NONPARAMETRIC METHODOWGY FORINCORPORATION OF SURROGATE

IN CLINICAL TRIALS

byHabib Elias EI-Moalem

Department of BiostatisticsUniversity of North Carolina

Institute of StatisticsMimeo Series No. 2149

July 1995

NONPARAMETRIC METHODOLOGY FOR

INCORPORATION OF SURROGATES

IN CLINICAL TRIALS

by

Habib Elias EI-Moalem

A dissertation submitted to the faculty of the University of North Carolina at Chapel

Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Biostatistics.

Chapel Hill

1995

~ABIB ELIAS EL-MOALEM. Nonparametric methodology for incorporation of

surrogates in clinical trials. (Under the direction of Dr. Pranab Kumar Sen.)

ABSTRACT

In clinical trials, as well as in medical research, it is often the case that the

variables of interest, known as true endpoints, are either hard to measure or are too

costly, or simply require considerable time for completion. Hence we search for end­

points that are correlated with the true endpoints and that are easier and less costly

to measure or perhaps can be measured at an earlier time. These endpoints are called

surrogate endpoints. There is a controversy over the definition of a surrogate endpoint

due to the lack of formal methodology for making inference regarding a true endpoint,

Y, when a surrogate,S, is used intstead.

Prentice (1989) defined a valid surrogate as an outcome variable that would

"yield a valid test of the null hypothesis of no association between treatment and

true response". Thus, the surrogate should be informative about the primary end­

point, and fully capture the effect of treatment on true endp?int.

A surrogate is used in this work to mean a substitute for a true response. It

differs from a surrogate as a substitute for a true covariate dealt with in measurement

error or in Latent-class models Sen (1992) considers design and analysis of a general

class of incomplete multiresponse designs (imd) that take into account use of sur­

rogate variables. He formulates appropriate rank based procedures to test the null

hypothesis of no treatment effect in a robust manner using only one true endpoint

and one surrogate.

The purpose of this study is to generalize Sen's approach into a multi-response,

multi-surrogate setup. A vector of primary response variates is partitioned into vari­

ous subsets on which measurements are obtained for different number of experimental

units in each subset. A vector of surrogate response variates is partitioned simillary.

The set of the validation samples consists of measurements on a subset of primary

11

~ well as surrogate responses taken along with concomitant variates. On the other

hand, the set of surrogate samples consists of measurements on a subset of the sur­

rogate variates along with covariates. Hence, we have an incomplete multiresponse

design scenario. A rank-based test is developed to detect differential treatment effects

on primary variates while incorporating information from surrogates also. The test is

developed in two design settings: the randomized block design, and the balanced in­

complete block design. In the latter recovery of ineterblock information is considered

in detail.

III

Acknowledgement

I thank God for giving me this wonderful oportunity to study at the hands of masters

of statistics. I thank my advisor Dr. Pranab Kumar Sen for inspiring me and for his

constant encouragement. I thank all my committe members Drs. Q~ade, Kupper,

Helms and Vine for their helpful comments and syggestions. Also, I am indebted

to Dr. Bahjat Qaqish for all the training he gave me during my employment as a

graduate research assistant at the Linberger Cancer Center. The experience I got

by working on consulting projects is invaluable. Many thanks go to the staff in the

Department of Biostatistics for their technical support; in particular I thank Betty

Pounders and Betty Owens for the administrative help and Bruce Walter and Corey

McEntyre for their computing support. I also thank my fellow students, particularly

my past and present office mates: Ralph Demasi, Joseph Galanko, Antonio Pedroso,

and all the other wonderful fellow students; they have been a constant source of en­

couragement and given me positive feedback throughout my stay here. I am thankful

to my brother Basim and my sister Rose and my brother in law Basil for helping me

financially and emotionally.

IV

Contents

1 General Introduction 1

1.1 Introduction. . . 1

1.2 Literature Review . 3

1.2.1 Surrogate Endpoints: Uses and Abuses 3

1.2.2 Prentice Criteria for Surrogacy .... 5

1.2.3 Designs of Clinical Trials with Surrogate Endpoints 7

1.2.4 Parametric and Semi-parametric Approach ..... 10

Advantages and Disadvantages of the Parametric and Semi-

Parametric Approaches 14

1.2.5 Nonparametric Approach. 15

Hypothesis Testing 16

Estimation. . . . . 24

1.3 Synopsis of The Work Done 28

2 Methodology In Randomized Block Design (RBD) 33

2.1 Introduction . . . . . . . . 33

2.2 Randomized Block Design 34

2.2.1 RBD For The Surrogate Set 1 34

2.2.2 RBD For The Validation Set 1* . 43

2.2.3 RBD For The Set 1° . . . . . . . 45

2.2.4 Construction of the Test Statistics 46

VI

2.2.5 Asymptotic Non-null Distribution of LO· 47

3 Nonparametric Intra-Block Inference 51

3.1 Balanced Incomplete Block Designs 51

3.1.1 BIBD for Surrogate Set 1 . 51

3.1.2 BIBD For The Validation Set 1* 63

3.1.3 BIBD For The Set 1° . . 64

3.2 Construction of the Test Statistic 65

3.3 Asymptotic Non-null Distribution of LO· 67

4 Recovery of Inter-Block Information (RIBI) 71

4.1 RIBI for Surrogate Set 1 72

4.1.1 RIBI For The Validation Set 1* . 81

4.1.2 Construction of The Test Statistics 82

4.1.3 Asymptotic Non-null Distribution of LO· 83

5 General Case of a Vector of Primary Variates 87

5.1 Intra And Inter-Block inference . . 88

5.2 Construction of The Test Statistic . 89

5.3 Asymptotic Non-Null Distribution of LO· 90

6 Properties of The Test Statistics And an Illustration 92

6.1 Introduction............. 92

6.2 ARE for the Complete Block Case. 92

6.3 Example: The Effect of Zidovudine on Survival in Patients with AIDS 97

6.4 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102

VII

Chapter 1

General Introduction

1.1 Introduction

In clinical trials, as well as in medical research, it is often the case that the vari­

ables of interest, known as true endpoints, are either hard to measure, are too costly,

or simply require considerable time for completion. Hence a search for endpoints that

are correlated with the true endpoints and that are easier and less costly to measure,

or perhaps can be measured at an earlier time, results in what are called surrogate

endpoints. There is controversy over the definition of a surrogate endpoint due to the

lack of formal methodology for making inferences regarding a true endpoint, Y, when

a surrogate,S, is used instead.

Prentice (1989) has defined surrogacy and established some operational criteria

with which to choose a surrogate outcome variate in the context of clinical trials.

Basically Prentice (1989) defined a valid surrogate as an outcome variable that would

"yield a valid test of the null hypothesis of no association between treatment and

true response". This simply means that for a surrogate to be a valid substitute for

a true endpoint, the treatments being studied should affect the true endpoint only

through the surrogate. In other words "the surrogate endpoint must have precisely

the same relationship to the true endpoint under each of the treatment strategies

1

b.eing compared".

Pepe (1992) considered settings where, for a random subsample of the subjects

being studied, some validation data are available to study the association between

one true endpoint and one surrogate variable. She considers a completely randomized

design (treatment wise). Let P{3(Y I Z) denote the regression model relating the true

endpoint Y to the covariate Z. Pepe showed that the maximum likelihood estimate

of (3 is nonrobust to misspecification of the conditional distribution of the surrogate

X giventhe unobserved true outcome Y and the covariate Z, P(X IY, Z). Moreover,

she considers as an alternative a semi-parametric model where she lays no structure

on P(X I Y, Z). This partial likelihood approach leads to an estimate of f3 which

behaves much better than the fully parametric estimate, yet remains sensitive to any

departure from the structure imposed on P{3(Y I Z) which would affect the validity

and efficiency of the statistical analysis.

Sen (1994) considers design and analysis of a general class of incomplete multire­

sponse designs (IMD) that take into account the use of surrogate variables. The Pepe

(1992) setup is a special case of an IMD with one primary response variate and one

surrogate. The model that Sen considers here is a nonparametric model that avoids

the stringent conditions needed in a parametric or semi-parametric model. He formu­

lates appropriate rank based procedures to test the null hypothesis of no treatment

effect in a robust manner using only one true endpoint and one surrogate. More­

over, he tackles the problem of estimating the conditional regression quantiles of the

conditonal distributions (given concomitant variates) of the primary and surrogate

variables.

The purpose of this study is to generalize Sen's approach into a multiresponse,

multisurrogate setup and to study the properties of the proposed nonparametric pro­

cedures in both finite and infinite samples. Although the focus is on use of surrogate

variables for primary response variables, covariates may also need surrogates. There

has been much work published recently on the so called measurement error models

2

for continuous as well as discrete covariates. For instance, see Kupper (1984), Fuller

(1987), Carroll (1989), Carroll and Ruppert (1988), Pepe and Fleming (1991), Satten

arid Kupper (1993) among others.

1.2 Literature Review

1.2.1 Surrogate Endpoints: Uses and Abuses

The term surrogate comes from the Latin 'surrgare' meaning 'to substitute'.

Thus a surrogate endpoint simply means a substitute measure for some other variable.

Ellenberg and Hamilton (1989) suggest that "investigators use a surrogate when the

endpoint of interest is too difficult and/or expensive to measure routinely and when

they can define some other, more readily measurable endpoint that is well correlated

with the first to justify its use as a substitute". They present some surrogate endpoints

for survival time in cancer studies such as tumor response, time to progression, or

time to reappearance of disease. However, Piedbois, et al. (1992) warn that tumor

response should not be considered a valid surrogate endpoint for survival in patients

with advanced colorectal cancer.

Wittes, Lakatos and Probstfield (1989) define a surrogate endpoint in cardiovas­

cular clinical trials as an "endpoint measured in lieu of some other so-called 'true'

endpoint." Ejection fraction is given as an example of a surrogate for total mortal­

ity. Hillis and Seigel (1989) define a surrogate as "an observed variable that relates

in some way to the variable of primary interest, which we cannot conveniently ob­

serve directly." In clinical trials that study ophthalmologic disorders, they examined

the use of the "status of one eye as a surrogate for the (unobservable) status of the

opposite eye in the same individual."

Machado, Gail and Ellenberg (1990) mention the use of CD4+ lymphocyte levels

as surrogates for the development of Acquired Immune Deficiency Syndrome (AIDS)

or death. However, Choi, Lagakos, Schooley and Volberding (1993) conclude that

3

CD4+ lymphocytes are an incomplete surrogate marker for progression to AIDS in

asymptomatic HIV infected persons taking Zidovudine. Gruttola, Wulfsohn, Fischl

and Tsiatis (1993) observe that although CD4 lymphocyte counts are associated with

improved survival in patients with AIDS and AIDS-Related Complex, "they account

for only a small portion of the survival benefit of Zidovudine". Lagakos and Hoth

(1992) echo the same conclusion. Baccheti, Moss, Andrews and Jacobson (1992)

found that CD8 counts and changes in hemoglobin and WBC during therapy could

add independent predictive power to the CD4 count and thus would produce more

valid and useful surrogates than the CD4 count alone.

The urgency for developing cures for terminal illnesses, such as cancer and AIDS,

coupled with the long period of patient follow up necessary to obtain valid data for

estimating survival time has led scientists to consider surrogate endpoints that could

be useful for several reasons. Clinical trials that use a surrogate endpoint require

a much shorter follow up time than a true endpoint. Surrogates often require less

invasive techniques of measurement than true endpoints, and hence are easier to

measure. Sometimes a disease might be rare, which makes study of the true endpoint

very difficult.

These benefits should not overshadow the possible pitfalls that abound in using

surrogate variables. Fleming (1992) writes that "one rarely can establish that sur­

rogate endpoints are valid". DeMets, in his commentary on Fleming (1992), refers

to the reliance on surrogate markers as "the most disturbing, even threatening, issue

today in clinical trials". Prentice (1989), after generalizing his operational criteria,

mentions that "in spite of the hope for such extension I am somewhat pessimistic

concerning the potential of the surrogate endpoint concept, as it is interpreted in this

paper." Ellenberg (1991) emphasizes that there should be a "strong biological ratio­

nale" for a valid use of surrogate markers in AIDS trials, added to a strong predictive

ability of survival time at any given point in time.

The fact that treatment might affect the 'true' clinical outcome through path-

4

,

..

..

'Yays or biological processes other than the surrogate marker a raises lot of doubt

as to the markers' validity in predicting the true outcome. In such a situation, false

positive conclusions are misleading. A recent example often cited in the literature is

provided by the Cardiac Arrythmia Suppression Trial (CAST, 1989). Arrhythmias

were accepted as a surrogate for sudden death since it was believed that suppression

of arrhythmias would reduce sudden deaths. The FDA approved the antiarrhythmic

drugs, ancainide and flecainide, and these were widely used. Later on the CAST trial

established that these drugs nearly tripled the death rate relative to placebo.

False negative conclusions might also arise when treatment has no effect on

the surrogate marker but is beneficial with respect to the true clinical endpoint.

Thus reliance on the surrogate marker in this case would deprive patients of effective

treatments. Fleming (1992) cites the Chronic Granulomatous Disease (CGD) clinical

trial. Gamma interferon led to noticable reduction in the rate of serious recurrent

infections in CaD patients, while there was no detectable effect on either superoxide

production or bacterial killing, both of which were used as surrogates for the life­

threatening infections.

Thus, Machado et al. (1990) say that understanding the mechanism of action of a

new treatment is an important a priori condition for selection of a surrogate endpoint,

and whenever such mechanisms are unclear, data on the true endpoint seems to be

the essential source of information. However, they add that "assuming that other

agents in the same class have similar mechanisms of action, it may be possible to rely

on surrogate endpoints in later studies to evaluate these agents" ..

1.2.2 Prentice Criteria for Surrogacy

In light of the above complications of choosing a valid surrogate, formal statistical

methods to aid in this process are urgently needed. Unfortunately, they are also

scarce. Prentice was the first to adopt criteria for valid surrogacy. He defines a

surrogate endpoint as "a response variable for which a test of the null hypothesis

5

~f no relationship to the treatment groups under comparison is also a valid test of

the corresponding null hypothesis based on the true endpoint." The term valid here

means that both hypotheses are parallel in the sense they would either both reject

the null hypothesis or both would accept it. Note that the surrogate endpoint in this

definition depends on the treatments being compared, and cannot be assumed to be a

universal substitute for the true endpoint. In symbols, this means that if t denotes the

time of enrollment in a clinical trial, and if Y denotes a true time-to-failure endpoint,

and if we let z = (Zl,' .. , zp) be indicators for p of the p+ 1 treatments to be compared

with respect to the corresponding hazard functions Ay(t I z), then with X(t) as the

candidate surrogate we should have

P(X(t) I z, F(t)) - P(X(t) I F(t)) {:} Ay(t I z) - Ay(t) (1.1 )

where F(t) consists of the failure and censoring histories prior to t for the true end­

point, and P(.) stands for the probability distribution. This means that for a surro­

gate to be valid, a test of the null hypothesis of no treatment effect on the surrogate

should be parallel to the test of no treatment effect on the true endpoint. In other

words, the two tests would either both reject or both accept the null hypothesis.

Prentice established two criteria to check the validity of (1.1), namely

Ay(t IX(t), z) =Ay(t IX(t)) (1.2)

which states that the surrogate fully captures the effect of the treatment z on Y, and

Ay(t I X(t)) -I Ay(t) (1.3)

that is, the surrogate is informative about Y . .In order to establish (=» in (1.1), it is

necessary in addition to condition (1.3), to restrict the class of alternatives to the null

hypothesis of no treatment effect on the surrogate endpoint to "those for which the

treatment effects on the surrogate response distribution have some impact on average

true endpoint risk".

Although time to response was the true endpoint here, Prentice (1989) considered

6

generalizations where the true response was a stochastic process {Y(t), t 2:: O}, where

t is the time from begining of follow-up, and Y(t) = {Yi(u), Y2(u),···; 0 ~ u < t}.

Moreover, instead of considering comparison of two or more treatments in a clinical

trial setup, a surrogate response is sought in order to replace a true response in

"respect to its dependence on some general exposure or covariate history". Thus if

we denote the possibly time-dependent covariate by W(t) = {W1 (u), W2(u),···; 0 ~

u < t}, then condition (1.2) can be extended to

Pr{Y(t) IX(t), W(t), F(t)} =Pr{Y(t) IX(t), F(t)}. (1.4)

It is well known that in actual practice, condition (1.2) rarely is known to hold

since it is hard to verify or simply is not true. Moreover, measurement error in the

surrogate variable may invalidate criteria (1.2 or 1.4) and (1.3). It is possible also to

use a later response as a surrogate for an earlier true endpoint if the latter is hard to

measure, and the same criteria above can be employed to validate surrogacy.

1.2.3 Designs of Clinical Trials with Surrogate Endpoints

The information gathered in a clinical trial usually involves a number of re­

sponses, some of which may be more important than others, and these are what Sen

(1994) calls primary variates. It is essential that the statistical relationship between

the primary variates and the surrogate endpoints be studied before any inference is

made about the treatment effects on the primary endpoints. But then it is often

difficult and/or costly to record data on all the primary variates (as well as surro­

gate variables and concomitant variables) for all experimental units, thus creating

the need for simultaneous measurements of the primary and surrogate variables for

different subsets of experimental units. This is the genesis of incomplete multire­

sponse designs (IMD) in clinical trials, as Sen (1994) has so elaborately explained.

In situations where the primary variates can be ordered in terms of their importance

or relevance, a hierarchical design (HD ) can be adopted as follows.

7

Assume that Y = (Yl ,"', Yp ) is a vector of response variates such that Yl is

the most important primary variate, and the other responses have a decreasing or­

der of importance. Assume also that Yo is a q-vector of surrogate variates that

may be recorded for all subjects with relative ease. If S = So denotes the set of all

experimental units, then a possible hierarchical scheme would be:

S = So ::J Sp 2 ... 2 SI, (1.5)

where on SI, all the p + q responses are recorded, on S2 \ SI (Y2,"', Yp and Yo)

are recorded, and so on; on Sp \ Sp-l, Yp and Yo are recorded, and on So \ Sp only

Yo is recorded. Here Sp \ Sp-l means all elements of Sp that are not in Sp-l' Data

pertaining to treatment as well as design variates are also recorded in all of the above

subsets. Now for the specific subsets,

(1.6)

efficient designs {D} like those discussed in chapter 9 of Roy, Gnanadesikan and

Srivastava (1971) may be adopted to draw statistical conclusions.

However, sometimes all the primary variates can be equally important and no

ordering may apply. Here the hierarchical design loses its appeal and other response

wise incomplete blocking may be applied to Y. To see this, Sen (1994) formulates

a general class of IMDs in the following manner. Let P denote {I"", p} and

.consider the totality of 2P subsets of P, defined by i r = {il,"', iT}' for all possible

1 ~ i l < '" < iT ~ p and r = 0,1"" ,p; io = O. Consider a proper subset Po of P,

determined by clinical and other factors, such that:

Po = {ir : (r,ir ) E lo}. (1. 7)

where 10 is the set corresponding to Po. Then the set of all experimental units is

partitioned into a system

S(Po) = {Sir: (r, ir) E PO}.

8

(1.8)

w;here for the subset Sir' the primary variate(s) Yi l ,· •• , li r along with the surrogate

variates Yo and design and concomitant variates are recorded. For the subset Sir' an

appropriate design Dir (for the treatment as well as design variates), can be adopted

leading to the design sets

D(Po) = {Di r : (r,ir ) E Po}. (1.9)

Thus, an incomplete multiresponse design that incorporates surrogate endpoints can

be formulated in terms of the dual design sets, responsewise design set and treatment-

wise design set, namely

{S(Po), D(Po)}. (1.10)

Sen (1994) stresses that the choice of optimal designs of the type (1.10) may depend

heavily on the cost and/or difficulty of measuring the responses, whereby a cost­

benefit approach will be needed.

In setups where there is more than one surrogate endpoint, and data, on all of

them for all experimental units cannot be recorded easily, Sen (1994) suggests a more

general class of IMD designs. Here the set P will be replaced by

P* = {1 ... p·1 ... q}, '" , , (1.11)

and ir by (ir , is) = {i l ,···, iT; i~,·.·, i:}, for all possible 1 ~ i l < ... < iT ~ p,

1 ~ ii < ... < i: ~ q; 1 ~ r ~ p, 1 ~ 5 ~ q, and include io or ioin this system. Also

Po is replaced by Po = (ir , i~) : (r, 5, iT' i:) E 10. Then (1.8) is extended to

S(Po)= {S(ir,is) : (r, 5, ir , is) E 1O}, (1.12)

where in the subset S(ir,is)' the primary response variates Yill ···, Yir , and the sur­

rogate response variates Yi~, ... ,Yi~ are to be recorded on the experimental units.I s

However, Sen (1994) does not foresee usage of such general designs since it requires

large cardinality of S, and a sophisticated statistical analysis that may not be ap­

pealing to the clinical scientist.

9

1.2.4 Parametric and Semi-parametric Approach

Prentice (1989) used his criteria to draw inferences based solely on surrogate

data. The design that he considered is a special case of an IMD wit?- p = 1, q = 1;

Po refers to the set of no primary index and 5(Po) = 50 is the entire set of experimen­

tal units where only the surrogate variate is recorded. Pepe (1992) considers settings

where some validation data on both the true endpoint Y and the surrogate variable

5 is available. The design here is also a special case of an IMD with p = 1, q = 1

and io = {O}, h = {I}. Thus one subset contains data only on the surrogate variate,

while the other subset contains both the primary and surrogate variates (validation

sample).

