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RESAMPLING METHODS FOR ADAPTIVE
DESIGNS

By

Hui Zhang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

2005

ABSTRACT

RESAMPLING METHODS FOR ADAPTIVE
DESIGNS

By

Hui Zhang

In clinical trials, the estimation of effects difference is often of primary importance.
Proper resampling methods will provide second order correct estimates, which will
outperform the traditional normal approximation. Bootstrapping has been known
for a long time for i.i.d. random variables. Unfortunately, traditional bootstrapping
methods are not appropriate because the observations of adaptive designs are depen-
dent due to adaptive allocation. We address this problem by developing and studying
resampling methods for dependent data in adaptive designs including theoretical re-

sults and simulations.

© 2005

Hui Zhang

All Rights Reserved

To my parents and Guoren

iv

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my thesis advisors Professor Vin-
cent Melfi and Professor Connie Page for their invaluable assistance and insights
leading to the writing of this dissertation. I am thankful for their constructive sug-
gestions that significantly improved my unprofessional presentation of this work and
brought it. up to the present form. Their kindness and patience are very much ap-
preciated. My sincere thanks also go to the members of my graduate committee,
Professor L. Post and Professor R. Ramamoorthi, for their understanding and helpful
suggestions.

Many thanks to Professor James Stapleton for always being there for me, providing
help when I needed most and pulling me through my difficult. times.

My research career is impossible without the patient love from my family. My
parents VVenyan Zhang and Guoliang Zhang, my sister Rui Zhang have continually
offered me support and encouragement.

Especially, I would like to give my special thanks to my husband Guoren Cheng,
a genius in mathematics. for helping me trouble-shoot the simulations, stimulating

suggestions, and turning my worries into solutions.

TABLE OF CONTENTS

List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Introduction to Adaptive Designs .................... 1
1.1.1 Notation .............................. 3
1.1.2 Allocation Adaptive Designs ................... 5
1.1.3 Response Adaptive Designs ................... 6
1.2 A Short Review of Resampling Methods for Adaptive Designs 9
2 Non-overlapping Block Bootstrap 13
2.1 Introduction of Bootstrap ........................ 13
2.2 Weak Dependence and Stationary ................... 15
2.3 Introduction for Non-overlapping Block Bootstrap ........... 17
2.4 Theoretical Properties of NBB Estimator ................ 20
2.4.1 Resampling Consistency of NBB Variance Estimator for Sample
Proportions ............................ 20
2.4.2 Resampling Consistency of N88 Estimator for the Distribution
of Sample Proportions ...................... 24
3 Martingale Based Bootstrap 27
3.1 Introduction of Martingale Based Bootstrap .............. 27
3.2 Central Limit Theorem for Martingales ................. 28
3.3 Theoretical Properties of MBB Estimator ................ 35
3.3.1 Resampling Consistency of MBB Variance Estimator for the
Binomial Difference ........................ 35
3.3.2 Resampling Consistency of MBB Estimator for the Distribution
of the Binomial Difference .................... 36
4 Sequential Likelihood Resampling 38
4.1 Introduction of Resampling ....................... 38
4.2 Introduction of Sequential Likelihood Method ............. 40

vi

5 Simulations and Results

5.1 Constructing Confidence Intervals ....................

5.2 Resampling Procedure and Simulation Results .............

5.3 Summary and Conclusion

6 Conclusion and Eiture Work
6.1 Conclusion .........
6.2 Future Work ........

Appendices
Appendix A: Definition Index

Bibliography

vii

45
45
47
50

71
71
72

74

75

76

List of Tables

5.1 NBB method, coverage probability of p A ................ 54
5.2 NBB method, interval length of p A ................... 54
5.3 NBB method, coverage probability of p A — p B ............ 55
5.4 NBB method, interval length of p A — p B ................ 55
5.5 MBB method, coverage probability of p A ............... 56
5.6 MBB method, interval length of p A ................... 56
5.7 MBB method, coverage probability of p A — p B ............ 57
5.8 MBB method, interval length of p A — p3 ............... 57
5.9 SLR method, coverage probability of p A ................ 58
5.10 SLR method, interval length of p A ................... 58
5.11 SLR method, coverage probability of p A — p B ............. 59
5.12 SLR method, interval length of p A — p3 ................ 59
5.13 IID Bootstrap method, coverage probability of p A .......... 60
5.14 IID Bootstrap method, interval length of p A .............. 60
5.15 IID Bootstrap method, coverage probability of p A — p B ....... 61
5.16 IID Bootstrap method, interval length of p A -— p B .......... 61

viii

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16

List of Figures

NBB method, RPW(1,0,1) rule, coverage probability of p A .......
NBB method, RPW(1,0,1) rule, interval length of p A ..........
NBB method, RPW(1,0,1) rule, coverage probability of p A — p B- . . .
NBB method, RPW(1,0,1) rule, interval length of p A — p3 .......
MBB method, RPW(1,0,1) rule, coverage probability of p A- .....
MBB method, RPW(1,0,1) rule, interval length of p A ..........
MBB method, RPW(1,0,1) rule, coverage probability of p A — p B-
MBB method, RPW(1,0,1) rule, interval length of pA — p3.
SLR method, RPW(1,0,1) rule, coverage probability of pA. ......
SLR method, RPW(1,0,1) rule, interval length of p A- .........
SLR method, RPW(1,0,1) rule, coverage probability of p A — p3. . . .
SLR method, RPW(1,0,1) rule, interval length of pA — p3. ......
IID Bootstrap method, RPW(1,0,1) rule, coverage probability of p A- .
IID Bootstrap method, RPW(1,0,1) rule, interval length of p A- . . . .
IID Bootstrap method, RPW(1,0,1) rule, coverage probability of p A — p B-
IID Bootstrap method, RPW(1,0,1) rule, interval length of p A — p13. .

ix

63
63
64
64
65
65
66
66
67
67
68
68
69
69
7O
70

Chapter 1

Introduction

In clinical trials, it is often desirable that design be adaptive using past information
to allocate present subjects. For better statistical inference, resampling methods are
often used to provide second order correct estimates. This dissertation is motivated by
these two issues. The dissertation will focus on introducing and analyzing resampling
methods for adaptive designs. Adaptive designs will be introduced in Section 1.1, and
a short review of resampling methods for adaptive designs will be given in Section

1.2.

1.1 Introduction to Adaptive Designs

The main focus of this section is to introduce adaptive design and to review some
of the adaptive designs that have already been proposed in the literature. Consider
a clinical trial to evaluate the relative effectiveness of two treatments, A and B.

It is assumed that patients arrive sequentially and each of the patients must be

assigned to exactly one of the two treatments. We assume that the response will be
observed immediately. It may be desirable that the treatment assignment takes into
consideration the information obtained from the past observations. A design that
incorporates in the allocation rule the information obtained from past observations is
called an Adaptive Design (For the benefit of readers, an index of definitions is given
in Appendix A). A common use of adaptive designs is to compromise between two
major yet conflicting goals: (i) to draw reliable statistical inference for the benefit
of future subjects, which is the utilitarian goal; (ii) to maximize the total number of
patients receiving the better treatment, which is the individualistic goal. Some early
adaptive designs can be found in Thompson (1933) and Robbins (1952).

Adaptive designs can be divided into two groups: allocation adaptive and response

adaptive designs.

0 In allocation adaptive designs, the allocation rules are based only on the allo-
cation of previous patients. For example, the Biased Coin Design, proposed by

Efron (1971), is an allocation adaptive design.

0 In response adaptive designs, the allocation rules are based on the responses
as well as the allocations of previous patients. The allocation rules are either
based on intuitive motivation, such as the Randomized Play-the-wz’nner Rule,
proposed by Wei and Durham (1978), or based on optimal target. allocations,
for instance, the Doubly Adaptive Biased Coin Design, proposed by Eisele(1994)

and Eisele and VVoodroofe (1995).

Applications of adaptive designs in many different. disciplines are discussed in

Flournoy and Rosenberger (1995).

In the remainder of this dissertation, unless otherwise noted, we will consider an
adaptive model in which two treatments are compared and the responses are binary.
In Section 1.1.1, notation is introduced. Allocation adaptive designs are covered in

Section 1.1.2, and response adaptive designs are covered in Section 1.1.3.

1.1.1 Notation

To better explain later work, it is necessary to introduce notation. In the setting of
clinical trials, suppose there are two competing treatments, A and B. Each one of k
sequentially arriving patients must be allocated to either treatment A or treatment B.
Let X_,- and Y]- represent the jth patient’s immediate potential responses to treatments
A and B, respectively, even though in practice only one of them will be observed. For
simplicity, here and below, unless otherwise mentioned, we assume binary responses.
Let p4 and p B be the underlying probabilities of success of treatments A and B,
respectively. It. is assumed that the vectors {X j.YJ-}f=1 are i.i.d., where X1 ~
Bernoulli(pA),Y1 ~ Bernoulli(pB). Note that there may be dependence within
pairs. A sequential allocation procedure is given by a sequence of random variables

6‘ I: ,where
{ .7}_}_l

1 if X j is observed;

(ij = (1.1)
0 if Y]- is observed.
The observation at stage j is given by ZJ- = (5ij + (1 — (SJ-)1? Let {Uj }§=1

be a sequence of i.i.d. uniformly distributed random variables on [0, 1]. independent

of the other variables {61-}le and {X j,Yj}§:1. The sequence of Uj’s is used to
achieve randomization in the allocation. Let fj be the sigma-algebra generated by
X1,...,Xj,Y1,...,YJ-,61,...,6j, here .70 is the trivial sigma algebra. It is useful to
consider the sigma-algebra

Q’j =ij0{Uj+1}. (1.2)

where Uj+1 is the auxiliary randomization. Hence, (91°, j 2 1} is an increasing
sequence of sigma-algebras such that {X1313} is g,- measurable for every j 2 1. Note
that (SJ-+1 is 93- measurable and the random vector {X j+1,Yj+1} is independent of
g,-. Let N A.k and N 33k denote the numbers of patients allocated to treatment A and

B through stage k, then

1.:
NA}: = Zdj (1.3)
j=1
and
k
NB}: =20 —<5j) =k-NA,1.~- (1-4)
j=1

Note that N A.k / k, N 8,): / k are the proportions of patients allocated to treatment
A and B, respectively, by stage k. In practice, adaptive designs typically have nonzero
equal initial sample sizes, so that NA,k and N13,}, are not equal to zero. Also note
that, due to adaptive allocation, N A,k and N 3,], are random variables.

Further let S Air and SB", denote the numbers of successes from treatment A or

B through stage k. Then

k
SAJc = Edi/Y]. (1.5)
j=1

and
k
53,, = 2(1— (my, (1.6)
j=1

Hence the maximum likelihood estimators of p A and p B are

 

SA k
13A,}: = —’ (1-7)
NAJc
and
513 k
A = ’ 1.8
PBJc NBJ: ( )
respectively.

