in, Section 2.2, thle probabilistic parametric mdel we will thus be using is 21 2
Yn=M+En (3. 1 .2)
where the elements of Yn and M are defined by
K >;
Y i ini (3 1 .3)
1
I i < r r < k
and
k
T n f E (3. f sry
Th for of Enwl flosoty
40
Justification of the model. proposed in (3. 1.1) will now be given. A brief discussion of classical methods will first be given.
Classical methods of analyzing quantal response
assays are applicable when there is a linear relationship between the logdose and the transformed fraction of positive responses. Probit analysis, for example, produces such a linear relationship when tolerances (measured in the logdose scale) have a normal distribution. The tolerance of a subject is the dose level at which that subject would exhibit a positive response. Needless to say, not all quantal response assays have a normal distribution of tolerances. For assays such as these, probit analysis is not appropriate.
Although for a nonnormal distribution of tolerances another transformation might produce a linear relation ship, it would be desirable to find one method of analysis which would be appropriate for a wide class of tolerance distributi ons .
In many quantal response problems, the, fraction of
positive response is montomically increasing, with respect to dose in the area of experimentation of interest. This, ,of course, implies that dose is monotonically increasing with respect to the fraction of positive responses. Both logdose and sin I lpl/2 ) are monotone functions. Thus, p1 < ~P2 and d1 < d2 if anid only if log d1 < log d2 and si"1 (p1/2) < sin 1 (p 1/2) Thus, dose is a monotonically increasing
function of the fraction of positive responses if and only if logdose is a monotonically increasing function of 2>2
. 'm {]
sin l(P12 In this case, it would b~e appropriate to moe
the relationship between 1og~dose and sin 1p'112) by a polynomial, which isd the model given in (3.1.1).
In conclusion, the model given in (3.1.1) is appropriate regardless of the distribution of tolerances, as long as the fraction of positive responses is an increasing function of the dose. Thus, the inverse regression approach is applicable to a much wider class of quantal response assays than classical methods.
Theorem 3.1.1 (Mean Value Theorem)
If f is continuous on [a, b] where a< b and differentiable on (a, b), then there exists a point c c(a, b) such
that A
f(b) f(a) = (b a) f' (c). (3.1.6).
We will now use Theorem 3.1.1 to prove other useful results. ..
Theorem 3.1.2
Let a random sample of size m be taken from a binomial population with parameter, p. Then
ml/2Ein l ~/2J sin' [p /2]l L N(O, 1~) (3. 1 .7)
where sin pl /2) and sin' (p are measured in radians.
42
Proof
Let
f(x) =sin" i1 (x112 (3.1.8)
Thus,
f'(x) 1 [x(bxfl" 112, (3.1.9)
From Theorem 3.12., there exists a c such that Ipcl !PpI and 4
1l/2Ei n 1 [,1 /2) si n {p 1/21j
= nm (P~p)Lc(1 c)P1 (3,2.10)
Si nce~ cp < 'p I 1and
11
icpi 0m 1/2) (3.1.12)
thenA
m12[sn 1 r1I2 s 1 [1/2j1
=i112 1~~ p(~p]12 ~ /
m~~~~~~~ >PP PI), +O 12 (..3
S i n c.
3 43~43
(1.L N(T 1)72 (3. 1 14)
1/2
2[p(lp)]
Thus, by Lemma 2.3.2
m1/[L ^ 112] si n 1 p/2J L ~> N(0, 1/4).(3.1..16)
This completes the proof.
Theorem 3.1.3
Let a random sample of size m be taken from a binomial population with parameter, p. Then for any given {C.}
r~
1 (1/2] l 1 2
m /2 cj{Jin1(l/2 tsin1 [If
j~l L
L r r1/ k (3 1.17
> N 10, 1/4 Z jl(ICC 1
j=l k=l kS
44 4~4
Proof kw
Let
f(x) m in12r C. x(3.1.18) j=1 J:
1/2 r
f'(x) m zn U(1).C. xi2~3..
j= 1 3
From Theorem 3.1.1, there exists a u such that
ju sin1 (p112) < Isin 1 (P&12) sin 1(p 1/2) (3. 1.20) and 44j
12r12j1 1j 1l/2 jsn1[ 112] si 1tpl/23m12~(~ ) rujSince I^PJ O(m 112) by the Mean Value Theorem
4<4 1k s [in1 ~ = O(m~t) (3, 1 22)~
where I < k
Iu sn'I~EI~j< in(~p sin (p/?j (3.1.23)
u [sin 1 (p 1/'2 A O(m' 1/2) (3. 1.24)
Thus,
r j1
m 12Ec { in p in i( l
j1~~< 2 S) S
[,11 ) 1 2 1/ r [sinl(pl/2 j 2 M,12/ L P p m j~l01)
(3. 1 25
Fromthe esul of heorm 3..2 w canthu
condetha
j=1
m1 /2 r /jin( / J 1 n l 1/2 jj 1 I
r r 1 .. ..(. 1 2
L> N(O, 1/4 (j. (k )C Ck inj=1 k=l S 12
This completes the proof.
In the model given in (3.1.2), En is a random
error matrix to explain the asymptotic variability of Yn as compared to M. With Yn and M as defined by (3.1.3) and (3.1.4), respectively, and using the results of Theorem 3.1.3, we are now ready to justify the form of
En'
The rows of Y. are independent and we have shown in Theorem 3.1.3 that
r j1 j 1
1/2 1 C i 1/2i 1 /2
spi s i pi
r r j+k4
> N(0, 1/4 r r (j 1)(k )C .Ck si n2') ( 27)
j=1 k=l k
k n.
With n = n./k, we assume that as n 1 A
i=11 n
i = 1, 2, k. Also, k, Thus, as n+, each
n +O. Thus every linear combination of n 1/2 i '
is asymptotically normal as n+ where Y and M1 are given in (3.1.3) and (3.1.4), respectively. It then follows that the ith row of En is a random vector such that
.n1/2 e (i) L > N (0, V.), as n+ (3,1.28)
n r 1
where
Ss+t 4
V = (s1 (t1 i p 2 l
The element V. is found directly from the variance component given in (3.1.17).
Thus, we see that by using the angle transformation we can apply weighted least squares and form a model which complies with that proposed in Section 2.2. We are now ready to use the general results of Chapter II for this particular case.
3.2 Estimation of LD(50)
We will now discuss, the estimation ofLD(50) by inverse regression with he specific use of the angle transformation. Recall that LD(50) isthe median lethal
>48
dose. Since LD(50) is.often of prime interest in quantal response assays, the main emphasis of th is section will be the discussion of estimating LD(50). Since estmation of LD(lO0p) may also be of interest (when p .5), we will give results concerning this also.
We will first recall the notation andt define (or redefine) the matrices appropriate for this problem.
The classical quantal response assay consists of independently sampling n 1 subjects at dose, di, i = 1, 2, .,k. The observed response frequency,.
Piis calculated for each dose level, d. i*The true
response probability for dose, di, is represented by p i
Let :~~%;
n.. ..... (3 2 1
The gen eral deterministic model is given by
log d = ; l 1
Let wA: 4
V>AI j4 IS< :A
where r < k. We will employ weighted least squae an~rd~ write the probabilistic model as .
l~ogd :ik,; < '
n 2
.. .. <
,. #<>11
K(i !{
S49
We will define Yn M, and En as follows,n 1n2
Si~i .1./.2............1
F s i n' I i !!! !'!ii ... ..... ,,, N i !"iii
Ynij n si n pi/ J ,(3. 2. 4a)
1< i < k, 1 < j < r.
= ~ si 1/pi2 (3.2.4b)
Also, we assume that
x = MB (3.2.5a) where x is a k x 1 transformed dose vector with ith element
1/2
x. i log d., (3.2.5b)
n 82
_ = (3.2.5c)
L
where E n is a matrix of independent random vectors. In
Section 3.i we justified that the ith row of En is such that
1/2 (i) L .
n ern > NrN(0, Vi), as n (3, 2.6)
In Section 3.1, we.gave the form of Vi when si 1 1'2
 1 11/2
and sin (pi ) are measured in radians. Sice
... ..... 1 8 0 0 .
1 radian = 1800 57.29580, (3.2.7)
IT
it follows from (3.1.26) that when sin1 !i) and
1 1/2
sin (p ) are measured in degrees that the stth element of Vi is given by
Vist 820.7(s1)(t1) in Pi, (3. 2.8).
n n n n
r
1 s < r, '," 7 if .?]f ,
In order to predict the transformed dose, xji, 0
at which pi of the subjects respond, we would use the ~i
weighted least squares prediction equation 1/2
S[in (3 2. 1i0
Rather than estimate the transformed dose, xi, it is preferred to estimate the dose in the logdose scale Thus, the estimation can be given by
r r J
log d E jin' i12 (3.2.11)
To form a (1ct)lOO% nominal confidence interval
v.
for log di we may apply Corollary 2.3.2. Th~e confidence interval is given by
~n Z2%~ 1/2 ( 3.12
where
s n 1 p 1
= p~ (3.2 13)
ri
1/20
[sn',p
and anis given by Corollary 2.3.1. 4A4
If it is desired to estimate LD(50), pi .5.
Thus with O < sin(pi'2 < Tr/2.
454
(3.2. 15)
(45)r 1
We will now discuss in detail the estimation of
LD(5O) when it is assumed that the relationship between>jjfr+ log di and sin 1(p 1/'2 is linear. We will use the following notation:
n. 4
Siceweare assuming a linear relationship, we are usingFJ~/ the models given in (3.2.2) with r = 2. In this case, the weighted least squares estimate of can be ex pressed as
_ (3.2. 17)
n 2,
?
' > c ' ..".s : : L ". ..> : L .>, ; .. : ' i 6 4
where
2 Y~y (3 2,. 18) )
yy7
and
wi th V
k
=y (x Cm ,) (3.220?)
k k
Z W.x.iW
k k
and4
k k
From~~~~ (32.) e bsrv ta
54
n1/2 e (i > N2(, V.). (3.2.23)
n 2(01
From (3.2.8) we see that
T) 0
V = (3. 2.24)
0 820.7
In order to predict LD(50) in the logdose scale, we use (3.2.11) and (3.2.14) to obtain
A
LD(50) = + 45 2
= + 2 (459). (3.2.25)
L'.~~2 1 / .... .. .. ..,5; ,) ] ;
The asymptotic variance, a2 of' n12 LD(50) can be obtained from (2.3.24). By applying Corollary
2.3.1, a consistent estimator of a2 may be obtained. When a linear relationship is assumed a simplified expression for n can be obtained. This will be shown in the n~
following theorem.
Theorem 3.2.1
A consistent estimate of a, the asymptotic variance of nl 2[LD(50)LD(50)], is given by
^"2 = 2.[4~~ 2 ~ 2 1 ~ ~
a 820.7(45) S + k (3.2.26)
n xx yy
Proof
In (2.3.24) a2 was given in terms of M.. To find
2
a consistent estimator of a M is replaced byYn. To
2
find ,we will define the following vectors:
n
a_ = I (Y'Y ) Y'}
n n n n
x Y' (3. 2.27)
n n
= (Y y'n )1Z (3.2.28)
c = I '(Y 'Y ) Y (3.2.29)
n n n
= (YY )1Y' x
n n n
(3.2.30)
where x, Yn' n are defined in (3.2.5b), (3.2.4a), and (3.2.17) respectively.
(3.2.31)
45
Since a', b,; c', and aare thie same as a', b, c'~, and d of Theorem 2.3.1 with M replaced by Y n5 by Corollary 2.3.1 an E (a.i b'Vb c I. dVcl ac.vo,7 '323)
where Viis defined in (3.2.24) is a consistent estimate o f a2
=112 1/2^ 112
1/2 ~ 11 1/11i + W 2 w Y 2
2(2
k 2 'ky ,;
(Y'yn)' (3.2,34
i y 80.7 .(3. 2.35) ryy j 27.2
i 2
wij 22 (
[ 4 45(Y..y y Y w y. (3.2.36)
d' V.i d 2 820.7.(32.37
2^a.Z.bV.d =2a ic.5] 820.7, (3. 2. 38)
1 1 1 f P,
1 *I 1
where a. and 6 are given in (3.2.33) and (3.2,6) respectively.