Assume that P/3(Y I Z) is the regression model that relates the true response

to the vector of covariates Z, and let P/3,o(X I Y, Z) be the model that relates the

surrogate X to Y and Z. A validation subset V of experimental units on which Y,

X, and Z are recorded is considered so that the strength of the relationship between

X and Y could be studied. Pepe (1992) discusses two approaches in order to draw

inferences about (3.

In the first approach, inference for the true outcome is based on maximum likeli­

hood theory using a parametric model for P(X IY, Z). Maximum likelihood estimates

of (3 and () are based on the likelihood

L((3, ()) = II P/3(Yi I ZdP/3,o(Xi IYi, Zi) II P/3,o(Xj I Zj),iEV jEV

where V indicates the nonvalidation set of observations on which only the surrogate

and concomitant variables (as well as design variates) have been measured, and where

P/3,O(X I Z) = JP/3(y I Z)P/3,o(X I y, Z) dy.

Pepe (1992) demonstrates by an example that the maximum likelihood estimate of (3

is very sensitive and nonrobust to misspecification of P/3,o(X IY,Z).

In the second approach, no structure on P(X IY, Z) is imposed while P(Y I Z)

10

i~ still parametric in nature, hence the method is termed semi-parametric. Since

P(X I Y, Z) is nonparametric in nature it can be assumed to be independent of ;3.

If zs denotes the components of Z which are thought to be informative with respect

to the association between X and Y, then an empirical estimate of P(X I Y, ZS) is

found using the validation sample:

where in the discrete case

1- ~ I[X· = X 1': = Y ZS = ZS]vL...J t ,t 't ,

n iEV

~ EI[Yi = Y,Z: = ZS]n iEV

Here 1[.] is the indicator function and n V is the number of subjects in the validation

sample. If X, Y, or ZS is continuous, then suitable smooth kernel type estimators of

the probability ~ensities P(X, Y, ZS) and P(Y, ZS) are used for those components.

Define P(3(X I Z) = f P(3(Y I Z)P(X I y, ZS) dy, or the corresponding sum if Y

is discrete. The inference about ;3 is based on the estimated likelihood

L(;3) = II P(3(Yi I Zi) II P{3(Xj I Zj)iEV jEV

which is an estimate of

L(;3) = II P{3(Yi I Zd II P{3(Xj I Zj)iEV jEV

(1.13)

had P(X I Y, Z) been known exactly. Pepe (1992) showed that, provided the valida-

tion sample fraction nV/ n has a nonzero, positive limit pV, and the usual regularity

conditions of Cox and Hinkley (1974) hold for both P{3(Y I Z) and P{3(X I Z), the

maximum estimated likelihood estimate ~ converges in distribution to a normal ran­

dom variable with zero mean and a variance made of two components. The first,

I- 1 (;3)/n, is the variance of the maximum likelihood estimate based on (1.13), and

the second, K(;3), is the penalty induced by estimating (1.13) which is a decreasing

11

f':lnction of the validation sample fraction pl/.

Although it is true that in large samples there is no loss in using uninformative

nonvalidation set data in the analysis of validation set only, the use of informative

surrogates does increase the efficiency over estimates based solely on true outcome

data. The estimated likelihood method is fully efficient if the surrogate is perfect,

that is if P(Y I X, Z) = 1. In other words, if X is perfect then the variance of the

maximum estimated likelihood estimate is equal to the variance of the maximum like­

lihood estimate based on true outcome data for all subjects and hence the surrogate

would be as informative about (3 as the true endpoint. Pepe (1992) showed that when

(3 is a scalar parameter, the maximum estimated likelihood estimate is more efficient

than the maximum likelihood estimate based on the validation set alone if and only

if the validation fraction nl/ In is greater than !.Fleming (1992) discusses use of biological markers as auxiliary variables rather

than surrogates in order to strengthen the clinical efficacy analyses and to avoid

the risk of making false conclusions when surrogate endpoints are used. Three ap­

proaches are reviewed to deal with auxiliary variables: "variance reduction", "aug­

mented scores", and "estimated likelihood". The variance reduction approach was

explored by Kosorok (1991), and the latter two approaches by Fleming (1992). In

all three approaches, two conditions are needed for efficient analyses employing aux­

iliary variables: in the first the auxiliary variable and the true endpoint should be

highly correlated; and in the second "one pool of patients having longer follow-up,

and another pool of patients with auxiliary information but with relatively short-term

follow-up on the clinical endpoint".

In their commentary on Fleming (1992), Farewell and Cook discuss their method,

which estimates the correlation p between the surrogate and the true endpoint from

the asymptotic covariance matrix of Lin (1991) and uses it for appropriate weights of

the test statistics. To see this, let VI (t) and V2 (t) be the standard log-rank statistics

at time t for the treatment effect on the surrogate endpoint and the true endpoint,

12

r~spectively. A global test statistic is defined as

R(t) = pl(t)Vi(t) +P2(t)V2(t),

where Pl(t) and P2(t) are possible data-dependent weights that may be of the form

Pl(t) jl + p(t)

P2(t) j2 - p(t)

where jl and i2 are chosen to reflect the relative weighting of the marginal test

statistics assuming zero correlation. Their simulation studies found that even with

such weights, possible discordant treatment effects hamper the validation of surrogacy.

Also, p was found to be insensitive for assessing a surrogate candidate, whereupon

the weights simplified to a proper choice of jl and h.

Louis, in his commentary on Fleming (1992), postulates a parametric model that

relates a surrogate (X) to treatment (Z) and a true endpoint (Y) as follows

Pe(Z, X, Y) = P(Z)Pe(X IZ)Pe(Y I Z, X).

-Now, condition in two different orders and use the Fisher information decomposition

in Louis (1982) to get

Equating the right-hand sides and solving for I(TIZ) (0)

The term in square brackets determines the cost/benefit of using a surrogate. If it

is negative, the surrogate will be more efficient than the true endpoint in drawing

inferences about O. Otherwise, if it is positive and one still insists on using the

surrogate a larger sample will be needed.

13

4dvantages and Disadvantages of the Parametric and Semi-Parametric

Approaches

A sketch of the general theory and statistical analysis of IMDs is f~)Und in Mona­

han (1961), Srivastava (1966), and Srivastava (1968). In multiresponse situations the

multivariate general linear model (MGLM) is often adopted when there are no surro­

gate variables. Two assumptions in an MGLM are questionable here. The response

variables as well as the error vectors are assumed to be multivariate normal. These

are risky assumptions that cannot be verified in practice. Usually, one checks to make

sure that the marginal distributions are univariate normal. If they are found to be

skewed, appropriate transformations are made to bring about symmetry. But then to

claim that the transformed variables are multivariate normal would not be reasonable.

In addition, sometimes some of the response variables may be categorical in nature

and they would then require other treatment, for example logistic regression. In such

a case the MGLM cannot be applied for all responses. The MGLM is valuable if all

the responses are continuous and the multinormality and other assumptions are valid,

and if our interest lies in the covariance structure of the responses, or in a secqndary

parameter and/or hypothesis that involve more than one dependent variable in an

essential way, the two main reasons for employing a MGLM.

Generalized linear models (GLM) of McCullagh and Nelder (1989) can be adopted

in clinical trials when only one response is analysed. But here also there are drawbacks

that arise due to non-identifiability of such GLM's, high dimension of the "asymptotic

covariance" matrix, and large sample sizes.

The use of Cox (1972) partial likelihood by Fleming (1992) should be viewed

with caution. In clinical trials, censoring (Type I, II or random) often occurs, and

with moderate to high censoring, the partial likehood will not have enough informa­

tion for a powerful test. Moreover, if there are many covariates with a scatter that

is not very localized then any departure from the proportional hazards assumption

would invalidate the inference about the covariate parameters by introducing possible

14

bias and nonrobust standard errors.

In summary, the parametric setup should be "viewed with caution" as Sen (1993)

elaborates primarily because "any departure from the assumed functional forms may

cause considerable damage to the validity and efficacy of statistical analysis based

on the assumed model." Although the semi-parametric models offer more flexibility

by allowing some arbitrariness of the distribution functions (for example the baseline

hazard in Cox's proportional hazards model is nonparametric in nature), yet they are

not robust to qepartures from the assumed models. For these reasons it seems more

safe to assume a broader model that is less restrictive, and that will preserve consis­

tency of the estimates while providing a reasonably efficient test statistics. Hence the

need for "nonparametric models" which are discussed in detail in the next section.

1.2.5 N onparametric Approach

Nonparametric formulations are more flexible but also more complex. Sen (1994)

states that the reason for this is that "any reduction of the statistical information

through only a few summaritative measures merits a much more careful consideration,

and often, a finite number of such measures may not suffice the purpose." Sen (1994)

lays down the foundation of nonparametric inference using surrogate variables by

considering a uniresponse model with a true outcome, Y, a set of covariates denoted

by Z, and a surrogate variable, X.

For the majority of experimental units, data are recorded only on (Xi, Zi), i E 1.

The set 1 is called the surrogate set. On another subset of experimental units, J*,

disjoint from 1, measurements are made on Y, X, and Z so as to allow assessment of

the relation between the true outcome and the surrogate. Thus, the set of all exper­

imental units would be 50 = 1 U J* and 51 = J*, which is a hierarchical design as

in (1.2.3). An IMD can also be considered by including a third subset, 1°, with data

recorded on (Y, Z), but not on X.

Denote the conditional distribution function (d.f.) ofY, given Z = z, by F(y I z),

15

apd the corresponding conditional survival function (sJ.) by F (y I z) [= 1-F (y I z)].

Also, let the corresponding functions of X, given Z = z, be denoted by G(x I z) and

G (x I z) respectively. Let H(y, x I z) be the conditional joint dJ. of (Y, X).

Two main goals of the analysis of an IMD are discussed by Sen (1992). The

first is when a test of the null hypothesis of no treatment effect is desired, and the

second relates to estimation of the treatment effects.

Hypothesis Testing

2.5.1.1 ~urrogate Sample Analysis

For the sake of simplicity, Sen (1992) deals first with the surrogate set, I, alone,

and then extends the method to the validation set, 1*. The Prentice (1989) criteria

for a surrogate translate here as concordance of F (. I z) and G (. I z), viewed as

functions of the concomitant variate z. The term concordance is used in·the usual

sense of concordance between two random variables, and the term concomitant variate

is used interchangeably throughout this work with the term ~ovariates. The method

proposed makes use of nonparametric analysis of covariance (ANOCOVA) tests that

were reported in Puri and Sen (1971) and Puri and Sen (1985). For extensions of

ANOCOVA tests to survival analysis see Chapter 11 of Sen (1981).

Let the vector of concomitant variates Z be expressed as Zj = (ci, ~D' , i 2: 1,

where the Ci are non-stochastic r(2: 1) vectors mostly relating to the design variables,

and the ~i are stochastic covariates. Note that random assignment of treatments to

subjects ensures the independent and identically distributed nature of the ~i's, which

is the basic assumption of ANOCOVA (in the parametric as well as the nonparametric

setups). Now, let II(~) be the marginal distribution of ~i, and let

16

l'he null hypothesis of no treatment effect is formulated in a nonparametric way as

Ho : G i (. I ~) = G(. I~)' i E I (1.14)

where G is a suitable yet unknown d.£. which is assumed to be continuous everywhere.

Let ~i's be p x I-vectors, and define the (p + 1) x I-vectors Xi = (Xi,~D', i E I. If

the cardinality of I is n then Xi leads to a (p + 1) x n matrix. Arrange the elements

in order of magnitude within each row of this matrix, and denote the corresponding

ranks by Rji,j = 0,··· ,p; i = 1,···, n. This (p + 1) x n matrix is called the rank

collection matrix. Define the r-row vectors linear rank statistics (for each j = 0"" ,p

separately) as

T nj = l)Ci - c)anj(Rji ), j = 0,··· ,p,iEI

(1.15)

where c = C£iEI ci)/n, and the anj(k), k = 1,···, n, are suitable scores (for example

Wilcoxon scores: anj(k) = k/(n + 1), k = 1,·'·, n). Let V n be the (p + 1) x (p + 1)

matrix with elements Vnjjl given by

Vnjjl = n-1I: anj(Rji)anjl(Rjli)- an/injl,iEI

for j,l = 0,·,· ,p, and anj= n-12::k=1 anj(k),j = 0,,·· ,p. Also define

n

C n = I:(Ci - C)(Ci - c)',i=l

(1.16)

(1.17)

and assume that Rank(Cn ) = r(2:: 1), and as n increases n-1C n converges to a

positive definite matrrix C. Let T~ = (T~ll'" , T~o)' be an p x r matrix, and write

((vnjjl))j,jl=l,... ,P as V nOO , and let Vno = (VnOl,···,Vnop ). Then proceeding as on page

365 of Sen (1981), fit a linear regression of the surrogate variate rank statistics on the

concomitant part in order to eliminate the effects of the concomitant variates. Define

the residual rank statistics-vector

and let

* (V )-1'VnOO = VnOO - VnO nOO V nO •

17

(1.18)

(1.19)

~inally, let

(1.20)

Sen (1994) proposes L no as a test statistic for testing the nul! hypothesis in

(1.14). Moreover, under (1.14), Lno is a permutationally (conditionally) distribution-

free test, and hence for small n, one may use an exact test, whereas for large n, L no

can" be approximated by the central chi squared distribution with r degrees of free­

dom under Ha. When r = 1, one can perform a one-sided test if one pleases based

on T* J(v* )1/2nO nOD .

2.5.1.2 Censoring

A little adjustment is needed in the above discussion if the design incorporates cen­

soring. In particular, if there is Type II censoring, i.e., if only data on the kn (out

of n) smallest ordered values of the Xi variable are available, where n-1 kn is close

to some pre-fixed a (0 < a < 1), then as in Chapter 11 of Sen (1981), the censored

version of (1.15) should be considered by replacing anO(Roi) in (1.15) by bnO,kn(Rod,

where

•

,i = 1,"" kn ,

for j > kn .

(1.21)

The censored version of Vnjj' also has to be considered, and proceeding as in (1.18-

1.20), one can define the statistics T~o(kn), v~ao(kn) and Lno(kn ), for every kn(1 ::;

kn ::; n). When kn is large, the exact (permutational) conditional distribution of

Lno(kn) can be approximated by the chi-squared distribution with r degrees offreedom

under Ho.

When there is Type I censoring (i.e., truncation at a prefixed timepoint T), then

if kn(T) = L,iEII(Xi ::; T), where I(.) is the indicator function, then under Ha, as

n ~ 00

-lk (T) a.sn n ~ aT,

18

0< aT < 1. (1.22)

(see Chapter 11 of Sen (1981)). The test based on Lno(kn(T)) is also a conditionally

(given kn(T) = kn) distribution-free test under Ho that has a large sample chi-squared

approximation as well. Moreover, if the Prentice (1989) criteria for a surrogate are

fulfilled, then to circumvent the loss in efficiency brought about by censoring, time­

sequential tests (such as the progressive censoring scheme (PCS)) of Chapter 11 of

Sen (1981) could be employed. Group sequential testing (GST) in the context of

repeated significance tests could be adopted also. However, if the Prentice (1989)

criteria can not be justified then validation sample analysis will be urgently needed if

valid inference about differential treatment effects on the primary endpoint is desired.

2.4.1.3 Validation Sample Analysis

Let the set of experimental units on which we have data for the primary, surrogate,

and concomitant (as well as design) variates be denoted by Iv with cardinality nv.

If C stands for the concordance between Y and X, then one way to incorporate the

validation set into the analysis is to perform first a preliminary test of

Ho : Pr(C) ~ 1/2. (1.23)

Now, corresponding to Xi, we have a (p + 2)-variate observation (Yi, Xi, ~i)', i E Iv,

and T no becomes here T nvo which is a 2 x r matrix with r elements for each of Y and

X. Also, V n is now V nv, a (p + 2) x (p + 2) matrix that can be partitioned as

where V nvOO is a 2 x 2 matrix, V nvO+ is 2 x p and V nv ++ is a p x p matrix. Then,

proceeding as in (1.15) through 1.19, let

T nvo - Vnvo+(Vnv++)-l~v'

V nvOO - VnvO+(Vnv++tlv~vo+

19

(1.24)

(1.25)

'I;he permutational covariance matrix of T~"o is given by Cn" ®Vn"OO, where Cn" IS

defined as in (1.17) with n replaced by nv and I by Iv. Then 1.23 can be reframed as

Ho : V~O,XY 2: 0 ver.sus HI: V~O,XY < 0 (1.26)

where V~O,XY is the population counterpart of V~.oo. An asymptotically normal test,

L~", of (1.26) can be performed using the theory developed in Chapter 8 of Puri

and Sen (1971). Now, if Z; is the critical level of L~. with TJ (0 < TJ < 1) being

the level of significance for this test, then if L~" < Z;, we reject the null hypothesis

of concordance, and we do not proceed to test the original hypothesis of treatm,ent

effects based on the surrogate set only. Alternative designs (for example, adaptive

designs) will then be needed to test the basic hypothesis in a valid and reliable man­

ner. In such a situation, Sen (1994) considers another test statistic, L~", of a parallel

hypothesis constructed from Iv such that the component of Tn"o corresponding to Y

is the dependent variable, and the other component corresponding to X, along with

T~" serve as the covariate rank vector. A simple linear regression is done as before

and L~" will be permutationally (conditionally) distribution-free with a chi-squared

large sample approximation with r degrees of freedom.

If, on the other hand, the null hypothesis of concordance is accepted, that is if

L~. 2: Z;, then Sen (1994) suggests combining statistical evidence from both I and Iv

to get a test statistic to test the original hypothesis of no treatment effect. One such

combination can be L~~" = L no + L~., and this test is also permutationally (condi­

tionally) distribution-free with a large sample central chi-squared distribution with

2r degrees of freedom under Ho. However, the fact that the degrees of freedom are

2r instead of r causes concern regarding the efficiency of L~~" in terms of asymptotic

power. Hence, Sen (1994) suggests combining the covariate adjusted rank statis­

tics from I and Iv before constructing the quadratic forms, and then constructing a

quadratic form in such a way that under Ho, the resulting test is permutationally

conditionally distribution-free with large sample chi- squared approximation with r

degrees of freedom.

20

Before outlining the details of such a test, it is worth mentioning at this stage

that if the Prentice (1989) assumption holds (based upon clinical considerations),

performing a preliminary test actually complicates matters due to the multiple tests

involved. Sen (1994) illustrates this point by pointing out that two tests are per­

formed, the preliminary test at significance level 1] , and the second stage test at level

(Y2, say. Then the overall level, say (Y, is given by

(1.27)

where 1~2 and l~~ are the critical levels of L~v and L~~v respectively. The basic prob­

lem here is that the pair (L~v' L~v)' and the pair (L~v' L~~v) may not be stochastically

independent, hence evaluation of (1.27) will be quite complicated (even in the asymp­

totic setup). That is why performing a preliminary test of concordance may not be

very appealing.

Sen (1994) considers analogues of the standard parametric procedures discussed

in Roy et al. (1971), though in a nonparametric setup. In the normal theory setup,

specifying the mean and covariance structure completely specifies the distribution of

the test statistic, whereas here, the covariate adjusted linear ~ank statistics, although

asymptotically multivariate normal, include some unknown functionals in the mean

vector and covariance matrix that need to be known to be able to construct the

test. One option then would be to use adaptive procedures, but the large sample size

required eliminates such an option. Hence, Sen (1994) prescribes the unique combi­

nation procedures which we shall describe in detail.

First, partition the residual vector T~vo in (1.24) and the dispersion matrix in

1.25 as follows

(1.28)

Also let

(1.29)

21

The statistics T~vo(x) and T~vo(Y:x~ are statistically independent under the permu­

tational model for large n, and permutationally uncorrelated for finite n. Moreover,

each of them, when normalized, is asymptotically multinormal. This prompted Sen

(1994) to consider the combination

T~:o = [(v~vxxt 1 + (vnvY:x t 1t 1{ (vn"xx )-lT~vo(x)+ (v~"(y:x))-lT~,,o(Y:X)}·

(1.30)

The permutational mean of T~:o is 0 and its permutational dispersion matrix is

(1.31)

The final step is to combine T~:o and T~o in (1.18) to get the suggested test statistic.

Denote the r x r matrix in (1.31) by AI, and let A 2 = v~oo.Cn where C n and v~oo

are defined in (1.17) and 1.19 respectively. Let

and let

W. - [A-1 + A-1]-1 A-It- 1 2 i' i = 1,2 (1.32) •

(1.33)

TO' is conditionally distribution-free, under the permutational model, with 0 mean

and covariance matrix [A~l + A21]-1, and the permutational multivariate central

limit theorem of Sen (1983) applies here. Hence the motivation for the overall test

statistic

(1.34)

The exact permutational distribution of LO' can be obtained by enumeration if n

and nv are small to moderate. But for large nand nv , the distribution of LO' is

asymptotically a chi-squared with r degrees of freedom. For local (i.e., Pitman-type)

alternatives, the asymptotic distribution of LO' is a noncentral chi-squared with r

degrees of freedom and noncentrality parameter t:,.L which depends upon the alter­

native and the score functions. The test based on LO' has, at least asymptotically,

22

~etter power than the two stage test in (1.27) if the Prentice (1989) assumption holds,

where it is not as favorable if there is reason to doubt the concordance between the

surrogate and the primary variate.

Moreover, both Type I and Type II censoring can be incorporated into these

test just like before in (1.21) and (1.22). The V matrices would be modified and we

proceed as in (1.28) through (1.34). The extension to repeated significance testing is

also conceivable albeit more complex.