1.1.2 Allocation Adaptive Designs

In allocation adaptive designs, the allocation of each patient depends only on the
allocations of the previous patients. These designs do not consider the response of the
patients, so the individualistic issue is not addressed. A common goal of allocation
adaptive designs is to achieve some degree of balance in terms of the number of
patients assigned to each treatn‘ient.

Complete Randomization consists of assigning each patient to either of the two
treatments with equal probe-ibility. As discussed in Efron (1971), complete random-
ization is used as a baseline for statistical inference while minimizing the possibility
of conscious or unconscious selection bias. Sometimes, especially when the number of
patients in the trial is small, complete randomization may result in some unpleasant
imbalances.

The Biased Coin Design proposed by Efron (1971) is a modification of complete

randomization, which allocates patients to one of the two treatments according to
a biased coin-tossing. Let p be a constant in [0.5,1). Let Dj denote the difference

NAJ/j — NBJ/j at stage j. Then the rule is described by:

p if Dj < 0
P(5j+1=1)= 1/2 if 13,- = 0 (1-9)
1 — [2 if Dj > 0.
The allocation rule tends to balance the number of patients allocated to both treat-
ments.

Wei (1978) noted a disadvantage of this procedure in that the allocation rule
neither takes into consideration the number of patients treated thus far, nor does
it discriminate between small or large absolute values of Dj. He proposed a new
procedure of the biased coin type, Adaptive Biased Coin Design, that takes these
issues into consideration. This design allocates patients according to the following
rule. Let h : [—1,1] ——> [0, 1] be a non-increasing function such that [1(1) = 1 — h(-—3r)
for any a: 6 [—1,1]. Then P(6j+1 = 1) = h(DJ-). The allocation rule will force an

imbalanced experiment to be balanced in the limit.

1.1.3 Response Adaptive Designs

Recall that response adaptive designs are such that the allocation rules are based
on the responses as well as the allocations of previous patients. Such designs are
used when some compromise is sought between both the individualistic goal and the

utilitarian goal or when some other considerations make it desirable to have unequal

numbers of patients assigned to treatments. For example, it is very natural in clinical
trials to want to assign more patients to better treatment out of the two competing
ones.

Zelen (1969) proposed the Play-the-winner Rule, where a success on one treatment
results in the next patient’s assignment to the same treatment, and a failure on one
treatment results in the next patient’s assignment. to the opposite treatment.

The allocation of Play-the-winner Rule is deterministic, while randomness is of
importance in adaptive designs. Randomization not only guards against researcher
bias, but also provides probabilistic basis for an inference from the observations (See
Rosenberger and Lachin, 2002). Wei and Durham ( 1978) incorporated randomness
into the design by proposing the Randomized Play-the-winner Rule ( RP W Rule). We
consider an urn model with initial composition of u balls of two different types, A
and B. When a patient comes in, a ball is drawn and replaced. If the ball chosen is
of type i = A, B, treatment i is assigned. The response is observed immediately, a
success results in the addition of .13 balls of the same color and (1 balls of the opposite
color, a failure results in the addition of 6 balls of the opposite color and (1 balls of
the same color, where B Z a 2 0. This design is denoted by RPIVUL, a, B). Different
choices of the triple (u, a, ,8) give different levels of compromise between balance and
allocation to the better treatment. In simulation, RFD/(1,0,1) is popular because
of its simple implementation. There is one major disadvantage of RPI/V(l, 0, 1) rule,
the initial urn composition ,u = 1 is small. it is an important parameter whose effect
can be explored by simulation. As Rosenberger and Hu (1999) addressed, starting

with just one ball of each color in the urn may lead to a higher chance that the urn

could be overwhelmed by a treatment that is very successful early on. Having a few
more balls of each color to start will lead to more stable results.

So far, the designs we discussed are established with intuitive motivation, but
not in terms of a target. A target is typically unknown and expressed in terms
of limiting proportion, but the limit designed is motivated in different ways, e.g.
precision. (See Rosenberger and Lachin, 2002). So another approach for adaptive
design is based on an optimal allocation target. A large class of such rules are based
on an estimate of such target by current stage. Let u A and VB = 1 — VA denote the
desired (limiting) allocation proportion of treatment A and B, and 19/” and 198,3’ be
the estimates at stage j. Eisele (1994) and Eisele and Woodroofe (1995) introduced a
Doubly Adaptive Biased Coin Design , where the allocation rules depend on both the
current proportions on each treatment and the current estimate of desired allocation

proportion. The allocation rules can be generally described by:

. NA} .
5j+1=1 Uj+1<¢>(-].—‘,VA.3‘) , (1-10)

where a), the allocation function which satisfies certain regularity conditions, is a
function from [0,1]? to [0,1], so that the (j + 1)“ patient is allocated to A with
probability ¢(NA,J- /j, DAJ).

Another example is the Randomized Adaptive Design (Melfi, Page (1998) and
Melfi, Page, Geraldes (2001)), where (f)(.1?,y) = y in this design. In the spirit. of
Neyman allocation, the target allocation is proposed to be I/ A = W/(m +
M) to minimize the variance of 15A,]: — 1331,.

Recently, Hu and Zhang (2004) modified Eisele and VVoodroofe’s design with

weaker and simpler conditions on the allocation function ¢ and proposed a family

of allocation functions aimed at minimizing the variation of proportion N A}: / k.

1.2 A Short Review of Resampling Methods for

Adaptive Designs

This dissertation is a study of resampling methods for adaptive designs. We had a
literature review of adaptive designs in the previous section. We will focus on a short
review of resampling methods, which we will use for adaptive designs, in this section.

The resampling method has been applied to a variety of statistical problems and
often outperformed other statistical methods, more specifically the normal approxi-

mation. Resampling methods have two major advantages:

0 The resampling principle allows estimation of the sampling distribution without
obtaining full knowledge of the underlying population distribution. Hence, it
can be applied to any statistic, not just the sample mean. For example, if we
want to estimate the variance of median, the traditional normal approximation
does not work for this problem because we don’t have a formula. like s/fi to
provide estimated standard errors. Instead, we can calculate the variance of
median from R resample observation vectors as an estimate. (See Efron and

T ibshirani (1993) for more examples).

0 The resampling estimate is second order correct. Singh (1981) was the first

to show the second order correctness property of resampling, i.e. the rate of

resampling approximation to the sampling distribution is faster than the rate
of the traditional normal approximation. In this milestone paper, he derived an
almost sure Edgeworth Expansion for the distribution of the resample statistic.
The work showed the resample statistic corrects the skewness of the underly-
ing distribution and thus attains a better approximation than the normal law
provides. Hence resampling estimates are more accurate compared to normal

approximation.

Resampling methods are particularly useful in the context of adaptive designs. In
a clinical trial, we assume patients will come in sequentially. We will allocate patients

to two competing treatments A and B.

0 First, note that the observation units are patients. So usually the sample size
is small. For small sample size, normal approximation may not be so efficient.
Resampling methods often have better results than normal approximation due

to second order correctness property.

0 Second, usually we are evaluating two competing treatments. Resampling meth-
ods are particularly successful when effects of treatment A and B are close.
Babu (1989) illustrate this point in details by examining the second term of

Edgeworth Expansions.

0 Third, traditional resampling methods usually do well for i.i.d. cases, but they
are not appropriate for adaptive designs. The observations of adaptive design

are dependent due to adaptive allocation. We will show later in Chapter 5

10

that the confidence interval constructed by resampling method assuming i.i.d.
responses has lower coverage probability than interval based on normal approx-
imation. So we are looking for appropriate resampling methods to account for

the dependent structure in adaptive designs.

Resampling methods for dependent data are developed as a consequence of the
rapid growth of dependent data studies. Lahiri (2003) gave a review of resampling
methods for dependent data. Politis, Romano and Wolf (1999) discussed subsampling,
which is a special case of resampling, where the resample size is smaller than the
sample size. Second order Properties were shown by Lahiri (2003) for normalized and
studentized statistics under weak dependence.

In the context of adaptive design, we have some early work. Rosenberger and
Hu (1999) showed for the first time some parametric resampling results in construct-
ing confidence intervals for proportions in adaptive design. Their method does not
involve resampling the original data. Instead, by computing observed estimates of
the success probabilities from a clinical trial, they simulated additional trials using
these estimates as the underlying probabilities. Hence, this method is called Naive
Parametric Resampling.

In this dissertation, we will study resampling methods broadly. The major con-

tributions of this dissertation are:

0 We examine three resampling methods taking the dependence structure of adap-

tive design into consideration.

0 Resampling consistency of resampling estimators are shown for the distribution

11

of sample binomial difference.

0 Our simulations show that. confidence intervals constructed from these resam-

pling methods often outperform the intervals based on normal approximation.

Two basic assumptions that are in force throughout this dissertation, and will be

repeated as needed, are that:

(Al) As is —> 00, NA]: —+ 00, NB}: -—) 00 almost surely .

(A2) As k —> oo, NAJC/lc —> VA, NBJc/k —+ VB almost surely, where VA,VB E (0,1),

and VA-f-VB =1.

The rest of the paper is organized as follows. In Chapter 2, 3, and 4, we describe
three proposed resampling methods for adaptive designs. They are Non-overlapping
Block Bootstrap, l\/Iartingale Based Bootstrap, and Sequential Likelihood Resam—
pling. Consistency of Non-overlapping Block Bootstrap and Martingale Based Boot-
strap resampling estimators will be proved respectively. Simulation results are pre—
sented in Chapter 5. We make concluding remarks and discuss future work in Chapter

6.

12

Chapter 2

Non-overlapping Block Bootstrap

2.1 Introduction of Bootstrap

We will first introduce some notation for resampling methods. Let Zk --'= {Z 1, ..., Z k},
be a vector of random variables with joint distribution Gk. The observed data is a
realization of Zk. Suppose we like to estimate the population mean 6 based on the
observations Zk. Let 6k be the sample mean, and H k denote the sampling distribution
of the centered and scaled estimator Tk = fi(Rk — 6). If {Z1, ..., Zk}’ are i.i.d. with

finite mean and variance, it is well known that

«119,, — 9) i N(0. 02), (2.1)

2 is the population variance. This is so called Normal Approximation for

where a
sampling distribution. The statistical inference of 6, such as constructing a confidence

interval for (9, is based on precise estimation of the sampling distribution H k- Since

13

the joint distribution Gk is unknown, Hk remains unknown. Resampling methods
typically apply to estimation for Hk.

The general procedure for resampling can be described as following:

0 First, an estimator Ck of the joint distribution Gk is constructed from the

observations Zk.

0 Second, we simulate R resample vectors, which are i.i.d. distributed as Ck. We
denote the generic resample vector by Z; i {Z 1*, ..., Z k*}’ , which is the sample
for the resampling version of the original problem. Then we draw statistical

inference for the sampling distribution H k based on R. resample vectors.

Bootstrap is a particular type of resampling method, where the resampling dis-
tribution Ck is the product of estimators of a single marginal distribution Bk, such
that

(3“,, = E, xx Fk. (2.2)
\_‘,_J
k
Hence, the components of the resample vector Z“, ='= {Z1*, Zk*}’ are i.i.d. Rk. A

common choice of F k is the empirical distribution function

at.) a k_IZI(Zj g .). (2.3)

In this case, resamples are simply with replacement samples from the original obser-
vations.