Thus, by substituting (3.2.33), (3.2.35), (3.236), (3.2.37), and (3.2.38) in (3.2.32), we obtain
2=820.7 1 Rx. (45s)
n 2 i
i= yy~
+2 (45 )2(y~ 2 ^ 2
+_95)A2 2 + 2+2 2+ (45 2(i k iy 2al
58
+ LO 2 k 2 go 2(yi )Yi
k 2(yiy) Z'W iyi 2 Y
2 22 k go(x.,R)(Y
Y E W (45
k Yi i=l iyl Y)R2
2 k 2
(x i )(45YW 2 E W iyi + 2(xi,)Yi 9(45 )R 2
2 2 k
2 2 2 2
+ 90 (yiY) (45 ) + (y y
2 k i9)(45 ) E W i i
i
2 (YiY)(459)yi l (3. 2.39)
2 I
Performing the summation we obtain
^2 820.7[ 2 + 2
Un 2 45y) Sxx .2 (45Y)2 yy
yy
2 (45 )2Sx + (45)2 2S + 2[k w iy2j 2
R2 y 2 yy
,,,2 2 2 go 2 2 2 2
P E Wiyi 2yS 2 2y W y
yy
59
+ 2 Y(45, )Sx
90(45 )^2 xy 2 y
+ go 2 (45Y)S 2 2(45 ) S yyl (3. 2.40)
2 yy 2
Since from. (3.2.18)
S xy =.Y yy (3.2.41
,,2 820.7 )2 + ^2 2an 2 4.5 9 xx 2 (459) S yy
S
yy
2 k
12 2 2 + 2 22 2 (45 ) S yy + (45)2 2 yy iyi S yy
go 2 90(45 ) 2 S + 2 2
2 yy 2 yy .2 yy
,N2 ^2(4
+ 90 2 (45 )S yy 2 2 YYI
2 k
821D 7 2 + 2 2
1(45y) xx k iy i yy
S L
yy
^2 2y
]
820.7L45 )<~x S( 2
xyy k2(..2
This completes the proof. Kii
Corollary 3.2.1
LD(50) z2a n:L a/2 n
forms a nominal (1c)10of% confidence interval for LD(50) where LD(50) is given in (3.2.25), and an is given in ~~ (3. 2. 26) .
Proof
Corollary 3.2.1 follows directly from Corollary
2.3. 2. K
We will now derive some results for the KnudsenCurtis [5] method.for analyzing quantal response data. We can thus compare the inverse regression approach their. method. 2
KnudsenCurtis use classical weighted leas squa
to fit the model.
Y b1+ b2 x i+(.
where
2 ; o ;T?
: 77 7",.1
.61
x ..= log di, 3.2,46)
and from an argument similar to that used n Section 3.1 we can assume
n1/2 C> N( 820,7), as ni ( 2.4
and c. is independent of ce. for if t .
1 34
Since weighted least squares will be employed, the model to be fit could be expressed as
1 2 n 1 /2 1/ 1/2
1 i n b I+ b n X.+ E. (3.2.48)
n 2 n n
:'. :
Letti ng
we see from (3.2.47) that. e e
n1/2E > N(0, 820.7)., (3.2.50)
1 11
an s.i indepedentof.. for i i:ii
? 44
7'62
Using the notation of this section the weighted least squares estimates are
82 Sxy / xx (.25
and
b= 82 (3. 2.52)>
Thus, to estimate LD(50) in the logdose scale, KnudsenCurtis would use
45 8'7'7i~
LD*(50) B2 *
R + (45y)/S2. (3.2.53)
Theorem 3.2.2 4
n 1/2 (c* 1 l LD*(5o) LD(5O)] L  .2.5
where <'
2 2^ 4117
*2 820,7Fc45 )2 i +kb4 3 2.55)
Proof ~{N
Frm(3.2.48), N
kp~
b2 xx
x Nx
Substtutin yi fom (3(.442yi5l)
S xx
k
z (x~ 1 1 *
b2+i ~ l.3..l7
2S
xx
n (b2 +1=) > 11,
In a like manner, it can be shown that~'t '
n112 b) L > N(0, u2) (3.2.59) A
and any linear function of n 112(81b1 and l/ 12 b 2)
is asymptotically normal.: 1i
From (3.2.44) it can be seen that.
b 24
[LDA')0 LD( ) 45 45 b3 2. 1
45bl (45bl b
bL(0 b + (32.0
26 bb b
2.
Now,
6 465
Thus, "
[LD*(50) LD(50)]
(45b) b b
1 ofb2 2bl 1 1
= b
+ 4 +' b (3. 2;.63)
2 j=1 2,7
Thus, from (3.2.58) and (3.2.59)
1/2 j L 2
n/ [LD*(50) LD(50)] > N(. (3
2.. . 2 m + 1 1,, L"
It still r emains to find 62. .
In essence we need the asymptotic variance of
nl/2LD(50). We will first find the asymptotic variance of some other variables.
m" =. ";.6
I 2e, i 1 (x.i )w n 1/2 Y
AVar(n b /2 AVar = sxx
Z~ (x.iZ)2w.A~a n 1/2ii
2 1 k 2 Yi>4
z (xi R) 2Wi 820.7
x1 1
820.7(3.2.65)
XXs 4,
x>A
1/21/2
bw.ar'rfl y) 4
= 1 k
2 1 w. 820.7 >4
k i=i
820.7 (3 2. ~66)
k>4 ''
y 21 =AVar[n 112 (45Y)/I?21L(,.7
If we let j~be such that 4~>
 > N(O,y 2 thenA
45j 1 _(451i) + (P~ ~)
b2 2 b 2 2
22 1
 E4~l)+lJ~T+~22 (3. 2. 68)4
+ j tb2 i
Since 2and 62 are asymptotically independent, an
terms b 8b
2 2
where j > 2, are of order n' we obtain tha
AVar[n 112 LD*(5O)]
2A~ b A
b'2 2
68
= b 4(450)2 820,7 g&I + 2 820,7 kr1 (3.2.69)
(2 :x 2~
0F2 =AVar{nl/2[LD*(50 LD(50)]}
820.7[(45p) 2 b2 (3.2.70)
By Lemma 2.3.3, (a*)2 is a consistent estimator
o 2
of a2 As in Corollary 2.3.1,
n 1/2 (a*) 1[LD*(50) LD(50)] N( 1). (3.2.71)
This completes the proof.
Corollary 3.2.2
21 1/
S 2.Assuming a linear model for sin against
logdose,
Thig os copee throf
^ P'
C /a* > 1(3.2.72) inverse method to the KnudsenCurtis method is unity.
Proof '
From (3.2.26) and (3.2.55) we obtain that
(45Y)2 S k Ir2 2
2 2 kxyS SYY
rn (459)2S 6 4 + k S3
xxxyxxy xy x
r 2[(45 ) 2 S 6 S4 S + k'143 S ]Y
xy xx yy xy xx yy
r 4 (3.2.73)
where,
r=)/ (3.2.74)
(XX)12( yy12
is the sample coefficient of correlation. ~'i~ 4
S2.
Since a linear model is assumed, ~'
r >P 1, (3.2.76)
and thus
70
P4
r n' Cr( 7
<' A:
This completes the proof.
It should be noted that the KnudsenCurtis method is itself asymptotically efficient Thus, the method of inverse regression is asymptotically efficient.
Confidence intervals with nominal (1a)100% coverage 'are obtained by either
A.A
(5 1 z: G n' 1/2
LD(50) 2 z on (3.2.78)
or
LD*(50) + z a a n (3.2.79)
Since the choice of method cannot be made on the basis of asymptotic efficiency, we shall examine the two methods on the basis of robustness.
Consider the set of points in R
1 1/2.
i = 1 2
S = [log di, sin (pi 2)],i 1, 2, 80
The inverse regression method consistently estimates a weighted least squares line which minimizes the horizontal deviations for the deterministic S. Assuming
n i1kn > thep
ni/kn > as n + m then the point Si. carries weight
ci.Similarly,, the KnudsenCurti s approach consistentlyi estimates a weighted least squares line which m~inimizes the vertical deviations for the deterministic S..
Since the error statements abou t LD50), relative potency, etc., are made in the horizontal scale, he inverse method, when linearity is false, should tend to have a smaler.asymptotic bias than the KnudsenCurtis method.
Corollary 3.2.3
AVar[LD(50)] < AVar[LD*(50)1. (3.2.81)
Proof
From (3.2.75),
a~~~ ~~ 2 (*)2 r 3..2
4 744: '4
n4
Thu s ,ee
where P is the population coefficient of cor for a bivariate random variable with mas funtio
P[X = Si] o,0 i 1, 2, k. (3.2.84)
1 if and only if there is truly a linear relation
ship. Otherwise 1 < p < 1. Thus, from (3.2.70) w~e see that
AVar[LD(50)] < AVar[LD*(50)], (3.2.85)
with equality holding when the linear mode is correct.
This completes the proof.
Thus we can conclude that when estimating LD(50) in the logdose scale inverse regression seems to yield a more reasonable estimate than the KnudsenCurtis method as far as robustness is concerned. As a further bonus, LD(50) will tend to be a better estimate than LD*(50) in terms of variances of the asymptotic distributions.
3.3. Estimation of Relative Potency
If two drugs are involved in a quantal~ respne assay, it is often of interest to estimatethe reatie potency of the two drugs. The relative pote t
ratio of equally effective doses. It should b V ,e
that relative potency is a valuable measure o i the
quantal response curves are parallel. Th thruhout this section we will assume the response cuve are parallel.
In order to estimate the relativ potency, we illJ use the results of Chapter II and Section 3.2. 'To achieve
this end we will first give the notation nd
this section, Table 3. 1 '
Notation Chart Drug 1 Du
Dose levels d d k d k, +, 1 9 ',d k
Sample sizes n, nk n k........n
Response probability p, "k Pk1+1 9 P
Observed responds P P k
frequencyP1 Weight, n i/n wk W +1 . k
Recall that
k
n En /K.(3.1 i
Since the curves are assumed tobe r
deterministic model can be expressed as
log d. E j[sin' [P/2j i
1 : 1~
r F
In this case, r+l k. It can thus be seen ttfj2222 j2~
relative potency = ~ e2 (3..4
The probabilistic model will again b2e 2of' theform~
The response matrix Y n has 222224
1/ 122/
13i 1 ~i nl{Ij P l
and 2 222 24224 22,
The, mari M4j~ iso h am ma Ynwt i
replaced~~242 bysn ,p / lo
i$ 24412
x~~ (3.3.9
75
where the transformed does vector, x, has th component
x 2 log d 1 < i < k, (3. 10)
1 i
and
2
(3.3. 11)
r+1
In the same manner as was employed in Section 3.1, we can assume that E is a matrix of independent random vectors with
Z4
1/2 i) L
n e > N (0, V.). 3.3.2
n r+1 I
It again follows that Vi has the form given in (3.2.8) That is, V. has entries
s+t4
Visit (820.7)(s1)(t1) sin 1
= 0 otherwise (3.3.14)
and
= (Y'yn Y ny' _X ( .3 5
is the weighted least squares estimate of R.
We can now employ the results of Chapter II to estimate the relative potency. We will perform this estimation in the logdose scale. Thus, we will estimate r+l by means of a (1a)100% confidence interval. If
we let' (0, 0, 1), then from Corollary 2.3.2
a'~n ZCX2 a n 12(3. 3.16)
forms a nominal (lu )l00% confidence interval for and thus the relative potency. a n is obtained by applying Corollary 2.3.1. ..?
We have thus estimated the relative potency (in the logdose scale) of two drugs. It was assumed t hat the quantal response curves are parallel. In th ext se we will give a test for parallelism.
3.4 Test for Parallelism
If two drugs are involved
77
We will use the same notation as that given in Table 3.1. The deterministic model is now given by
r
log d j in 1 /2 1 (3.4 1)
3=1
2 r 1 p1/2] r1
E sin p. ,i>k ,(3.4. 2)
jr+1 j I
where r < min(k1, kk 1).
The probabilistic model is again given by
Yn = M + En (3.4.3)
The response matrix Y has entries
n
1 /2 1 1 /2 (3.44
Y .. = W sin Pi ,1
nij i i PJ I 1~I
= l 1/ 2 k
i i 1= 0 elsewhere. (3.4.6)
M is defined in a manner similar to Y with sin ( )1/2
n Pi
replaced by sin 1 (p 1/2 Again we have
x M ~(3.4.7)
where the transformed dose vector, x, has ith component
xi l o / 1 < k,'< k, (3.4.3)
E n is a g aiJn' a m atr. ix o.f ra d m ve t r En ca no b
andn
] ., ; ;' ( : K ; ,
rhepresEnted by 2 r lr n kk1)r epciey
E (1) and E (2) are thus comprised of independendnt random vectors, and we again have
n1/2 e > N (0, V i),
n r i
From (3.2.8) we see that Vi has entries
 s+t4
V. =t 820.7(s1)(tl)[)sin1 1/2 1st< ,i 1 .4. 12)
L
F s+t2r4
= 820.7(sr1)(tr1) Lin [pi2 r+14s,t<2r,i>k
I (3.4.13
= 0 elsewhere (3.4.14)
S (iY n) Y x (3.4.15)
n nn n
is the weighted least squares estimate of .