In simple designs (e.g., two-sample or multi-sample models), it is possible to

construct functionals of the distribution functions F(. I z) and G(. I z) that help in

drawing statistical inferences about the treatment effects. Consider the functional

O(z) = O(F(. I z), and the functional ~(z) = ~(G(. Iz)). Examples of such functionals

could be the nonparametric regression quantiles (e.g., the conditional median), or the

conditional mean, although the latter is not preferred due to its sensitivity to the

tails of the F (. I z) and will not be considered here.

The dependence between the surrogate and the primary variates is reflected in

some suitable nonparametric functional, 'ljJ(.), where

O(z) = 'ljJ(~(z)), z E 3 z ; (1.35)

Note that in a hierarchical design, ~(z) may be estimated from the set 1, and 'ljJ(.) from

the set 1*. In an imd design, O(z) may be estimated from 1°, ~(z) from 1, and 'ljJ(.)

from 1*. Let 0(.) and ~(.) be conditional quantiles and consider the simplest model

of placebo versus treatment. The Prentice (1989) assumption may be tested here as

follows. Partition the concomitant vector Z as (Z(l), Z(2)), with Z(1) containing the

treatment and design variates, and Z(2) has the other possibly random concomitant

variates. In the placebo vs. treatment case, Z(1) takes only the values 0 for placebo,

and 1 for treatment.

The null hypothesis of no treatment relationship to the true outcome can be

formulated as

(1.36)

23

f9r all Z(l) and Z(2). Similarly, the null hypothesis of no relationship of the treatment

to the surrogate can be formulated as

(1.37)

for all Z(l) and Z(2). Testing the validity of the Prentice (1989) assumption is equiv­

alent here to requiring (1.36) to be testable through (1.37). Thus if O(Z(l), Z(2»)

depends on Z(l), then the concordance condition would have to be verified before

making any conclusions. If Zo and Zl are two distinct values of Z(1), then we may

require that for every Z(2)

(1.38)

This brings us to the domain of estimation of the functionals 0(.), e(.) and 'ljJ(.) in a

nonparametric way which we shall deal with in detail in the next subsection.

Estimation

Consider the simplest case of placebo versus treatment when ZP) = 0 or 1,

and when there is no concomitant variate, i.e., when Z(2) = o. For the set I, we

have two subsets corresponding to the placebo and treatment groups, from which

we estimate the quantiles e(O) and e(l) as the sample quantiles of the respective

distribution. For the validation subset, J*, Sen (1994) suggests considering the two

bivariate distributions of (X, Y) for the placebo and treatment groups, and then

estimating the sample quantiles from the marginal distributions for each sample alone.

Thus, from the validation subset, J*, we have the estimators

(1.39)

(1.40)

Also, from the surrogate set, I, we have the estimators ts(O), ts(1). By standard

multivariate nonparametric methods (see Chapter 5 of Puri and Sen (1971)), the

asymptotic distribution of the two bivariate points (tv(O),Ov(O)) and (tv(l), Ov(1)) is

~ ( tv(O) - e(O) ) '" Ar (0 r )nvo ~ .IV2 , 0,Ov(O) - 0(0)

24

where nvo is the cardinality of the subset of 1* for which Z = O. Also, if nvl stands

for the cardinality of the subset of 1* for which Z = 1, we have

(1.41)

where ro and rl are to be consistently estimated from the validation subsets, though

with a better rate of convergence if we assume that ro = r1= r. Similarly, if nso

and n sl are the sizes of the subsets of I for which Z = 0 or Z = 1, then

(1.42)

(1.43)

where /50 and /51 are consistently estimated from the respective samples, also with

a better estimate if we let IsO = /sl = /5. By (1.40) and (1.41), E(Ov(O) I tv(O))

and E( Ov(1) I tv (1 )) are both linear in tv(O) and tv (1 ) respectively, and hence the

motivation for the following estimators of 0(0) and 0(1)

(1.44)

(1.45)

The regression estimates, ~o and ~I, are to be obtained from classical linear inference

procedures by using estimates of r o, rI, lOs and /ls' This procedure applies also

when Z(I) takes more than two values.

Now, with the introduction of the concomitant variate, Z(2), we may have to

deal with three possible situations. The first arises when Z(2) is categorical in nature.

The same method of estimation discussed will still apply but based on the subsets

corresponding to the possible values of Z(2). So, if these values are denoted by ar, r =

1,' .. ,I<, then the functionals to estimate are

O(Z(I), a r ), e(Z(I), a r ), r = 1,· .. ,I<.

25

(1.46)

1;'he second case may arise when Z(2) is a continuous random variable. Consider a

simple model when Z(2) is a scalar random variable with a continuous distribution.

To estimate O(Z(l),XO) for a given Xo, we let

Wi = IZ?) - xol, i E I(and i E 1*).

Now, within each subset of I (and 1*) we consider a subset of observations for which

the Wi have the smallest k values, where k is not small but k/ nvo (or k / nvl, etc.)

is small. Based upon these k observations we proceed as in (1.40) through (i.45)

and estimate O(Z(1), xo). The theory for this is explained in Gangopadhyay and Sen

(1992a) and Gangopadhyay and Sen (1992b). It should be noted that the rate of

convergence in ((1.40))-(1.43) was .,jn, whereas here it is of the order n a for some

a < 2/5.

The third case is when some covariates are discrete and others are continuous. A

combination of the methods for the first and second cases will be a good prescription.

2.5.2.1 Concomitance Assumption

Up till now we have been assuming that the (Xi, Zi), i E I, has the same distribution

as (Xi, Zi) i E 1* so that O(z) and e(z) could be estimated consistently. But this may

'not be tenable in practice due to planned censoring of the surrogate or concomitant

variates that may damage the homogeneity assumption of the covariates made in

(1.14),' and hence, all the analysis made so far may be invalid. Sen (1994) gives

us some hope here by introducing some modifications that will extend the method

discussed to such situations.

Consider the case when 1= {i : Xi :::; xo} for some real xo. Here G(. I z) will be

right truncated at xo. Define

G(x I z)crx;;rz)'

26

x:::; Xo,

x> Xo

and consider the functional

which is different from e(z). On the other hand variates in 1* have values that

correspond to values of Xi exceeding Xo. Thus, H(x, y I z), for this upper tail, will

be truncated from the left at Xo with respect to Xi. Define

H (x y Iz) - H(x, y I z) - H(xo, y Iz) for x > Xo, Y E R+.L, - 1 H( I) ,- Xo,oo z

Then the corresponding marginal distribution of X is

G ( I ) - H(x,oo Iz) -H(xo,y Iz)L x Z - , x > Xo.

1 - H(xo, 00 I z)

and parallel to eR(Z), let

Thus, n s and n v are nonnegative integer random variables that add up to a

known n, and such that

n s P ( ) d nv p ( )- ---t Pr X ::; Xo an - ---t Pr X > Xo, as n ~ 00.n n

The asymptotic normality results of (1.40)-(1.41) on the estimates of fh(z) and

eL(Z), and ~L(Z) [as in (1.42-1.43)] will not lead to better estimates of O(z) [as in (1.44­

1.45)], unless eL(Z), eR(Z), OL(Z), and O(z) are related by some estimable functional

forms. Sen (1994) contends that although it is tempting to consider semiparamet­

ric models (such as the proportional hazards model of Cox (1972)) to enforce such

estimablity conditions, these models should be avoided here on account of lack of

robustness. Sen (1994) suggests as a partial solution, extension of the validation sub­

set, 1*, by including additional observations for which Xi ::; Xo. This will permit the

nonparametric estimation of the relationship between edz) and ~R(Z), and also the

relationship between O(z) and ~L(Z).

27

1.3 Synopsi~ of The Work Done

There are two main issues that need to be addressed when incorporating surrogate

variables in clinical trials. The first issue is economic in nature, while the second is

statistical. A reduction in the cost of a clinical trial, although desired, should not

be at the expense of unreliable statistical inference. This could happen, for example,

if one abuses surrogate endpoints by using them without considering carefully their

association with the true endpoint. Thus, in the design stage of the clinical trial, one

has to try and balance the practicality of primary and surrogate response measure­

ments (in terms of cost and difficulty), with the minimization to the extent possible

of the bias of the estimates and inefficiency of the tests involved. Hence, a proper

choice of the validation subset, 1*, seems to be essential.

Although this study will not focus on design issues of clinical trials that handle

surrogate responses, we will consider statistical analysis in incomplete multiresponse

design (IMD) settings of which the hierarchical design is a special case. The reason

for this is that it allows investigators greater flexibility than if we consider hierarchical

designs only. The contribution of this work consists in generaJizing Sen's method into

the multivariate case in a randomized block setting (complete as well as incomplete),

and to combine inter and intra-block information in a nonparametric way. This has

not been done before in the literature. Above all it is the introduction of surrogates

that is also new.

There are two scenarios that we will study in detail which I shall now describe.

As we have seen in (1.2.1), one surrogate may not suffice to draw valid inference

about the treatment effect on the primary variate. One major concern of scientists

that puts the validity of the surrogate endpoints in doubt is the possibility that treat­

ment may affect the true endpoint via other pathways than the surrogate. Thus, if

the investigators think that this is the case with a certain surrogate, they can use

measurements of other surrogates to account for these other pathways of treatment

effects on the primary endpoint. In this way we will have more information about dif-

28

f~rential treatment effects on the true endpoint and our inference would be valid. For

example, we mentioned in (1.2.1) that Baccheti et al. (1992) found that CD8 counts

as well as changes in hemoglobin and WBC during therapy add more information to

the effect of treatment on AIDS or death than CD4 counts alone. Thus, in many

cases use of more than one surrogate to predict treatment effect on a true outcome

is better than relying on only one surrogate.

The first scenario can thus be formulated as follows. A primary variate, Y, is

considered along with a q-vector of surrogates, X, and measurements on the concomi­

tant and design variates, Z, are obtained as well. An IMD here pertains to designing

three subsets of experimental units; the first subset, I, has measurements on X and

Z, and is termed the surrogate set. The second subset, called the validation subset,

1*, has measurements on Y, X, and Z. The third subset, 1°, contains measurements

on Y and Z.

The second scenario that we will focus on deals with a vector of primary vari­

ates, Y, a vector of surrogates X, and accompanying concomitant variates, Z. Here

surrogate and validation subsets will be considered also.

The methodology proposed here will be an extension of the methods in (1.2.5).

The tests proposed by Sen in (1.2.5) dealt with hierarchical designs only, and they

accomodate use of one surrogate for a primary variate. Keeping in mind that hier­

archical designs are a special case of an IMD, this research generalizes these tests

to the IMD setting. As pointed out in (1.2.5), statements were made regarding test

statistics and their properties without any detailed proofs. Supplying proofs in the

multivariate setting of the two scenarios above will be one of the tasks to be carried

out during this study.

The second chapter will focus on analysis of designs of the first scenario type.

Since it would take too much space here we will not go into the technical details, but

rather we content ourselves with only a motivation. The null hypothesis of interest is

the same as (1.14) when we consider the surrogate sample alone with the difference

29

that G(x I ~) is now a multivariate distribution function. The aim is to formulate

test statistics, based on linear rank statistics in (1.15), in each of the subsets 1,1*,

and 10, and then to come up with a unified test that combines information from all

these subsets similar to the test suggested in (1.34).

To be more specific, if we assume that all the surrogate variates can be recorded

with relative ease, then if we have reason to believe that the Prentice (1989) criteria

hold true, we will proceed with the surrogate sample analysis as in (1.2.5). Thus, we

will have now

which leads to a (p + q) X n matrix. The linear rank statistics are constructed as in

(1.15), but now T no will be a qr-row vector with r elements for each of the q surrogate

variables. V n is now a (p+q) x (p+q) matrix. We fit a multiple linear regression with

the q surrogates as the dependent variables and the p concomitant variables as the

independent variables. An appropriate residual rank statistic, say T~o, correspond­

ing to (1.18) will then be constructed. Censoring can also be incorporated here by

replacing the censored version of the score functions as in (1.21).

Simillar modifications have to made when constructing a residual rank statistic

for the validation subset, J*. T nvo is now a (q+ l)r-row vector with r elements for Y,

and r elements for each of the q surrogate variables. V nv is now a (p+ q+1) x (p+ q+1)

matrix, where V nvOO is (q + 1) x (q + 1), V nv O+ is (q + 1) x p and V n v++ is a p x p

matrix. We regress Y and X on the p concomitant variates, and then we consider

the residual rank statistic, say T~vo' corresponding to (1.24). We then partition T~vo

and the corresponding covariance matrix V~voo as in before. V~voo now becomes

where vnvYY is variance of Y, and v~vYX is a 1 x q vector, and v~vxx is a q x q matrix

of the covariance of X. We then construct a statistic similar to T~:o in (1.30).

An analogous treatment to the subset 10 would yield a similar residual rank

30

s}atistic, say T~oo, where no is the cardinality of ]0. Here we would regress Y on

the p covariates to obtain T~oo. We shall use T~oo along with the previously defined

residual rank statistics for] and 1*, namely T~o and T~:o respectively, to construct

an overall test statistic similar to (1.34). Thus, if we let TO·, as in (1.33), be

T O· W T**' W T*' W T*'. = 1 nv O + 2 nO + 3 noO (1.47)

where the weights Wi, i = 1, ... ,3 are to be defined as in (1.32), then the final step

consists of constructing the quadratic form similar to (1.34) that will be permuta­

tionally distribution-free, and will have a large sample chi-squared approximation.

The third chapter will be devoted to studying the second scenario. Here we will

make use of the nonparametric multivariate techniques in Puri and Sen (1971) in

order to come up with the combined test. We consider residual rank statistics based

on regressing the q surrogate variates on the p covariates in the surrogate sample.

Moreover, if Y is an s x 1 vector, then we regress the q + s variates on the p co-

variates to get the residual rank statistic in the validation subset, 1*. In the subset

]0, we regress the s primary response variates on the p concomitant variates to get

the corresponding residual rank statistic. Then we construc,t the combined test by

constructing the appropriate quadratic form as in (1.34).

It maybe noted in (1.47), that if, based on other considerations, the weights

are prespecified, then although n1/2To· will still be asymptotically multinormal, the

quadratic form, LO·, based on TO· and the prior weights may not converge to a chi­

squared distribution because the discriminant of such a form may not be equal to the

covariance matrix of the combined test-statistics, and thus Cochran's (1934) Theorem

may not apply. Here it would be appropriate to consider resampling methods, like the

bootstrap, when dealing with the distribution theory in the multivariate case. Further

note that a hierarchical design is a special case of an incomplete multiresponse design

so that the methods developed in this work will apply to such designs as well. Hence,

they will not be considered in detail.

Finally, the last chapter will study the power function of the test developed and

31

it,s asymptotic relative efficiency. An example will be based on a completed double­

blind placebo-controlled trial conducted by Burroughs-Wellcome, which treated 281

patients with advanced HIV disease. Of these, 137 patients were randomized to receive

placebo and 144 patients were randomized to receive a 250-mg dose of Zidovudine

(ZDV) every four hours. In this study CD4 counts were determined prior to treatment

and approximately every four weeks during therapy. The median duration of follow­

up was 120 - 127 days, at which point the study was stopped due to the superior

results of the ZDV arm in decreasing mortality.

32

Chapter 2

Methodology In Randomized

Block Design (RBD)

2.1 Introduction

This chapter will be devoted to studying the first scenario mentioned in (1.3) in

which a primary variate Y, a q-vector of surrogates X, and a p-vector of concomitant

variates, Z, are considered. As mentioned earlier the new treatment may affect the

primary endpoint through possibly more than one path. This model allows us to

account for these different paths.

An IMD here pertains to designing three subsets of experimental units: the

first is the surrogate set /, which has measurements on X and Z; the second is the

validation set, /*, which has measurements on Y, X, and Z; and the third subset is

/0, contains measurements on Y and Z.

Although this study will not focus on design issues of clinical trials that handle

surrogate responses, statistical analysis is considered in two design settings: the ran­

domized block design (RBD), and balanced incomplete block design (BIBD). The

main reason for considering factorial designs is to reduce the variability of the esti­

mates by eliminating the effect of one or more nuisance variables. Each of the above

33

designs will be considered separately for the subsets f, 1* and fO. The next section

will deal with the simplest design, namely the randomized block layout in each of the

subsets f, 1* and fO.

2.2 Randomized Block Design

In classical linear models theory, the analysis of data from a randomized block setup

makes the following assumptions when estimating parameters:

i) The block and treatment effects are additive.

ii) The errors are independent and homoscedastic.

iii) Blocks and treatments do not interact unless each cell contains two or more

observations.

Moreover, the errors are assumed to be normally distributed in case confidence inter-

vals or significance tests for the parameters are desired. Here, in the nonparametric-

setup, these assumptions are relaxed for the most part. We do not assume additivity

of blocks and treatments; we drop the homoscedasticity assumption of the errors,

rather we assume that the errors are independent random vectors each of which has

a continuous distribution that is symmetric in its arguments. The normality assump-

. tion is completely relaxed.

2.2.1 RBD For The Surrogate Set I

Consider a two-way layout with s blocks of r plots each where r different treatments

are randomly assigned to the plots. Assume there are no replicates for simplicity.

Thus the cardinality of f is N(= rs). The response in the ith block receiving the

jth treatment is a p + q-vector V ij = (xg), ... ,xi1),zg), ... ,Zi1))' = (Xij' ZiJ' of

34

measurements corresponding to the p covariates and the q surrogates. Consider the

model

V ij = J.L + ai +Tj + fij, j = 1,"', r, i = 1,"', S, (2.1)

where J.L is the mean effect, 01, ... ,as are the block effects, TI, ... , T r are the treat-

ment effects, and En,··', Esr are all p + q-vectors. The following assumptions are

made:

a) Vi = (Vill ···, Vir) has a continuous r(p+q)-variate c.d.£. Gi(u), u E Rr(p+q), i =

1,··" s.

b) The joint c.d.f. of Zi = (Zil,"', Zir) is symmetric in its r p-vectors. This

is the concomitance assumption of the covariate distribution in the analysis of

covarIance.

We wish to test the null hypothesis,

Ho : Tj = 0 j = 1,' .. ,r.

while the set of alternatives relates to shifts in location due to treatment effects.

The linear rank statistics discussed in (1.2.5) will be based upon the method of

ranking after alignment described in detail in Chapter 7 of Puri and Sen (1971). The

alignment procedure eliminates the block effect by subtracting from each observation

in a block a translation invariant symmetric function of the observations like the

block average, block median, the Winsorized or trimmed mean, etc. Let Vi. be such

a function. Define the aligned observations as

J. = 1 ... r. '" i = 1,'" ,So (2.2)

For the kth variate, rank the observations ug)*, ... , U};)* in ascending order of magni­

tude and denote by R~J) the rank of Ui~k)* in this set, for i = 1," . ,S, j = 1,' .. ,r, and

k = 1,'·· ,p+ q. Thus, there is a rank vector R ij = (RU),···, R~f+q))', corresponding

to V~· = (U(~)* ... U~~+q)*), i = 1 '" S J' = 1 ... r~J ~J'.' ~J " , " •

35

For each N(= rs) and each k = 1,' .. ,p + q, define suitable rank scores a~) =

(a~~ll"" a~:N)' where a~:i = J};)(jj(N + 1)), 1 :::; j :::; N. Moreover, J};)(u) is

defined in accordance with the Chernoff-Savage convention, that is J};)(u) satisfies

the following conditions:

(a) limN ---+ ooJ};)(u) = J(k)(u) exists for 0 < u < 1 and is not constant,

(b)

where

(k) 1[ (k)*]GN[i] (x) = -; number of Uii :::; x, k = 1, ... ,p + q, j = 1, ... , r,

and

(k)( 1~ (k) ( )HN x) = - LJ GN[jl x , k = 1, ... ,p + q.r i=l

Define

(k,k') 1 [ ( (k)* (k')* . ]GN[j.il(x, y) = -; number of Uii ,Uii ):::; (x, y) ,

for k, k' = 1,' .. ,p + q, j, 1= 1,' .. ,r with either j -=/:- I or k -=/:- k' or both.

Let a~)= N- 1 L~=l a~!a' Also let C}j~ = 1 ifthe a th smallest abservation among

the N values of Ui~k)* belong to the ph treatment, and let C}j~ = 0 otherwise, for

a = 1,' .. ,N, j = 1,' .. ,r. Now construct the linear rank order statistic for the kth

variate and the ph treatment, j = 1,"', r, k = 1,'" ,p + q

N

T (k) _ -1 "" CU) (k)N,i - s LJ NaaN,o.,

0.=1

This leads to the r x (p +q) matrix

TN = ((T~;))i=l,. .. ,T, k=l"",p+q

36

(2.3)

(2.4)

Define the rank collection matrix of order (p + q) x N by R';v

Partition RjV into s submatrices of order (p + q) X reach

(Rn ,' .. ,Rsr ).

R* = (R(p+q)xr '" R(p+q)xr) . . D. = (R. ". D.)N 1 "S ,.LLi ~1, ,.LLir, i = 1-"", s

Under the null hypothesis the distribution of U71'···' U7r is symmetric in the r

vectors and hence remains invariant under any permutation of the r vectors. Thus

the joint distribution of

remains invariant under the finite group 9s of transformations {9s} which maps the

sample space onto itself. The cardinality of 9s is equal to (r!Y. Typically a 9s is such

that

where (U?1'···' U?r) is a permutation of U71" . " U7n etc. i = 1,"', s. Let R~

denote the rank collection matrix corresponding to U~. Note that for every 9s E 9s,

there exists a R~ = 9sRN which is permutationally equivalent to RjV.