The bootstrap method has been proposed in the context of dependent data.

Different. from i.i.d. case, the population is not characterized by the identical

14

marginal F only, but rather depends on the joint distribution G k of the whole vector
Zk i {21, Z k}, . The Block Bootstrap methods take care of the dependence struc—
ture by keeping the dependence within the block and taking the blocks as resampling
units. For simplicity, we will focus on Non-overlapping Block Bootstrap and apply

NBB method for estimation of p A and p3 separately.

2.2 Weak Dependence and Stationary

Lahiri (2003) discussed in details the resampling methods for dependent data. Two
basic conditions are needed for applying Non-overlapping Block Bootstrap in adaptive
designs: The observation sequence is weakly dependent and strictly stationary.

Let (X n, n E N) be a sequence of random variables. Note that these X’s are
general notation for introduction of definitions. Weak Dependence essentially says
that the dependence of the process decreases as the distance m between the two
segments {Xi : i S k} and {X1- : i Z k + m + 1} increases. V'Ve first introduce the
most commonly used standard measures of weak dependence: Strong Mixing. Let
(9,13,?) be a probability space and let A and B be two sub 0 fields of .7.

Definition 2.1: The measure for strong mixing or a mixing is given by

a(A,B) = sup{|P(A F) B) - P(A) - P(B)[ : A E A, B E B}. (2.4)

Definition 2.2: Let (Xn, n E N) be a sequence of random variables on (9,3277).

Let .73 = 0({X2- : a S i < b}), 1 S a S b S 00. The strong mixing coefficient of

15

{X,-}§>:1 is defined by

a(m) = st1p{a(ff+l,fj3:m+1):k E N}, m 2 1, (2.5)

where a(-, ) is defined above. The process {X i},- _>_1 is called Strong Mixing if a(m) —+
0 as m —> 00.

Definition 2.3: A stochastic process {Xt,t E T}, whose index set T is linear, is
said to be

(i) strictly stationary of order k, where k is a given positive integer, if for any k

points t1, tk in T, and any h in T, the k-dimensional random vectors
(X(t1), ...,X(tk)) and (X(t1+ h), ...,X(tk + h)) (2.6)

are identically distributed;

(ii) strictly stationary if for any integer k it is strictly stationary of order k.

In the context of adaptive design, since Non-overlapping Block Bootstrap will be
done for NA). treatment A observations and N 8,]: treatment B observations sep-
arately from the original sample Zk, we will check stationarity assumption for se-
quences of treatment A and B respectively. Let 3% be the sigma-algebra generated
by X1, Xj, Y1, Yj, 61, ..., 61-, here To is the trivial sigma algebra. It is useful. in

the proofs that follow, to consider the sigma—algebra
gj' =ij0{UJ'+1}, (2-7)

where U341 is the auxiliary randomization we mentioned in Chapter 1. Hence,

{Qj,j Z 1} is an increasing sequence of sigma-algebras such that {XJ-,Yj} is g,-

16

measurable for every j 2 1. Note that 6j+1 is g,- measurable and the random vector

{X j+1, Yj+1} is independent of 9,. We need a theorem from Melfi and Page (2000).

Theorem 2.2.1. Suppose that (Xj+1,YJ-+1) is independent of QJ- for 61)er Z 1,
Then

(i) (X1,X2, ...) are i.i.d. with common distribution FX;

(it) (Y1,Y2, ...) are i.i.d. with common distribution Fy;

(iii) The above two sequences are independent of one another.

Hence, weak dependence and stationarity assumptions are both satisfied. Note
that in the context of adaptive design, dependence structure is induced by adaptive
allocations. Hence, NA]; and N qu are random numbers, where N A,k + N 8,1: = k.
Once the sequence (X1, X2, ...) is truncated as (X1,X2, ..., XNA,k)’ the components
are no longer i.i.d.. However, strong consistency and asymptotic normality for un-
known parameter 6 X = pA still hold. (See Melfi and Page (2000)). Thus, this
wouldn’t hurt in proving consistency of N BB estimator. We will demonstrate this in

Section 2.4 in proofs of Theorem 2.4.2 and Theorem 2.4.4.

2.3 Introduction for Non-overlapping Block Boot-

strap

We restrict our discussion to the case of Non-overlapping Block Bootstrap (NBB)
method in the context of adaptive designs for estimation of p A- Similar results will

hold for estimation of p3. Estimation of pA — p3 follows from asymptotic inde-

17

pendence of sample binomial difference estimator p,” — 13ng. (See Melfi and Page
(2000)).

First, we obtain the sample vector Zk, where N A,k: N13,;c are the numbers of
observations from treatment A and B respectively. Let KN A, 1: denote the subgroup
of Zk, which is composed of N A), treatment A responses. Let 0A,]: denote the exact
distribution of XNAJc' Let 13A,]: = SA,k/NA,k be an estimator of pA based on the

sample X Statistical inference of p A is based on approximating the sampling

NAJc'
distribution of TAN/1,]: = NAJJ/QQLL,c — pA).

Under NBB method, the given vector of observations XNAJc é {X1"”’XNA,l.-},
is partitioned into non-overlapping blocks. Let I denote the block length, b denote
the total number of blocks. And suppose that. l is an integer such that both I and
N A,k /l are large, and 1 tends to infinity with N AJ: but at a slower rate. For ex-
ample, l : [N Aka-j for 0 < 6 < 1. Let b 2 1 be the largest. integer satisfying
lb S N14,}? Then, let Bl, ..., 3;, denote the b blocks of length I under the NBB, given
by Bl = (X1,...,Xl)’,...,Bb = (X(b—1)l+1v---1Xbl),- A set of b blocks are resam-
pled with replacement from these observed blocks to generate the resample vector
Xbl* = ( ’1“, ..., B;)' = {X*, ...,X;,}’. Let Sift denote the numbers of successes from
resamples XI,” for treatment A. Let 13AM denote the sample proportion of the first

bl observations of X then

NA,k’

u
[but = (Ml—12L (28)
i=1

which equals to 15A,]: if NAJ: is a multiple of l. The NBB version of TANA k is defined

18

as Tia a mesh — 13AM), where 157,, = 551m.

The idea of Non-overlapping Block Bootstrap is: because of the strict stationar-
ity, each block has the same l-joint distribution G 1; because of the weak dependence,
these blocks are approximately independent for large values of 1. Hence, we could
take these blocks as approximately i.i.d. units. Let G[’ E G; x x 0;, which is close
to the exact joint population distribution GAJc' By resampling from (81,...,Bb)
randomly with replacement, the relation between xNAJc = {X 1,...,XN A,k }’ and
the exact joint distribution GAJ; can be reproduced by the relation between X}; =
(8*, ...,Bg‘)’ = {X’f, ...,Xgfl’ and Clb, where C? denotes the empirical joint distri-
bution of (81,...,Bb)'. Let E*,Var* denote the expectation and variance of the
resampling distribution, which are conditional on observation vector XN A, k'

Let p A k = N A k—1 2141’]: X j be the sample proportion. The bootstrap version

, , J
is 137,3, = Iii-1231:, 3:.
Note that the resample blocks {B:‘ }[-’:1 are i.i.d. conditional on data XN A, k’ with

distribution

P*(Bi‘ = B.) = (2.9)

for i = 1, ...,b.

The NBB method for estimation of [2A can be described as the following steps:

and Y

0 Obtain the sample vector Zk, divide it into two vectors X

NA,k New

where XN A 1: contains the N A,k treatment A responses.
0 Partition XNA is into b blocks of length l, 81,..,Bb.

19

Resample b blocks from Bl, .., Bb with replacement.

Calculate pj, k from the resamples.

Repeat steps 3 and 4 R times.

Draw statistical inference based on R 15:, k‘

2.4 Theoretical Properties of NBB Estimator

2.4.1 Resampling Consistency of NBB Variance Estimator
for Sample Proportions

Let XNA k be the vector of responses from treatment A. In this section, it is shown
the variance of NBB estimator 13:, k is strong consistent. Similar results extend to

[3% k' Let TA, N 4 k denote the centered and scaled sample proportion, such that

TANAJ, = y/NA,k(RA,k - 10/1)- (2-10)

Suppose b = [N A,k /l[ blocks are resampled, thus the resample size is bl. The

bootstrap version of T A, N A k is given by

Thu 5 mffiht — 13.4.“). (2-11)

20

The bootstrap estimator of Var(TA, N A,k ) is given by var*(7:i,bl)° Let W,- =
(X(,j_1)j+1 + + X,1)/l,1 S i S b be the average of the ith block and IV," =
(X84),+1 + + Xm/LI S i S b be the average of the ith resampled block under
NBB method. Let W1,- = \flav, —pA) = \/l((X(,-_1),+1+...+X,-l)/l—pA).1 g i g b
be the scaled and centered sample proportion of the ith block. Note that my“, =

b—1 259:1 III/"i. Since Bf, 1 S i S b are i.i.d. conditional on Zk and

% lSiSb (in)

We have 13;, = b-1 22;, W,*. Thus,

Waugh.) Vane/H032), — mt»
blVar*((b‘IZI-’=1(Wi* - PA» — (13AM - ml) (2-13)

= l(;1;Z$-’=1(I’Vi - 17/02 - (13A,k,b - 19,02)-

To prove consistency, we need the following lemma. Under mild moment condition,
which is satisfied for Bernoulli responses, and two major assumptions we mentioned
in Chapter 1, lylelfi, Page and Geraldes (2001, Theorem 2) showed a central limit

theorem for sample proportions, such that

Lemma 2.4.1. As I; —> oo,

(\/NA,k{I3A,k_PA}) $N([O[ [PA‘IA 0 D (2.14)
\/NB,kfRB,k_PB} 0 ’ 0 P808

Hence we have

21

lim Var(TA,NA k) = pAqA. (2.15)
kf—ew 2

To prove consistency, essentially we need to show that the bootstrap estimator

Var..(T/’"1 bl) is an estimator of the population parameter p Aq A- We claim

Theorem 2.4.2. In the context of adaptive designs, ifl_1+l\7;il = 0(1) as k —* 00,

then
Var*(T/';'bl) —> pAqA as, as k —) oo. (2.16)
Proof:
Recall that
1 b
Var . (T); u) =1( (3 Z(W - (13A,k,b — PA)2)- (2-17)

Since , /NA,,,(;5A,,, — pA) 1» N(0,pAqA), in addition, RA,k,b is the sample propor-

tion of the first bl observations of X it follows that 106,,“ — 12,4)2 = 0(1/b) —»

NA, is”

0 as. Hence, it remains to show that

b
l1
l5 21(“f2bl—2p‘4) —-> pAQA- (1.8.. (2.18)

Note that by definition. IV- = \/l W'- — p is the scaled and centered sample
Ii 1 A

22

proportion of the it" block. In addition, note that l —> 00 as k —+ 00. Hence, for each

i, 1 S i S b,
d
Wii —’ NULPAQA) (2-19)
Thus,
1 b
M, 2 W13) —+ pig... (2.20)
i=1

Equation (2.18) holds because (IV,- — p A)2 2 0. This finishes proof of theorem

2.4.2. C]

This theorem shows the consistency of var*(TA,bl)’ as long as l tends to infinity
with N A,k but at a slower rate. There are many research work Show that the optimal
block length, which minimizes the asymptotic MSE of Var...(TZ,bl), is of the form
I = CNXSU + 0(1)) as NAJc —> 00 as, where C is a constant depends on some
population parameters. For estimating the distribution function and quartiles of
TAvNA,k’ the optimal block length is of the form I = CNX:(1 + 0(1)). (See Hall,
Horowitz and Jing (1995), Lahiri (1999c, 2005)).