Since we desire to test for parallelism, the test of interest can be expressed by the hypothesis
Ho: r+j
This hypothesis is thus of the form
p /
A, ,.. ,
H0: A $3 0, (3.4.17)~
where the entries of A an rx2r matrix are a = .. o. 3 8
a11 a22 a (3.4.rr18)
a I r+l1 2,r +2 ar2r(3.4. 19 (
a0 elsewhere. (3.4.20). >
Thus, we can apply Corollary 2.3.3 and tes t H0 by the test statistic
n(A~n )' (A l A~n x 2 (3.4.21) ,
n (A A<, (A j!
where A, 13n, and Q are defined in [(3.4.18),( 3.41) (3.4.20)], (3.4.15), and Theorem 2.3.2, respectivel
If xris sufficiently large to reject HO: A we can conclude that the response curves are t
3.5 Summary
We have now discussed the applic o e
transformation to inverse regression f
assays. We have shown that inverse rersso i11 'g i v
better asymptotic results than the Knudsen method
when the relationship between log di and sin 1/2 i
jf< 81
not truly linear. IAnverse regression maybe used to fit models other than the linear model wherea the~ KnudsenCurtis method is not appropriate. We have gven methods of
CHAPTER IV
NUMERICAL APPLICATIONS
4.0 Preamble
In this chapter we will apply the results obtained in Chapter III. Several numerical applications will be given. Section 4.1 will give the exact probabilities that 95% nominal confidence intervals cover LD(50). Eight probability schemes will be considered which satisfy a linear relationship between logdose and sinl(pl/2). In Section 4.2 we will compare the use of inverse regression to other methods of analysing quantal response assays. Section 4.3 will be a summary of the chapter. /
4.1 Exact Coverage Probability (95%~
Nominal Confidence Interval)
In this section we will investigate small( smple results for eight probability schemes satisyn the liea
model
i + sinl(pv/2),a i 1, 2, 3. (4.1.1)
The logdoses, x.i, i = 1, 2, 3, 4 were fixed at four equally spaced values. Equ~al sample sizes of five, ten, .. .. )k8
and fifteen were considered. We ran all possible assays for the model given in (4.1.1) with the conditions described. For all realizations, we then computed nominal 95% confidence limits for LD(50). These limits were found by use of the results given in Corollary 3.2.1.
For the eight different cur ves, we then computed the exact probability that the true LD(50) lies in the confidence interval.
Based ~on a pilot study we replaced pi by
(a) ^i + (2n) ifpi < 1/2, and. (4.1.2)
(b) pi (2n) i > 1/2 (4.1.3)
or in We recommendthis continiy
correction whenever the sample sizes are relatively small. This continuity correction has no affect on he aipot distribution.
The following table summarizes the r obtained.
<' k:,~kD I
q~jb> ; T ]{
. .. ,
L 7!:1 I1,
;if: ]f:,C?,7>< ;111i=:j
"," it, "' ila! !! ii;)
,'I
84
TablIe 4. 1
Exact Coverage Probability
(95% Nominal Coverage)
Run Number 1 2 3 4 5 6 7 8
l 1.5) .039 .029 .152 .087 .230 .319 .230 .152 = .5) .319 .206 .415 .230 .415 .415 .319 .230 = .5) .708 .485 .708 .415 .614 .515 .415 .319 P4(x4 1.5), .971 .770 .928 .614 .794 .614 .515 .415
1og LD(50) .04 .55 .21 .93 .07 .35 1.35 2.35
Sample Size Coverage
all, n = n ,5 .999 .980 .997 .880 .999 .882 .660 .557 all n1 = 10 .994 .954 .986 .912 .995 .962 .74 63
all n = 15 .979 .954 .94 .922 .987 .99.779 71.
The first four lines of Table 4.1 gi~v e prbilt
of response, pi, at logdose, xi, for t eh
considered. The fifth line gives the true LD(50). The last three lines give the eact coverage probability for the various equal sample sizes. VW were limited
to relatively 'small samples in this investigation since
 ~~< ~ 85iV'
there are, for example, more than 65,000 possible realizations (each of varying probability) associated with n. i15.
As was previously stated, the above examples are all linear in terms of logdose against sin1 (pl/2). Four different slopes were used. Run 1 had the smallest slope, runs 2 and 3 the next smallest, runs 4 and 5 the second largest, and runs 6, 7, and 8 the largest. Runs 1 through 6 provide excellent small sample approximation, while runs
7 and 8 do not. For runs 7 and 8, there is substantial probability that all pi's ar'e less than .5, and hence LD(50) must often be estimated by extrapolation. We conjecture that convergence is slow whenever extrapolationis highly probable.
4.2 Estimation of Relative Potency
by Various Linear Techniques
In this section we will compare the estimation of
relative potency by various methods of analyzing quna response assays. The data we will analyze reaexml presented by Finney E3]. The data are the resut of a of i nsul i n. Mice were injected with vary or with a test preparationand the numbers the symptoms of collapse or convulsionswr reordd For
the data of Finney [31, page 477, we obtaind 95 nominal confidence intervals for the relative oencyof the insulin as compared to th e test preparation. Excellent linear f~it
was obtained for all methods. A summary of the analyses is given in Table 4.2.
Table 4.2
Estimation of Relative Potency
Method R LCL 2C lP
P r obi t 13.4] 11. 11 16.12 .28
Logit 13.38 11.04 16.16 .35
Angle (MLE) 13.50 11.31 16.05 .17
KC 13.70 11.58 16.20 .35
IRL 13.72 11.66 16.18 .19
Legend: R = estimated relative potency
LCL = lower 95% confidence limit UCL = upper 95% confidence limit
X 2(P =chisquare statistic for parallelism, on~e1
Degree of freedom.
MLE = maximum likelihood estimation1
KC = KnudsenCurti s method
IRL = Inverse regression: linear. =' i
Chapter III)
While the five methods give virtual
results, the KnudsenCurtis and inverse regesso mehos
require the more elementary computations an rersi
to explain to nonquantitative scienti.sIn situations where parallelism reasonaebu
linearity is not, we can use inverse r.... 2,
w the o m a
Me tho'l ;;, Iii' ,,4'~ Ir,,,... p
L o i~ 1 .:3 :;;:iI, ?. gl ;::; <.A
Lege d]: I! :;: ,:e tim ;.t d ii~i! e po ;,: i~~ii...
<4 87
4.3 Summary
By use of numerical examples we have shown the applicability of inverse regression in analyzing quantal response assays. Since other methods of analysis are restricted to linear models, we have compared inverse egression to some other methods when a linear fit is excellent. As has been stated before, inverse regression can also be applied to quantal response assays when linearity is doubtful.
Finally., another reason to use inverse regression to
analyze quantal response assays is the computational simplicity and ease of explaining the results, to nonquantitative scientists.
s t ; d ;77 ;< ]g~ir 01'" "+7 ,7 7. ; ,<. /j/ ii 7 !D ; 7 77 .< 7 g < .1' 4d
/ ,.7 "
 , / S : ,;': < "
" ,' ) .>
f < ?
); !:i~ . ...
:7 ,77I
, i += ,'+i "" .'llI II
BIBLIOGRAPHY
[1] Bliss, C. I. (1939). The toxicity of poisons applied jointly. Ann. Appl. Biol. 26, 585615.
[2] Finney, 0. J. (1971). Probit Analysis. 3rd Ed .
Cambridge: University Press.
[3] Finney, 0. J. (1964). Statistical Method in Biological Assay. 2nd Ed. London: Griffin and Co.
[4] Halperin, M. (1970). On inverse estimation in linear regression. Technometrics 12, 72736.
E5] Knudsen, L. F. and Murtis, J. M. (1947). The use of the angular transformation in biological assays.
J. Amer. Statist.Assoc'" 42, 889902.
E6] Krutchkoff, R. G. (1967). Classical and inverse regression methods of calibration. Technometrics
9, 42539. T
[7] Moore, R. H. and Zeigler, R. K. (1,967). The use of nonlinear regression methods for analysing 1i<>4
sensitivity and quantal response data. Biometrics~
23 56566.
[8] Nelder, J. A. (1968). Weighted regression, quantal response data, and inverse polynomias Bio:: ,. metri,:c~s 24, 97:9_,85 i ii, ~ !ii.
9] Rao C. R. (1965).. Linear Statistical Ifencad Its Applications. New York John Wile n
Sons, Inc.
[ 10] Saw, J. G. (1970). Letter to the editor. Tcnomtr
12, 937.
11 Williams. E. J. (1969). A note on regresion methods in calibration. Technometrics 11 1992.
88
<~Additional References
Berkson, J. (1944). Application of the logistic function to bioassay. 3. Amer. Statist. Assoc.
39, 35765.
Krutchkoff, R. G. (1969). Classical and inverse regression methods of calibration, in extrapolation.
Technometrics 11, 6058.
Litchfield, J. T. and Wilcoxon, F. (1949). A simplified
method of evaluating dose response experiments.
J. Pharmacol. Exp. Therapeutics 96, 99113.
Patel, K. M. and Hoel, D. G. (1973). A generalized
Jonckheere ksample test. against ordered alternatives when observations are subject to arbitrary
right censorship. Comm. Statist. 2, 37380.
Steel, R. G. and Torrie, J. H. (1960). Principles and
Procedures in Statistics. New York: McGrawHill Book Co.
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BIGAPIA SKETC
Frank I HanDerc Iwsbono uut9 95
burgFank High Schetich June was3 bonn SAugt9,1945f
that year he enrolled in Wilkes College, receiving the degree of Bachelor of Arts with a major in mathematics in June, 1967. In September of that year hie enrolled in Bucknell University, receiving. the degree of Master of Arts with a major in mathematics in January, 1970.
The writer also taught high school for the school A
year 19681969 at Selinsgrove Area High School. He entered the University of Florida Graduate School ~in~ September, 1970. Mr. Dietrich has worked as a teaching j4'
assistant for the Department of Statistics~ siethat time, simultaneously pursuing his work towards th4ege of Doctor of Philosophy../4
90
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Jonathan J. Shuster, Chairman Associate Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
(3,...... "'j
Pejaver V. Rao
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and' quality as a dissertation for the degree of Doctor of PhilosQphy.
Stratton H. rerr
Professor of Entomology
This dissertation was submitted to the Graduate Faculty of the Department of Statistics in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy..
August, 1975
Dean, Graduate School

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mods:dateIssued 1975
marc 1975
point start 1975
mods:recordInfo
mods:recordIdentifier source sobekcm AA00062827_00001
mods:recordCreationDate 770325
mods:recordOrigin Imported from (ALEPH)025344667
mods:recordContentSource University of Florida
marcorg fug
FUG
mods:languageOfCataloging
English
eng
mods:relatedItem original
mods:physicalDescription
mods:extent viii, 90 leaves : ; 28cm.