The distribution of R';v over its (N!)(p+q) possible realizations will depend on the

unknown c.dJ Gi , even when the null hypothesis holds. However, under Ho, U~ has

the same distribution as UN for all 9s E 9s, and hence, the conditional distribution of

U';v over {U~ = 9sU';v; 9s E 9s} will be uniform, each realization having the common

conditional probability (d)-s. This leads to the probability law, g:Js:

Under Ho, the conditional distribution of R';v over the (r!)S realizations {R~ =

9sR';v;9s E 9s} is uniform, each realization having the conditional probability (d)-s.

Since g:Js is completely specified, the existence of conditionally distribution-free tests

for Ho is thus established.

Letr

-(k) -1"'" (k)a (k)= r ~a (k)N,R N,R•. j=1 'J

be the intrablock averages for the kth variate, k =1,,·' ,p + q.

37

Theorem 2.1 Let V N be the (p + q) x (p + q) matrix with elements Vkk' given by

[1 ~~( (k) :;(k)

Vkk' = s(r -1) LL aN,R(k) - aN,R(k»)1=1 t=1 It l.

(k /) :;(k/)]x(a (k') - a (kl))

N,Rlt N,RI.(2.5)

k, k' = 1,'" ,p + q. Moreover, let eN to be the r x r matrix with elements Cjjl given

by

1 (C 1)'" 1C' 'I = - 0 "/r - ),) = ,"', rJJ sr JJ

where 8jj , is the usual Kronecker delta. Then,

(2.6)

E(TN)

Var [Vec (TN)]

(~)" ", aX;+q)) 0 J

where J is an r X 1 vector of ones and Var [Vec (TN)] is a r(p+q) x r(p+q) dispersion

matrix.

Proof: Note that

and

E ((k) )2 1 ~ ( (k) )21". aN,R(k) = N L aN,ex .

'J ex=1

Also, for all i,i'= 1,···,s j,l = 1,···,r and k,k' = 1,···,p+q

To justify the first equality above note that 1can take any of 1, ... ,s with probability

~, m can take any of the remaining s - 1 numbers with probability S~1' t, u can take

38

-apy of 1, ... ,r with probability ~, and under ps the blocks are independent. On the

other hand if i = i',j =I j', then

1 ~ ~ (k) (k)

( 1) L.... L.... aN R(k)aN R(k)sr r - 1=1 t:;eu=1 'It 'I"

1 {~(~( (k) ))2 ~ ~((k) 2}sr(r -1) ~ ~ aN,R~;) - ~~ aN,R~;)) .

_ 1 {~r2(a(k»)2 _ ~(a(k) )2}sr(r _ 1) L.... N,R(k) L.... N,OI

1=1 l. 01=1

and

1 s T

"'" "'" (k) (k') if k -I- k',)' -I- ).,-(--1-) L.... L.... aN R(k)a (k') r rsr r - 1=1 t:;eu=1 'It N,RI "

and

1 ST

"'" "'" (k) (k'). ,-L....L.... a (k)a (k') If k =I k,sr 1=1 t=1 N,R/t N,RIt

Thus, we have

and

. (k»)Var p , (TN,j

39

(2.7)

Moreover,

The expected value in the first term in (2.7) above contains products of scores in the

same block and same treatment. Such a product occurs wjth probability ~ under

~S. The expected value in the second term consists of products of scores in different

blocks but same treatment. Keep in mind that the blocks are independent under ~S'

Also,

E [(8-1 ;'" C(j) a(k) )(8-1 ;... C(j') a(k/»)]

Ps L...J NOI N,OI L...J NOI N,OI01=1 01=1

8-2 [;... C(j) C(j')E (a(k) a(k')) + ;... C(j) C(j') E (a(k) a(k'))] (2.8)L...J NOI NOI Ps N,OI N,OI L...J NOI N{3 Ps N,OI N,{301=1 0If:{3=1

-2 [ 1 ~ ~ (k) (k') ~ E (k) (k') )]8 r(r _ 1) L...J L...J aN,R(k)aN R(k') + L...J Ps aN,R(k)aN R(k')

1=1 tf:u=1 It 'Z" If:m=1 I)' m)'

-2 [ 1 ~ ~ (k) (k') ~ 1 ~ (k) ~ (k') ]8 r(r -1) ~ t/:::l aN,R~~)aN,R~~/)+1~1 r 2 ~aN,R)~) ~ aN,R~2

-2 [ 1 ~ (~~ (k) (k') ~ (k) (k') )8 r(r _ 1) (:: 8 ~ aN,R~~)aN,R~~') - ~ aN,R~~)aN,R)~/)

40

The expected value in the first term in (2.8) above contains products of scores in the

same block but different treatments. Such a product occurs with probability T(T~l)

under ps. The expected value in the second term consists of products of scores in

different blocks and different treatments. Hence,

(k) (k'))Covps(TN,j' TN,j

-(k)-(k')-aN aN

Also, for j =1= j' we have

•Denote the marginal c.d.f. of Ui~k)* and of (Ui~k)*,USk')*) by G~VJ(x) ~nd G~V.l](X,y)

respectively, for j,l = 1,·· . ,r, k, k' = 1,· .. ,p + q, with at least one of j =1= 1, k =1= k'

41

~eing true, and let

1 S T

jj~) (x) = - L L G~VJ(x), fork = 1,'" ,p + q.sr i=l j=l

Define the monotone transformation

W " - (W(l) W(p+q)),·· - 1 . - 1 .tJ - ij' ... 'ij ,J - ,,"', r, Z - ,"', s,

bkkl 'I' - E(W(k)W(k' )) k k' - 1 ... p + q J' - 1 ... r'.J ,t - ij iJ ' , -" ,-",

S

b(s) - -1" b k k' - 1 . - 1 .kk'.jl-S .L..J kk'.jl,i , - ,"',p+q, J- ,"',r,i=l

..

Finally let

I T 1 T T

" (s) "" (s)Vkk',N = - .L..J bkkl. jj - 2' .L..J.L..J bkkl. jl ,r j=l r j=l 1=1

k,k'= 1,"',p+q; (2.9)

VN = ((Vkkl ,N))k,k'=l'''',P+q' (2.10)

Now by Lemma 7.3.10 of Puri and Sen (1971) V N :e.., VN, and V N is positive definite in

probability when VN is positive definite. Now, using this fact and conditions (a), (b),

and (c) we arrive at the following thoerem.

Theorem 2.2 For each j = 1,"', r, the 1 x (p + q) row- vector of TN in (2.4)

is asymptotically normal with mean equal to (aW,· .. ,~+q)), and dispersion matrix

Cjj V N, where Cjj is the diagonal element of eN.

For proof see Theorem 7.3.3 of Puri and Sen (1971). For each j separately, j =

1,"', r, let us partition the corresponding 1 x (p + q) row vector of TN in (2.4) into

two components after subtracting the mean; the first component, TNo,j, is a q x 1

vector of the centered linear rank statistics corresponding to the surrogate variables

that is

TNo . = (TN(1) - a}N1) ... TN(q) - a}Nq)),,J ,J ",J '

and the second component, Tt,j, is a p x 1 vector of centered linear rank statistics

corresponding to the concomitant variates that is

TO . = (T(q+1) - a(q+l) ... T(p+q) - a}p+q)),N,J N,J N" N,J N .

42

..

S;imillarly, we partition the matrix V N as follows

(VNOO VNO+)V N =

. V~ci+ V N++(2.11)

where V NOO is a q x q matrix, V NO+ is q x p and V N++ is a p x p matrix. Then, from

the classical normal theory, (see Theorem 2.5.1 of Anderson (1984)), for large N we

have

Define the q x 1 residual rank-statistics

V NOO V V V (-l) V'NOO - NO+ N++ NO+ (2.12)

Now, for each j = 1,···, r, obtain similarly a residual vector, consider the q x r

residual matrix corresponding to the different treatments

(2.13)

where

and

Moreover, the dispersion matrix of sl/2Vec (TNO ) is the qr x qr matrix eN ® V NOO .

It maybe noted that censoring can also be incorporated in the design and we would

then use the censored version of the score functions as in (1.21).

2.2.2 RBD For The Validation Set 1*

Consider a two-way layout with n* blocks of r plots each where r different treat­

ments are randomly assigned to the plots. Since there are no replicates the cardinality

43

o,f 1* is N v (= n*r), say, thus allowing the number of subjects in the validation set to be

different from the number in the surrogate set. The response in the ith block receiving

h ·th t . 1 U·· - ("l-":. X(I) X(q) Z(I) Z(p))' ft e J reatment IS a p + q + -vector tJ - .I iJ' ij'···' ij' ij'···' ij 0

measurements corresponding to the primary variate Yij, the p covariates and the q

surrogates.

Consider the same model and assumptions as in (2.1) with the only difference

that now Ell,·· . ,En • r are all p + q + 1 vectors. That is to say we regress the pri­

mary variate and surrogates on the concomitant variates. The method of ranking

after alignment will also be used here. Proceeding as in proof of Theorem 2.1 with

N replaced by N v , and s by n*, so that k ranges now from 1 to p + q + 1 and hence

V N v is a (p + q + 1) x (p + q + 1) matrix, then, corresponding to (2.11) we have

V NvO+ )

V Nv ++

(2.14)

where V NvOO is ~ (q + 1) x (q + 1) matrix, V NvO+ is (q + 1) x p and V N v++ is a p x p

matrix. Moreover, corresponding to (2.13) let the (q + 1) x 1 residual rank-statistics

and dispersion matrix be

T*NvO,j

V V V(-I) V'NvOO - NvO+· N v++ NvO+ (2.15)

Now, for each j = 1,···, r, we obtain similarly a residual vector. Consider the

(q + 1) x r residual matrix corresponding to the different treatments

(2.16)

Partition the residual matrix in (2.16) and the (q +1) x (q + 1) dispersion matrix in

(2.15) as follows:

44

'Yhere vNvYY is 1 xl,. vNvYX is 1 x q, and V NvXX is q x q. Also let

(2.17)

(2.18)

Note that the dispersion matrix of the 1Xr vector n*1/2 TNvO(Y:X) is given by vNv(Y:X)CNv '

where the elements of CNv are given by (2.10) with N replaced by Nv .

2.2.3 RBD For The Set 1°

Consider a two-way layout with nO blocks of r plots each where r different treat-

ments are randomly assigned to the plots. Since there are no replicates the cardinality

of 1° is N°( = nOr). The response in the ith block receiving the jth treatment is a

p + I-vector Uij = (lij, zg), ... ,Zi~»)' of measurements corresponding to the pri­

mary variate lij, and the p covariates.

Consider the same model and assumptions as in (2.1) with the only difference

that now en,···, enO r are all p + 1 vectors. The method of ranking after alignment

will also be used here. Proceeding as in the proof of Theorem 1 with N replaced by

N°, and s by nO, also with k ranging from 1 to p+ 1 and hence V NO is (p+ 1) x (p+ 1)

matrix which is partitioned as we partitioned V N Next partition V NO as

where vNOOO is 1 xl, vNoo+is 1 x p, and V NO++ is P x p. Moreover, corresponding to

(2.12) let the (p + 1) x 1 residual rank-statistics and its dispersion matrix be

(2.19)

Now, for each j = 1,· .. ,r, obtain similarly a residual rank statistic. Finally, let the

1 x r residual vector corresponding to the different treatments be

(2.20)

45

penote by VNoooCNo, the dispersion matrix of nOl/2TNOO' where CNo is the r x r

matrix whose elements are given by (2.6) with N replaced by N°.

2.2.4 Construction of the Test Statistics"

The next step consists of combining information from all subsets. Since the tests

developed in the different subsets are independent, we will take a weighted linear

combination of these tests with the weights being the inverses of the corresponding

dispersion matrices. Note that for such a combination to make sense the dimensions

of the tests should be the same.

One possible way to get around this difficulty would be to reduce the order of

the q x r matrix in 2.13 into a 1 x r vector by taking a linear combination of this

matrix, say aTiVo, where a is of order 1 x q, subject to the two restrictions, i) a'a = 1,

and ii) the variance given in (2.12) is minimum. Although by doing this we are losing

information, the minimum variance condition maximizes the noncentrality parameter

of the chi-square distribution of the quadratic form based on the transformed residual

vector, which in turn maximizes the power of this test. Moreover, a can be ordered

if there is inherent ordering in the surrogate vector. Note that

..

TNO(X) aTNO

(aTNO,I' ... , aTNO,r)

.with dispersion vNXCN, where VNX = aViVooa'. To see this

Cov(aTNO,j' aTNO,jl) aCov(TNO,j' TNO,jl )a'

cjj'aVNOOa'

where cjj' is the corresponding element of CN. Let Al = VNXCN, A2 = VN"(Y:X)CN,,,

and A 3 = vNO OO CNO. Let .

W · = [A-I +A-I + A-I]-IA~I . 1 2 3t 1 2 3 t' Z= , ,

46

(2.21 )

~ow, noting that the mean of each of the three statistics TNO(Xl' TNvO(Y:X), and T NOO

is 0, following the Gauss-Markov Theorem we consider the linear combination

T O· W T*' W T*' W *'= I NO(X) + 2 NvO(Y:X) + 3T NOO (2.22)

Assume that :. ~ PI, and :0 ~ P2 for positive and finite real PI and P2. Then SI/2To·

is conditionally distribution-free, under the permutational model, with °mean and

covariance matrix

and the permutational multivariate central limit theorem of Sen (1983) applies here.

Thus, for large N, Nv , and N°, SI/2To· "" N,. (0, A). Finally consider the overall test

statistic

(2.23)

The exact permutational distribution of LO· can be obtained by enumeration if N,

Nv , and N° are small to moderate. It maybe noted that rank(CN) = rank(CNv ) =

rank(CNo) = r - 1. Thus for large N, Nv , and N° the null distribution of LO· by

Cochran (1934) is asymptotically a chi-squared with r -1 degrees of freedom since it

is a quadratic form of asymptotically normal variates with discriminant of rank r - 1.

2.2.5 Asymptotic Non-null Distribution of LO·

For the study of the non-null distribution of LO· we will consider local alterna­

tives only. Fixed alternatives lead to consistent tests in the sense that the power

goes to one in large samples, which makes it hard to compare them. Thus they

will not be considered further. Let us first work with the surrogate set I. Write

Gi(u) = Gi(X,Z), x E n:q, z E Rrp

, and consider the following sequence of alterna­

tive hypotheses:

(2.24)

47

where ..x = ((.\;k»)) stands for an r x q matrix of treatment effects. Let G~~]

S-1 Li=1 G~~~, 1 :s; j :s; r, 1 :s; k :s; q, and let

Thus, under {HN }

j=l,···,r, k=l,···,q;

Assume that G~VJ(x), j = 1,"', r, k = 1,"', q, i = 1"·,, s are all absolutely

continuous. Moreover if each G~k) ~ G(k) then lims-+oo S-1 Li=1 G~k)(x) = G(k)(x).

Also, asymptotically jj~) (x) is equal to G(k)(x), which is the limit of G~~](x). Let

g(k) (x) be the density function corresponding to G(k) (x).

Now expand G(k)(X - N-1/2>..Y») in a Taylor series around x for a fixed x. We

have for large N

(k)J1N,j JOO (k) -(k) -(k)

-00 J [HN (x)]dGs[j] (x),

1: J(k) [G(k)(x)] dG(k)(x - N-1/2>..;k») + O(N-1/2)o

i: J(k) [G(k)(x)] dG(k)(x) +i: J(k) [G(k)(x)] d [G(k)(x - N-1/2>..;k») - G(k)(x)]

+o(N-1/

2)

•

Integrate by parts the second integral in the last term

J1~!j lo1 J(k)(u)du -1: [G(k)(x - N-1/2>..;k») - G(k)(x)] :x J(k)(G(k)(x)) + o(N-1/2)

[1 J(k)(u)du + (N-1/2)>..;k) JOO ~J(k)(G(k)(x))g(k)(x)dx + O(N-1/2)Jo -00 dx

[1 J(k)(u)du + (N-1/2)>..;k) joo ~J(k)(G(k)(x))dG(k)(x) + O(N-1/2)Jo -00 dx

where the last two steps resulted from a Taylor series expansion of G(k)(x_(N-1/2>..Y»)

around x for a fixed x. Thus, if we let

48

then from the above we have

N1/2[J1~!j - fa1 J(k)(U)dU] -+ A~k)B(G(k)), j = 1,···,r as N -+ 00

By Theorem (7.3.12) of Puri and Sen (1971) [N1/2(T~} - J1~!J, j = 1,···, r, k =

1, ... , p+q] has asymptotically a multinormal distribution with null mean and disper­

sion matrix that converges to the same dispersion matrix under the null hypothesis

namely, VN. Also by condition (a) of (2.2.1) we have

Thus in the light of the above discussion

(2.26)

Now, by Lemma (7.3.10) and by the convergence model in Section (7.2.4) of Puri

and Sen (1971), 'VN '" v defined in (2.10) with bkkl.jjl all defined for the limiting

distributions of the average c.d.f. 'so Let v I be the version of v in the surrogate set.

Partition VI as in (2.11) and corresponding to (2.12) and (2.13) let

T * T -1 TONO = NO - VI0+ V l++ N

Thus, let 111 = ((1])k))) where 1])k) = A)k)B(G(k)), j = 1,···,r, k = 1,···,q. Partition

11 I into 1110 corresponding to the surrogates, and 11~ corresponding to the covariates.

Let TNO(x) = aTNO' then for large n

E ( 1/2T* IH ) ( -1 0)' *S NO(X) N -+ a 1110 - vlo+vI++111 = #-tID

and its dispersion matrix is vixeN where vix = avjooa'.

In a similar fashion, define {HN} for the validation set 1* such that ((A)k))) is

now an r x (q + 1) matrix. Also, 111. is defined in the same way as 111 with the

49

difference that k = 1,· .. ,q + 1. V Nv rv VI* which is the version of VI defined in the

validation set ]*. Thus, corresponding to (2.14) we have

Partition the r x (q + 1) expected value matrix "71* and the (q +1) x (q +1) dispersion

matrix V 1* as follows

* ( vj*yyv I • OO =

*'vI*YX

where vj*yy is 1 x 1, vj.yX is 1 x q, and vj.xx is q x q. Hence, for large n*

d h d·· . f *1/2 T * . * C han t e IsperSlOn matrIX 0 n NvO(Y:X) IS vI*(Y:X) 1* were

Finally, along the same lines, we define {HN } for the subset ]0 such that .\ is now

an r x 1 vector. Also, "710 is defined in the same way as "7-[. V NO rv V 10 which is

the version of V I defined in the subset ]0. Thus, for large n the expected value of

no1

/2Tl'voo under {HN } converges to p,joo = "710 with dispersion matrix VIOOOCIO.

Consider the weights Wi defined in (2.21) where now we have Al = vjXCN,

A 2 = vj*(y:X)CNv ' and A 3 = vjoooCNo. Thus, under {HN }, the mean of n1/

2To* in

(??) is for large n

Therefore, from Theorem (7.3.12), Lemma (7.3.10), and Theorem (2.8.2) of Puri and

Sen (1971) and under {HN}, LO· defined in (2.23) has asymptotically a noncentral

chi-squared distribution with r -1 degrees of freedom and the noncentrality parameter

50

Chapter 3

Nonparametric Intra-Block

Inference

Blocking in experimental design is .done in order to use experimental units as nearly

homogeneous as possible. Complete block designs lose their efficincy due to the fail­

ure to eliminate heterogeneity among units when the number of treatments to be

compared is not small. In such cases, incomplete block designs are used where ex­

perimental units are divided into blocks containing fewer units than the number of

treatments to be compared. Comparisons of treatments with equal accuracy may re­

quire equal number of replicates and other constraints leading to balanced incomplete

block designs (BIBD).

3.1 Balanced Incomplete Block Designs

3.1.1 BIBD for Surrogate Set I

Consider n replications of a BIBD consisting of s blocks of constant size r(~ 2) to

which v treatments are applied such that:

(i) No treatment occurs more than once in any block,

51

(ii) The ph treatment occurs in rj(~ s) blocks j = 1,··· ,v, and

(iii) The (j, j')th treatments occur together in rjj'(> 0) blocks (j =I- j' = 1, ... ,v).

Let Si stand for the set of treatments occurring in the ith block, i = 1; ... ,s. For the

a th replicate, the response in the ith block receiving the jth treatment is a stochastic

V (X(l) X(q) Z(l) Z(p))' (X' z' )' fp + q-vector exij = exij, .•. , exij, exij,···, exij = exij, exij 0 measurements

corresponding to the p covariates and the q surrogates. Consider the model

V exij = JL ex + f3 exi + Tj + Eexij, j E Si, i = 1,···, s, a = 1,···, n (3.1)

where JL ex is the replicate effect, f3 exi is the block effect, TI,···, Tv are the treatmentI

effects, and Eexij are the error vectors. The following assumptions are made:

a) V exi = (Vexij,j E Si) has a continuous r(p + q)-variate c.d.f. Gi(u), u E

Rr(p+q), i = 1,··· ,So

b) The joint c.d.f. of [Zexij, j E Si] is symmetric in its r p-vectors. This IS the

concomitance assumption in ANOCOVA.

c) [Eexij,j E Si] have a jointly continuous cumulative c.d.f. G(2:I,···,2: r ) which is

symmetric in its r vectors. This includes the i.i.d. assumption of the distribution

of [Eexij,j E Si, i = 1,··· ,s] as a special case.

d) The joint distribution of [Zexij, Eexij,j E Si] is exchangeable and is the same for

all i = 1,··· ,So

Note that the Tj is a (p + q)-vector with two components: Tjl is a q-vector of

treatment effects corresponding to the surrogates, and Tj2 a p-null vector of treatment

effects for the covariates, since we assume no interaction between treatment and

concomitant variates. We wish to test the null hypothesis, How,

How: Tjl = 0 j = 1,·",v,

52

..

while the set of alternatives relates to shifts in location due to treatment effects. To

adopt the method of ranking after alignment [see Chapter 7 of Puri and Sen (1971)],

define the aligned observations by

U:ij = U Olij - r-1 L U Olil j E Si, i = 1,"', S, a= 1"·,, n.