Our final goal is to estimate the binomial difference p A — p3. We prove the
resampling consistency for 13:1,]; — 13;“. Let b A, b B denote the total number of blocks
from vectors XNAJc and YNBJc respectively. Let T; = WHITE}; —;3*qu) — (pAykfiA —

13 8.1.7123» be the centered and scaled resample binomial difference. we then have:

23

Theorem 2.4.3. Under same conditions of Theorem 2.4.2, then

——->pAqA+pBQB as, as k—+oo.

VA ”8

 

V(lr*(\/E((I5:4,k“l5h,kl‘(RA,k,bA —AB,k,iB>>>

(2.21)

Proof:
Result follows from Theorem 2.4.2, assumption 2, independent separate resam-

pling for p A and p3, and Slutsky’s Theorem. Cl

2.4.2 Resampling Consistency of N BB Estimator for the Dis-
tribution of Sample Proportions

Recall that H k = P(TA, N A k S 1:) denotes the sampling distribution of TA, N A k' Let
If k = FATE,“ S 2:) denote the distribution of T2,“. We attempt to approximate H k
by Hk.

We establish the consistency of the NBB estimator for sampling distribution of

TA, N A, k by following theorem:

Theorem 2.4.4. Under the same condition of Theorem 2.4.2,

sup [PAT/Z bl S r) — PfTANA k S :r)| —+ 0, as. as k. -—+ 00. (2.22)
:rEIR I I ’

Proof:

Since TAN/1 k converges in distribution to N(0,pAqA), which is continous, by

24

Polya’s Theorem, we have

sung IP(TA,NA k S x) — @(r;pAqA)| ——1 0 uniformly, as k —> oo. (2.23)
x6 ’

Thus, it suffices to show

sup |P...(TZ bl S :17) —- <I>(a:;pAqA)| ———> 0, as. as k -—> oo. (2.24)
xER I

Let W: = (Xfl-l)i+1 + + XE)/l,1 S i S b denote the average of the ith
resampled block under NBB method. Note that l/Vl’", ..., WE" are i.i.d. conditional on

Zk,and
1 b b l
TAM = WWI“ ‘fiA,k.b) = mfg Z ”3* —I5A,k,b) = 2 £04”? ~fiA,k,b)- (225)
i=1 i=1

For all 5 > 0, let ANN, = gm, E...(li",-* — aAkabPIa/Qw; — 15,4,ka > 6).
Note that 8:,1 S i S b are i.i.d. conditional on Zk. As I) —> 00, by asymptotic

normality of 15,4“, we have

iZLl E*(I’V,* - 13A,k,b)21(\/f|W,-* - 15A.k,b| > (5)
= 1% Zi=1(Wzi — W(fiA,k,b - PA))21(|Wu - ”(13AM - PA)| > b<5) (2.26)
= (W11 - i/lffiAJct - PA))21(|W11 — ”(15AM — PA)| > M)
——> 0 a.s..

By the Central Limit Theorem for independent random variables, the distribution
of TAM converge to N(0,pAqA) almost surely as k —» 00. This finishes the proof of
Theorem 2.4.4. C]

Finally, we will show the resampling consistency of NBB centered and scaled
binomial difference T12“. Let Tk = flap/1,), — pA) — (1533,, — pB)) be the centered and

scaled binomial difference. We have:

Theorem 2.4.5. Under the same condition of Theorem. 2.4.2,
sup |P*(T,: S :r) -— P(T;C S :r)| ——+ 0, as. as k ——+ 00. (2.27)

IER

Proof:
Result follows from Theorem 2.4.4, independent separate resampling for p A and

p B assumption 2, and Slutsky’s Theorem. Cl

26

Chapter 3

Martingale Based Bootstrap

3.1 Introduction of Martingale Based Bootstrap

Martingale Based Bootstrap was introduced by Lin et al ( 1993) for checking the
normality of Cox model. Later, Lin and Spiekerman (1996) also applied it for model
checking in a parametric regression. Wang and Jing (2000) applied this method to
inference for a class of functionals of survival distributions and termed it Martingale
Based Bootstrap, abbreviated as MBB method. Recently, Wang and Wang (2001)
applied it to inference for the mean difference in the two sample random censorship
model.

Compared with other resamplng methods, an obvious advantage of MBB method
is its simple implementation involving only resampling from a normal distribution.
Suppose we want to estimate the binomial difference p A — p3. Typical resampling

methods are conducted by resampling with replacement from the original sample

27

observations, and then calculating p2 k — 1333 k based on resamples as an estimate of

p A — p B- Martingale Based Bootstrap follows following steps:

0 First, based on the asymptotic normality of the scaled and centered sample
binomial difference 15A,}; — 153,,“ we construct an estimate for the asymptotic

variance Of 15A,}; — 158*.

0 Second, the martingale based bootstrap estimates will based on simulations

from a normal distribution with mean zero and this variance estimate.

In this chapter, we will show that MBB method works well in the setting of adaptive
design if we can show the martingale structure of the 15.4,}: — p37,, and prove the

asymptotic normality accordingly.

3.2 Central Limit Theorem for Martingales

We first demonstrate the martingale structure of the sample binomial difference [3A,]: —
133$. Recall from Section 1.1.1, all the moments of X fs and Yj’s are finite because
they are binary responses. we want to estimate p A, p3 and the binomial difference
p A — p B- we consider 1) A — p B and note that it is easy to extend the results to p A or
p B- In this section, it is shown that asymptotic normality of 13.4.1: — 133,), holds when
the following two assumptions are satisfied.

As we mentioned in Chapter 1, two basic assumptions are in force throughout:

0 Assum tion 1: N —> 00, k —— N -> 00 almost surely as k —> 00.
p A,k A,k

28

o Assumption 2: NA'k/k —» VA, NBA/ff —> VB, almost surely as k —+ 00, where

VAaVB E (0,1),VA + VB =1.

Recall that 7-3 is the sigma-algebra generated by X1, ..., Xj, Y1, ..., Yj, 61, 61-. It.

is useful, in the proofs that follow, to consider the sigma-algebra

QJ- =ij0{Uj+1}. (3.1)

Hence, {Qj, j 2 1} is an increasing sequence of sigma-algebras such that (X j, Yj)
is g measurable for every j Z 1. Note that (SJ-+1 is g,- measurable and the random
vector (X j+1, Yj+ 1) is independent of Q.

The sample proportions are defined by :

. 2215 X
W = J ,, J J (3.2)
23:16.)
and
- 2: —1(1— (5 )Y
PBJ; = J), J J (33)

 

Melfi, Page and Ceraldes (2001) proved a central limit theorem in the context of
adaptive designs for general difference of sample means 2? — 17. Follow their spirit of

proof, we have
Theorem 3.2.1. Under assumptions 1 and 2, as k —+ 00,

. - i P q P ‘1
firm —pB.t> - (pi —pB>1 ‘—> No... A A + B B). (3.4)

1’A ”B

 

29

Proof:

Fix real constants a. and b. define for each k 2 1 and j = 1, k

=(1/fl {a<X —pA<>‘ +b<Y—p3)<1—6j>}. (3.5)

Note that kaj is g,- measurable. In addition, note that 63- is gj_1 measurable

and X j, Y]- are independent of 9,4, then

Ekajlgj—il Elfl/flfaijj — PAl5j + b(Yj_PB)(1" 5jllgj—1l
(1/\/—) )jfa5 E(X -PA) + b(1- 5 j)E(Yj — 1913)} (3-6)

=0.

Hence, {I'ij : k 2 1,1 S j S k} is a martingale difference array. Let Ski =
23-21 l/ij for i = 1,...,k. Therefore, {Ski : k 2 1,1 S i S k} is a zero mean
martingale array with differences {If k J :k > 1,1 S jS ’k}. Note that E(Sk,-)2
a2 + b2 < oo, hence {Ski : k 2 1, 1 S i S k} is square integrable.

By the martingale central limit theorem, see Theorem 3.2, Hall and Heyde (1980),

it will follow that

k
, 1
2 WM La N(0. aszQAl/A + b2PBQBVB) (3.7)
i=1

if the following three conditions are satisfied:

max [ii/M) 3» 0. (3.8)
J

30

, p 2
Z ”33- -+ a2PAQAVA + 5 P303149. (39)
j

E(maxI/Vk?j) is bounded in k. (3.10)
.7

Note that E(Sk,-)2 S a2 + b2 implies condition (3.8). For condition (3.10), note
that
mijwkjl g 1/\/h(|a|+ lbl) £0, as k—>oo. (3.11)
Finally, for verifying condition (3.9), note that
23' WE]- - (a2PA‘IAVA + bzpeqsl/B) = 0'2ij 2921ij - PAl25j — PAQAVA}

+b2ff 252103 - PB)2(1 - 53') - PBQBVBl-
(3.12)

It suffices to show that both terms on the right converge almost surely to 0.

Now, write the first term as

2

k
a NA}:
7; foxj - PA)2 - PAQAl‘Sj + GQPAQ/if—k— — VA)- (3.13)
i=1

By the assumption 2, the second term in this equation converges almost. surely to

For the first term, define, for each k 2 1,

31

k

1
M. = 2 3M,- — pi)"2 — mm). (3.14)
i=1

Note that {Ailw k 2 1} is a martingale. In addition, all the moments for binary

response are finite. Then we have

I:
1
E<M§> < El((X —pA)2 — 12.4qu 2 .—2< (3.15)
i=1

Hence, sup;c E(.M,E) < 00. By L2 convergence theorem, Mk converges almost
surely to an almost finite limit. Kronecker’s lemma implies the first term converges
to 0 almost surely. The term involve b2 in turn converges to 0 almost surely.

Leta =1/1/A,b = —1/1/B, we have

 

WM: (1/\/—‘) k3)X{(( —PA)<5j )/VA ((33 -PB)(1- (fill/VB}, (3-16)
such that,
k.
2 ij 1» N(O, “(1’4 + quB). (3.17)
. ”.4 ”8
Note that

VEKPAJC -PB,i-) — (PA -PBl =Zj=1(1/\/_’){((XJ -)'PA)5)‘V—

(3.18)
-((Y-PB)(1— 51))N—l-

Hence, By assumption 2 and Slutsky’s theorem, we finished this proof. El

32

This is so called normal approximation of the sampling distribution of sample
binomial difference 13,”, — p 3,1,. The above normal distribution can used for general
statistical inference, e.g. constructing a confidence interval for p A — p B

In Martingale Based Bootstrap method, this normal distribution can be justified
as our resampling distribution. We will look at the asymptotic variance of sample
binomial difference, i.e. p Aq A / VA + quB/z/B, construct an estimate of asymptotic
variance, and then resample from a normal distribution with mean zero and this vari-
ance estimate. Note that, the second condition of central limit theorem for martingale,
i.e. equation (3.9), provides us an natural estimator of the asymptotic variance.