mods:subject SUBJ690_1
mods:topic Statistics thesis Ph. D
SUBJ690_2
Dissertations, Academic
Statistics
mods:geographic UF
mods:titleInfo
mods:title Quantal response assays by inverse regression
alternative displayLabel Added title page
Inverse regression, Quantal response assays by
Regression, Quantal response assays by inverse
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sobekcm:statement UF University of Florida
sobekcm:SortDate 720988
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QUANTAL RESPONSE ASSAYS BY INVERSE REGRESSION By FRANK HAIN DIETRICH II 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 UNIVERSJTY OF FLORIDA 1975
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To my Mother and Father for their love and faithful support
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ACKNOWLEDGMENTS I wish to express my deepest thanks to Dr. J. J. Shuster for his expert and helpful guidance in this effort. I also wish to thank Dr. J. T. McClave for many helpful discussions and comments. Finally, I wish to thank Mrs. Nancy McDavid for the outstanding job of transf0rming the rough draft I gave her into this typing masterpiece. i i i
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CHAPTER IV (cont.) TABLE OF CONTENTS (continued) 4.2 Estimation of Relative Potency by Various Linear Techniques 4.3 Summary BIBLIOGRAPHY BIOGRAPHICAL SKETCH V 85 87 88 90
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LI ST OF TABLES Table 3. 1 Notation Chart 73 4. 1 Exact Coverage Probability 84 4.2 Estimation of Relative Potency 86 vi
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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 QUANTAL RESPONSE ASSAYS BY INVERSE REGRESSION By Frank Hain Dietrich II August, 1975 Chairman: Dr Jonathan J. Shuster Major Department: Statistics Numerous methods are available to analyze quantal response assys. Some of the more popular methods of analysis are discussed The general aspects of inverse regression are also discussed. A general inverse regression procedure for estimating dose response curves in quantal response assays is presented. Asymptotic distributional properties are developed. Procedures to form (la.)1OO% nominal confi dence intervals for quantities of interest are given. Methods of testing hypotheses ~f interest are also developed. The particular method of applying inverse regres sion to quantal response assays by use of the angle transformation is presented. The special case is given where, after application of the transformation, the dose response curve is linear. The inverse method has a decided advantage over the more classical methods in Vi i
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this case, both in flexibility and in ease of applica tion. The procedure will be shown to be fully efficient in the asymptotic sense. Numerical examples are presented to demonstrate the applicability of inverse regression to quantal response assays. The numerical examples deal with linear response cu~ves since cla~sical methods of analysis are only ap plicable in this case. Inverse regression may also be used when the response curves are nonlinear. V i i i
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CHAPTER I STATEMENT OF THE PROBLEM 1.0 Preamble In Chapter I the general problems of quantal response assay and inverse regression are presented. In Section l. l we will introduce the classical quantal response problem along with different objectives of a quantal response assay. Section 1.2 gives a history of frequently used methods of an~lyzing quantal response curves. In Section 1.3 we ~ill discuss the general topic rif inverse regression. Jnherent in this discussion is a comparison with classical regression. In Section l .4 we will give a summary of Chapter I along with the results obtained in the remaining chapters. l. l Introduction In the classical quantal response problem subjects (plants, insects, patients, e tc.) are subjected to a stimulu~ (fungicide, insecticide, physical therapy, etc.), and an all or nothing response is recorded. Al though it would usually be desirable to measure the response quantitatively, it is often only possible to measure a l
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response as occurring or not occurring. It is this type of response we will be interested in analyzing. The stimulus is often referred to as a dose, and the dose is administered at different levels. Generally, we independently sample n. subjects at dose, d., l l 2 i = l, 2, ... k. For each dose, di' we are interested in the true fraction of positive respon~es, pi. Thus, for each dose level the number of positive responses ob served in a sample of ni subjects is a binomial random variable with prob~bility of success equal to pi. For each dose, d., we calculate the observed fraction of posi1 tive response~, ~i, the maximum likelihood estimator of Pi A quantal response curve is then fit. The response curve is basically found by fitting the fraction of positiv~ responses observed against dose. Usually both the fraction of re~ponses and the doses are transformed before the curve is actually fit. This type of analysis is often used to assess the potency of drugs of all types when it is either impractical or impossible to determine the potency by chemical analysis. The actual objective of a quantal response assay may be the solution of one of a number of related problems. An objective of many assays is to estimate LD(lOOp), the true dose at which 100p% of the subjects have a positive response. In particular, LD(50), called the median lethal
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dose, is often of prime interest One reason for this is that it is used in an attempt to classify drugs as to their effectiveness. At one time it was attempted to classify drugs by a minimal lethal dose or a maximal 3 lethal dose. The minimal lethal dose would be the smallest dose at which a positive response is attained for at least one subject. The maximal lethal dose would be the smallest dose at which all the subjects would exhibit a positive response. Needless to say, it would be very difficult to estimate these quantities. For a fixed number of sub jects LD(50) can be estimated more accurately than a minimal lethal doie or a maximal lethal dose Thus, LD(50) is now often used to attempt to measure the effectiveness, or potency of a drug. There are however instances, such as toxicological problems, where doses producing 100% response are of more intere st than LD(50). If two or more drugs are to be compared, it is often done in terms of the relative potency, the ratio o( equally effective doses. Even if a new drug is to be com ~ pared to a standard, the tolerance of the population may change, and both drugs must be experimented with at the same period of time. Thus, an estimate of relative potency is obtained rather than measure the performance of the new drug singly and measure its effectiveness in relation to the standard as an absolute effect. Reiative potency is a
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valuable measure only if it is found that the quantal response curves are parallel, Thus, the ratio woul d be the s~me at all equally effective doses; The relative potency is therefore usually measured as the ratio of median effective doses. 4 Whenever two or more drugs are under consi~eration in a articular problem, it is desired to know if a mixture of the drugs might be more effective than applying the drugs individually. In general, the joint action of a mixture of drugs can be classified in thr~e categories. The three categories as given by Bliss [l] are independent joint action, similar joint action, and synergistic action. If drugs have independent joint action, they act indepindently and have different modes of action The drugs may or may not be crirrelated in terms of the sus ceptibility of one component as compared to another. The potency of the mixture can be predicted from the fitted curve for each drug alone and the correlation in suscepti bility to the drugs. The potency of the mixture can be computed on this basis whatever the relative proportions of the individual constituents Drugs are classified as having similar joint action ff they produce similar effects so that one component can be substituted at a constant proportion for the other. Variations in individual susceptibility to the drugs are
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5 completely correlated or parallel. The potency of a mixture is predictable from the relative proportions of the individual components. The last classification is synergistic action. The potency of the mixture cannot be assessed from a knowledge ~the individual potencies. It must be based upon a study of their combined potency when used in different propor tions. If the potency of the mixture is greater than that expected by studying the mixtures singly, the drugs are said to synergize. One drug antagonizes another if the ~ixture has a smaller potency than expected. We have now stated the basic problems of interest in a quantal response assay. The next section will deal with a history of methods for analyzing quantal response curves. 1.2 HistoryPr~vious Methods of Analyzing Quantal Response Curves Although numerous methods have been proposed for analyzing quantal response curves, the most frequently used method is probit analysis. A thorough discussion of probit analysis is given by Finney [2]. In the classical quantal response problem, we inde~ pendently sample ni subjects at dose, di' and obtain ~i' the fraction of positive responses, i = 1, 2, ... k.
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6 In order to use prob ft ana Tys is in analyzing quan tal response curves, the probit of p Z., is found by 1 l the following transformation, l/2x 2 d e x, i = l 2 k.(l.2.1) Once the probits have been determined, a linear response curve is fit against log dose by iterative weighted least squares. A procedure similar to probit analysis was sug gested by Knudsen and Curtis [5]. Rather than use the probit transformation given in equation (l 2.1), Knudsen and Curtis suggest the use of the angle transformation Zi = Arcsine (pi 112 ), i = l, 2,. ; ., k, (l.2.2) where Zi is recorded in degrees. Once the ang1e transfor mation has been performed, a linear response curve is fit against log dose by ordinary least squares if the sample sizes are approximately equal, and by weighted least squares otherwise. For all practical pur~oses the angle transformation is a linear function of the probit transformation. Mo o r e a n d Z e i g l e r [ 7 J d i s c u s s t h e u s e o f n o n linear regression methods for analyzing quantal response
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7 curves. They demonstrate that any methods based on maximum likelihood estimation of appropriate parameters may be formulated as nonlinear regression problems. It should be noted that both probit analysis and the angle trans f0rmation are based on maximum ltkeli hood principles, and thus fall in this_category. Moore and Zeigler conclude that a reasonably general least squares computer program could replace several specialized quantal response analysis programs. It has also been pointed out by Nelder [ s J that there is an important class of estimation problems which leads to a form of solution which is closely arialogous to linear rather than nonlinear regression; Basically, the conditi6n which must be satisfied to be in this class ~estimation problems is that the first derivative of the likelihood can be put in a form where p, the true fraction of positive responses, is a linear function of the unknown parameters. Again, probit analysis and use of the angle transformation fall into this class of problems. Thus, a wellconstructed linear regression program co~ld be adapted to cope with this type of problem. Although quan tal response assays usually involve discrete distributions, Nelder also shows that the same iterattve linear regression procedure can be used on a class of nonlinear models which inv6lve continuous rather than di~crete distribu tions.
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8 1.3 HistoryInverse Regression Krutchkoff [6] discusses the general problem of inverse regression,and in particular as it applies to the problem of calibrating an instrument. He uses the example of calibrating a pressure gauge. To calibrate the gauge, one subjects it to two or more controlled pressures, and notes the gauge markings. From these data, the calibration parameters are estimated, and the gauge is calibrated, Unknown pressures are then estim~ted by reading the cali brated markings. If x represents the controlled variable, and y represents the measured variable, then the relationship between x and y can be expressed by the usual linear model y =Cl+ SX + (1.3,1) The classical approach to calibration using model (1.3. 1) with k values of x, and independent identically distributed errors with zero mean, uses the usual least squares estimates of a and S. These estimates are found by A s = and k I (x.x)(y.y) l l l l = k I: (x.x) 2 1 l l = (1.3.2)
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where ,.., a = y "\ BX X = k EX. i = l l k and y = k E y. i = l l k The least squares Jine is then represented by A I', y =a+ Bx, and the calibration equation is A v a X = ,L_.::::___ 6 9 (l.3.3) (1.3.4) (1.3..5) (1 .. 3 6) Thus, from a gauge reading of Y, the classical estimate, Xe, for the pressure is Y~ A f3 Using the inverse regression approach, model (1.3.1) is rewritten as x = y + oy + E' where y = a /B, o = 1/B, and E'= E /f3. (l.3.7) (1.3.8) Again, the usual least squares estimate of y and o are
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found by k 8 = _r (xix)(yiy) 1 = l k 2 L (y .... y) l 1 = (1 .3.9) ahd y = X (l .3.10) The least squares line is now one and the same as the calibration equation and is expressed by A "\ X = y + oy. (1.3.11) Thus, using inverse regression, for a reading of 10 Y of the gauge, the inverse r~gression estimate, i 1 of the pressure is X 1 = y + 6Y. (1.3.12) The estimates given by equations (1.3.6) and (1 .3.11) are not generally the same. It is therefore of interest to judge which estimate is better by the use of certain criteria. Krutchkoff uses the criterion of mean square error to judge the relative effectiveness of the estimates.
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1 1 Krutchkoff concludes on the basis of a Monte Carlo study (in which values of l l<.001 were replaced by +:001 as appropriate) that the mean square error of the inverse estimate is uniformly less than that of the classi cal estimate. The Monte Carlo study involved different values of a and S, different variances, different designs, and normal as well as non r normal error distributions. Thus, on the basis of mean square error, it appears that the inverse estimate is more de5irable than the classical estimate. Williams [11] points out that under the assumption of normally distributed errors the classical estimate has undefined expectation and infinite variance, and hence infinite mean square error, Under the same assumption the inverse estimate has finite mean square error. Thus, Williams concludes that the inverse estimate is better than the classical estimate from the mean square error point of view. Williams goes on to point out, however, that this conclusion is not very satisfying. He reaches this conclu sion because ~11 that was shown is that the mean square error of the inverse estimate i~ less than infinity. He questions using mean square error at all as a criterion for comparing the two estimators.
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1 2 It is also of interest to note that Williams shows that there is no unbiased estimator with finite variance. In somewhat the same spirit as William s, Halperin [4] notes that a random drawing from any distribution with finite variance would provide a better estimate than the c 1 a s s i c a 1 e s t i m a t e in th e m e a n s q u a r e e r r o r s e n s e Rather than dwelling on the mean square error argument, Halperin considers the criteriori of relative l cl o sen es s II of two estimators 1 and 2 to X Here "closeness" is in the Pitman sense That is, x 1 is ft closer estimate of X than 1 2 if, for all X, (l .3.13) Halperin shows that the inverse estimate is a closer esti~ mater than the classical estimate for all values in a closed interval of X. This interval depends on quantities such as S, cr, x, and the sample size. It turns out that if IPI is large, where p = S/cr, Y is well determined, or the val u es of the i n dependent var i a b 1 e a _re w i de l y d i s per s e d the estimates are indistinguishable. Saw [10] shows that for any distribution on the errors, the slo~e of the inverse regression line is always of the same sign, but greater modulus, than the slope of the classical line. Thus, at X = x, the inverse estimate is closer to X than the classical estim.ate with probability one.
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Saw goes on to point out that any line t~rough (y, x) with slope of the same sign, but greater modulus than the classical regression line,will perform better (as an estimate of X) than will the classical estimate within some nefghborhood of x. A similar statement can be made in reference to the inverse regression 1ine. Thus, there exists no best way to estimate X uniformly over an interval of X. This being the case, Saw concludes the specific use of inverse calibration is unappealing. 1.4 Sum~ary of Results l 3 In CMapter I we have presented the general problem of quantal response assay. We have also discussed the general method of inverse regressiun. We have chosen to apply i~verse regression to quantal response assays for a number of reasons. In quantal response assays we usually seek solutions of F{dose) = p (1.4.1) and relationships among such solutions, for two or more drugs. The following criticisms are levelled against the classical approach: 1) the least squares process minimizes the residual sum of squares in the transfcirmed probability ~cale (verti cal), while estimates of the solutions of (1.4.1) have errors measured in the log,dose scale (horizontal).
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14 2) Serious problems occur in estimating solutions to (1.4. 1) when we model 1 F (Probability of Response) for mtl. m = r Br (log~dose)r r=O (1.4.2) The main problem is that the solutions to (1.4.1) may not exist or may not be unique when they do exist. Thus, the classical approach is pretty well limited to situations where a linear rela tionship exists (that is to situations when m=l in (1.4.2)). In Chapter II we will develop the general theory necessary to apply inverse regression to find solutions to (1.4.2) when mfl. The solutions will minimize residual sum of squares in the logdose scale. In Chapter III we will use the angle transforma tion with inverse regression to develop a particular method of analysis. Chapter IV will give some numerical applications of the methods developed in the preceding chapters. By actual application of our results it is seen that inverse regression offers the most elementary computations as compared to oth~r methods.