IESi

Let

-1 "" . S . 1Tj,i = Tj - r L..J TI, J E i, Z = ,.", S.

IESi

-1 ""eOlij = EOlij - r L..J Eoil,

IESi

Then we have

j E Si, i = 1, ... ,S, a = 1,' .. ,n.

U:ij = Tj,i + eOlij, j E Si, i = 1,,·, ,S, a = 1",· ,no

Thus if F(zl,···, zr) is the c.d.f. of eOlij,j E Si, and if Fr(p+.q) stands for the class of

all G(Zl,···, zr) which are symmetric in their r (p + q)-vectors, then

Moreover, the joint distribution of (Z:ij' eOlij, j E Si) is als0 exchangeable. For the

kth variate, rank the observations ugi*,···, u~~~* in ascending order of magnitude

and denote by R~~} the rank of Ui7]* in this set, for i = 1, ... ,S, j E Si, a = 1, ... ,n,

d k 1 Th d· U* (U(l)* U(p+.q)*), han =, ... ,p + q. us, correspon mg to Olij = Olij,···, Olij we ave a

k R (R(l) R(p+.q)),· 1 . S d - 1ran vector Olij = Olij, ... , Olij , Z = ,... ,s, J E i, an a - , ... , n.

For each N(= nsr) and each k = 1,'" ,p + q, we define suitable rank scores

a (k) - (a(k) ... a(k) ) where a(k). - fk)(J'j(N + 1)) 1 < J' < N Moreover J(k)(u)N - N,l' 'N,N' N,] - N , - -' , N

is defined in accordance with the Chernoff-Savage convention, i.e., we assume that

JJ.P (u) satisfies the conditions a, b, c of Section 2.2.1. For notational simplicity, let

TJ (k) - a(k) J' E S· ,; - 1· S '" - 1 ... n'Olij - (k)' ., • - , •• , , ~ -, "N,R"'iJ

s n

TJ~:~ = r-1 L TJ~:;, TJ~~! = S-l L1J~:~, and 1J.\~) = n-

1 L 1J;:'!jESi i=l 0l=1

53

f<?r k = 1,' .. ,p + q. The proposed test is based on the statistics

(k) _ 1~" (k) . _ _TN,j -;; ~.~ T/aij' J -l,···,v,k -l,···,p+q.

a=l zEPJ

where Pj = {i : j E Sil, j = 1," " v. Let TN = ((T~~))j=l,. .. ,v,k=l'."'P+q.

Tests based on aligned ranks are only conditionally distribution free. Define

U~i = (U~ij,j E Si) ,Ua = (U~l""'U~s), a = 1,···,n, and URr = (Ui,···,U~).

Hence, by (3.2) we have the following three arms of the permutational law:

(i) The joint distribution of U~i remains invariant under any permutation of the r

vectors among themselves, there being r! such permutations, i = 1,' .. ,s,

(ii) As U~i' i = 1,' .. , s are i.i.d., the joint distribution of U~ remains invariant under

any permutation of the s sets U~i' i = 1," " s, among themselves, there being

s! such permutations,

(iii) As replicates are independent, the joint distribution of URr remains invariant

under any permutation of the n sets U~, a = 1,"', n among themselves.

Thus, if we define (In to be the compound group of transformations {9n} by

U o . U o* [(l)U* (n)u*] (a) r. 1On N = N = 9n 1"" ,9n n' 9n E ~n, a = ,"', n.

Then (In contains [s!(r!yt transformations, and under How it leaves the joint distri­

bution of U~ invariant. Let S(U~) = {U% = OnU~ : gn E (In}. Then, it follows

from the above discussion that

P [U?v = uo; IS(U?v), How] = l/N*;N* = [s!(rwt,

for all uo; E S(U?v), whenever G E Fr(p+q). Let us denote this conditional (permu­

tational) probability measure by rn. As rn is completely specified the existence of

conditionally distribution-free tests for How is thus established.

54

"

Theorem 3.1 Let

I ns(1) _ "" "" "" [(k) (k)] [(k') (k')]

VN,kk' - -s- L.J L.J L.J "lOlij - "lai. "lOlij - "lai. ,n r 01=1 i=l je8i

and

k, k' = 1, ... ,p + q,

I ns(2) _ "" "" [(k) (k)] [(k

l

) (kl)]

VN,kk' - - L.J L.J "lai. - "la.. "lai. - "la.. ,ns 01=1 i=l

k k' = 1 .. , p + q', " ,

Y (l) (( (1»)) d y(2) (( (2) ))N = VN,kk' an N = VN,kk' .

Also, let

A(l) = ((aJ~~)) and A(2) = ((aJ~~))

where

aJ~~ = [srjj' - rjrj'] /(s - 1), j,j' = 1,"', v,

where hjj , is the usual Kronecker delta. Finally, let

BN is a (p + q)v x (p + q)v matrix. Then

Proof: Note that for a given 0:',0:' = 1," . ,n, under rn the probability that i will be

any of 1" .. ,s is ~, and the probability that j E Si is ~, hence

1 ~ "" ( (k»);:; L.J L.J E pn "lOlij. 01=1 iePj

H; i~ Ur t, j~, ~~~l)r .71(k) i = 1 .. , s J' = 1 ... v k = 1, ... ,p + q.J'/... , , " ,

55

¥oreover, let A = Hi, i') : i E Pj, i' E Pj" i =I- i'}, and note that then the cardinality

of A is rjrj' - rjA= rj(rj - 1) if j = j'). Then

nE [r(k) - E (r(k))] [r(k') - E (r(k'))]Pn N,) N,) N,) N,)

[(1~"( (k) (k))) (1 ~" ((k') (k/)))]nEpn - L- L- 1]OIij - 1]01.. X - L- L- 1]{3i'j - 1]{3 ..

n 01=1 iEPj n {3=1 i'EPj

n

n-1 I: I: E pn [(1]~:} -1]i~~) (1]~:; - 1]i~:))]01=1 iEP}

n

+n-1 " " E [(1](~). _1](k)) (1](~;) _1](k' ))]L- L- Pn at) 01.. 01' ) 01 ..01=1 (i,i')EA

n-1 "" [( (k) (k)) ((k') (k'))]+n L- L- E pn 1]OIij - 1]01.. 1]{3ij - 'fJ{3 ..

0I#{3=1 iEPjn-1"" [( (k) (k)) ((k') (k'))]+n L- L- E pn 'fJOIij - 'fJ0I .. 'fJ{3i'j - 1]{3 ..

0I#{3=1 (i,i')EA

Note that the last two terms are null because the replicates are independent and

E pn ('fJ~:; - 'fJi~:) = O. The first term yields

n

-1 " " [( (k) (k)) ((k') (k'))]n L- L- E pn 'fJOIij - 'fJ 0I .. 'fJOIij - 1]01 ..01=1 iEP)

56

..

The second term gives

n-1 "'" "'" E [( (k) (k)) ((k

/) (k

/))]n ~ ~ Pn TJOIij - TJ OI .. TJOIilj - TJOI ..

01=1 (i,i/)EA

-1 ~ "'" E ((~) (~/).) _ rj(rj - 1) ~ ( (k) (k /))n ~ ~ Pn TJOI,)TJOI,I) ~ TJOI .. TJOI ..

01=1 (i,i/)EA n 01=1

- n-1t 2: [ 1 t (~2: (k)) (~L (~/))]01=1 (i,i/)EA S(S - 1) i#i

'=1 k jEBi TJm) k jEBi, TJOI,,)

_ rj(rj - 1) ~ ( (k) (k /))~ TJ OI .. TJOI ..

n 01=1

-1 ~ "'" [ 1 (~~ (k) (k/) ~ (k) (k

l))]

n ~ ~ S(S _ 1) ~ ~ TJOIi. TJOIi ' . - ~ TJOIi. TJOIi.01=1 (i,i/)EA i=1 i ' =1 i=1

_ rj(rj - 1) ~ ( (k) (k /))n ~ TJ OI .. TJOI ..

rj(rj - 1) [~82 (k) (k /) _ ~ ~ (~) (k /)]

( 1) ~ TJOI.. TJOI.. ~ ~ TJOI'. TJOI'.n8 S - 01=1 01=1 i=1

_ rj(rj - 1) ~ ( (k) (k /))n ~ TJOI .. TJ OI ..

rj(rj - 1) ~~ ( (k) (k /) _ ~ ~ (~) (~/))]( 1) ~ TJ OI .. TJOI.. ~ TJOI .. TJm.

n 8 - =1 8 i=1

rj(rj - 1) ~ ~ ( (k) (k /)) (k) (k /)- (8 _ 1) ~ ~ TJOIi. TJOIi. - 8TJOI .. TJOI ..

n8 01=1 i=1 .

_ rj(rj - 1) {~ ~~ [ (~) _ (k)] [ (~/) _ (kl)]}

( 1) ~~ TJw. TJOI.. TJOI'. TJOI ..8 - n8 01=1 i=1

r·(r· -1) (2))) v- (8 - 1) N,kk'

Hence, the sum of the two terms give

C (T (k) T(k/))n ovPn N,j' N,j

(1) (1) (2) rj(rj - 1) (2)ajj'vN,kk' + rjVN,kk' - (8 _ 1) vN,kk'

(1) (1) (2) (2)a· "vN kk' +a· 'IvN kk')), )),

On the other hand, let B = {(i, i') : i E Pj, i' E Pjl, i = i'}. The cardinality of B is

rjj', Thus

nE [T(k). - E (T(k))] [T(k'), - E (T(k"~)]Pn N,) N,) N,) N,)

_ [(1~ "'" ((k) (k))) [1 ~ "'" ((k/) (k

/)))]- nEpn ;:; ~.~ TJOIij - TJOI.. x;:;~ .,~ TJ{3i l j' - TJ{3 ..

01=1 'EP) {3=1 , EPj'

57

n

-1 "" "" E [( (k) (k») ((kl

) (kl

»)]n L- L- Pn 'rJo:ij - 'rJ0:.. 'rJo:i'jl - 'rJ0: ..0:=1 (i,i/)EB

n

+n-1 L L Epn [( 'rJ~:} - 'rJi~!) (7]~::}, - 7]i~~»)]

0:=1 (i,i')EA .

n

-1 "" "" E [( (k) (k») ((kl

) . (kl

»)]+n L- L- Pn 7]o:ij - 'rJ0:.. 7]f3i' jl - 7]13 ..o:f:.f3=l (i,i/)EB

n

-1 "" "" E [( (k) (k») ((kl

) (kl

»)]+n L- L- Pn 'rJo:ij - 7]0/.. 7]f3i ' jl - 7]13 ..o:f:.f3=l (i,i/)EA

Note that the first term in this last equality is the sum of products in blocks where

pairs of treatments (j,j') occur together, while the second term consists of the sum

of products in blocks where treatments (j, j') occur in different blocks. The last two

terms are null due to the independence of replicates. The first term then yields

n

-1 "" "" E [( (k) (k») ( (k' ) (kl

»)]n L- L- Pn 7]o:ij - 'rJ0:.. 7]o:il jl - 7]0: ..0/=1 (i,i / )EB

n nn-1"" "" E ((~), (k:~/) _ rjjl "" (k) (k' )L- L- pn 7]0:1J 'rJ0:1 J L- 'rJ0/ .. 'rJ0: ..

0/=1 (i,i/)EB n 0:=1

rjjl ~ [ 1 ~ "" (k) (k l),] _ rjjl ~ (k) (k')

n L- sr(r _ 1)~ ,L- 7]0:1)7]0:1J n L- 'rJ0: .. 7]0: ..0:=1 1=1 Jf:.J/ES; 0/=1

rjjl 1 ~~ ["" "" (k) (k' )' (k) (k l)]

---;:;: sr(r - 1) L- L- L- ,L- 7]o:ij7]o:ij' - L 7]O/ij'rJo:ij0/=11=1 JES; J/ES; JESi

n_ r jj' "" (k) (k' )

n ~ 7]0/ .. 'rJ0: ..

I ns nrjjl _ "" "" r 2 (~) (~') _ rjjl "" (k) (k')

1 L- L- 'rJm. 'rJ0:1. L- 'rJ0: .. 7]0: ..r - nsr 0:=1 i=l n 0:=1

_ rjjl 1 ~ ~ ["" (k) (k ' ) (k) (k' ) + (k) (~/)]r _ 1 nsr L- ~ L- 7]O/ij7]o:ij - r'rJo:i. 'rJo:i. r7]o:i.7]o:i.

0/=1 1=1 JESi

r' 'I [~ ~ (~ (~) (~/) _ s (k) (k l»)] _ rjjl V(l) IJJ ns L- ~ 7]0:1. 'rJm. 7]0: .. 7]0:.. _ 1 N,jj

. 0/=1 1=1 r

rjjl (1) (2)---VN "I + r'J'IvN "Ir - 1 JJ J JJ

And the second t~rm simplifies into

n

-1 "" "" E [( (k) (k») ((kl

) (kl»)]n L- L- Pn 7]o:ij - 7]0:.. 'rJO/iljl - 'rJ0: ..

0:=1 (i,i/)EA

58

..

..

•

n , n .

n-l'" '" E (~~ (~:~/) _ TjTj - Tjjl '" (k) (k /)L...J L...J l"n "lellt) "lat ) L...J "la .. "la ..a=l (i,i/)EA n a=l

TjT; - Tjjl ~ [ 1 ~ (~) (~/)] _ TjT; - Tjjl ~ (k) (k /)n L...J s(s _ 1) L...J "l0it. "l0it. L...J "lau "la..

a=l i:;t:i'=l n . a=l

TjT; - Tjjl ~ [S2 (k) (k/) _ ~ (~) (~/)] _ TjT; - Tjjl ~ (k) (k/)- ns(s - 1) L...J "lau "la.. ~ "lat. "lat. n L...J "la .. "la ..

a=l t=l a=l

_ (TjT; - Tjj/) [~ ~ (~ (~) (~/) _ (k) (k/»)]- S - 1 ns L...J ~ 1]OH. "'Ctz. TJa .. TJa ..

a=l t=l

(TjT; - Tjj/) (2)- s -1 VN,kk'

Therefore, the sum of the two terms gives

Hence the theorem.

Define the following•

•

I ns

(k,k/)( ) _ '" '" '" [( (k)* (k

/)*) ]HN,l x,y - N L...J L...J L...J I Uaij ,Uaij :::; (x,y)

a=li=ljE~ .

I ns

(k,k') ( ) '" '" '" [( (k)* (k/)*) ]

H N ,2 X, Y = nT(T _ 1)~ f:t #7es; I Uaij ' Uaijl :::; (x, y)

Moreover, denote the marginal c.dJ of ul;j* by Fi)k)(x) and note that under How it is

equal to F(k)(x) which is independent of i and j. Also, for every i = 1,,", s, denote

(k,k /)( (_ (k,k /)( ) d H ) . (k)* (k /)*) .by F ij x) - F l x, y un er ow the margmal c.dJ. of Uaij ,Uaij . Fmally,

for j =1= j' = 1,'" ,v, let Fj~~;k')(x)(= F?,k/)(x,y) under How) be the marginal c.d.f.

f (U(k)* U(kl)*) D fio aij, aij' . e ne

i = 1,2, for k,k' = 1,"',p+q, where Ilk = J~J(k)(u)du. Note that Ffk,k)(x,y)

reduces to the univariate c.d.£. F(k)(x) as x = y almost everywhere. Further let

. _ (( (kk /»)) . _ .V t - Vi k,k'=l,.u,p+q, Z - 1,2,

59

A = [(r ~ 1)] A(I) + [(s:. 1)] A(2); ,

B = [(r ~ 1)] A(I) _ [(s -ll~r - 1)] A(2);

{} = A ® VI - B ® V2

Theorem 3.2 Under How, B N as defined in Theorem 3.1 converges in probability

(as n -+ (0) to {} defined above.

Proof: Note that R~~~ = NH<;) (U~7]*). Hence, TJ~~} = J<;) [~IH<;) (U~7]*)]. It

follows that

Let us rewrite V~!kk' as follows

..

(1 )vN,kk'

1 n s

- L L L [TJ~~} - TJ~:~] [TJ~::) - TJ~~?]nsr 0=1 i=1 jESj

n~r ttL TJ~:}TJ~:j - :s t t [~ L TJ~:}] [~ .L TJ~~:~]0=1 t=1 JESj 0=1 t=1 JESj J'ESj

r - 1[_1 ~~ '" (~). (~')] _r - 1[ 1 ~ ~ '" (~). (~')]L..J L..J L..J TJOtJTJOtJ ( 1) L..J L..J L..J TJOtJTJOItJ ,

r nsr 0=1 i=1 jESj r nsr r - 01=1 i=1 j=f::j'ESj

r - 1Jf fk) [ N H(k)(x)] f k') [ N H(k')()] dH(k,k')(x )_r JR2 N N + 1 N N N + 1 N Y N,1 , Y

r - 1Jf fk) [ N H(k)(x)] f k') [ N H(k')()] dH(k,k')(x )r JR2 N N + 1 N N N + 1 N Y N,2 ,Y

p r - 1 [kk' kk']-+ -- lI1 - lI2r

where R2 is the the Euclidean plane, and the last term follows from Theorem 5.4.2 of

Puri and Sen (1971). Note that H<;) -+ H(k) (= p(k)under How), where H(k) is the

population combined c.dJ. Hence we have

60

..

L 't (2).et US rewn e VN kk' as,

(2)VN,kk'

The first term in the last equality gives

1 n s

- L L [(1]~~~ -1]~.~)) (TJ~~:) - TJ~~'))]ns 0'=1 ;=1

1 n s

- L L TJ~~~TJ~::) - TJ~~)TJ~.~')ns 0'=1 ;=1

2-~ ~ [~" (~).] [~" (~'>,] _ (k) (k')L.J L.J L.J TJm) L.J TJm) TJ ... TJ ...ns 0'=1 ;=1 r jESi r j'ESi

1 [ 1 ~~" (k) (k')] r - 1 [ 1 ~~" (k) (k')]- - L.J L.J L.J TJO';j1]O';j + -- (. 1) L.J L.J L.J TJO';j1]O';j'r nsr 0'=1 ;=1 jESi r nsr r - 0'=1 ;=1 j1=j'ESi

-TJ.(.~)TJ~.~')

~Jr J(k) [ N H(k)(x)] J(k') [ N H(k')(y)] dH(k,k')(x y) +r JR2 N N + 1 N N N + 1 N N,I'

r - 1 Jr J(k) [ N H(k)(x)] J(k') [ N H(k')()] dH(k,k')(x )r JR2 N N + 1 N N N + 1 N Y N,2 ,y

-TJ.\~)1].\~')

p 1 kk' r - 1 kk'-+ -VI + --V2r r

where the last term follows from Theorem 5.4.2 of Puri and Sen (1971). The second

term gives

61

_..!.. {_I~ ~ " [( (~). _(k)) ( (~') _ (k'))]}- LJ LJ L.-i 'fJcx~J TJ... TJO:~J 77...sr nsr 0'=1 i=1 jESi .

_ r - 1 { 1 ~~ " [( (k). _ (k)) ( (k') _ (k'»)]}sr nsr(r - 1) L.J~ . ~ 'rJQ<J 'rJ... 'rJQij' 'rJ ...

0'=1 <=1 J::f:.J'ESi

S - 1 { 1 ~ ~ [( (k) (k») (k') (k'»)] }--s- ns(s _ 1) L.J .~ 'rJQi. - 'rJ... 'rJQi'. - 'rJ ...

0'=1 <::f:.<'=1

p 1 kk' r - 1 kk'-+ --v - --v

sr 1 sr 2

Note that the last term in the last equality converges to zero in probability. To

see this, remember that {'rJ~~~} are independent for every Q - 1,"', n, i = 1,"', s.

Moreover, if we let FnQi(x) = ~ 'LjESi I(Ui7J* $ x), then

( 'rJ(~) - 'rJ(k») = joo J(k) [ N H(k)(x)] dFnQi(x) _ 'rJ(k)w. ... -00 N N + 1 N ...

~ fal J(k)(u )du - J.l(k) = o.

Thus, if we let ('rJ~~~ - 'rJ~.~)) = Mi7) ,then E pn Mi7) ~ 0, and .

1 ~ ~ [( (k) (k)) (le') (k'))]( _ 1) L.J L.J 'rJQi. - 'rJ... 'rJQi'. - 'rJ ...

ns s 0'=1 i::f:.i'=1

((k) (k'))E pn MQi MQi, .

p-+ O.

Therefore,

(2) P [(s - 1)] (kk') [(S - l)(r - 1)] (kk')vN kk' --+ VI +. V2's sr

Hence

vW~ [s:' 1] [VI + (r - 1)V2] .

Thus it is clear now how under How, B N ~ n.

62

•

Theorem 3.3 LetVe:c(TN)' a (p+t 1 )vx1-vector, denote the rolled out form ofTN.

Then n ~ (Vec (TN) - 1/) converges, in probability, to a multivariate normal distribu­

tion with null mean vector and dispersion matrix B N.