By equation (3.18), replace p A, p B with their consistent estimators 13A,}: and p B, 1:-
Strong consistency of 15A,k and P3,]: have been shown by Melfi and Page (2000), hence
the error introduced here is of 0(k‘1/2), which is negligible as we calculate the variance

estimator. Let

W)..- =<1NE>{((X.- —m,aa>fi — (<13- —1sB,k)<1 — (saw/’31}. (3.19)

Then we have,

2521 I’ll?)- = NZ3,kZ§.—.15j(Xj - PA,k)2 + Ngik Z§=1(1_ 51W”) - Pat)2
= kNXiKl — 15A,k)2SA,k +1534’k(NA,k - 5A,k)l
+ kNgilU - PB,k)2SB,k +1523,k(NB.k ‘ SBJr-ll'

(3.20)
By Theorem 3.2.1, as k —> 00, we have
k P
2 W13) -+ PAQA/VA + PBQB/VB- (3.21)
i=1

33

From now on, we let Tk denote the centered and scaled sample binomial difference,

such that

Tk = WEl(PA,k - PB,k) - (PA — PB)l (322)

with asymptotic variance 030 = pAqA/VA + quB/VB.

Denote the bootstrap version of Tk by

T1: = VEIIIXAA — Phi.) - (PA,k - Pail] (3.23)

where p2,: — 1513.1. are based on resamples from the normal distribution, i.e. N ( 13A,, —
RB,k~N,Zil(1“PA,k)2SA,k +1331,k(NA,k - SAM] “LAG-5,21. [(1 -PB,k)2SB,k +13%,kUVBJc —
S B,k)l)- In the spirit of bootstrapping, we assume the relation between sample and
population can be reproduced by the relation between resample and sample. Hence,

we hope that T; converges to the same normal distribution as Tk does, i.e.

 

-... .1. - - d PAQA PBQB
\/EI(PA,k-I)B,kl‘(PA,k—PB,A~,)I—*N(0» u... + VB ). (3.24)

We will prove this in Theorem 3.3.2.
Let 5,: be a random variable from the resampling distribution which is conditional

on Zk, such that

E}: N N(0. Njilfl - PA.k)2SA,k +R2A,k(NA,k — SA,k)l

_2 . 2 -2 (3.25)
+NB,kl(1 — par) 53.1. + Par-(NBA — SBA-ll)-

34

Clearly, 152 k — pg k can obtained as

13:1,}; — EBA = RAJ: — 158,): “I” 5; (326)

This will be used in the proofs in Section 3.3.

3.3 Theoretical Properties of MBB Estimator

3.3.1 Resampling Consistency of MBB Variance Estimator

for the Binomial Difference

Let Var... denote the variance of the resampling distribution, which is conditional 011

observation vector Zk. The bootstrap estimator of Var(Tk) is given by Var...(T,:).

Note that,
T; = fil(fi:1,k - 1313,19 — (PA,k - Pail] (3 27)
= flag).
Hence we show the consistency of Var*(T; )2
Theorem 3.3.1. If NAjc/k —) I/A almost surely as k —> 00, then
vm~,(:r,:) E. “(M + “”3. (3.28)

 

VA 1”B

35

Proof:

Var,.(TAf) = f’a.r*(\/h(£;))
= A7(Var*(£;)) (3.29)

L PARA + P3113.
”A ”8

3.3.2 Resampling Consistency of MBB Estimator for the
Distribution of the Binomial Difference

Theorem 3.3.2. Under assumptions 1 and 2,

sup [PAT]: S :r.) — P(Tk S r)| E» 0 as k —> oo (3.30)
IEIR

Proof:

Since Tk converges in distribution to N (0, 03,0), it suffices to show

sup [PAT]: S :r) — @(rwgcfl gr 0, as k —> oo. (3.31)
:rElR

Note that,

T; = ms). (332)

where E]: is a normal variable. It suffices to show the consistency of MBB variance

36

estimate for the binomial difference. Followed by Theorem 3.3.1, we have

var..(r;;) 3» ago. (3.33)

This finishes the proof of Theorem 3.3.2. Cl

37

Chapter 4

Sequential Likelihood Resampling

4.1 Introduction of Resampling

In this chapter we will focus on Sequential Likelihood Resampling, abbreviated as
SLR method. Recall that, as we mentioned in Chapter 2, when the observations
are dependent, the underlying joint population distribution Gk is not equal to the
product of i.i.d. marginal distributions F. Let Ck be an estimator of the joint
population distribution Gk constructed from the observations Zk é {Z1,...,Zk}'.
Resampling generalizes bootstrapping by eliminating the requirement that Ck be
the product of identical marginal estimators. It allows the estimator Ck to be the
product of a group of conditional distribution estimators based on the observations
Zk. Resampling methods aim at capturing the underlying data. generating process,
which gives the dependence structure of the observations.

Recall g,- = .7) V 0(Uj+1),1 S j S k are the nested increasing sigma-algebras

as we defined in Chapter 1, such that (SJ-+1 is g,- measurable and the random vector

38

{Xj+1,YJ-+1} is independent of gj. Let Zk i {21, Z2, Zk}I be the vector of sample
observations with joint population distribution Gk. We write the joint distribution

Gk as the product. of the one step ahead conditional distributions F j, 1 S j S k.

Gk(Zii Zkl = Eifzil H522 ijzjlgj—il

4U
k (

= Hj=1 Fj.

Let Fj = Fj(ZJ-[Qj_1),1 S j S k be an estimator of one-step ahead conditional

distribution F j. Accordingly,

ék<zi.....zk) =a(zi)n§=2 Iii-(2.19.4)
k ~

Let. Z; i (Z; , Zk*}’ denote a resample vector from {CJ- }§=1. Resamples are

obtained by simulating from F} sequentially and independently, i.e.

ISjSk. mm

The major idea here is that we will update the estimate of the conditional distri-
bution F J- based on the information we obtained by stage j. The actual conditional
distribution F j is hard to estimate, we are going to construct the estimator Fj intu-
itively based on the above idea. We propose to resample from a particular choice of

an

F j, that imposes the conditional moment restrictions.

39

4.2 Introduction of Sequential Likelihood Method

Sequential Likelihood Resampling method applies empirical likelihood method at each
stage j, 1 S j S k, sequentially. Empirical Likelihood Method was first introduced
by Owen (1988). The idea of empirical likelihood is very natural and appealing.
Let p AiPB be our parameter of interest. Let {Z 1, Zg, ..., Zk}' be observations from
joint distribution Gk. Let F (ZJ) be the probability mass of observation Z j. The
empirical distribution function F k assigns equal probability mass to each observation
and is often considered a nonparametric maximum likelihood estimate of F k because

it maximizes the likelihood function

k
Lem) = H{F<zj)} (4.4)
j=1

over all distribution functions F. Hansen in his milestone General Method of Mo-
ments (Gh’Ih/I) paper (1982) addressed that for small samples or dependent process,
it is often advantageous to profile the maximum likelihood estimator by conditional
moments which are based on current observations. The empirical likelihood method
is used to numerically calculate the probability mass under linear constraints, i.e.
237:1“? (Z 1)} = 1 and the conditional moments constraints. These are so called the
Profile Maximum Empirical Likelihood Estimators.

With this in mind, we define the empirical likelihood function in the context of
adaptive design. Note that in the context of adaptive design, we have 4 distinct

outcomes:

0 patients assigned to treatment A and we observe a success.

40

a patients assigned to treatment A and we observe a failure.
0 patients assigned to treatment B and we observe a success.

0 patients assigned to treatment B and we observe a failure.

At stage j, define multinomial distribution of 23- with cell probabilities IV]- =
(r1 j,r2j,r3j,r4j). Here j is fixed and rlj is the probability of observing a suc-
cessful treatment A, rgj is the probability of observing a failed treatment A, r3]-
is the probability of observing a successful treatment B, r4j is the probability of
observing a failed treatment B, such that for each 1 S j S k, 2?:1 rij = 1. Let
nz-j,i = 1,2,3,4 be the number of assignment to four cells by stage j. Note that
nlj = SA’k, 7in = NA). — 8A).,n3j = SB,k?n4j = N13,;c — 38,]: in previous notation.
Our goal is to estimate p A and p B, which are given by
T1}

73‘
PA = —, and PB = J

—. (4.5)
7‘1) + 7‘2j 7‘31' + T4i

Note that since {Xj,Yj} is independent of 9,4, rij’s depend on j, but r1j/(r1j +
7‘2j), r3j/(r3j + r43) do not depend on j. We impose a conditional moment condition
to incorporate the information from the current observations. For each 1 S j S k,

let tj be martingale difference

U = (ijXj - PA) + (1 - 5j)(Yj - PB)- (4-6)

41

We have shown in Chapter 4 that. the tj satisfy the conditional moment restriction

E(tjlgj—1): 0. (4-7)

We will include this as a linear constraint as our method of profiling for finding
the maximum empirical likelihood estimator (MELE) of F j, 1 S j S k.
Note that our observations fall into four distinct cells, accordingly let 9,- denote

the four possible values tj could take, such that
91:1—PA. 92 = ‘PA: 93 =1-PB, 94 = -P3- (4-8)

Thus, for each 1 S j S k, the profile maximum empirical likelihood estimator of
(rlj, rgj, r3}, r4j) will maximize

4
EmpiricalLikelihoodj(pA,pB) = H rig-"U (4.9)
i=1

subject to linear constraints

4

4
Zr.” =1, and Zgfl‘U = 0. (4.10)
i=1

i=1

Let E = Ej(Zj[QJ-_1),1 S j S k be an estimator of one-step ahead conditional

distribution F j, which can be described as a multinomial distribution with cell prob-

42

abilities:

4

I’Vj = (7713'. 7:23“. 7‘3]; 1:43) = argmarrljflgj,1.3j.,.4j Z n,J-log(r,-j), (4.11)
i=1

where 2:le rij = 1. Numerically, the profile MELE for rij’s are given by

nij

 

1 g i < 4. (4.12)

r--=. . _
U 7+)‘9i

where A is the Lagrange multiplier, which solves equation

 

4
ginn-
. 4.13
g .7 + Agiz ( )

Note that, RAJ and pBJ- are given by

RAJ = g—T, (171d 73ng = (4.14)

773} +T~‘4j

As a nonparametric method, the empirical likelihood method is best known for
its advantage of conveniently accommodating auxiliary information containing in the
adaptive allocation (SJ. See, for example, Qin and Lawless (1994, 1995), Chen and

Qin (1993), and Chen and Sitter (1999).