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CHAPTER II INVERSE REGRESSION OF QUANTAL RESPONSE ASSAYS: ASYMPTOTIC THEORY 2.0 Preamble In Section 2.1 we wil 1 introduce the basic reasons for studying the asymptotic theory. Section 2.2 will deal with developing a parametric model along with estima tion of population parameters. In Section 2.3 we Will develop the asymptotic distribution of the estimators. We will also develop methods of forming confidence inter vals and testing hypotheses of interest. Section 2.4 will be a summary of the results. 2. 1 Introduction As previously stated, the classical quantal response assay consists of independently sampling n. subjects at l dose, d., and ~~taining the fraction of positive responses, l A p., l = 1, 2, l k. If only one drug is of interest in the assay, it is often of interest to estimate LD(lOOp), the true dose of which lOOp percent of the subjects exhibit a positive response. In particular, LD(50) is a quantity often estimated. 1 5
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If more than one drug is involved in the assay, other aspects of the arialysis may be of interest. It is often desirable to compare Lo(50) values in terms of relative potency, the ratio of the true LD(50) values. If the assay involves drug mixtures, it is of interest to know if one drug synergizes or antagonizes the other. 1 6 In order to use inverse regression to analyze a quantal response assay, a model for the problem is neces sary. We will develop a parametric model for the classical quantal resporise assay. Once the model has been fdrmulate~, estimators of population parameters will be developed. Since confid~nce intervals for, or test hypotheses about, population quantities are of interest, the asymptotic distribution of the estimators will be studied. The re sults of the asymptotic theory will be stated in terms of ltnear combinations of the estimators. From thi~, confi dence intervals and tests of hyp0theses of interest will follow. 2.2 Parametric Model and Estimator~ We will now develop a parametric model to express the relationship between the observed fraction of positive responses at different doses and the corresponding true fraction~ In order to do this certain matrices and their relationships will be defined.
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1 7 Let M beak x r matrix with r < k. M will be of rank rand will usually consist of two different types of elements. M will contain elements which are functions of the true fraction of positive response. M may also contain dummy variables. A k x r matrix Y will be of a form n similar to M. If M contains a dummy variable in position m .. Y will contain the same element in position y ... l J n l J The remaining elements of Yn will be the maximum likeli~ hood estimates of the ccirresponding elem~nts of M. Thus, rather than containing functions of the true fraction of positive responses, as M does, Yn will contain the cor responding furictions of observed fraction of positive responses. For a k x 1 (transfotmed} dose vector,!, we hypothesize the following relationship: X = MS, (2.2.1) where Sis an r x l vector of parameters. Let En be a k x r matrix such that {~~i)}, the rows of En' are inde pendent random vectors. If we let n ~ea linear function of the n i l = 1 2, k, we will assume that 1/2 (') L n e 1 > N (0 v.) as n + 00 n r ' 1 (2.2.2) where Nr represents an rvariate normal random variable, and Vi is a continuous matrix function of M.
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18 With all matrices defined as above, we propose the following model for the relationship between the ob served fraction of positive responses and the true fraction of positive responses: Y = M + En. n (2.2.3) Multiplying equation (2.2.3') on the right by~ yields (2.2.4) Using the relationship given in (2.2.l) we see that (2.2.4) can ~e rewritten as or equivalently as E 8 n X = Y 8 E 8 n n ( 2 2 5 ) (2.2.6) Thus, using the unweighted least squares estimator of ..@_, we obtain 8 = (Y' y )'"l Y' ~n n n n x. (2.2.7) 2.3 Asymptotic Theory Now that a parametric model has been developed with the estimat0rs of these parameters, we will obtain the limiting distribution of the quantity
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l 9 (2.3.1) where l is an r x l specified vector. Before we actually find the asymptotic distribution of T we will first introduce some lemmas needed in later n proofs. The first three lemmas may be found in Rao [ 9 ]. Lemma 2.3. l Let {Xn, Yn}, n = 1, 2, ... be a sequence of pairs of random variables. Then p > L > L > Y, (2.3.2) that is, the li~iting distribution of Xn exists and is the same as that of Y. Lemma 2.3.2 Let {Xn' Yn}' n = l, 2, .. be a sequence of pairs of random variables. Then: (a) X _L_> X' y _P_> 0 ::::;> xnYn _P_> 0. n n (2.3.3) ( b) X _L_> X' Yn _P_> C ::::;> xn +Y _L_> X+c n n (2.3.4) X y L cX :::;:,. > n n (2.3.5) ='> X / Y l> X / c, if ct O. ( 2. 3. 6) n n
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20 Lemma 2.3.3 Let g be a continuous function. Then; ( a ) xn L > X L g ( X) (2.3.7) g(X ) > n ( b) xn p X g(Xn) p g ( X) (2.3.8) > > p > L 0, Y n > Y p > 0.(2.3.9) Lemma 2.3.4 Let g be a continuous matrix valued function of Y n a matrix. Then p > g ( M) (2.3.10) Proof Since g is a continuous matrix valued function of Y we can let >0 be arbitrary and let o >0 be such that n t IIY Mil< 0 =? lg(Y) g(M) I<. since p y > M. n (2.3.11) (2.3.12)
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21 p Since Eis arbitrary, g(Yn) > g(M). This completes the proof. Lemma 2 3.5 Let E = (2.3.13) beak x r matrix of random variables such that the asymptotic distribution of. n 112 e., the i~th row vector of 1 l/ 2 E h .. t V l 2 k n as variance covariance ma r,x ., 1 = 1 Assume that ACov(n 112 e .. n 112 e ) = 0, 'u'H=i, where ACov 1 J 1 J _. is the covariance of the asymptotic distribution. Let a I = (al a2, b = (bl b2' c = (cl c2' and d = ( d l d2' be vectors of constants Then, AVar[n 112 ~ E b 1/2 n C E ~_] = a k) b r) ck) dr) 2 (5 (2.3.14) (2.3.15) (2.3.16) (2.3.17) (2.3.18)
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where k [ 2 b 1 V b + c 2 .d V.d r. a ,,, ,1 = l 2a c.b 1 V d] l 11(2 3.19) 22 and AVar refers to the vartance of the asymptotic distri. bution. Proof k r = n 112 E E a.b.e .. i=l j=l l J lJ (2.3.20) and 1/2 1/2 k r n ~'Ed= n E E c.d.e .. i=l j=l l J lJ (?..3.21) Thus, 2 [k a = AVar E i = l r E (a.b j = l l J 1/2 c.d.)n e .. l J l J k r = E E (a b ~ i=l j=l 1 J 2 1/2 c.d } AVar(n e .. ) l J l J 1/2 1/2 + E ( a b c d ) ( a b. 1 c ,d ) A C o v ( n e j n e ) H l J l J l J l J l l J
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23 where H = { ( i j i j ) : ( i j ) t ( i 1 j ) ; i i" = 1 2 ... k ; j j = 1 2 ... r} .( 2 3 2 2 ) Since ACov(n 112 e .. n 112 e., ., } = 0, Vifi', equation l J l J (2.3.22) can be written as a 2 = (a.b. ~ c.d.) 2 AVar(n 112 e .. ) i=l j=l l J l J lJ 1 /2 1 /2 + L (a.b.c.d.)(a.b., .. c.d.,)ACov(n e .. ,n e .. ,) W l J l J l J l J lJ lJ k = L [a~b'V.b+c 2 1 d'V.d2a.c.b'V.d], 111l 111 = l where H'={(i,j,j'): (i,j)t(i,j'), i=l,2, ... ,k; j,j'=l,2, ... ,r}. (2 3.22) This ccimpletes the proof. We are now ready to derive the asymptotic distribu tion of Tn, as given in (2.3. 1). Theorem 2.3. 1 Under the conditions specified in section 2.2, T _L_> N(O 2) n a a s n + 00 (2.3.23)
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where 2 (J = with k 2 2 E {a.bLv.b+c.d~V.d~2a.c~~ ~ V ~ d} 1 1111l 111 = b = (M'M)'"" 1 l c' = (c 1 ... ck)= l' (M' M).,. 1 M 1 and d = (M' M)l M' x Proof From equation (2.2 1) we see that 1 f = (M'M) w ~Using equations (2.2.7) and (2.3.29) we obtain l ( t ~) = l { ( y y ) .,. 1 y, X .,. 01'' M ) .,. 1 M l X } ., n n n24 (2.3.24) (2 3 25) (2.3.26) (2.3.27) (2.3.28) (2.3.29) = l' {(Y'Y )'" 1 .. (M'Mr 1 }Y 'x + l (M'M)'"" 1 (Y'M')x. (2.3.30) n n nn
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25 From equation (2.2.3) we observe that Y' = M' + E' n n (2.3.31) and Y'Y = M'M + M'E h +E L M+ E ~ E n n n n n n (2.3.32) Substituting (2.3.31) and (2.3.32) into (2.3.30) yields ,e_'(B 8)=,t'{(M'M+M E +E'M+E' E f 1 ~(M'M)"" 1 }(M'+E )x+l'(M'M)lE' x.(2.3.33) '11 n n n n n n By making use of the identity (2.3.34) and letting U = M'M V = M'E + E'M + E'E n n n n (2.3.35) (2.3.33) can be written as + l' (M'Mf 1E'x. n(2.3.36)
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(I+ V) (I .. V) = I v 2 implies that ( I V ) = ( I + V ) l ( I + V ) "" l V 2 Recalling that n 112 e~i) L > N(0, V.) as 1 26 (2.3.37) (2.3.38) n+oo, and letting V be as defined in (2.3.35), we observe that p > 0, 'uo>0. Thus, from Lemma 2.3.3 l 2 o V 2 P 0 n > 'tfo > o. (2.3.39) (2.3.40) Combining the results of (2.3.30), (2.3.39), and (2.3.40) and applying Lemma 2.3.2 we obtain (I V) = (I + V)l + O(nl) (2.3.41) or (I + V)l = (I V) + 0(n1 ). (2.3.42) Using the re~ationship given in (2.3.42) and apply ing Lemma 2.3.2, equation (2.3.36) can be written as
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I + ,e ( M I M ) l E I x l' { ,M I M ) l E I x n n~(WM)1 (M'E +E 1 M+E 1 E )(M'M)1 M 1 x n n n n l'(M'M)1 (M'E +E 1 M+E 1 E )(M'M)lE'x n n n n n27 (2.3.43) Since any matrices involving E~En are of order n1 application of Lemma 2.3.2 reduces equation {2.3.43) to l' ( B s) = l 1 (M'M)1 WE (M'M)1 M 1 x n n = x'[IM(M'M)lM']E (M'M)l,e_ n
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where~, Q, ~, and d are given in (2.3.25), (2.3.26), (2.3.27), and (2.3.28) respectively. Thus, 28 n 1I2 ,e. (8 B) = n (2.3.45) From Lemma 2.3.l, we observe that both sides of equation (2.3 45) have the same limiting distribuiion. Thus, from the asymptotic properties of En a nd the appli cation of Lemma 2.3.3 and Lemma 2.3.5, we observe that L 2 Tn > N(O,o ), (2.3.46) where o 2 is as given in (2.3.24). This completes the proof. In Theorem 2.3. l the asymptotic variance component, o 2 was given in terms of M. El~ments of M involve the true fraction of positive responses. Since the true frac tion of positive responses is unknown in a practical situati9n, we will wish to estimate them and obtain a con~istent estimator of o 2
PAGE 37
Corollary 2.3.l By substituting Yn for Min equation (2.3.24), I" 2 including V terms, we obtain an, a consistent estimator 2 of' a a n d h e n c e 29 A1 L an Tn > N(0, 1). (2.3.47) Proof Chebyshev's Inequality states that for any random variable X with mean,, and variance, a 2 P(IXj~>.a) < l 2' >.> 0. (2.3'.48) A For a binomial random variable, p, the maximum likelihood estimator, of p, has meanJ p, and variance, Q_ E) < 0. n+oo (2.3.50) A Thus, p converges top in probability.