The proof is found in Theorem 4:1 of Sen (1969). Now; construct the resid­

ual rank statistics as we did in the complete block design. Partition [Vec (TN) - 1/]

into two components: the first component, TNO, is a qv X 1 vector of the centered

linear rank statistics corresponding to the surrogate variables, and the second com­

ponent, T~, is a pv X 1 vector of centered linear rank statistics corresponding to the

concomitant variates. Simillarly, partition the matrix B N as we partitioned V N in

(2.11)

(

BNOOBN=

B'rvo+

B NO+ )

B N ++

(3.3)

where B NOO is a qv X qv matrix, B NO+ is qv X pv and B N++ is a pv X pv matrix.

Then, from the classical normal theory, (see Theorem 2.5.1 of Anderson (1984)), for

large N we have

Define the qv X 1 residual rank-statistics TN-O' and BN-OO' the covariance matrix of

n1/2T'No as follows

B'Noo B B B (-l) B'NOD - NO+ N++ NO+ (3.4)

3.1.2 BIBD For The Validation Set 1*

Consider n* replications of a BIBD consisting of s* blocks of constant size r*(2:: 2) to

which v treatments are applied such that the conditions i, ii, and iii in Section 3.1.1

are satisfied.

63

Let Si stand for the set of treatments occuring in the ith block, i = 1, ... , s*. For

the a th replicate, the response in the ith block receiving the jth treatment is a stochas-

. U (lJ X(l) X(q) Z(l) Z(p))' ("t.r X' Z')' ftIc P+q + 1-vector OIii = .L OIi;"l exij'" • , exij' exii'·'" exii = .L i;"l OIij' exij 0

measurements corresponding to the primary variate, p covariates and the q surrogates.

Consider the the same model in (3.1) and the assumptions following it with

the difference that here we regress the primary variate and the surrogates on the

concomitant variates. We basically repeat what we have done in the surrogate set with

N replaced by N v(= n*s*r*). Thus B N in Theorem 3 is now a (p+q+l)v x (p+q+l)v

matrix ,BNv ' which we partition as in (3.3). Then, corresponding to (3.4) we have

B NvOO B B B(-l) B'NvOO - NvO+ Nv++ NvO+ (3.5)

Partition the (q + l)v x I-vector T NvO ' and the (q + l)v x (q + l)v matrix B NvOO as

follows

T NvO = (T~vo(Y), T~vo(X))'

(BN.,Yy B NVY.X ).B NvOO =

B~vYX B NvXX

where B NvYY is v x v, B NvYX is v x qv, and B NvXX is qv x qv. Also let

TNvO(Y:X)

BNv(Y:x)

T* B* B*(-l) T*NvO(Y) - NvYX NvXX NvO(X) ,

B * B* B* B*'NvYY - NvYX NvXX NvYX·

(3.6)

(3.7)

Note that the dispersion matrix of v x 1 vector n*1/2 TNvO(Y:X) is given by the v x v

matrix BNv(Y:X).

3.1.3 BIBD For The Set [0

Consider nO replications of a BIBD consisting of SO blocks of constant size

rOC~ 2) to which v treatments are applied such that the conditions i, ii, and iii in

(3.1.1) are satisfied.

64

Let Si stand for the set of treatments occuring in the i th block, i = 1,"', so.

For the a th replicate, the response in the ith block receiving the jth treatment is a

. ((1) (p)) I ( ') IstochastIc p + I-vector U aij = Yaij,"', Zaij, .. " Zaij = }ij, Zaij of measure-

ments corresponding to the primary variate, and the p covariates.

Consider the same model in (3.1) and the assumptions following it with the

difference that here we regress the primary variate on the concomitant variates. We

basically repeat what we have done in the surrogate set with N replaced by N°(=

nOsOrO). Thus BN in Theorem 3 is now a (p + l)v x (p + l)v matrix ,BNo, which is

partitioned as in (3.3)

where B'N0oo is v x v, BNoO+ is v x pv, and V NO++ is pv x pv. Thus, corresponding

to (3.5) we now have the v x 1 residual rank-statistics T'N0o and the v x v dispersion

. B* f Ol/2 T * f 11matnx NOOO 0 n NOO as 0 ows

B'N°oo (3.8)

3.2 Construction of the Test Statistic

The next step consists of combining information from all subsets. Since the tests

developed in the different subsets are independent, we will take a weighted linear

combination of these tests with the weights being the inverses of the corresponding

dispersion matrices. Note that for such a combination to make sense the dimensions

of the tests should be the same.

We follow a similar approach as in the complete block case. Write the qv x 1-

vector, T'No, obtained from the surrogate set I, as a q x v matrix. Then take a linear

combination of this matrix, say aT'No, where a is of order 1 x q, subject to the two

restrictions, i) a'a = 1, and ii) the variance given in (3.4) is minimum. Although by

65

qoing this we are losing information, the minimum variance condition maximizes the

noncentrality parameter of the chi-squared distribution of the quadratic form based

on the transformed residual vector, which in turn maximizes the power of this test.

Moreover, a can be ordered if there is inherent ordering in the surrogate vector.

Note that if we partition vW, and vW as we did for B N in (3.3), then we get

V~bo and V~bo each of which is a q x q matrix. Now write

TNO(X) - aTNO

(aTNO 1"'" aTNO v), ,

where {TN-O'i' j = 1, ... , v} are the q x 1 columns of the q x vmatrix T NO' Moreover,

Cov(aTNO,i' aTNO,i') aCov(TNO,j, TNO,j/)a'

[ (I)V(I) (2)V(2)]'a ajjl NOO + ajjl NOO a

Thus, denote the dispersion matrix of n I/2TNO(x) by B NX which is a v x v matrix.

Let Al = B NX' A2 = BNv(Y:X), and A 3 = B NOOO ' Let

W· . - [A-I + A-I + A-I]-IA:- I . 1 2 3,- 1 2 3 t' z= , , (3.9)

Now, noting that the mean of each of the three statistics TNO(x), TNvO(Y:X), and T NOO

is 0, following the Gauss-Markov theorem we consider the linear combination

(3.10)

Assume that nn. -t PI, and ~ -t P2 for positive and finite real PI and P2. Then nI/2To'

is conditionally distribution-free, under the permutational model, with °mean and

covariance matrix

and the permutational multivariate central limit theorem of Sen (1983) applies here.

Thus, for large N, Nv , and N°, nI/2To' ,..., Nv (0, A). Finally consider the overall test

statistic

(3.11)

66

The exact permutational distribution of LO· can be obtained by enumeration

if N, Nv , and N° are small to moderate. Thus for large N, Nv , and N° the null

distribution of LO· by Cochran (1934) is asymptotically a chi-squared with v-I

degre.es of freedom since it is a quadratic form of asymptotically normal variates with

discriminant of rank v-I.

3.3 Asymptotic Non-null Distribution of L O*

For the study of the non-null distribution of LO· we will consider local alternatives

only. Let us first work with the surrogate set I. Write Gi(u) = Gi(x, z), x E Rrq , z E

Rrp , and consider the following sequence of alternative hypotheses:

(3.12)

where ~ = ((Ay))) stands for an v X q matrix of treatment effects. Thus, under {HN}

Fi\') (x);" F(') [x - n-1/

2 (A\') - ~ ~ AI'») ], j E S" k= 1,· .. ,q.

Assume that all Fi~k) (x) are absolutely continuous. Let f(k) (x} be the density function

corresponding to F(k) (x). Let

(3.13)

where1 s

H(k)(x) = - I: I: Fi~k)(x), k = 1,"" q.sr i=1 jeSi

Now expand F(k) [x - n-1/

2 (A)k) - ~ LIes; A~k))] in a Taylor series around x for a

fixed x. We will have

Thus, for large n we have

67

I: J(') [F(')( x)] dF(') [x - n-1/' (Aj') - ~ I~ Ai'))]i: J(k) [F(k)(x)] dF(k)(x) +

I: J(') [F(') (x)] d {F(') [x - n-II' ( Aj') - ~ I~ Aj')) ] - F(')(x) }

+o(n-1/2

)

Now integrate by parts the second integral in the last term

(k)I-ln,ij r1

J(k)(u)du _jOO {F(k) [x _ n-1/ 2 ().)k) _ ~ L ).~k))] _ F(k)(x)} ~J(k)(F(k)(X))Jo -00 r IESi dx

+ o(n-1/

2)

r1J(k)(u)du + (n-1/ 2 ) ().)k) _ ~ L ).}k)) JOO ~J(k)(F(k)(X))f(k)(x)dx + o(n-1/ 2 )

Jo r IESi -00 dx

r1J(k)(u)du + (n-1/ 2 ) ().)k) _ ~ L ).}k)) JOO ~J(k)(F(k)(X))dF(k)(X) + o(n-1/ 2 )

Jo r lESi -00 dx

where the last two steps resulted from a Taylor series exapnsion of

around x for a fixed x. Thus, if we let

then from the above we have

Now, let Q~~lj = ~ I:~=l 7]~:}, j E Si, {= 1,"', s, k = 1,'" ,p + q. Note that

r;;3 = I:iEPj Q~~lj' Then by Theorem 5.1 of Sen (1969) [n 1/

2( Q~~lj -I-l~~lj)' j E Si, i =

1,"', s, k = 1,'" ,p + q] has asymptotically a multinormal distribution with null ..mean and dispersion matrix whose entries for i # if are zero, and for j = jf converge

to V~kl, and for j =I- jf converge to v;kl. Thus, by condition (a) of Section 2.2.1 and

68

Theorem 5.1 of Sen (1969) we have

lim E [n1/2 (T(k) - r'''l(k l ) I HN]n-+oo N,J J ...

lim E[n1/2 (Tjp - " /1(k).) IHN]+ lim E[n1/2 (" (/1(k). - '1l(k»)) I HN]n--+oo ,J L..J n,tJ n--+oo L...J n,tJ 'I ...

iEPj iEPj

lim E [n1/2 " (Q(k). _ I/(k).) I HN] +n--+oo L..J n,tJ rn;"J

iEPj

n~~E [nIl' (~(I'~:!j - 1.' J(kJ(U)dU)) I HN]

- J~ E[n1/2(.L("l.~~l _fa1 J(kl(U)dU)) IHN]~EP}

EH~ Jim n'l' (I'~:!j - /.' jlkJ(U)dU)] IHN}

B(F(kl ) .L [>.Yl - ~ L >.}kl]~EP} lESi

B(F(kl ) " [>.(kl _ ~>.(kl _ ~ " >.(kl]LJ J rJ rLJ IiEP) I#jESi

B(F(kl ) {[rj(r -1)] >.;kl _ t rjjl >.;~l}r j':f.j=l r

Let (J = ((eY»)) j=l, ... ,v,k=l,. .. ,q , where

Then, by Theorem 5.2 of Sen (1969), {n1/

2 (T;t,} - rj"l~~»), k= 1, ... ,p + q, j = 1,' .. ,v}have jointly asymptotically a multinormal distribution with mean (J and dispersion

matrix fl. Denote by fl I the version of fl pertaining to the surrogate set, I, and by

(JI the version of (J in I. Partition (JI as ((JIO, (J~). Also partition flI as we partitioned

B N in (3.3) and corresponding to (3.4) let

1")* _ I") I") 1")(-1) 1")1uIwOO - UIwOO - UIO+UI++UIO+'

69

Then for large n

and its dispersion matrix is njwx which is the population version of B'Nx.

In a similar fashion, define {HN} for the validation set 1iv such that ((>.~k))) is

now an v x (q + 1) matrix. Also, (J [tv is defined in the same way as (J [ with the

difference that k = 1,,' . ,q + 1. B Nv rv n[w which is the version of n defined in the

validation set 1iv. Thus, corresponding to (4.5) and (4.6) we have

TN-vo(y:X)

OJ,(y:X)

Hence for large n*

T * £")* £")*(-1) T*NvO(Y) - U ['YX U ['XX NvO(X) ,

n* £")* £")*(-1) £")*'HiI*YY - "~['YX"~['XX"~['YX'

E[ *1/2T* IH] (J £")* £")*(-1) (J' *n NvO(Y:X) N ---+ ['(Y) - U['YXU[,XX ['(X) = J..t['O(Y:X)

and the dispersion matrix of n*1/2TN-vO(Y:X) under {HN } is nj,(y:X)'

Finally, along the same lines, define {HN } for the subset 1° such that ~ is now-

an v x 1 vector. Also, (J [0 is defined in the same way as (J [ . Denote the expected

value of n01

/2 TN-oo under {HN } by /Ljoo, and its dispersion matrix is njoo'

Now, we consider the weights Wi defined in (3.9) where now we have Al = nN-X,

A 2 = nj,(y:X)' and A 3 = njoo' Thus, under {HN }, the mean of TO' in (3.10) is

Therefore, from Theorem 5.3 of Sen (1969) and under {HN }, LO' defined in (3.11)

has asymptotically a noncentral chi-square distribution with v-I degrees of freedom

and the noncentrality parameter

70

•

Chapter 4

Recovery of Inter-Block

Information (RIBI)

The model introduced in (3.1) and the analysis following it extracts intrablock in­

formation. However, when block comparisons are taken into account, the varietal

effects of treatments become more accurate; see Rao (1947). This section deals with

combined inference based on intra- and interblock information. A full treatment of

the parametric case is also found in Scheffe (1959).

The need for the recovery of inter-block information was first felt by Yates (1940)

in the context of incomplete block designs. Since the treatments are randomly al­

located to incomplete blocks, it is reasonable to assume that the block effects are

random variables instead of fixed. Moreover, Yates (1940) noted that if the experi­

mental material is fairly heterogeneous, treating block effects as fixed will result in

loss of information contained in the block totals.

71

4.1 RIBI for Surrogate Set I

The model considered in (3.1) assumes that the block effects f301i, a = 1"", n, Z =

1,' .. ,s, are fixed. In some instances this may not be viable. It may be more rational

to assume that blocks are random variables. For example, consider blocks made

up of litters of cats where each litter, and hence each block, come from a different

mother. The mothers are assumed to be a random sample from an infinite population.

Consider the same setup as in (3). Denote the block totals by hOli, a = 1, ... ,n, i =

1,"', s. Then, (3.1) gives

where

hOli - L XijjESi

L [ltOi + f301i + Tj + EOIij]jESi

rltOi + L Tj + f OIijES;

rltOi + T OIi + f OIi (4.1)

fOli = rf301i + L EOIij, i = I,···,s, a = I,···,n.jESi

For each a = 1,' .. ,n, the block effects f3 01i' i = 1,'" ,s are' assumed to be random

variables with a joint distribution that is symmetric in its s arguments. Moreover,

in addition to the assumptions following (3.1), f3 01i and LjES; EOIij are assumed to

be independent. T OIi is a p + q-vector with two components, T OIil a p x I-vector of

treatment effects corresponding to the surrogates, and T OIi2, a q x I-zero vector of

treatment effects for the covariates since we assume no interaction between treatment

and concomitant variables. We wish to test the null hypothesis

HOB: TOIil = 0 i = 1,"', s.

while the set of alternatives relates to shifts in location due to treatment effects. Let

hOi = (hOlI,"', hOls). Also, let h N = (hI,"', h n ). Under HOB, we have the following

(i) The joint distribution of hOi remains invariant under any permutation of the s

vectors among themselves, there being s! such permutations.

72

(ii) As replicates are independent, the joint distribution of h N remains invariant

under any permutation of the n sets hOI, a = 1,' .. ,n.

Thus, if we define On to be the compound group of transformations {gn} by

9 hN - h* - [g(l)h ... g(n)h] g(OI) E t: rv - 1 ... nn - N - n 1, 'n n, n ~n, \...< -, ,.

Then On contains [s!]n transformations, and under HOB it leaves the joint distribution

of hN invariant. Let S(hN) = {hN = 9nhN : gn E On}. Then, it follows from the

above discussion that

for all hNE S(hN). Denote this conditional (permutational) probability measure by

~~. As ~~ is completely specified the existence of conditionally distribution-free tests

for HOB is thus established.

For the kth variate, we rank the observations h~~), ... ,h~k) in ascending order of

magnitude and denote by S~~) the rank of h~~) in this set, for i = 1," . ,s, a = 1,' .. ,n,

and k = 1,'" ,p + q. Thus, corresponding to hOli = (h~l/, ... : h~+q»)' we have a rank

vector SOli = (S~~),· ., ,S~+q»)', i = 1,' .. ,s, a = 1,' .. ,n.

For each N'(= ns) and each k = 1,'" ,p + q, we define suitable rank scores

b~~ = (b~~,l,···,b~~,NI)' where b~~,j = J;;)(j/(N' + 1)), 1 :::; j :::; N'. Moreover,

J;;)(u) is defined in accordance with the Chernoff-Savage convention, i.e., we assume

that J;;) (u) satisfies the conditions a, b, c of Section 2.2.1. For notational simplicity,

we let

t(k) - b(k) ~ - 1 ... s '" - 1 '" n'"'OIi - I (k)' 0 -, ." \...< -, "N ,sa;s n

ei~) = S-l Le~~), e~.k) = n-1 Lei~)i=l i=l

for k = 1, ... ,p +q. The proposed test is based on the statistics

T (k). = _1 ~"t(~) . 1 k 1N',) . L.J L.J "'0/1 , J = ,"', v, = ,'" ,p + q.

nr) 0'=1 iEPj

73

where Pj = {i : j E Sd, j = 1,' .. ,v. Let TN, = ((T;;'~J)j=I ..",v, k=I"",p+q,

Theorem 4.1 Let

(1) = ~~~ [t(~) _ dk)] [t(~') _ d k')]wN',kk' L-J L-J ~m ~o. ~o~ ~o. ,

ns 0=1 i=1

and

k k' = 1 ... p + q, " ,

W(2~ I = ~ ~ [d k) _ d k)] [dk' ) - t(k')] k k' = 1 ... p + q'N kk L-J ~o. ~.. c"o. c".. " " ,, n 0=1

(1) (( (1) )) (2) (( (2) ))W N' = wN,kk' and W N' = wN,kk' .

Also, let

where

d(l) _ srjjl - rjrjl .., .jjl- 2( 1)' J,J=l,···,v;

rj s-

d(~), = rjrjl .., 1JJ 2 ' J,J = ,"',v,

rj

Finally, let

C - V(l) iOI W(I) + V(2) iOI W(2)N' - '<Y N' '<Y N" -

eN' is a (p + q)v X (p + q)v matrix. Then

where J is a v x I-vector of ones.

Proof: Note that for a given Q, Q = 1,'" ,n, the probability under p~ that i will be

any of 1 ... s is 1. hence, , s'

74

¥oreover, let A = {(i, i') : i E Pj, i' E Pj, i =I- i'}, and note that then the cardinality

of A is rjrjl - rjj/(= rj(rj - 1) if j = j'). Then

nEp ;. [Tt~j - E (Tt~j)] [T~~:~ - E (Tt:n]

nEp ;. [(~ t L: (e~~) - e~.k))) x (~ t L (e~::) - f~kl)))]nrJ a=1 iEPj nrJ f3=1 i'EPj

n~2 t L: Ep ;. [(e~~) - f\k)) (e~~/) - e~kl)) ]J a=1 iEPj

+n~~ t L: Ep ;. [(e~~) - e~.k)) (e~::) - e~kl)) ]J a=1 (i,i/)EA

+_1 ~ "E. [(e(~) - e(k)) (e(~/) - e(kl))]nr2 L.. L.. Pn at .. f3t ..

J a#f3=1 iEPj

+~ t L: Ep ;. [(e~~) - f~k)) (eh~:) - f\k l

))]

nrJ a#f3=1 (i,i/)EA

Note that the last two terms are null because the replicates are independent and

Ep ;. (e~:) - e~k)) = o. The first term yields

~ t L Ep ;. [(e~~) - e~k)) (e~:/) - e~.kl)) ]nrj a=1 iEPj

n~~ t .L: E p ;. (e~~) e~:/)) - r1 (e~.k) e~.k'))

J a=1 tEPJ J .

n~2 t L: [~t e~:) e~~/)] - :. ((~k) e.~kl))J a=1 tEPj t=1 J

_1_ t [~ te~:)e~~') - d~)d~/) + ei~)ei~')]nrj a=1 S i=1

-~ t (e~k)e~kl))r J a=1

_1_. t t [e~:) - ei~)] [ei~/) - d~/)]nsrJ a=1 i=1

+_1 t ei~)ei~') - ~ (f(.k)e~kl))nrj a=1 rj

_1_ ~~ [e(k) _e(k)] [t(~/) _ t<kl)]nsr' L.. L.. at a. ~at ~a.

J a=1 t=1

+n~' t [d~) - e~.k)] [e!:') - e~.k')]J a=1

1 [ (1) (2)]- - WN' kk' +WN' kk'rj' ,

75

The second term gives

Hence, the sum of the two terms give

((k) (k')) S - rj (1) (2)

nCov p • TN' j' TN' j = ( 1) wN' kk' +wN' kk"n , , rj S - ' ,

On the other hand, let B = Hi, i') : i E Pj, i' E Pj', i = i'}. The cardinality of B is

rjj', Thus

[(k) ( (k))] [(k') ((k') )]nEpi. TN',j - E TN',j TN',j' - E TN',j'

76

•

nEp:. [(~t I: (~~~) - ~~k))) X (~t I: (~}j;:) - ~~.kl)))]nr) a=1 iEPJ nr) {3=1 i/EPJI

_.1~ " E. [(~(~) _ d k )) (t(~;) _ d k/ ))]nr2 LJ LJ Pn m ":... <"m ":. ..

) a=1 (i,i/)EB

+_1~ " E. [(t(~) _ ~(k)) (t(~;) _ t(k'))]nr~ LJ LJ P n ":.at .. ":.at ":. ..

) a=1 (i,i/)EA

+_1 ~ "E. [(t(~) _ d k)) (t(~:) _ d kl ))]nr~ LJ LJ P n ":.at ":... ":.{3t ":. ..