Sequential Likelihood Resampling method can be described in some simple steps:

0 Calculate fig, 1 S j S k, by the numerical 1'1'1etliod in equations (4.11) and

(4.12) described above.
0 Simulate resamples Z; sequentially and independently from Ej,1 S j S k.

43

0 Calculate 15:1,}: and Elik from the resamples, where 13:1,), = SEA/NAM and
~ _ =1: :1:
Pale - SBJc/NBJc'

0 Repeat steps 2 and 3 R times.

0 Draw statistical inference based on R 13:1, k and 13*3, k‘

To prove the resampling consistency of the Sequential Likelihood Resampling es-
timator 15:, k — pg k1 we need to solve for the Lagrange multiplier A. The solution is
complicated, see equation (4.13), so we will explore the theoretical property of SLR

estimator in our future work.

44

Chapter 5

Simulations and Results

5.1 Constructing Confidence Intervals

Many statistical inferences can be drawn from resampling methods. In this chapter,
we will focus on constructing a two—sided 100(1 — a)% confidence interval for the
binomial proportion p A: and the binomial difference p A —— [23. We want to compare
the simulation results of normal approximation with each of these the resampling
methods we examined in our previous chapters respectively.

There are two criteria to evaluate a confidence interval:

0 Coverage Probability: Given a nominal confidence level, the coverage probability
of the interval should be close to the nominal. Given deviance from the nominal,
we prefer conservative confidence intervals rather than anti—conservative ones.
Conservative means that the coverage probability of the interval is at least as

large as the nominal.

45

0 Internal Length: Fix the coverage probability, the shorter the interval length

the better the confidence interval.

Hence, an ideal confidence interval should have at least nominal coverage proba-
bility and shorter interval length.

Many intervals based on normal approximation have been proposed. In particu-
lar, for binary responses, the Agresti Interval is recommended as the gold standard
confidence interval for binomial proportions. See Brown, Cai and DasGupta (2001).
Denote EAJ; = SA”), + 2, S3,;c = SBJc + 2, NA]: = NA,k + 4, NBJc = N13,). + 4. We

recenter the maximum likelihood estimators 13A,), and 133,), as:

C131
C131

Ak, and 15qu: ~B’k. (5.1)

A,k Nai-

lI

 

PA,k =

K

We will compare confidence intervals constructed from our proposed three resam-
pling methods with Agresti Interval to show the merit of resampling. In the context
of adaptive designs, the procedures for constructing confidence intervals can be de-
scribed as following:

Agresti Interval

e We obtain our sample vector Zk.

o A 100(1 - (1)70 confidence interval for p A — p3 is given by

 

PAMAJ: + PEA-(73k
NA,k N13,].-

 

(PA,k — PB,k) i Za/2\/( ), (5-2)

where Z042 is the a / 2 cutoff point of a standard normal distribution.

Confidence Interval Using Resampling Methods

46

0 Based on Zk, simulate from the resampling distribution R times, obtaining R

sequences of treatment responses Z2.

Compute 13:1. k — pg k from the resamples, these are R resampling estimates of

PA ‘PB.

Order these R 15:4,, — 153,, as (15:4,, — 153p“), (152,1. — 1373*)(3).

o A 100(1 — a)% percentile confidence interval is given by

((132,... — pairs/21.033), - Psalm—“0””). (5.3)

5.2 Resampling Procedure and Simulation Results

Simulations are done for various adaptive designs, such as Randomized Play-the-
winner Rule, Adaptive Randomized Design, Doubly Adaptive Weighted Design, etc.
Since the simulation results are similar, we only include here RPW(1,0,1) rule for its
simple implementation. Recall that, RPW (1,0,1) rule can be described by an urn

model, such that
e We have initial composition of 1 ball of each of two different colors.
0 When a patient comes in, a ball is drawn and replaced.

0 If the ball chosen is of color A, treatment A is assigned. If the ball chosen is of

color B, treatment B is assigned.

0 A success results in addition of 1 ball of the same color, a failure results in

addition of 1 ball of the opposite color.

47

We keep allocating patients with this rule. we obtained our sample vector Zk.
Then we will apply our three resampling methods based on this sample.

Simulations are run for a wide range of values of parameters p A and p3. Results
are reported for p A and p3 = 0.1 through 0.9 in increments of 0.1. The sample
size k = 100, and for each sample we choose resample size R = 2000. We calculate
coverage probability and interval length based on N = 10000 iterations.

Tables 5.1-5.12 show the results of coverage probability and average interval length
of pA and pA — p3 using NBB, MBB, and SLR methods. The numbers in the
parentheses are coverage probabilities of Agresti method. For each combination of
parameters, three dimensional plots for coverage probability and interval length of
p A and p A — PB are included here (Figures 5.1-5.12). The dark surface indicates
resampling method, the light surface indicates Agresti method.

In viewing the simulation results, it is useful to recall the numerator 6 of the
coefficient K3 of the second term of Edgeworth Expansion for sample proportions.

See Brown, Cai and DasGupta (2002). We have

6 _ ((IA - PAlPAqA _ ((13 - PBlpBQB
_ V2 V2 .
A B

 

 

In particular, K3 goes to zero when pA ~ p3 or pA ~ 1/2, p13 ~ 1/2. The
resampling methods will perform better in these cases.
The simulations support the expected large sample behavior. In terms of coverage

probability, the resampling methods have higher coverage probability than the Agresti

48

Interval when p A and PB are close or when p A and p B are moderate or large. Note
that this explains that resampling methods are especially useful in clinical trials,
where we commonly evaluate two competing treatments, so that p A and p B are close.

In terms of interval length, note that NBB and SLR methods have uniformly
shorter interval length than Agresti. While MBB method has similar interval length
as Agresti, the difference is in the third decimal place.

It is worth noting that from the three dimensional plots, the Agresti Interval is
very stable for all combinations of p A and p3. MBB behaves similarly to Agresti
because MBB is a resampling method based on normal approximation. NBB and
SLR do not behave well on the boundaries, i.e. when p A or p B are extremely small
or extremely large. In addition, boundary behavior is different for NBB and SLR.
For instance, NBB method does not do well when the difference p A — p3 is large.
SLR method is bad when p A ~ 0 and p3 ~ 0. This can be explained in terms of two

factors: the nature of the design and the nature of the resampling methods.

0 The nature of RPW rule. The Randomize Play-the-winner Rule, RPW/(11,0, )3)
is designed to account for the ethical issue. The allocation rule will assign more
patients to better treatment. If p A or p B are extremely small or extremely large,
especially when the difference p A — pB is large, observations from the better
treatment will dominate the inferior treatment, which will cause bad inference
for the inferior treatment. For example, in Table 5.1, when p A = 0.1, p3 = 0.9,
the coverage probability of p A is low. We propose two possible solutions: (i)

increase the initial urn composition. In general, starting with few more balls of

49

each color will lead to more stable results; (ii) increase the sample size.

0 The nature of Sequential Likelihood Resampling method. SLR method is based
on estimation of sequential multinomial probabilities. The Profile Maximum
Empirical Likelihood Estimator is based on Maximum Likelihood Estimator
with conditional moment corrections. The case when pA and p3 are both
small will lead to a higher chance that the conditional moment restriction may

overcorrect the Maximum Likelihood Estimator.

5.3 Summary and Conclusion

In summary, in the context of adaptive design, under two major assumptions, al-
though the components of observation vector 2,, are not i.i.d., the sample binomial

difference converges to a normal distribution

. . d P. (I. P (1
flip/1,1. — p31.) — (pi — pm} —> No. —j—A‘1 + 438—3). (5.5)

Let 132 k and 15*3 k be the resampling estimators, if

 

- . . - d PAQA PBQB
VEKP’IAJ, -P*B,k) - (PA,k ’PBJcll —* N01, VA + VB ). (5-6)

the resampling estimator is resampling consistent in distribution. Hence, the corre-
sponding resampling method is theoretically applicable.

\Nith this in mind, we can give a list of possible resampling methods. Note that,

50

the first three methods are not discussed in this dissertation:

o I.I.D. Bootstrap. This traditional resampling scheme is not appropriate for
adaptive designs, because the treatment assignments and response are not ex-
changeable. The dependence structure is not accounted for. Especially, when
sample size is small, the rate of convergence is too slow to lead to reasonable

results. (See Tables 5.13-5.16, Figures 5.13-5.16.)

0 1.1.0. Bootstrap for p A and p3 separately. Dependent. structure is accounted

for by bootstrapping vectors of responses from treatment A and B separately.

0 Naive Parametric Resampling. Proposed by Rosenberger and Hu (1999). the
dependence structure is accounted for by simulating the adaptive rule R times
using 13A,), and 138,]: as the underlying success rates. They demonstrated by

simulation that this resampling method works well for adaptive designs.

0 Non-overlapping Block Bootstrap. Blocking technique is applied. Dependency

is kept within the blocks. (See Tables 5.1-5.4, Figures 5.1-5.4.)

o Martingale Based Bootstrap. Martingale technique is used to estimate the vari-

ance in the limit. (See Tables 5.5-5.8, Figures 5.5-5.8.)

o Sequential Likelihood Resampling. Dependency information is captured by re-
sampling sequentially from a group of conditional empirical likelihood. (See

Tables 5.9-5.12, Figures 5.9—5.12.)

To compare the performance of these resampling methods, we also include the

51

simulation results of I.I.D. Bootstrap, the only resampling method in the above list
ignores the dependence structure of adaptive design.

Keep the setting of simulation same, Tables 5.13-5.16 and Figures 5.13-5.16 present
the simulated coverage probability and average interval length of I.I.D. Bootstrap
method comparing with Agresti method. In most cases, the coverage probability is
lower than Agresti Interval. So the estimate of MD. Bootstrap is not reasonable for
adaptive designs.

In conclusion, there are many resampling methods that are theoretically applicable
in the context of adaptive designs. Resampling methods that appropriately account

for dependence structure usually will outperform the others.