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Recalling that Yn is identical to M except that where M contains functions of p, Yn contains the same A function~ of p. Thus, from lemma 2.3.3 we observe that each element of Yn converges in probability to the cor responding element of M. That is, 30 p Yn > M. (2.3.51) By application of Lemma 2.3 4, we observe that "' 2 P 0 2, a > n (2.3.52) where cr~ is found by substituting Yn for Min equation (2.3.24). Since (2.3.53) Lemma 2.3.2 justifies that A 1 L an Tn ~N(O, 1). (2.3.54) This completes the proof. Now that the asymptotic distribution of th e estimators has been developed, we will give a nominal (la)l00% confidence interval for f' ~We will give a confidence interval of this form because many of the
PAGE 39
estimation problems of interest can be phrased in terms of linear combinations of the~ parameters. Corollary 2.3.2 o ,.,B +_ z ,., n1/2 .:h n a/2 n (2.3.55) forms a nominal (la.)100% confidence interval for .f. 1 B, where za/ 2 is such that 31 (2.3.56) w h e n Z i s t he s ta n d a rd n o r m a l r a n d om v a r i a b l e Proof Corollary 2.3. 1 implies that T h u s a s n + 00 Therefore in the asymptotic sense A l' B n (2.3.57) (2.3.59)
PAGE 40
forms a nominal (la.)100% confidence interval for l' This completes the proof. Although estimation is often of prime importance, it may also be of interest to test hypotheses of the gen era 1 form 32 H: Ai=O, 0 (2.3.60) where A is a q x r matrix with rank q (1 q r). We will now develop a test statistic appropriate for this general hypothesis. In order to achieve this end, we will consider the asymptotic distributiort of & in n terms of a multivariate normal framework. We will first give a definition and two lemmas from Rao [ 9 ]. Definition 2.3. 1 A pdimensional random variable U, that is, a random vari able u taking values in EP (.Euclidean space of pdimension) is said to have a pvariate normal distribu tion~ NP, if ancl only if every linear function of Uhas a univariate normal distribution. Lemma 2.3.6 If Uhas a pvariate normal distribution, then the joint distribution of q linear functiorts of U is Nq. Let U have mean vector,~, and dispersion matrix, L. If
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Y = CU, where C is (q x p), represents the q linear func, tions, then Y has mean vector, C~, and dispersion matrix, ...... CLC I Lemma 2.3.7 33 Let Ube pvariate n0rmal with mean vector,~' and dispersion matrix, L. Then the necessary and sufficient condition that Q = ( u .E,) I R ( u (2 3.61) has a chisquared distribution with k degrees of freedom is L ( R L R' R) L = 0 (2 3.62) in which case k = trace (RL). (2.3.63) Theorem 2.3.2 nl/2 (~ ~) _L_> N (O, a) n r (2. 3. 64) where
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34 a .. = l/2(coefficient of 1.1.) l J l J (2.3.65) in (2.3.24), and a .. = (coefficient of 1~) (2.3.66) l l l A A2 in (2.3.24). an is similarly obtained by using an as defined in Corollary 2.3.1. 6n is a consistent estimator of a. Proof In Theorem 2.3.l we proved that every linear combination of normal. Thus, n 1 / 2 (s ~) is asymptotically univariate n since n 112 (s ~) is an rdimensional n random variable, by Definition 2.3.1 in the asymptotic sense n 112 (Bn : ~) has an rvariate normal distribution. T h e m e a n v e c t or a n d d i s p e r s i o n ma t r i x f o 1 1 ow d i re c t 1 y from Theorem 2.3.1. It should be noted that the elements of a are defined as they are because in (2.3.24), coefficient t.1. = 2ACov( 0 (i) g(f)) 1 J 1=..n n (2.3.67) where B(i) is the ith element of the vector S and n n coefficient 1~ = AVar(S(i)). l n ( 2. 3 6 8)
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35 It again follows from Lemma 2.3.3 that each element of Qn converges in probability to the corresponding element of n. Thus, p !G > Q, n and is thus a consistent estimator. This completes the proof. Corollary 2.3.3 To test the general hy pothesis H : A.@_= 0, 0 the test statistic is Proof L > From Lemma 2.3.6 and Theorem 2.3.2, L > if A.@_= 0. Thus, from Lemma 2.3.7, if A.@_= 0 (2.3.69) (2.3.70) (2.3.71)
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L > since in the notation used in Lemma 2.3.7 I: = An A' and Thus, and 36 (2.3.72) (2. 3. 73) (2.3.74) trace(RI:) = trace(I:1 I: ) = trace (I) = q. (2.3.75) Application of Lemma 2.3.2 implies that if A~= 0, L 2 > X. q This completes the proof. (2.3.76)
PAGE 45
37 2.4 Summary We have now developed the general asymptotic theory necessary to solve problems of interest in classical quantal response assays. In Chapter III we will apply tbese general results to the particular use of the angle transformation. In particular we will discuss estimation of LD(lOOp) and relative potency. We will also discuss testing the hypothesis of parallelism.
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3.0 Preamble CHAPTER III APPLICATION Tb TH~ ANGLE TRANSFORMATION In Section 3. l we will discuss the rationale behind choosing the angle transformation to apply the method of inverse regression. We will also justify the propoied model. In Section 3.2 we will discuss the esti~ mation of LD(50). We will include results for the case when the relationship between logdose and sinl (p. 1 1 2 ) 1 is linear as well as the case of nonlinearity. Sections 3.3. and 3.4 will deal with estimation of relative potency and a test for parallelism, respectively. Section 3.5 will be a summary of the results. 3. l Introduction In this chapter we will discuss using inverse re gression to fit the general model (given here as deter ministic) log d. c: 1 E S. sin r 1 j = l J j1 r l / 2 1 l pi J (3.1.1) where r < k. We will employ weighted least squares to 38
PAGE 47
39 fit the model. Appealing to the notation developed in Section 2.2, the probabilistic parametric model we will thus be using is Yn = M + E n (3.1.2) where the elements of Yn and Mare defined by l/ 2 ~. l[,l/2 j 1 Y n i j = ["nil l] (3.1.3) s, n p. l 1 < i < k, 1 < j < r, r < k, and = (nij 1 1 2 f:. 1 [ 1;2 )~jl Mij n L,n pi (3.1.4) 1 < i < k, 1 < j < r, r < k, and k n = n I k 1 l l = (3.1.5) The form of En will follow shortly. /
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40 Justification of the model proposed in (3.1.1) will now be given. A brief discussion of classical methods will first be given Classical methods of analyzing quantal response assays are applicable when there is a linear relationship between the logdose and the transformed fraction of positive responses. Probit analysis, for example, produces such a linear relationship when tolerances (measured in the log dose scale) have a normal distribution. The tolerance of a subject is the dose level at which that subject would exhibit a positive response. Needless to say, not all quan tal response assays have a normal distribution of tolerances. For assays such as these, probit analysis is not appropriate Although for a nonnormal distribution of tolerances another transformation might produce a linear relationship, it would be desirable to find one method of analysis which would be appropriate for a wide class of tolerance distri butions. In many quantal response problems, the fraction of positive response is montomical1y increasing with respect to dose in the area of experimentation of interest. This, of course, implies that dose is monotonically increasing witA respect to the fraction of positive ~esponses. dose and sinl (p 112 ) are monotone functions. Both 1 ogThus, pl < p 2 .... 1( 1/2) and d 1 < d 2 if and only if log d 1 < log d 2 and sin p 1 < sin ~ 1 (p~ 12 ) Thus, dose is a monotonically increasing
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41 function of the fraction of p6sitive responses if and only if log.dose is a monotonically increasing function of .. l( l/ 2 ) I h 1d b sin p n t 1s case, tt wou e appropr i ate to model the relationship between log"dose and sin .. 1 (p 112 ) by a polynom i al, which is the model given in (3.l.l). In conclusion, the model given in (3.l.l) is appro .. priate regardless of the distribution of tolerances, as long a~ the fraction of positive responses is an increa~ing function of the dose. Thus, the inverse regression approach is applicable to a much wider class of qwantal response assays t~an classical methods. Theorem 3 1.l (Mean Value Theorem) If f is continuous on [a, b] where a < b and differ entiable on (a, b}, then there exists a point c s(a, b) such that f(b} f (aj = (b a) f' (~). ( 3. l 6 ) We wi 11 now use Theorem 3. 1.1 to prove other useful results. Theorem 3. 1.2 Let a random sample of size m be taken from a binomial population with parameter, p. Then m ,12r:. _, ["112 1 J .... [ 112n _b_> t '(a 1) n p ... s 1 n p JJ 4 (3.1 .7) where sin .. 1 (p 112 ) and sin .. 1 (p 112 ) are measured in radians.
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Proof Let f(x) ,..,1/ 112) = sin \.x (3.1.8) Thus, l[ ) '"'l/2 f 1 ( X) = 2 X ( l ,,X ] (3.1.9) From Theorem 3. l, l, there exists a c such that Ipcl s_ Ippl and 1/2~. 1 [ "' 1121 m s 1 n p .1( 1/2]11 s, n p LJ __ 1/2( (\ )lL (l )]"'l/2 m ~p 2 C ... c Since lepl 2. lp~pl and then IPPl = O(m 112 ), lepl = O(m11 2 ), Since (3,1.10) (3.l.11) (3.1.12) (3.1.13) 42
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43 1 / 2 A m (p"p) L t 1 (0 1) 1/2 > l [p(lp)] (3.1.14) 1 / 2 A m (pp) 2 [ p ( 1 p ) J 1 / 2 L > N(O, 1/4). Thus, by Lemma 2.3.2 l lr;) l :.) l/2ir. 1 ["1/2) m L, n p 1 [ 1/2]~ sin p I I L > N(O, 1/4). (3.1. 16) This completes the pro~f. Theoreni 3. 1.3 Let a random sample of size m be taken from a binomial population with parameter, p. Then for any given {C.}, J _l_> Nfo, 1/4 I (jl)(kl)C.ckfsin1 r p 1I2 'j l ( 3 .1. l 7) r uj+k4 l j=l k=l J L l J
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Proof Let f ( X) f 1 ( X) r = m 112 IC. j = l J X j l r = m 112 I (jl)C. j=l J 44 (3.l.18) j2 X (3.1.19) From Theorem 3.1.l~ th~re exists au such that and ~ 1 r"l/2] 1 ( 1/2]~ 1/2 ( l )C j2 = sin p sin p m L., J. u \. j = l J (3.l.21) Since !~Pl = O(m112 ), by the Mean Value Theorem,
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~ ilk ,.., [ "J 12 )11 sin p I, .. l ( l / 2}~ k = 0 ( l / 2 ) c,n p m (3, 1.22) where l < k j "" l. Since (3.1.24) Thus, (3.1.25) From the result of Theorem 3.1.2 we can thus conclude that 45
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46 (3.1.26) This completes the pr6of. In the model given in (3. 1.2), En is a random error matrix to explain the asymptotic variability of Yn as compared to M. ~vith Yn and Mas defined ~Y (3.1.3) and (3.1.4), respectively, and using the results of Theorem 3. 1.3, we are now ready to justify the form of The rows of Yn are independent and we have shown in Theorem 3. 1.3 that n /2 C { i n 1 [P I 2 ]~ j l l j=l J 1 r _L_ > N(O, 1/4 I: j =l k n. With n I: n./k, we assume that as n +oo 1 1 1 n l = + >... l (3.1.27)
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47 k i = 1, 2, .. k. Also, L. \i = k, Thus, as n+ 00 each l = 1 n 1 +co. Thus every linear comb1nat1on of n I2 (Y .. ,.. M .. ) nlJ lJ i s a s y mp t o t i c a 1 1 y n o rm a 1 a s n + QO, w h e r e Y n1J and M are 1 J given in (3.1.3) and (3.1.4), respectively. It then follows that the ith row of En is a random vector such that n 1 / 2 i ) _L_ N r ( Q_, V i ) a s n + oo, (3. 1. 28) where 1 1 ( 1 2) I :7S+t ... A Vist = 4 (s1)\tl) sin [P / JJ 1 ~s, t
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48 dose. Since LD(50) is often of prime interest in quantal response assays, the main emphasis of this section will be the discussion of estimating LD(50). Since estimation of LD(lOOp) may also be of interest ~hen p I .5), we will give results concerning this also. We will first recall the notation and define (or redefine) the matrices appropriate for this problem. The classical quantal response assay consists of independently sampling ni subjects at dose, di' i = 1, 2, ... k. The observed response frequency, p i s ca 1 c u 1 ate d f.o r ea c h dos e 1 eve 1 d The tr u e l l response probability for dose, di' is represented by pi. Let 1 k n = k En 1 l l = (3.2.1) The general deterministic model is given by (3.2.2) where r < k. We will employ weighted least squares and write the probabilistic model as Y = M + E n n (3.2.3)
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50 where En is a matrix of independent random vectors. In Section 3. l we justified that the ith row of E is such n that 1/2 n ( i ) e n L > + oo, (3.2.6) In Section 3. l, we gave the form of Vi when sin1 (p~ 12 ) 1 l / 2 and sin (pi ) are measured in radians. Si~ce l radian= 180 7T 57.2958, (3.2.7} it follovJs from (3.1.26) that when sin1 (p~ 12 ) and 1 l / 2 sin (pi ) are measured in degrees that the stth element of v. is given by l [ s+t4 Vist = 820.7(s1Xtl)[in1 H 12 )J (3.2.8) l < s t < r, $, 2 ('\ (Y' Y )'"' 1 y l ( 3 2 9 ) In = X = n n n
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In order to predict the transformed dose, X. l at which pi of the subjects respond, we would use the weighted least squares prediction equation 1/2 j~l 51 r rni) 1 r l /2'U x. = ESsin p. J 1 j=l J \ n \ (3.2.10) Rather than estimate the transformed dose, xi, i t is preferred to estimate the dose in the logdose scale, Thus, the estimation can be given by ( 3 2 l l ) To form a (la)l00% nominal confidence interval for log d., we may apply Corollary 2.3.2. The confidence l interval is given by l' fn 1/2 n (3.2.12) where l l 1 ( 1/2') S l n pi ( 3. 2 l 3) ~ llr l/2l~rl sin p. ) J