) a#={3=1 (i,i/)EB

+_1 ~ "E. [(t(~) _ d k)) (t(k;) _ d kl ))]nr~ LJ LJ P n ":.at ":... ":.{3t ":. ..

) a#={3=1 (i,i/)EA

The first term then yields

_1 ~ " E. [(~(~) _ ~(k)) (t(~I) _ dk/))]nr~ LJ LJ P n at .. ":.at <" ..

) a=1 (i,i/)EB

_1 ~ " E. (~(~)~(k')) _ rjjl (dk)dkl))nr~ LJ LJ Pn O!t O!t 2 ":... ":. ..

) a=1 (i,i/)EB r)

~ t I: [~t~~~)~~:/)]- rj~' (~~.k)~~.kl))nrj a=1 (i,i/)EB S i=1 rj

rjjl ~ [~~ ~(~)~(~/) _ ~(k)~(kl) + ~(k)~(kl)]nr2 LJ s ~ at at a. a. a. a.

) a=1 t=1n

_ r:~, I: ((~k)(~kl))) a=1

rjjl [~ ~ ~ [~(~) _ ~(k)] [t(~I) _ d kl )]]r~ ns LJ~ at a. ":.at ":.a.

) a=1 t=1

+r:r [~~ d~) ~~~/) - (~~.k) ~~.k/))]

rjjl [~ ~ ~ [~(k) _ ~(k)] [~W) _ ~(kl)]]r2 ns LJ~ at a. at a.

) a=1 t=1

+r:r [~~ [~~~) - (~k)] [~~~') _ ~~.kl)]]rjjl [ (1) (2)]r~ WNI,kk' +WNI,kk'

)

The second term gives

_1~ " E. [(~(~) _ d k)) (t(~:) _ d kl ))]n ~ LJ LJ P n at ":... ":.at ":. ..r) a=1 (i,i/)EA

77

The details were skipped because they were done earlier. Therefore, the sum of the

two terms gives

C (T (k) T(k')) ,(1) (1) d(2) (2)n OV P:; N',j' N',j' = ajj'wN',kk' + jj'WN',kk'

Hence the theorem.

Define the following

•

1 n s

H};i,;') (x, y) = ns L ~ I [(h~~)*, h~~')*) ~ (x, y)]a=l,=l

(k,k') . 1 ~ ~ [( (k)* (k')*) ( )]HN',2 (x,y) = (-1) L..J L..J I hai ,hai ~ x,yns s a=l i#i'=l

Moreover, denote the marginal c.d.f of h~~)* by F}k)(x) and note that under HOB it

is equal to F(k)(x) which is independent of i. Also, for every i = 1,"', s, denote by

F?,k')(x)(= F1(k,k')(x,y) under HOB) the marginal c.d.f. of (h~~)*,h~~')*). Finally, for

i =I i' = 1,···,s, let F}:"k')(x)(= FJk,k')(X,y) under HOB) be the marginal c.d.f. of,

(h(k)* h(k')*) D fiai , ai' . e ne

z = 1,2, for k,k' = 1,"',p+q, where J-lk = f~J(k)(u)du. Note that F1(k,k)(x,y)

reduces to the univariate c.d.f. F(k) (x) as x = y almost everywhere. Further let

(kk') ) .Wi=((Wi )k,k'=l, ..·.,P+q, z=1,2;

78

•

E = (8 - 1) D(1) _ ~D(2).,8 8

Theorem 4.2 Under HOB, eN defined in Theorem 3.3, converges in probability (as

n --+ 00) to 1P defined above.

Proof: Note that S~7) = NHc;) (h~~)). Hence, ~~)

follows that

Rewrite w~J kk' as follows,

(1 )wN',kk'

where R2 is the Euclidean plane, and the last term follows from Theorem 5.4.2 of

Puri and Sen (1971). Note that Hc;J --+ H(k)(= p(k)under HOB), where H(k) is the

population combined c.d.f. Hence we have

(1) P s - 1W N' --t -- [WI - W2] .

s

L . (2)et us rewnte wN',kk' as

(2)WN',kk' ~ f [~~~) - ~~k)] [~~~') _~~.k')]

0'=1

79

~ t t t [(~~~) -~~.k») - (~~~:) _~~.k'»)]ns 0'=1 i=1 i'=1

~ t t [(~~:) - ~~.k») _ (~~~') _~~.k'»)]ns 0'=1 i=1

+_1 ~~ [(~(~) _ ~(k») _ (~(~:) _ ~(k'»)]ns 2 LJ LJ at.. at ..

0'=1 i=1

~ ~ [w~k' + w;k']Hence we have

f

Thus the theorem. •Theorem 4.3 n1

/2 [Vec(TN') - eJ converges in probability to a multivariate normal

distribution with null mean vector and dispersion matrix C N ,.

The proof is found in Theorem 4.1 of Sen (1969). Now, construct the residual rank

statistics as we did in the intra-block case. Partition [Vec (TN') - eJ into two com-

ponents, the first component, T N'O, is a qv x 1 vector of the centered linear rank

statistics corresponding to the surrogate variables, and the second component, T~"

is a pv x 1 vector of centered linear rank statistics corresponding to the concomitant

CN,O+ )

C N ,++

variates. Similarly, partition the matrix C N' as follows

(

CN'OOC N ,=

C~I\T'o+

(4.2)

where C N'OO is a qv x qv matrix, C N'O+ is qv x pv and C N'++ is a pv x pv matrix.

Then, from the classical normal theory, (see Theorem 2.5.1 of Anderson (1984)), for

large n we have

Define the qv x 1 residual rank-statistics TN,o, and the covariance matrix of n 1/

2T N,o,

CN,oo, as follows

C C C (-1) C'N'OO - N'O+ N'++ N'O+

80

(4.3)

4.1.1 RIBI For The Validation Set 1*

Consider n* replications of a BIBD consisting of s* blocks of constant size r*(2:: 2) to

which v treatments are applied such that, all the assumptions made i~ the surrogate

set I are satisfied.

Let Si stand for the set of treatments occuring in the ith block, i = 1, ... , s*. For the

a th replicate, the response in the ith block receiving the jth treatment is a stochastic

p+q+l-vectorUoij = (lij,X~~J, ... ,X~~J,Z~~J, ... ,Z~~J)' = (Y;j,X~ij,Z~ij)' of mea-

suremeIits corresponding to the primary variate, p covariates and the q surrogates.

Consider the the same model in (4.1) and the assumptions following it with the differ-

ence that here we regress the primary variate and the surrogates on the concomitant

variates. We basically repeat what we have done in the surrogate set with N' replaced

by N'v(= n*s*). Thus CN' in theorem 6 is now a (p + q + l)v x (p + q + l)v matrix

,CN'v' which we partition as in (4.2). Then, corresponding to (4.3) we have

(4.4)

Partition the (q + l)v x I-vector T"N,v o' and the (q + l)v x (q + l)v matrix C"N,voo as

follows

(

C*C* _ N'vYY

N'vOO - ,CN'vYX

where CN'vYY is v x V, C"N,vYX is v x qv, and C"N,vxx is qv x qv. Also let

T * C* C*(-l) T*N'vO(Y) - N'vYX N'vXX N'vO(X),

C * C* C* C*'N'vYY - N'vYX N'vXX N'vYX·

(4.5)

(4.6)

Note that the dispersion matrix of v x 1 vector n*1/2T"N,vO(Y:X) is given by the v x v

matrix C"N,v(Y:X).

81

4.1.2 Construction of The Test Statistics

For the reasons mentioned in the begining of Section 3.2 we work with TN,o(x)

bTN,o instead of TN,o, where b is a 1 x q- vector subject to the two, restrictions, i)

b'b = 1, and ii) the variance given in (4.3) is minimum. Denote the v x v dispersion

matrix of n I/2T;,Tlo(x) by CN,x'

The next step consists of combining the interblock and intrablock information.

Since the tests developed in Chapte:s 3 and 4 are independent, we will take a weighted

linear combination of the four tests where the weights are inverses of the corresponding

dispersion matrices. Recall that we let Al = B NX' A 2 = BNv(Y:X) developed in

Chapter 3. Let A 3 = CN,x, and let A 4 = CN'v(Y:X) developed in this chapter. Then

let the weights be defined by

•

W · - [A-I +A-I +A-I +A-Ij-IA-I . 1 2 3 4,- I 2 3 4 i' z= , , , (4.7)

Now, noting that the mean of each of the four statistics TNO(x), TNvO(Y:X), TN,o(x),

and TN,vO(Y:X), is 0, consider the linear combination

Assume that ;:. -+ p for positive and finite real p. Then n I / 2To· is conditionally

distribution-free, under the permutational model, with °mean and covariance matrix

and the permutational multivariate central limit theorem of Sen (1983) applies here.

Thus, for large N, Nv , N', and N~, nI/

2To· '" Nv (0, A). Finally consider the overall

test statistic

(4.9)

The exact permutational distribution of LO· can be obtained by enumeration if N,

N v , N', and N~ are small to moderate. Thus for large N, Nv , N', and N~ the null

distribution of LO· by Cochran (1934) is asymptotically a chi-squared with v - 1

degrees of freedom.

82

4.1.3 Asymptotic Non-null Distribution of LO*

For the study of the non-null distribution of LO· we will consider local alternatives

only. Recall that we have found in Section 3.3 that the the covariance .matrices of the

linear rank statistics in both sets I and 1* under HN, converge to the same dispersion

matrices under How. On the other hand, the test statistics had a different mean under

HN than under How. The same results hold for the tests using block totals. To see this

let us first work with the surrogate set I. Write Gi(u) = Gi(x, z), x E Rq, z E RP,

consider the following sequence of alternative hypotheses:

(4.10)

where A = ((>.;k))) stands for an v x q matrix of treatment effects. Thus, under {HN}

y(k)(X) = F(k) [x - n-1/ 2 )..(k)] k = 1 '" qt J" ,.

We assume that F}k\x) are all absolutely continuous. Let j(k)(x) be the density

function corresponding to F(k)(x). Let

(4.11)

where

H(k)(X) = ~ t F?)(x), k = 1,"', q.S i=l

Now expand F(k) [x - n-1/ 2>.;k)] in a Taylor series around x for a fixed x. We will

have

Thus, for large n we have

J1S~l 1.: J(k) [H(k)(x)] dF?)(x)

1: J(k) [F(k)(x)] dF(k) [x - n-1/ 2 )..;k)]

1: J(k) [F(k)(x)] dF(k)(x) +

1: J(k) [F(k)(x)] d{F(k) [x - n- 1/ 2>.;k)] - F(k)(x)}

+o(n-1/2

)

83

~ow integrate by parts the second integral in the last term

(k)f-ln,i [1 J(k)(u)du -100

{F(k) [x _ n- 1/ 2 A)k)] _ F(k)(x)} ~J(k)(F(k)(x))k -00 dx

+ o(n-1/2)

[1 J(k)(u)du + (n-1/2)A)k)100

~J(k)(F(k)(X))f(k)(x)dx+ o(n-1/2)Jo -00 dx

[1 J(k)(u)du + (n- 1/2)A(k)100

~J(k)(F(k)(X))dF(k)(x)+ o(n-1/2)Jo J -00 ·dx

where the last two steps resulted from a Taylor series expansion of F(k) [x - n-1/ 2 A)k)]

around x for a fixed x. Thus, if we let

then from the above we have

Now, let Q~~l = ~ L::=1 ~~:), i = 1,···, s, k = 1,·" ,p + q. Note that Tj;3 =1.. L:iEP Q~k]. Then along the same lines of theorem 5.1 of Sen (1969) [nl/2(Q~kl_T J ), ,

f-l~~l), i = 1,···, s, k = 1,··· ,p + q] has asymptotically a IIlUltinormal distribution

with null mean and dispersion matrix whose entries for i = i' converge to wtkl, and

for i =I- i' converge to W~kl. Thus, by condition (a) of Section 2.2.1 and Theorem 5.1

of Sen (1969) we have

lim E [n 1/ 2 (T(~). - ~(k)) IH N ]n-+oo N ,J ..

84

- E {[:..2: J~~ n1/2(Ji~:l- fal ](k)(U)dU)] I HN}

J te~ 0

B(F(k)) :. 2: )..}k)J iePj

= B(F(k)))..}k)

Let I = ((,Y))) j=l, ...,v,k=l, ...,q , where IY) = B(F(k)))..}k) Then, by Theorem 5.2 of Sen

(1969), {n1/

2 (T;;'~j - ~~k)), k= 1,·,· ,p + q, j = 1,·'·, v} have jointly asymptot­

ically a multinormal distribution with mean I and dispersion matrix W. Denote by

WI the version of W pertaining to the surrogate set, I, and by II the version of I in

I. Partition WI as we partitioned CN' in (4.2) and corresponding to (4.3) let

.Tr* _ .Tr .Tr .TA-l).Trl'£' 100 - '£' 100 - '£' [0+ '£' 1++ '£' [0+·

Then for large n

and its dispersion matrix is 1P~x which is the population version of eN-,x.

In a similar fashion, we define {HN} for the validation set 1* such that (()..}k)))

is now an v x (q + 1) matrix. Also, 1[" is defined in the same way as T I with the

difference that k = 1, ... , q+ 1. CN'v f"V 1P[" which is the version of 1P defined in the

validation set 1*. Thus, corresponding to (4.7) and (5.1) we have

Hence for large n*

TNvO(Y:X)

W~"(Y:X)

T * .Tr* .Tr*(-l) T*NvO(Y) - '£' I"YX '£' I"XX NvO(X),

.T.* .Tr* . .Tr*(-l) .Tr*'~ ["YY - '.I!' I"YX '£' I·XX '£' I·YX·

d h d·· . f *1/2 T * . .Tr*an t e IsperSlOn matrIX 0 n NvO(Y:X) IS '.I!' I"(Y:X)'

85

For now, focus only on the surrogate and validation subsets. Consider the weights

Wi defined in (3.9) where now we have Al = .njx, A 2 = .nj.(y:X), A 3 = qi'jx, and

A 4 = qi'j.(y:X). Thus, under {HN }, the mean of n I/

2T o• in (4.9) is

Therefore, from Theorem 5.3 of Sen (1969) and under {HN}, La· has asymptotically a

noncentral chi-squared distribution with v-I degrees of freedom and the noncentrality

parameter

86

•

Chapter 5

General Case of a Vector of

Primary Variates

It is hard to find an ideal surrogate that fully captures the treatment effect on the

true endpoint, because a new treatment may affect the true endpoint through other

potential surrogate variates. Hence, there is a need to consider all possible surrogates

so as to avoid losing information about treatment effects on the primary variates.

Moreover, it is often difficult/costly to record data on all primary variates (as

well as surrogates and covariates) for all experimental units. Consider a vector

of primary response variates (y(l), ... , y(m))', and a vector of surrogate responses

(X(l),'" ,X(q))' that we partition into various subsets on which measurements are

obtained for different numbers of experimental units. Such a partition is done based

upon practical consideration of medical as well as economic factors. This is the genesis

of incomplete multiresponse designs (IMD).

Sen (1994) formulates a general class of IMDs in the following manner. Let

P denote {1,'" , q + m} and consider the totality of 2q+m subsets of P, defined by

(ir , i;) = {ill' .. ,ir ; ii', ... ,i:}, for all possible 1 ::; il < ... < ir ::; m, 1 ::; ii' < ... <

i: ::; q; 1 ::; r ::; m, 1 ::; s ::; q, and include the null sets io or i~ in this system.

Consider a proper subset Po of P, determined by clinical and other factors, such that

87

P.o = {(ir,i;) : (r,s,ir,i:) E Io}, where 10 is the set corresponding to Po. Then the

set of all experimental units is partitioned into the system

where in the subset S(ir,is) the primary response variates y(il),.", y(ir) , and the

surrogate response variates XUi), ... ,XU:) are to be recorded on the experimental

units.

For the subset S(ir,is ) efficient design D(ir,is) (for the treatment as well as design

variates) may be adopted leading to the design sets

Thus, an incomplete multiresponse design that incorporates surrogate endpoints

can be formulated in terms of the dual design sets, responsewise design set and

treatmentwise design set, namely {S(Po),D(Po)}.

Assume that there are II, ... ,hI surrogate subsets of sizes nI, ... , nkll that con­

tain measurements on surrgate and concomitant variates, and I;"" ,Ik2 validation

subsets of sizes ni',·· . ,nk2

, with measurements on both primary and surrgate end­

points along with covariates. Assume also that a p-vector of covariates is easily

measured on all subjects in the respective subsets.

5.1 Intra And Inter-Block inference

Consider n* replications of a BIBD consisting of s blocks of constant size r(~ 2) to

which v treatments are applied in subset II, such that all the previous assumptions

discussed in Chapter 3 apply. For the a th replicate, the response in the i th block receiv- ~

. h 'th . h' U .. - (Xi~ Xi: Z(I) ZIp))' -mg t e J treatment IS a stoc astIc vector en) - e<ij' ••• , e<ij" e<ij'···, e<ij -

(X~ij' Z~ij)' of measurements corresponding to the p covariates and the surrogate re­

sponses in {SOo,i:): (r,s,io,i~) E Po}.

88

The focus here is, on hypothesis testing. Moreover, since a BIBD design is a

connected design, testability of hypotheses is ensured (see Bose (1947)).

Apply the same method developed in Chapter 3 to create a test statistic in each

of the surrogate subsets. Simillarly, obtain a test statistic in each of the validation

sets. Moreover, obtain a test statistic in each of the surrogate as well as validation

subsets by modelling the block totals as explained in Chapter 4.

Thus, for within block inference we conceive of a vector of test statistics (T~ w,"', T~ w)1, kl'

in the surrogate sets, and another vector of test statistics,

(T~. w, .. "T~. w), in the validation sets. Also, for between block inference wel' k 2 '

have a vector of tests (T~ B"", T~ B) in the surrogate sets, and a vector of tests1, kl'

(T~. B,' .. ,T~. B) in the validation subsets. Let n = nl +... + nkl + n~ +... + n k .l' k2' . 2

5.2 Construction of The Test Statistic

The next step consists of combining information from all subsets. Since the tests

developed in the different subsets are independent, we will take a weighted linear

combination of these tests with the weights being the inverses of the corresponding

dispersion matrices. Note that for such a combination to make sense the dimensions

of the tests should be the same. Thus, each test is appropriately transformed so as

to allow a meaningful linear combination.

For simplicity of notation replace the within block tests in the surrogate and

validations subsets by their transformed version (Tl,w,"', Tn,w), and the between

block tests in all subsets by (Tl,B,"', Tn,B). Also replace their corresponding dis­

persion matrices by (Al,w,"', An,w), and (Al,B,"" An,B)

Define

for i = 1,' .. ,n, j E {W, B}. Consider the linear combination

89

Assume that ~ -+ Pi for j = 2,' .. ,kt, and !!t -+ PI for 1 = 1,'" ,k2 , where then J nz

p's are all positive and finite real numbers. Then n~/2To* is conditionally distribution-

free under the permutational model, and the permutational multivariate central limit

theorem of Sen (1983) applies here. Hence n~/2To* is multinormal with 0 mean and

covariance matrix

Finally consider the overall test statistic

(5.1 )

The exact permutational distribution of LO* can be obtained by enumeration if n,

is small to moderate, while for large n, the null distribution of LO* by the Cochran

(1934) Theorem is asymptotically a chi-squared with v-I degrees of freedom.

5.3 Asymptotic Non-Null Distribution of LO*

The dispersion matrix of the test developed in the different subsets converge, under

local alternatives to the same limit as under the null. The mean of the test-statistics

however, is shifted under local alternatives. The distribution of LO* under local alter-

natives is a non-central chi-squared. The details follow in the same way as has been

done in Chapters 4 and 5.

Assume that the mean of the respective tests under HN defined in Chapter 4 is

a 2n x 1- vector, (PI w,' ", Pn w, PI B,' .. ,Pn B)', " ,

Define the weights Wi using the population versions of the covariance matrices.

Thus, under {HN }, the mean of n~/2To* is

Therefore, from Theorem 5.3 of Sen (1969) and under {HN }, LO* has asymptotically a

noncentral chi-squared distribution with v-I degrees of freedom and the noncentrality

90

Chapter 6

Properties of The Test Statistics

And an Illustration

6.1 Introduction

This chapter discusses studies the asymptotic relative efficiency (ARE) of the test

statistics LO·, developed in Chapters 2,3, and 4 with respect to the parametric version

constructed in exactly the same way using the least squares estimates lse's of the

treatment effects instead of the linear rank statistics T;;'~.

Also, calculations for the complete block case will be done on the Burroughs­

Wellcome data mentioned at the end of Chapter 1 to illustrate the method developed.

6.2 ARE for the Complete Block Case

Consider the same setup in the surrogate set as in Section 2.2.1. The model in

2.1 assumes now the errors to be normally and independently distributed. This

means that fij has now a (p+q)-variate multinormal distribution with null mean and

covariance ~, independently for all i = 1,· .. ,s, j = 1,· .. ,r.

92

The least squares estimates of the treatment effects are defined by

which is the average of the aligned variable Ui~k) over blocks. The alignment is done

by subtracting the block average from Ui~k). Moreover, V N, the (p + q) x (p + q)

matrix defined in Theorem 2.1 has now elements VH' given by

k,k'=1,···,p+q,and

where eN is the matrix of coefficients defined in Theorem 2.1, and TN is the r x

(p +q) matrix of least square estimates corresponding to the different treatments and

variates.

Repeat the same method developed in the nonparametric setup. Partition V N

as in 2.11, then obtain TNo, a q x r matrix of residual 1se's in the surrogate set.

Similarly, obtain TNvO(Y:X)' a 1 x r vector of residual1se's in the validation set. Build

the weighted linear combination TO· as in (2.22). Finally construct the quadratic

form La· using TO· and its covariance matrix.