52

Tables

53

PA

 

 

 

 

pB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.804 0.929 0.949 0.962 0.963 0.955 0.944 0.928 0.856
(0.969 (0.957 (0.953 (0.953 (0.947 (0.952 (0.951 (0.952 (0.955)
0.2 0.80 0.92 0.95 0.96 0.96 0.95 0.94 0.92 0.854
(0.966% (0.957% (0.953) (0.952 (0.948 (0.952 (0.952) (0.955) (0.955
0.3 0.79 0.92 0.95 0.96., 0.96 0.95 0.944 0.924 0.84
(0.968 (0.959 (0.954 (0.954 (0.948) (0.952 (0.952) (0.953% (0.956
0.4 0.77 0.92 0.94 0.96 0.964 0.95 0.944 0.92 0.83
(0.970 (0.959 (0.954) (0.955% (0.945 (0.951 (0.951 (0.953 (0.956
0.5 0.73 0.92 0.95 0.96 0.96 0.95 0.94 0.92 0.83
(0.972 (0.959 (0.956 (0.954 (0.946 (0.950 (0.949 (0.953 (0.956)
0.6 0.68 0.91 0.95 0.96 0.96 0.95 0.94 0.92 0.824
(0.972 (0.962) (0.955 (0.955 (0.948 (0.951 (0.949 (0.955 (0.957
0.7 0.61 0.91 0.95 0.96 0.96 0.95 0.94 0.91 0.80
(0.975) (0.964 (0.960% (0.955) (0.949 (0.950 (0.950 (0.952 (0.958)
0.8 0.474 0.89 0.95 0.97 0.97 0.95 0.94 0.90 0.774
(0.975 (0.970) (0.963) (0.958 (0.954 (0.952 (0.950) (0.953 (0.955
0.9 0.16 0.85 0.95 0.97 0.98 0.96 0.934 0.88 0.70
(0.972) (0.972) (0.968) (0.963) (0.957) (0.953) (0.948) (0.951) (0.957)
Table 5.1. NBB method, coverage probability of p A
:1: Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.166 0.203 0.223 0.231 0.229 0.218 0.199 0.170 0.128
(0.176) (0.213 (0.272 (0.240 (0.237 (0.225) (0.205 (0.175 (0.232
0.2 0.17 0.20 0.22 0.23 0.23 0.22 0.20 0.17 0.12
(0.182 (0.239 (0.238) (0.245) (0.242 (0.230 (0.209 (0.178 (0.134)
0.3 0.17 0.21 0.234 0.24 0.23 0.22 0.20 0.17 0.13
(0.189 (0.226 (0.245 (0.252 (0.248% (0.235 (0.213) (0.181 (0.136
0.4 0.18 0.22 0.24 0.24 0.24 0.23 0.21 0.17 0.13
(0.199) (0.236 (0.254 (0.261 (0.256) (0.242 (0.219 (0.178) (0.138
0.5 0.194 0.23 0.25 0.25 0.254 0.24 0 21 0.184 0.13
(0.211 (0.248 (0.266 (0.272) (0.266 (0.251) (0.226 (0.191) (0.142
0.6 0.20 0.24 0.26 0.27 0.26 0.25 0.22 0.19 0.14
(0.228 (0.264) (0.282 (0.287 (0.280 (0.263 (0.236 (0.199) (0.148
0.7 0.22 0.264 0.28 0.28 0.28 0.26 0.23 0.20 0.14
(0.254 (0.288 (0.306 (0.309) (0.300 (0.281 (0.252 (0.211 (0.156)
0.8 0.25 0.29 0.31 0.314 0.30 0.28 0.25 0.21 0.161
(0.296 (0.327 (0.342 (0.343 (0.332) (0.310 (0.276) (0.231) (0.171
0.9 0.30 0.33 0.35 0.35 0.344 0.32 0.29 0.24 0.18
(0.380) (0.400) (0.408) (0.404) (0.388) (0.360) (0.321) (0.268L(0.200)

 

Table 5.2. NBB method, interval length of p A

=1: Numbers in the parentheses are Agresti results.

54

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.987 0.975 0.953 0.920 0.883 0.819 0.703 0.484 0.053
(0.982) (0.971 (0.961) (0.955 (0.954) (0.948) (0.941 (0.937g (0.931
0.2 0.97 0.97 0.97 0.95 0.94 0.914 0.87 0.81 0.68
(0.970 (0.966 (0.964 (0.963 (0.961) (0.956 (0.956 (0.948 (0.952
0.3 0.95 0.97 0.97 0.96 0.954 0.94 0.92 0.90 0.88
(0.964) (0.963 (0.962 (0.958 (0.953) (0.956 (0.956 (0.953 (0.958
0.4 0.924 0.95 0.96 0.96 0.96 0.95 0.94 0.94 0.93
(0.958 (0.958 (0.958 (0.956 (0.953) (0.951 (0.952 (0.951 (0.957)
0.5 0.88 0.93 0.95 0.96 0.964 0.95 0.95 0.95 0.954
(0.953) (0.955 (0.957 (0.955 (0.954 (0.953 (0.953 (0.955 (0.956
0.6 0.821 0.91 0.94 0.95 0.96 0.95 0.95 0.95 0.95
(0.952 (0.955 (0.957 (0.954 (0.954 (0.951 (0.953 (0.952 (0.954
0.7 0.70 0.87 0.92 0.94 0.95 0.95 0.95 0.94 0.94
(0.945) (0.954 (0.955 (0.951 (0.953 (0.951) (0.953 (0.953 (0.953
0.8 0.49 0.81 0.90 0.93 0.94 0.95 0.94 0.94 0.91
(0.941 (0.952 (0.953) (0.953) (0.953) (0.953 (0.952) (0.957 (0.955
0.9 0.05 0.69 0.88 0.93 0.95 0.95 0.94 0.92 0.86
(0.938) (0.954) (0.957) (0.953) (0.951) (0.952) (0.951) (0.959) (0.966)
Table 5.3. NBB method, coverage probability of p A — p B
at Numbers in the parentheses are Agresti results.
PA
my 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.240 0.270 0.288 0.299 0.305 0.307 0.309 0.317 0.344
(0.250 (0.281 (0.300 (0.312 (0.318 (0.322) (0.328 (0.347 (0.404
0.2 0.26 0.29 0.31 0.32 0.33 0.334 0.33 0.34 0.36
(0.281 (0.310 (0.329 (0.341) (0.347 (0.351) (0.358 (0.375 (0.424
0.3 0.28 0.31 0.33 0.344 0.34 0.35 0.35 0.36 0.38
(0.300 (0.329 (0.348) (0.359) (0.365 (0.369 (0.374 (0.389 (0.442
0.4 0.29 0.32 0.344 0.354 0.35 0.36 0.36 0.36 0.38
(0.312 (0.341 (0.359 (0.369 (0.374 (0.377) (0.381 (0.329 (0.430
0.5 0.30 0.33 0.34 0.35 0.36 0.36 0.36 0.36 0.37
(0.318 (0.347) (0.365) (0.374 (0.378) (0.378 (0.379) (0.386) (0.417
0.6 0.30 0.334 0.35 0.36 0.361 0.35 0.354 0.35 0.36
(0.322 (0.351 (0.369 (0.377) (0.378 (0.375 (0.371 (0.372) (0.394
0.7 0.30 0.33 0.35 0.36 0.36 0.35 0.34 0.334 0.33
(0.328 (0.357) (0.374 (0.380 (0.338 (0.370) (0.360) (0.353 (0.362)
0.8 0.31 0.344 0.36 0.36 0.36 0.35 0.334 0.31 0.304
(0.346) (0.374 (0.389 (0.392) (0.386 (0.372 (0.352) (0.333) (0.327
0.9 0.344 0.36 0.38 0.38 0.37 0.35 0.334 0.304 0.27
(0.404) (0.424) (0.432) (0.429) (0.416) (0.393) (0.362) (0.326L (0.297L

 

Table 5.4. NBB method, interval length of p A — p13

=1: Numbers in the parentheses are Agresti results.

55

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.944 0.951 0.954 0.959 0.955 0.956 0.952 0.945 0.931
(0.966) (0.958) (0.954 (0.955 (0.95 (0.951) (0.949 (0.945 (0.94
0.2 0.95 0.95 0.95 0.95 0.95 0.954 0.95 0.94 0.92
(0.967 (0.958 (0.954 (0.954 (0.951 (0.951 (0.949 (0.943) (0.938
0.3 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.944 0.92
(0.969 (0.958) (0.954 (0.952 (0.948 (0.951) (0.95 (0.944 (0.936
0.4 0.94 0.954 0.95 0.95 0.95 0.954 0.95 0.94 0.92
(0.969) (0.959 (0.956 (0.951 (0.947 (0.949 (0.948) (0.943 (0.936
0.5 0.944 0.95 0.95 0.95 0.95 0.95 0.95 0.94 0.92
(0.972 (0.959 (0.955 (0.952 (0.945) (0.947 (0.947% (0.942 (0.936
0.6 0.93 0.95 0.95 0.95 0.95 0.95 0.9 0.94 0.92
(0.973) (0.962 (0.957 (0.955 (0.944 (0.946 (0.943 (0.941) (0.936
0.7 0.94 0.95 0.9 0.9 0.95 0.95 0.9 0.94 0.92
(0.977 (0.966 (0.961 (0.956 (0.948) (0.944) (0.943) (0.939) (0.932
0.8 0.9 0.95 0.9 0.95 0.95 0.95 0.95 0.94 0.92
(0.978 (0.972 (0.963 (0.957 (0.946 (0.941 (0.94 (0.934) (0.928
0.9 0.98 0.95 0.9 0.95 0.95 0.95 0.94 0.94 0.92
(0.976) (0.976) (0.971) (0.957) (0.947) (0.936) (0.932) (0.929) (0.923)
Table 5.5. MBB method, coverage probability of p A
:1: Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.158 0.208 0.234 0.244 0.243 0.231 0.210 0.179 0.132
(0.173 (0.210) (0.230 (0.238 (0.236 (0.226 (0.207) (0.179 (0.138)
0.2 0.16 0.214 0.23 0.25 0.24 0.23 0.214 0.18 0.134
(0.179 (0.216 (0.235 (0.243 (0.241) (0.230 (0.211 (0.182 (0.140
0.3 0.16 0.22 0.24 0.25 0.254 0.24 0.21 0.18 0.13
(0.185 (0.222 (0.242% (0.249 (0.247 (0.235 (0.215)) (0.186 (0.142
0.4 0.17 0.22 0.25 0.26 0.26 0.24 0.22 0.19 0.13
(0.194) (0.231 (0.250 (0.257 (0.254 (0.242 (0.221 (0.190 (0.146
0.5 0.18 0.23 0.26 0.27 0.27 0.25 0.23 0.19 0.14
(0.205 (0.242 (0.261 (0.268 (0.264 (0.251) (0.228 (0.196) (0.150
0.6 0.19 0.25 0.28 0.29 0.28 0.271 0.24 0.204 0.14
(0.220 (0.256 (0.275 (0.281 (0.277 (0.262 (0.238% (0.204 (0.156
0.7 0.20 0.27 0.30 0.31 0.30 0.28 0.25 0.21 0.15
(0.241 (0.276 (0.294 (0.300 (0.294 (0.277 (0.251 (0.215 (0.164
0.8 0.22 0.29 0.33 0.34 0.33 0.31 0.28 0.23 0.16
(0.273 (0.306 (0.323) (0.326) (0.318 (0.300 (0.271 (0.232 (0.178
0.9 0.26 0.34 0.38 0.394 0.38 0.35 0.31 0.26 0.19
(0.331) (0.356) (0.369) (0.370) (0.359) (0.337) (0.303) (0.259) (0.199)

 

Table 5.6. MBB method, interval length of p A

* Numbers in the parentheses are Agresti results.