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"2 and a is given by Corollaiy 2,3.l. n 52 If it is desired to estimate LD(5b), pi = .5. Thus with O sin(p~ 12 ) TI/2. sinl [(.5) l/~ = 45 ,f_ = l 45 (45)rl (3.2.14) (3.2.15) We will now discuss in detail the estimation of LD(50) when it is assu~ed that the relationship between log ct. and sin1 (p~ 12 ) is linear. We will use the follow, l ing notation: w. = 1 n l n (3.2.16) Since we are assuming a linear relationship, we are using the model given in (3.2.2) with r = 2. In this case, the weighted least squares estimate off can be expressed as ~ l = ~lJ n S 2 (3.2.17)
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53 where (3.2.18) and (3.2.19) with k L ( X X) ( y ""Y) w l l l l l = (3.2.20) k k r w.x. r w.x. i = l l l i = l l l x = = (3.2.21) k k r W. i = 1 l and k k r w.y. r w.y. ~ i = l l l i = l l l ( 3. 2. 22) y = = k k r w. i=l l From (3.2.6) we observe that
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1/2 n ( i ) e n L > From (3.2.8) we see that V. = 1 0 54 (3.2.23) 0 (3.2.24) 820.7 In order to predict LD(50) in the logdose scale, we use (3.2.11) and (3.2.14) to obtain A LD(SO) = s 1 + 45 A = x + s 2 (45y). ( 3 2 2 5 ) The asymptotic. variance, cr 2 of n 112 [LD(50)LD(50)] can be obtained from (2.3.24). By applying Corollary 2.3.l, a consistent estimator of cr 2 may be obtained. When a linear relationship is assumed a simplified expression "'2 for crn can be obtained. This will be shown in the following theorem. Theorem 3. 2. l A consistent estimate of 2 cr the asymptotic l / 2 "' variance of n [LD(50)LD(50)], is given by
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55 2 = 820.7[(45~ ) 2 S S 2 + k'" 1 s 2 2 ] n Y xx yy (3.2.26) Proof In (2.3.24) 0 2 was given in terms of M. To find a consistent estimator of Q 2 Mis replaced by Yn. To find ~ 2 we will define the following vectors: n ;, = x'{IY (Y 1 Y )'" 1 v 1 } n n n n x "1 y (3.2.27) = ~n n 6 = (Y 1 Y )l,e. (3.2.28) n n 2 = 1 (Y 1 Y )1 y (3.2.29) n n n a = (Y' Y )ly X n n n A (3.2.30) = ~n where~' Yn' ~n are defined in (3.2.5b), (3.2.4a), and (3 2. 17) respectively. (3.2.31)
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Since ~t, b, ('\ and d are the same as ~, E_, ~, and~ of Theorem 2.3. 1 with M replaced by Yn, by Corollary 2.3. 1 56 A 2 k A 2 A A A 2 A A ~A~ A a = E (a. b'V b + t. d'V.d 2~.c.b V.d), n l 1l 1l 111 =l (3.2.32) where Vi is defined in (3.2.24) is a consistent estimate 2 of cr A 1/2 l/2A 1/2 a. = w. xw. Bw. Y B 2 l l l l l l l 1/2 1/2~ 1/2~ A 1/2 A = w. X w x + w. y B2 w. YB2 l l l l l l Since = w~ 12 [(x.x) s2(y.y)]. l l l k 2 E w.y. i = 1 l l (Y 1 Y )l = n n 1 ks yy ky y ~12 820 7. ky k (3.2.33) (3.2.34) (3.2.35)
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w ~/ 2 r k J c. =~I E w.y~ ... 45ky + 45ky. ,. kyy 1 1 kS i=l l l l yy w~/2 k 1 ~ l 2 ~ = ~ 45(y.~y) + k E w.y. p y.y. S l ._ 1 1 1 1 j yy ,(3.2.36) d I Vi d = ]~ 8 2 0. 7. (3.2.37) A A A A A A ~:y~"2a .c bV .d 2a.c. B 2 820.7, l 111 1 .) yy (3. 2. 38) where;. and 2 ~ are given in (3.2.33) and (3:2.36) 1 1 respectively. T h u s by s u b s t i t u t i n g ( 3 2 3 3 ) ( 3 2 3 5 ) ( 3 2 3 6 ) (3.2.37), and (3.2.38) in (3.2.32), we obtain &~ = 820. 7 A 2 2 2 2 ~ 2 .. 8 2 Gk A 2 2 ~ 2 + (45) s 2 (y.y) + 1 E w.y. + B 2 y.y 1 < i=l 1 1 1 57
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58 k 90 ~ 2 2 A 2( P + k ~ 2(y.~y) E. w.y ... 90 B2 y. ~ y}y.y 1 i=l 1 l l l 2 8 2 k + 90 s 2 2 (y 1 .Y) 2 (45y) + k 2 (y.y)(45y) E w.y~ 1 1 1 1 1 = 2 s 2 2 {y.y)(45y)y.~. 1 1 j (3.2.39) Performing the summation we obtain 6 2 Gk ~ 2 2 B 2(45y) 2 s y + (45} 2 @ 2 2s y + k 2 I w.y~ .. X y i=l 1 1 k .. 90 s 2 s,s 2 s 2 2 s, 2 E (JJ.y~ 2 yy i=l 1 1
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59 (3.2.40) Since from (3.2.18) (3.2.41) A 2 = 820.7~ ( 45 .. ~)2s + 8 2( 45 .., ~ )2s 0 n "' 2 Y xx f'2 Y YY s yy r ~ 2 k = 820.71 (45y)2S + _g_ L w.y~S S 2 L xx k i = l 1 1 yy yy
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60 (3.2.42) This completes the proof, Corollary 3.2. l A A 1/2 LD(50) za 12 ann (3.2.43) forms a nominal (l~a)l00% confidence interval for LD(50), where LD(50) is given in (3.2.25), and;~ is given in (3.2.26). Proof Corollary 3.2.l follows directly from Corollary 2 3 2 We will now derive some results for the Knudsen Curtis [5] method for analyzing quantal response data. We can thus compare the inverse regression approach to their method. KnudsenCurtis use classical weighted least squares t o f i t t h e m ode 1 x. + E:. l l (3.2.44) where
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. ~1( (\ 1/2) Yi = sin Pi (3 2.45) (3.2 46) and from an argument similar to that used in Section 3.1 we can assume 1/2 n. E:. l l L > N(0, 820 7), as n. + 00 l ands. is independent of s for i t j. l J (3.2 47) 61 Since weighted least squa~es will be employed, the model to be fit could be expressed as Letting (3.2.49) we see from (3 : 2.'47) that l /2 _L_> n E: l N(0, 820.7). (3.2.50)
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Using the notation of this section, the weighted least squares estimates are t"", "' 6 = S /S 2 xy xx (3.2.51) and bl = y (3.2.52) Thus, to estimate LD(50) in the log~dose scale, KnudsenCurtis would use LD*(50) = 45 .. bl 8 2 62 (3.2.53) Theorem 3.2.2 LD(50)] where L N(0,1),(3.2.54) (3.2,55)
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Proof Froni (3.2.48), k I ( X. X ) ( y y ) w A i=l l l l 62 = = 5 xx k I (x.x)y.w. i=l l l l "' s xx Sub~tituting yi from (3.2.44) yields k (x.i)(b 1 +b 2 x.+E.)w. i=l l l l k I (x.X)E.W i=l l l l = b2 + 5 xx (3.2.56) (3.2.57) Thus, using the properties of E given in (3.2.50) l implies that (3.2 .58) 63
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In a like manner, it can be shown that L > 2 N(O, u ), (3.2.59) and any linear function of n 112 (6 1 b 1 ) and n 112 (S 2 b 2 ) is asymptotically normal. Now, From (3.2.44) it can be seen that LD(50) = Thus, 45B [LD*(50) LD(50)] = l = l = 1G 6 2 (45b 1 ) + (b 1 6 1 ) b2 + 62 b2 (45bl) + (b,b,) l b2b2 L b2 (3.2.60) (3.2.61) 64
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(3.2.62) Thus, [LD*(50) LD(50)] = (3.2.63) Thus, from (3.2.58) and (3.2.59) L > 2 N(O. o ). (3.2.64) It still remains to find o 2 In essence we need the asymptotic variance of n 112 LD*(50). We will first find the asymptotic variance of some other variables. 65
PAGE 74
k ,.. 1/2 (x. n X)w.n y. l 1 1 l 1 = = AVar l k 2 ( 1/2 1 = ~ 2 E (x .x) w.AVar n. Y j l 1 1 1 1 sxx 1= l k 2 = :2 E (x.~x) w; 820.7 S l 1 = xx ,= 820.7 5 xx k 1/2 AVar(n 112 y) E w.n y. l 1 1 1 = = AVar k l k [ 1/2 1. = 2 E w.AVar n. Y;j k i=l 1 l 1 l k = 2 I: w. 820;7 k l = 1 = 820.7 k 66 (3.2.65) (3.2.66)
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then If we let be such that (y L > 2 N(O,y ), (45) + {y) b2.B2 b2 = b (45)+(y) l+ E l~ ~c 00 [b ... 6?]J 2 j =l 2 67 (3,2.67) (3.2.68) and 8 are asymptotically independent, and 2 terms where j > 2, are of order n"" 1 we obtain that AVar[n 112 LD*(50)] l ( )2 ( l/2r.) l AV ( 1/2~) = 4 45AVar n b 2 + 2 ar n y b2 b2
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= b~ 4 (45~) 2 820 7 + ~" 2 820 7 k~l 2 xx 2 (3.2.69) Thus, 0 2 = AVar{n 112 [LD*(50} LD(50)]} (3.2.70) By Lemma 2.3.3, (a*) 2 is a consistent estimator of 0 2 As in Corollary 2.3. l, This completes the proof. Corollary 3.2.2 L > N(O, l). (3.2.71) Ass u m i n g a l i near model for s i n 1 ( p 1 / 2 ) a g a i n st logdos e, /0* _P_> l, n .. (3.2.72) and thus, the asymptotic relative efficiency of the inverse method to the KnudsenCurtis method is unity. 68
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Proof From (3.2.26} and (3.2 ~ 55) we obtain that (45. "' ) 2 s s + y xx yy ( 0)2 "' 3 ~,.. 4 45 ..., ., sxx.:)xy + = r4 where, ~ s r = X is the sample coefficient of correlatfon Thus, lo* n 2 = r Since a linear m6del is assumed, 2 _P_ > l r and thus 69 (3.2.73) (3.2.74) (3.2.75) (3.2.76)
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d /a* _P _> l. n Ihis completes the proof. (3.2.77) It should be noted that the Knudsen~Curtis method is itself asymptotically efficient Thus, the method of inverse regression is asymptotically efficient. Confid~nce int~rvals wit~ nominal (la)lOO % coverage are obtained by ei ther A ,., l / 2 LD(5O) I', z a/2 on n (3.2.78) or LD*(5O) za/2 q ~l/2 n (3.2.79) Since the choice of method cannot be made on the basis of asymptotic efficiency, we shall examine the two methods on the basis of robustness. Consider the set of p6ints in R 2 : 70 The inverse regression method consistently esti mates a weight~d least squares line which minimizes the horizontal deviations for the deterministic S .. Assuming 1 n 1 /kn > 0 1 as n + 00 then the point s 1 carries weight
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0i. Similarly, the KnudsenCurtis approach consistently estimates a weighted least squares line which minimizes the vertical deviations for the deterministic S .. 1 71 Since the error statements about LD(50), relative potency, etc., are made in the horizontal scale, the inverse met hod, when linearity is false, should tend to have a smaller asymptotic bias than the KnudsenCurtis method. Corollary 3.2.3 AV a r '[ to ( 5 0 ) ] < AV a r [ L D ( 5 0 )] (3.2.81) Proof From (3.2.75), 4 = r (3.2.82) Thus, (3.2.83) where pis the population coefficient of correlation for a bivariate random variable with mass function P[X = S.] = 0., i 1 1 = l, 2, ... k. (3.2.84)
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p = if and only if there is truly a linear relation ship. Otherwise 1 < p < l. Thus, from (3.2.70) we see that A AVar[LD(50)] < AVar[LD*(50)], ( 3. 2. 85) with equality holding when the linear model is correct. This completes the proof. Thus we can conclude that when estimating LD(50) in the logdose scale inverse regression seems to yield a more reasonable estimate than the KnudsenCurtis method as far as robustness is concerned. As a further A bonus, LD(50) will tend to be a better estimate than LD*(50) in terms of variances of the asymptotic distri butions. 3.3. Estimation of Relative Potency If two drugs are involved in a quantal response assay, it is often of interest to estimate the relative potency of the two drugs. The relative potency is the ratio of equ~lly effective doses. It should be recalled that relative potency is a valuable measure only if the quantal response curves are parallel. Thus, throughout 72 this section we will assume the response curves are parallel. In order to estimate the relative potency, we will use the results of Chapter II and Section 3.2. To achieve
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73 this end we will first give the notation and model for thi.s section, Table 3 1 Notation Chart Drug 1 Dose levels d 1 ? Sample sizes n l Response probability p 1 ; I\ Observed response pl frequency Weight, n./n w, l Recall that k n = tn./K. l l l = Drug 2 dk d k + 1 dk 1 1 nk n k + l nk l l pk pk + l Pk l l r.,. t\ pk pk + l pk l l wkl w wk 1<,+1' (3.3.l) Since the curves are assumed to be parall~l, the deterministic model can be expressed as a
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74 ... 1 [ 1 / 2 1 r r j .l E B.is1n p. I i?k 1 j=lJL ,, (3,3,3) In this case, r+l < k. It can thus be seen that relative potency (3.3.4) The probabilistic model will again be of the form yn = M + En. (3.3.5) The response matrix Yn has ( 3 3 6 ) and y. l ,r+ = w~ 12 i < k 1 l (3.3.7) = 0 (3.3.8) Y "th l ("'l/2) The matrix Mis of the same form as n w, sin Pi replaced by sin~ 1 (p~ 12 ). Also, (3.3.9)
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where the transformed does vector,~' has ith component and l/2 x = w. log d. l < i < k, ( 3 3 l O) l l l (3 = r s, (3.3.11) In the same manner as was employed in Section 3.1, we can assume that En is a matrix of independent random vectors with l/2 (i) L n ~n > N r + l ( Q V i ) (3.3.12) It again follows that Vi has the form given in (3.2.8). That is, Vi has entries = 0 otherwise, (3.3.14) and 75
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76 s = (Y'Y ) 1 v x n n n n (3. 3. 15) is the weighted least squares estimate of~. We can now employ the results of Chapter Ii to estimate the relative potency. We will perform this estimation in the logdose scale. Thus, we will estimate Sr+l by means of a (la)l00% confidence interval. If we let .R' = (0, 0, l), then from Corollary 2.3.2 01 ; z A n1/2 N ~n a/2 an (3.3.16) forms a nomihal (la)l00% confidence interval for Sr+l, and thus the relative potency. ;~ is obtained by applying Corollary 2.3.l. Wi have thus ~stimated the relative potency (in the logdose scale) of two drugs. It was assumed that the quantal response curves are parallel. In the next section we will give a test for parallelism. 3.4 Test for Parallelism If two drugs are involved in a quantal response assay, it may be of interest to test for parallelism of the response curves. If it is found that the curves are not par~llel, it woul d be inappropriate to attempt to use the relative potency of the t wo drugs in any way.