Whenever the errors have a finite dispersion matrix (~), the 1se's are asymptot­

ically normal, and hence the null distribution of La· is asymptotically a chi-squared

with r - 1 degrees of freedom. However, under the sequence of alternative hypothe­

ses {HN } in (2.24), T = n-1/

2>., and thus the mean of T undergoes a location shift

leading to the following shift in the mean of T

(6.1)

Let Als be the version similar to A, the covariance of the weighted linear com­

bination of the test TO· under {HN }. Note that the limit of the expected value of VN

for large sis r;l~ under both Ho and HN . Thus, construct A as was done in Section

93

2.2.5 but using r-1 eN 0 ~ instead of v. Let /-t?: be the mean ofthe TO· under {HN }.. r

Then LO· has asymptotically a noncentral chi-squared distribution with r-1 degrees

of freedom and noncentrality

( 0·)' (-1)( 0·)~LO· = /-tIs A ls /-t Is .Zs

Thus, the ARE is given by

ARE = ~LO•.

~Lo·Zs

Rewrite /-to· = M1~*' and /-t?s· = M2~*, where M1 and M 2 are both 1 x (2q+ 1), and

~* is the (2q+ 1) x r matrix of the shifts in treatment effects in the surrogate set stacked

over the (q + 1) x r matrix in the validation set. Moreover, let B~ = M~A(-l)Ml'

and let Bg = M~A~;1)M2. Then the ARE becomes

In the univ~riatecase A*, B~, and Bg) are all scalars. The ratio depends only on

the scalars B? and Bg. No common value can be assigned for all possible directions.

In the multivariate case however, the ARE depends upon A*, B~, and Bg) and a

unique solution may not exist. Therefore, one considers the maximum and minimum

possible values over the variation of ~ *.

If cm(G,J) and CM(G,J) are respectively the minimum and maximum charac­

teristic roots of B~B~(-l), then by virtue of the well-known Courant theorem on the

extremum of the ratio of two non-negative definite quadratic forms we have

Cm(G,J) ~ e (A*,B~,B~) ~ CM(G,J), VA*.

It may be noted that if the normal scores are used for J, and if G is normal then

B(G(k)) = O"kk1/2

, where O"kk is the (k, kyh entry of ~ corresponding to the kth variate.

v defined in (2.10) is the same as ~. These facts imply that B~B~(-l) = I, and hence

cm ( G, J) = CM(G, J) = 1, and hence the normal scores LO· and L?: are asymptotically

power equivalent.

94

..

It may be noted that in such a multivariate setup there is no generally uniformly

most powerful test of the null hypothesis. The least squares test developed here is

asymptotically equivalent to the likelihood ratio test when the distribution of the

errors is normal. But any departure from normality makes it nonrobust. Moreover,

the least squares test is not invariant to monotone transformation of the data, e.g.,

the log transformation. This is not the case with the rank procedure developed in

this work which remains invariant under coordinate-wise strictly monotone transfor-

mation, although it is not affine invariant. Therfore, there may exist some situations

when rank statistics fair much better especially with heavy tails when the distribution

is strictly non-normal.

The same mechanism can be applied to the balanced incomplete block design to

find the ARE and the bounds on it. Consider the same model in (3.1). Assume that

the errors fOiij have a p + q multinormal distribution with null mean and covariance

!: which is assumed to be nonnegative definte.

The intra as well as the inter-block tests developed in Chapters 4 and 5 are re­

placed here with the least square version of these test. Thus, for intra-block inference,

the v x (p + q) matrix of least square estimates, TN,W has entries given by

which is the average of the aligned variable Ul7J over blocks containing the ph treat­

ment and over all replicates. The alignment is done by subtracting the block average

from Ul7J. Moreover, v~) defined in Theorem 3.1 has now entries defined by

(1) = _1 ~~ '" [U(~~* _ U(~)*] [U(~~)* - U(~/)*]vN,kk' L.J L.J L.J W) Oit. Oit) OiL'

nsr 0i=1 i=1 jESi

and vW defined in Theorem 3.1 has entries defined by

k, k' = 1, . ". ,p + q,

V(2) ,= ~~ ~ [U(~)* - U(k)*] [U(~/)* - U(k')*] k, k' = 1,·· . ,p + q",N,kk L.J L.J OiL Oi.. OiL Oi..'

ns 0i=1 i=1

where the dots in the subscripts indicate averages over the dotted subscript. Then,

95

wjth A (1) and A (2) defined as Theorem 3.1 we have

BN is a (p + q)v x (p + q)v matrix, and

nV[Vec(TN,w)] = B N.

The Expected value of vW for large n converges to (r~I)~. Similarly, The

limit of the mean of V~2) for large n is (S~I) (r~). Thus if A = (r~l) A(I) and

B = r (1 - ~) A(2), then the limit of the mean of B N for large n is (A + B) @ ~.

This limit can be now used to bulid the noncentrality parameter as was done in

Section 3.3 or in Section 4.1.3.

On the other hand, for inter-block inference, consider the model in (4.1) and

note that LjES; €aij has covariancer~. Let TN,B be the v x (p + q) matrix of least

square estimates with entries given by

A(k) 1 ~" (k) .TN' J" = - LJ LJ hai , J = 1,"', v, k = 1,'" ,p + q.

, nr" "J a=1 tEPj

The covariance matrix of f;;'~j is the same as that defined in Theorem 4.1 with

the ~~~) replaced by h~~). In such a case the limit of the mean of the matrix W~~

defined in Theorem 4.1 converges for large n to (S~I) (r~), and the limit of the

expected value of W~! converges for large n to (n~l) (;~) .

Define D = ~D(1) and E = r(n-l)D(2) where D(I) and D(2) are defined ins' ns'

Theorem 4.1. The limit of the expected value of the least square version of eN'

defined in Theorem 4.1 converges for large n to (D +E) @~. This limit can be used

to compute the noncentrality parameter under HN as done in Section 4.1.3 using the

least squares estimates. The ARE will be the ratio of the noncentrality parameters

as before and the same conclusions arrived at in the complete case apply here too.

96

..

6.3 Example: The Effect of Zidovudine on Sur­

vival in Patients with AIDS

The example uses data from a double blind placebo-controlled clinical trial (study

02) conducted by Burroughs-Wellcome in the year 1986 to evaluate the effect of

Zidovudine (ZDV) on survival in patients with acquired immuno-deficiency syndrome

(AIDS). A total of 281 patients were enrolled in the study; 144 patients were assigned

to the ZDV group and 137 patients were assigned to the placebo group (See Tsiatis,

Gruttola and Wulfsohn (1995)).

The primary response is the length of survival, but because there is a need to

evaluate new therapies in a shorter period of time, surrogate outcomes are of interest.

Entry into the study was staggered over a period of five months between 19 February

1986 and 7 July 1986. The study was stopped after seven months because of a

significant decrease in mortality in patients treated with ZDV.

At that time Burroughs Wellcome conducted a second study (study 08) where

patients receiving placebo were then offered ZDV and followed for clinical outcomes.

Entry into study 08 was staggered over time between 16 Sept'ember 1986 and 6 April

1987.

The survival time for this illustration was calculated differently for patients on

placebo and those on ZDV. (Tsiatis et al. (1995)). Thanks to Dr. Michael Wulfsohn

for sending me the data sets and the SAS program that calculates the survival time.

The SAS code that calculates the survival time is attached in the Appendix. A

description of the SAS code that calulates the survival time follows.

Since most of the placebo patients were dying of AIDS after approximately 80

weeks of entry into study 02, their sample size became too small to allow a powerful

comparison with the ZDV group. Hence the survival time was truncated at 80 weeks

of entry into study 02. Let a denote the time of entry into study 02 plus 80 weeks.

If the death date, the date of study termination, and the last date the patient

97

w;as known to be alive were misssing then the survival time was calculated as the

difference between 16 September 1986 and entry into study 02 irrespective of the

treatment group.

Let m stand for the minimum of entry into study 08 and 16 October 1986 where

the minimum is taken over nonmissing values of entry into study 08. By October

16 nearly 90% of placebo patients switched to ZDV. Let M denote the maximum of

the death date, the date off-study, the last date the patient was known to be alive,

and entry into study 08, where the maximum is taken over the nomissing values of

these variables. The date off-study for a patient is the date when that patient was

no longer followed because of AIDS complications. Patients died shortly afterwards.

For placebo patients whose death date was not missing and was less than or

equal to m, the survival time is the time from entry into study 02 to the death date.

Otherwise, if either the death date was missing or was greater than m, then the

survival time was calculated as the time from entry into study 02 to the minimum

of entry into study 08, M, and 16 October 1986. Placebo patients were censored if

either their death date was missing or if it exceeded entry into study 08 or 16 October

1986.

For patients receiving ZDV whose death date was not missing and was less than

a, the survival time is the time from entry into study 02 to the death date. Otherwise,

if either their death date was missing or was greater than a, then the survival time

is the time from entry into study 02 to the minimum of M and a. Patients on ZDV

were censored if either their death date was missing or if it exceeded a.

As mentioned in Section 1.2.1 both CD4 and CD8 cell counts (counts/Liter) have

been proposed as surrogates for survival in AIDS patients. The CD4-lymphocyte

count has been proposed as a potential surrogate for human immunodefficiency virus

(HIV), the cause of AIDS, in many studies because of its observed correlation with

the clinical outcome, survival. CD4 counts decrease soon after infection with HIV

because the virus attacks the immune system. They continue to decline during the

98

•

<l.9ymptomatic stage of the disease until full-blown AIDS develops and terminates its

victims.

Patients with advanced disease have lower CD4 counts than those in the early

stage of infection. In this study CD4 counts were determined prior to treatment and

aproximately every four weeks during therapy. The initial CD4 measurement was

used in this example. Moreover, as mentioned in the Section 1.2.1, Baccheti et al.

(1992) found that CD8 counts add predictive power to the CD4 counts and thus would

produce more valid and useful surrogates than the CD4 counts alone. The CD8 cells

act both as deterents of virus multiplication by a mechanism that is not very well

understood, and as accelerators of AIDS by expressing CD38, CD57, or HLA-DR. (

See Romagnani (1994) for a review of CD4 and CD8 cells in the progression of AIDS.

Thus, CD8 count is considered as another surrogate in this example, and again only

the initial measurement was used.

At the time of this analysis covariate data were not available. Patients were

assigned random values for age generated as psuedo-random numbers independently

and uniformly distributed between 18 and 55.

To illustrate the theory an artificial responsewise as well as treatmentwise design

were imposed. With regard to the responsewise design, only those patients who were

not censored were included in the validation set. The validation set data consist of

the survival time, CD4 count, CDS count, and age. There were 24 patients on placebo

that were not censored. These were matched randomly with 24 out of 55 patients

who were not censored and were on ZDV. Thus a randomized block treatmentwise

design with 24 blocks was formed for the validation set.

The surrogate set data consisted of the variables CD4 count, CDS count, and

age. There were 113 patients on placebo and 89 patients on ZDV who were censored.

Twenty four out of 55 patients who were not censored and were on ZDV were added to

the 89 patients re'sulting in 113 patients on ZDV who were censored. These were ran­

domly matched with the 113 patients on placebo who were censored. A randomized

99

b~ock treatmentwise design with 113 blocks was formed for the surrogate set.

Descriptive statistics for the overall data, surrogate set, and validation set IS

given in Table 1 in the Appendix. In the validation set, the overall correlation between

the survival time and the surrogate responses is given in Table 2 in the Appendix.

Moreover, the correlation between survival and the surrogates in each treatment group

in the validation set is given in Tables 3 and 4 in the Appendix. In Tables 3 and 4 the

negative correlation is due to chance fluctuations. The Spearman rank correlation

did not give negative values.

The mean of the survival time for placebo group is 109 days, whereas for the

ZDV group it is 360 days. A two sample t-test to test the equality of means of the

survival time of the placebo group and that of the ZDV group in the validation set

gave a p-value of 0.0001. The normal approximation for the Wilcoxon rank sum

test gave a p-value of 0.0001. This shows that the data support a more beneficial

treatment effect over placebo.

In the surrogate subset, CD4 count, CDS count, and age were aligned by sub­

tracting their block averages, then ranked and Wilcoxon scores were assigned to the

ranks. A 2 x 3 matrix of linear rank statistics was constructed by taking the mean

over blocks of the scores belonging to the ph treatment. Each of these statistics

were centered by subtracting its expected value under the null hypothesis and the

permutational model.

The scores were aligned by subtracting the block score average. The estimated

covariance matrix VN defined in Theorem 2.1 was computed yielding the following

3 x 3 matrix of sums of squares and cross products of the aligned scores summed

over all blocks and treatments and divided by 113(= s( r - 1)), where s = 113 is the

number of blocks, and r = 2 is the number of treatments.

The matrix of coefficients eN for the surrogate set defined in Theorem 2.1 IS

100

g~ven by

0.004424779

-0.004424779

-0.004424779 },

0.004424779

the entries being ±1/226, where sr = 226 is the cardinality of the surrogate set. The

estimated covariance matrix of the surrogates and age scores is

0.16519 0.076239 0.013159

VN = 0.076239 0.16520 0.010146

0.013159 0.010146 0.16502

where the columns and rows correspond to CD4, CD8, and age respectively. Next,

partition VN and calculate the residual statistics, TN-a defined in 2.13

* _ {-0.023765 0.0237655}T No -

-0.002536 0.0025356

where the rows correspond to the CD4 and CD8 rank statistics, respectively, and the

columns correspond to the treatments, placebo and ZDV.

A similar manipulation was done in the validation set. Thus a 2 x 4 matrix of

linear rank statistics was formed. In addition, residuals were calculated by regressing

the survival time on the surrogates after regressing both the survival time and sur­

rogates on age as described in Section 2.2.2. This resulted in a 1 x· 2 vector of linear

rank statistics of survival time corresponding to placebo and ZDV.

Weights were calculated as in (2.21), the weighted linear combination of the

test statistics, TO·, constructed as in (2.22), and the quadratic form La· in (2.23)

computed. The following table gives the values of La· for both the parametric and

nonparametric versions with and without surrogates.

Least Squares Test

Rank Based Test

With Surrogates

70

125

101

Without Surrogates

346

388

The value of the rank based Lo· was 125 when surrogates were incorporated, which

obviously leads to rejection of the null hypothesis of no treatment effect on the primary

endpoint, survival, when compared with the chi-squared distribution with 1 degree

of freedom.

Separate analyses were done using the validation set but ignoring surrogates in

order to compare the with and without surrogate analyses. The value of the quadratic

form was 388 which also leads to rejection of Ho : Tj = 0 j = 1"", r. But the

significance level is smaller when surrogates are ignored. Thus, if significance level is

used as a criterion to see what was gained by using surrogates, this example shows

that one gets a more conservative significance level by using surrogates thus guarding

against faulty conclusions.

Parametric analyses were done using the least square estimates for comparison

with the nonparametric analyses. The same surrogate and validation sets were used.

It may be noted that the Shapiro-Wilk test of normality for the survival time, CD4

count, and CD8 count led to rejection of the null hypothesis that the distribution of

each of these variables is normal (p-value=O.OOOl). Even after taking the logarithm

of these variables, the Shapiro-Wilk test gave a p-value less' than 0.0001. Analyses

were carried on the logged variables. The value of the quadratic form in (2.23) for

the with surrogate analysis was 70.15, and for the without surrogate analysis it was

345.57.

Note that both the parametric and nonparametric methods led to very high

values of the quadratic form when surrogates were ignored. The small significance

level in such a case may lead researchers to optimistic conclusions that may be faulty.

6.4 Further Research

The method developed in this work handles only continuous data. However, it is

common for researchers to be interested in using discrete response variates. If the

102

sample size is fairly large, one may repeat the same analyses described in this work for

each category and compare categories accordingly. However, if there isn't sufficient

data in each category this appraoch may not work. Further research is needed to

incorporate discrete data in the methodology expounded in this study.

The linear models considered in (2.1), (3.1), and (4.1) do not accomodate inter­

action between blocks and treatments. The method discussed here can be extended

to cover such a possibility.

Moreover, the replicates may not be independent in the balanced incomplete

block case. This may happen if the replicates are measured on the same person

over time. Appropriate changes in the covariance matrices should be made to allow

dependent replicates.

A more involved problem is that of estimation of conditional quantiles of both the

surrogate and primary variates (conditional on the covariates). Sen (1994) discussed

such an estimation problem. It is more complex than the hypothesis testing problem.

This has been touched upon in the introductory Chapter. More work in that direction

is needed.

This procedure does not handle missing data. It would be worthwhile to incor­

porate missing cells in such a methodology. Rubin (1976) formulates three conditions

on the process that causes missing data which enable the statistician to ignore this

process when making inferences about the distribution of the data. If () is the param­

eter of the data, and ¢ is the parameter of the conditional distribution of the missing

data indicator given the data, then the missing data mechanism is ignorable if the

following conditons hold:

a) the data are missing at random ifmissingness depends on the data only through

observed values

b) The observed data are observed at random if for each possible value of the missing

data and the parameter ¢, the conditional probability of the observed pattern

103

of missing data, given the missing data and the observed data, is the same for

all possible values of the observed data.

c) The parameters () and </> are distinct in the sense that their joint p?-rameter space

factorizes into a </>-space and a ()-space.

Moreover, Rubin (1976) defines data to be missing completely at random if the

indicator for missing data is independent of the observed data. However, sometimes

the process that caused missing data cannot be ignored for example when the missing

pattern depends upon the observed data.

A nice account of the parametric approach to missing data is found in Rubin

(1976), Little and Rubin (1987), Little (1993), and Little (1994) among others. Their

approach is based on maximum likelihood techniques and the assumption of normality

is essential in the models they select. However, as the example shows that even when

data are transformed, normality may not hold. This leads the researcher to investigate

other, more widely applicable nonparametric methods.

The nonparametric approach to missing variables in multi-sample rank permu­

tation tests for MANOVA and MANOCOVA is developed in: Servy and Sen (1987).

The multi-sample rank permutation tests for the complete data case are extended to

random missing patterns. Define the scores of the observed variables as they were

defined in the complete case (See Puri and Sen (1971)), disregarding the blanks, and

filling up any blank with the mean of the scores of the non-missing values of the

variable it belongs to. They show that under a random missing scheme, the rank­

permutation principle holds and yields conditionally distribution-free test statistics.

They study in detail the asymptotic relative efficiency of the proposed test statistics.

An extention of their methodology to the block case is planned to make this study

gain wide applicability.

104

* ENTRY02= DATE OF ENTRY INTO STUDY 02 (MMDDYY)

* ENTRY08= DATE OF ENTRY INTO STUDY 08. ALL PATIENTS ENTERING THIS

* STUDY WERE PUT ON RETROVIR THERAPY. (MMDDYY)

* DDEATH = DATE OF DEATH (MMDDYY)

* DTERM = DATE OFF STUDY (MMDDYY)

* DALIVE = DATE PATIENT WAS LAST KNOWN TO BE ALIVE (MMDDYY)

**********************************************************************;

data D;

set D;

last_al='160CT86'd;

last_azt=entry02+7*80;

if ddeath ne . then death=1;else death=O;

*placebo;

if trt=O then do;

cens=(death=1 and .<ddeath<=min(entry08,last_al));

if cens=l then survt=ddeath-entry02;

else survt=min(entry08,max(dterm,dalive,ddeath,entry08),last_al)-entry02;

end;

if trt=O and ddeath=. and dalive=. and dterm=. then do;

survt='16Sep86'd-entry02;

end;

*AZT;

if trt=l then do;

cens=(death=l) and ddeath<=last_azt;

if cens=1 then survt=ddeath-entry02;

106

else survt=min(max(dterm,dalive,entry08,ddeath),last_azt)-entry02;

end;

if trt=1 and ddeath=. and dalive=. and dterm=. then do;

survt='16Sep86'd-entry02;

end;

107

Table 1

Means and Standard Deviations

Survival

(days)

CD4

(cells/L)

CD8

(cells/L)

Age

(years)

Surrogate

(N=226)

276

(189)

130

(129)

556

(351)

37

(11)

Validation

(N=48)

235

(164)

75

(99)

492

(409)

38

(11)

108

Overall

(N=274)

268

(185)

120

(126)

545

(362)

37

(11)

•

Table 2

Overall Correlation Between

Survival Time and Surrogates

in The Validation Set (N=48)

Pearson Correlation

(p-value)

Spearman Correlation

(p-value)

Survival CD8 Survival CD8

CD4

CD8

0.0146

(0.9216)

0.193

(0.1885)

0.634

(0.0001)

109

CD4

CD8

0.0137

(0.9259)

0.2609

(0.0732)

0.5868

(0.0001)

Table 3

Correlation Between Survival

Time and Surrogates in The

Validation Set For the Placebo Group

(N=24)

\.

Pearson Correlation

(p-value)

Spearman Correlation

(p-value)

CD4

CD8

Survival

-0.2688

(0.2041)

0.2925

(0.1654)

CD8

0.3562

(0.0876)

no

CD4

CD8

Survival

0.0461

(0.8306)

0.2301

(0.2795)

CD8

0.7149

(0.0001)•

..

Table 4

Correlation Between Survival

Time and Surrogates in The

Validation Set For the ZDV Group

(N=24)

Pearson Correlation

(p-value)

Spearman Correlation

(p-value)

CD4

CD8

Survival

-0.0207

(0.9237)

0.0619

(0.7735)

CD8

0.7435

(0.0001)

111

CD4

CD8

Survival

0.1818

(0.3952)

0.2057

(0.3349)

CD8

0.5127

(0.0104)

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