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We will use the same notation as that given in Table 3. 1. The deterministic model is now given by (3.4.1) = 2 { s.lsin1 [p~ 12 ]~jrl i>k 1 ,(3.4.2) J=r+l JL lJ where r .:.. min( k 1 kk 1 ). The probabilistic model is again given by yn = M + En. (3.4.3) The response matrix Yn has entries I 2 r l[ "1 / 2 )~ j 1 Y n i j = wi L i n p i 1.:.. i ~_kl 1.:._j ~r ( 3 4 4 ) r+l~j_~_2r (3.4.5) = 0 elsewhere. (3.4.6) l("l/2) Mis defined in a manner similar to Yn with s,n P; 77
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replaced by sin1 (p~ 12 ). Again we have X = M (3.4.7) where the transformed dose vector,~' has ith component xi = w~/ 2 log di' 1 < i < k, (3.4.8) and = (3.4.9} En is again a matrix 6f random vectors. En can now be represented by 0 (3.4.10) where E(l) and E( 2 ) are k 1 xr and (kk 1 )xr respectively. 78
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E(l) and E( 2 ) are thus comprised of independent random vectors, and we again have 1/2 (i) L n ~n > Nr (0, Vi)' lk 1 (3.4.13) = Q elsewhere = (Y'Y )l Y' x n n n is the weighted least squares estimate of~(3.4.14) (3.4.15) Since we desire to test for parallelism, the test of interest can be expressed by the hypothesis H : O This hypothesis is thus of the form (3.4.16) 79
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Ho: A.@_= 0, where the entries of A an rx2r matrix are a .. l J = 0 = elsewhere. = a = 1 rr ar,2r = 1, (3.4.17) (3.4.18) (3.4.19) (3.4.20) Thus, we can apply Corollary 2.3.3 and test H 0 by the test statistic L > 2 X r (3.4.21) where A, ~n' and Bn are defined in [(3.4.18), (3.4.19), (3.4.20)], (3.4.15), and Theorem 2.3.2, respectively. 80 If x~ is sufficiently large to reject H 0 : AS= Q, we can conclude that the response curves are not parallel. 3.5 Summary We have now discussed the application of the angle transformation to inverse ~egression of quantal response assays. We have shown that inverse regression will give better asymptotic results than the KnudsenCurtis method 1( 1/2) when the relationsh ip between log di and sin pi 1s
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81 not truly linear. Inverse regression may be used to fit models other than the linear model whereas the Knudsen Curtis meth~d is not appropriate. We have given methods of forming confidence intervals for LD(lOOp) as well as relative potency. We have also given a test for parallel ism. In Chapter IV we will give examples of numerical application of the results developed in this chapter
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CHAPTER IV NUMERICAL APPLICATIONS 4.0 Preamble In this cha~ter we will apply the results obtained in Chapter III. Several numerical applications will be given. Section 4.1 will give the exact probabilities that 95% nomi nal confidence intervals cover LD(50). Eight probability schemes will ship between be considered which satisfy a linear relation logdose and sinl(~~/ 2 ). In Section 4.2 we 1 will compare the use of inverse regression to other methods of analysing quantal response assays. Section 4 3 will be a summary of the chapter. 4. l Exact Covera e Probabilit (95% Nominal Confidence Interval In this section we will investigate small sample results for eight probability schemes satisfying the linear model (4.1.l) The logdoses, xi, i = l, 2, 3, 4 were fixed at four equally spaced values. Equal sample sizes of five, ten 82
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83 and fifteen were considered. We ran all possible assays for the model given in (4.1 .1) with the conditions de scribed. For all realizations, we then computed nominal 95% confidence limits for LD(50). These limits were found by use of the results given in Corollary 3.2.l. For the eight different curves, we then computed the exact probability that the true LD(50) lies in the confi dence interval. A Based on a pilot study we replaced Pi by ( a ) p ; + ( 2 n ) 1 i f p i < l / 2 and (4.1.2) (b) Pi (2n)1 if ~i > 1/2 (4.1.3) prior to taking sin1 (~ll 2 ). We recommend this contin~itj correction whenever the sample sizes are relatively small. This co~tinuity correction has no affect on the asympototic distribution. The following table summarizes the results obtained.
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84 Table 4. l Exact Coverage Probability ( 95% Nominal Coverage) Run Number l 2 3 4 5 6 7 8 pl ( x l = l 5) .039 .029 l 5 2 .087 .230 .319 230 l 5 2 P2(x2 = 5 ) 319 .206 .415 .230 .415 41 5 3 l 9 .230 P3(X3 = 5) .708 .485 .708 .415 .614 515 .415 .319 P4(X4 = l. 5) .971 .770 .928 .614 .794 .614 51 5 415 log LD(50) .04 55 .21 .93 .07 .35 l. 35 2.35 Sam~le Size Coverage a 11 n = 5 .999 .980 .997 .880 .999 .882 .660 .557 a 11 n. = 10 .994 .954 .986 91 2 .995 .962 .749 .633 all n = l 5 .979 .954 .974 .922 .987 .959 .779 .701 The first four lines of Table 4.1 give the probability of response, Pi, at logdose, xi, for the eight linear models considered. The fifth line gives the true value of log LD (50). The last three lines give the exact coverage proba bility for the various equal sample sizes. We were limited to relatively small samples in this investigation since
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85 there ~re, for exa~ple, more than 65,000 possible realiza tions (each of varying probability) associated with n 1 = 15. As was previously stated, the above examples are all linear in terms of logdose against sin1 (p 1 1 2 ). Four different slopes were used. Run l had the smallest slope, runs 2 and 3 the next smallest, runs 4 and 5 the second largest, and runs 6, 7, and 8 the largest. ~uns l through 6 provide ex~ellent ~mall sample approximation, while runs 7 and 8 do not. For runs 7 and 8, there is substantial probability that all p/s are less than .5, and hence LD(50) must often be estimated by extrapolation. We conjecture that convergence is slow whenever extrapolat ~ ion is highly probable. 4.2 Estimation of Relative Potency by Various Linear Techniques In this section we will compare the estimation of relative potency by various methods of analyzing quantal response assays. The data we will analyze are an example presented by Finney [3]. The data are the result of an assay of insulin Mice were injected with varying doses of insulin o~ with a test preparation,and the numbers of mice showing the symptomi of collapse or convulsions were recorded. For the data of Finney [J], page 477, we obtained 95% nominal confidence inter~als for the relative potency of the insulin as compared to the test preparation. Excellent linear fit
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86 was obtained for all methods. A summary of the analyses is given in Table 4.2. Method Probit Logit Angle (MLE) KC IRL Legend: R LCL uc L Table 4.2 Estimation of Relative Potency R LCL 1 3. 41 11. 11 1 3. 38 11 0 4 13. 50 11 31 13.70 11 .58 13.72 11. 66 = estimated relative potency = lower 95% confidence limit = upper 95% confidence limit UCL 2 X 1 ( p) 1 6. 1 2 .28 1 6. 1 6 .35 16.05 17 1 6. 20 35 1 6. 18 1 9 2 x.(P) = l chisquare statistic for parallelism, one degree of freedom MLE KC IRL = maximum likelihood estimation = KnudsenCurtis method = Inverse regression: linear (p=2 in Chapter III) While the five methods give virtually the same results, the KnudsenCurtis and inverse regression methods require the more elementary computations, and are easier to explain to nonquantitative scientists. In situations where parallelism is reasonable, but linearity is not, we can use inverse regression with r > 2, whereas the other methods are inappropriate.
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87 4.3 Summary By use of numerical examples we have shown the appli cability of inverse regression in analyzing quantal response assays. Since other methods of analysis are restricted to linear models, we have compared inverse regression to some other methods when a linear fit is excellent. As has been stated before, inverse regression can also be applied to quantal response assays when linearity is doubtful. Finally another reason to use inverse regression to analyze quantal response assays i~ the computational simpli city and ease of explaining the result~ to nonqua~titative scientists.
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BIBLIOGRAPHY [l] Bliss, C. I. (1939). The toxicity of poisons applied jointly. Ann. Appl. Bio l 585615. [2] Finney, D. J. (1971). Probit Analysis. 3rd Ed. Cambridge: University Press. [3] Finney, D. J. (1964). Statistical Method in Biologi cal Assay. 2nd Ed. London: Griffin and Co. [4] Halperin, M. (1970). On inverse estimation in linear regression. Technometrics ]1_, 72736. [5] Knudsen, L. F. and Murtis, J. M. (1947). The use of the angular transformation in biological assays. J. Amer. Statist. Assoc .!, 889902. [6] Krutchkoff, R. G. (1967). Classical and inverse regression methods of calibration. Technometrics 1, 42539. [ 7 J [8] [9] Moo re R H a n d Z e i g l e r R K ( l 9 6 7 ) The u s e of nonlinear regression methods for analysing sensitivity and quantal response data. Biometrics _g]_, 56566. Nelder, J. A. (196~). Weighted regression, quantal response data, and inverse polynomials. Bio metrics!, 97985. Rao, C. R. (1965) Linear Statistical Inference and Its Applications. New York: John Wiley and Sons, Inc. [10] Saw, J. G. (1970). Letter to the editor. Technometrics ]1_, 937. [ l l ] W i l l i a m s E J ( l 9 6 9 ) A n o t e o n r e g r e s s i o n m e t h o d S in calibration. Technometrics l_l, 18992. 88
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Additional References Berkson, J. (1944). Application of the logistic func tion to bioassay. J. Amer. Stati~t. Assoc. 12_, 35765. 89 Krutchkoff, R. G (1969),. Classical and inverse regres sion methods of calibration in extrapolation. Technometrics ll, 6058~ Litchfield, J. T. and Wilcoxon, F. (1949). A simplified me t h o d o f e v a l u a t i n g d o s e r e s p o n s e ex p e r i m e n t s J. Pharmacol. Exp. Therapeutics 2.._, 99113. Patel, K. M. and Ho el, D. G. (1973). A generalized Jonckheere ksample test against ordered alterna tives when observations are subject to arbitrary right censorship. Comm. Statist. I, 37380. Steel, R. G. and Torrie, J. H. (1960). Principles and Procedures in Statistics. New York: McGraw Hill Book Co.
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BIOGRAPHICAL SKETCH Frank Hai n Dietrich II was born on August 9, 1945, in Lewisburg, Pennsylvania. He was graduated from Lewis burg Joirat High School in June, 1963. In September of that year he enrolled in Wilkes College, receiving ~he degree of Bachelor of Arts with a major in mathematics in June, 1967. In September of that year he enrolled in Bucknell University, receiving t he degree of Master of Arts with a major in mathematics in January, 1970. The writer also taught high school for the school year 19681969 at Selinsgrove Area High School. He entered the University of Florida Graduate School in September, 1970. Mr. Dietrich has worked as a teaching assistant for the Department of Statistics since that time, simultaneously pursuing his work towards the degree of Doct6r of Philosophy 90
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jonathan J. Shuster, Chairman Associate Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequat~, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. cClave Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. f Statistics I certify that I have read this study and that in my op inion it c0nforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pejaver V. Rao Professor of Statistics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stratton H. Kerr Professor of Entomology This dissertation was submitted to the Graduate Faculty of the pepartment of Statistics in the Col lege of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1975 Dean, Graduate School