56

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.965 0.956 0.953 0.949 0.94 0.933 0.925 0.912 0.914
(0.981 (0.967 (0.96 (0.953 (0.942) (0.938 (0.93 (0.919) (0.924
0.2 0.95 0.95 0.95 0.95 0.95 0.95 0.94 0.94 0.94
(0.968) (0.961 (0.958 (0.955 (0.953 (0.951 (0.949 (0.943) (0.947
0.3 0.95 0.95 0.9 0.95 0.95 0.95 0.95 0.95 0.94
(0.958 (0.957 (0.957 (0.953 (0.952 (0.952 (0.955 (0.95 (0.954)
0.4 0.94 0.95 0.9 0.95 0.95 0.95 0.95 0.95 0.954
(0.952) (0.955g (0.957 (0.952 (0.951) (0.954 (0.952 (0.953 (0.954
0.5 0.94 0.95 0.95 0.95 0.96 0.95 0.9 0.95 0.95
(0.944 (0.953 (0.953 (0.952 (0.953 (0.949 (0.953 (0.951 (0.951),
0.6 0.9 0.94 0.95 0.95 0.95 0.95 0.95 0.9 0.95
(0.936 (0.947 (0.952 (0.952 (0.951 (0.95 (0.951) (0.952 (0.945)
0.7 0.92 0.94 0.95 0.95 0.95 0.95 0.954 0.95 0.954
(0.933 (0.947) (0.951 (0.951 (0.95 (0.948 (0.945) (0.953 (0.95
0.8 0.91 0.944 0.95 0.95 0.95 0.95 0.954 0.95 0.95
(0.924) (0.947 0.952) (0.949 (0.948) (0.946) (0.947 (0.952 (0.951)
0.9 0.91 0.94 0.95 0.95 0.95 0.95 0.95 0.95 0.96
(0.925) (0.946) (0.955) (0.95) (0.947) (0.944) (0.946) (0.953) (0.961)
Table 5.7. MBB method, coverage probability of p A — p B
>1: Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.226 0.266 0.289 0.301 0.305 0.303 0.298 0.294 0.302
(0.246 (0.276 (0.296 (0.308 (0.313 (0.316 (0.319 (0.328) (0.360)
0.2 0.26 0.30 0.32 0.33 0.34 0.34 0.34 0.35 0.37
(0.276 (0.305 (0.324 (0.336 (0.342 (0.345 (0.348) (0.357 (0.384
0.3 0.28 0.32 0.34 0.36 0.36 0.37 0.374 0.38 0.40
(0.296) (0.324 (0.342 (0.354 (0.360 (0.363 (0.366 (0.373) (0.397
0.4 0.301 0.33 0.36 0.37 0.38 0.38 0.38 0.394 0.41
(0.308 (0.336 (0.354 (0.365 (0.370 (0.372 (0.373 (0.379) (0.398
0.5 0.30 0.34 0.36 0.38 0.38 0.38 0.38 0.39 0.41
(0.314) (0.342 (0.360 (0.370 (0.374 (0.374 (0.373 (0.375 (0.390)
0.6 0.304 0.34 0.37 0.38 0.38 0.38 0.38 0.37 0.39
(0.316 (0.345 (0.363% (0.372 (0.374 (0.3721 (0.367 (0.365 (0.373
0.7 0.29 0.34 0.37 0.38 0.38 0.38 0.36 0.35 0.35
(0.319 (0.349 (0.366 (0.373) (0.373) (0.367 (0.358 (0.349 (0.349
0.8 0.29 0.35 0.38 0.394 0.39 0.37 0.35 0.335 0.31
(0.329 (0.357) (0.373 (0.379 (0.376 (0.365) (0.349 (0.332 (0.319
0.9 0.30 0.37 0.40 0.41 0.41 0.39 0.35 0.31 0.27
(0.336) (0.384) (0.397) (0.399) (0.391) (0.374) (0.349) (0.319) (0.29%

 

Table 5.8. MBB method, interval length of p A — p B

:1: Numbers in the parentheses are Agresti results.

57

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.051 0.416 0.826 0.976 0.999 0.999 0.999 0.998 0.992
(0.968 (0.957 (0.952 (0.952 (0.949 (0.952 (0.950 (0.955 (0.957
0.2 0.18 0.64 0.91 0.98 0.99 0.99 0.99 0.99 0.98
(0.966 (0.956 (0.953 (0.951 (0.950 (0.952 (0.952) (0.954 (0.960
0.3 0.34 0.78 0.94 0.98 0.99 0.99 0.994 0.99 0.98
(0.966 (0.960 (0.952 (0.953 (0.948 (0.953 (0.952 (0.953 (0.956
0.4 0.46 0.85 0.95 0.98 0.99 0.99 0.99 0.98 0.97
(0.969 (0.960) (0.955) (0.955 (0.946 (0.951 (0.950 (0.953 (0.957)
0.5 0.58 0.89 0.96 0.98 0.98 0.98 0.98 0.98 0.97
(0.970 (0.958) (0.954 (0.955 (0.947) (0.950 (0.951 (0.953 (0.957
0.6 0.66 0.914 0.95 0.97 0.98 0.98 0.97 0.97 0.95
(0.972 (0.959 (0.955 (0.952 (0.947 (0.953 (0.948) (0.956 (0.957
0.7 0.73 0.92 0.95 0.96 0.97 0.97 0.97 0.97 0.94
(0.973 (0.962) (0.958 (0.951) (0.948 (0.949 (0.952 (0.955) (0.959
0.8 0.77 0.914 0.93 0.95 0.96 0.95 0.96 0.954 0.92
(0.973 (0.964 (0.960 (0.954 (0.951) (0.949 (0.955 (0.954 (0.959)
0.9 0.79 0.89 0.91 0.92 0.94 0.93 0.94 0.94 0.904
(0.974) (0.969) (0.962) (0.958) (0.954) (0.949) (0.951) (0.954) (0.961)
Table 5.9. SLR method, coverage probability of p A
* Numbers in the parentheses are Agresti results.
PA
pB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.147 0.188 0.213 0.226 0.229 0.222 0.206 0.178 0.133
(0.176 (0.214) (0.234 (0.242 (0.240 (0.229 (0.210 (0.180) (0.137
0.2 0.14 0.194 0.22 0.23 0.23 0.22 0.21 0.18 0.13
(0.181 (0.219 (0.239 (0.247 (0.245) (0.233 (0.213 (0.183 (0.139
0.3 0.15 0.20 0.22 0.23 0.24 0.23 0.21 0.18 0.13
(0.188 (0.226 (0.245 (0.253 (0.250 (0.239 (0.218 (0.187 (0.142
0.4 0.15 0.20 0.23 0.24 0.24 0.23 0.22 0.18 0.14
(0.196 (0.233 (0.253 (0.260 (0.257) (0.245 (0.223 (0.191 (0.145
0.5 0.16 0.21 0.24 0.25 0.254 0.24 0.22 0.19 0.14
(0.206) (0.243 (0.263 (0.270) (0.266 (0.252 (0.229 (0.196 (0.149
0.6 0.164 0.21 0.24 0.26 0.26 0.25 0.23 0.19 0.14
(0.218 (0.256 (0.275 (0.281) (0.277 (0.262 (0.238 (0.203 (0.154)
0.7 0.16 0.22 0.25 0.27 0.27 0.26 0.24 0.20 0.15
(0.236 (0.272) (0.291 (0.297) (0.291 (0.275 (0.249 (0.212 (0.161
0.8 0.17 0.234 0.26 0.28 0.28 0.27 0.25 0.21 0.15
(0.260 (0.295 (0.313 (0.317 (0.310 (0.292 (0.264 (0.224 (0.170
0.9 0.17 0.24 0.27 0.29 0.29 0.28 0.26 0.22 0.16
(0.299) (0.330) (0.345) (0.347) (0.338) (0.317) (0.265) (0.242) (0.184)

 

Table 5.10. SLR method, interval length of p A

* Numbers in the parentheses are Agresti results.

58

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.487 0.815 0.964 0.990 0.983 0.971 0.947 0.899 0.782
(0.983 (0.971) (0.961) (0.955) (0.954 (0.950 (0.943 (0.939 (0.930)
0.2 0.73 0.91 0.974 0.984 0.98 0.97 0.95 0.93 0.89
(0.969 (0.967 (0.963) (0.962 (0.961 (0.954 (0.955) (0.947 (0.949
0.3 0.82 0.93 0.974 0.98 0.97 0.97 0.964 0.94 0.92
(0.963 (0.961 (0.960) (0.956 (0.953 (0.953) (0.953 (0.955 (0.949
0.4 0.86 0.94 0.97 0.97 0.97 0.974 0.96 0.95 0.94
(0.955 (0.959 (0.959 (0.956 (0.955) (0.953 (0.949 (0.952 (0.950)
0.5 0.87 0.93 0.96 0.97 0.974 0.97 0.97 0.96 0.95
(0.953 (0.955 (0.959) (0.955 (0.957) (0.953) (0.954 (0.955) (0.951%
0.6 0.87 0.93 0.96 0.97 0.974 0.974 0.97 0.96 0.95
(0.952 (0.954 (0.957 (0.954 (0.954 (0.956 (0.954 (0.953 (0.954
0.7 0.84 0.91 0.94 0.96 0.96 0.96 0.96 0.96 0.95
(0.944) (0.952 (0.954 (0.951 (0.951) (0.953 (0.954 (0.957 (0.957
0.8 0.814 0.89 0.92 0.94 0.954 0.95 0.96 0.96 0.95
(0.940 (0.949 (0.950 (0.950 (0.952) (0.953 (0.954 (0.961 (0.960
0.9 0.74 0.86 0.89 0.92 0.934 0.94 0.95 0.95 0.95
(0.936) (0.951) (0.947) (0.949) (0.952) (0.952) (0.955) (0.961) (0.972)
Table 5.11. SLR method, coverage probability of p A — p B
* Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.223 0.250 0.269 0.281 0.284 0.281 0.272 0.257 0.235
(0.250 (0.281) (0.301 (0.312 (0.317 (0.318 (0.318 (0.319 (0.331)
0.2 0.25 0.284 0.30 0.31 0.31 0.31 0.31 0.29 0.28
(0.281) (0.310 (0.329 (0.340 (0.346 (0.347 (0.347) (0.350 (0.360)
0.3 0.274 0.30 0.32 0.33 0.34 0.33 0.33 0.32 0.304
(0.301 (0.329 (0.348) (0.359 (0.364 (0.365 (0.365) (0.366 (0.375
0.4 0.28 0.31 0.334 0.34 0.35 0.34 0.34 0.32 0.31
(0.312 (0.340 (0.358 (0.369 (0.374) (0.374) (0.373) (0.373 (0.379
0.5 0.28 0.31 0.33 0.34 0.354 0.35 0.34 0.32 0.30
(0.317 (0.346 (0.364 (0.373 (0.377 (0.376 (0.373 (0.370 (0.372
0.6 0.27 0.31 0.33 0.34 0.35 0.34 0.33 0.32 0.29
(0.318% (0.347 (0.365 (0.374 (0.376% (0.373 (0.366 (0.359 (0.356
0.7 0.26 0.30 0.32 0.34 0.34 0.34 0.32 0.30 0.27
(0.317 (0.347 (0.365 (0.372 (0.372 (0.366 (0.355 (0.342 (0.331
0.8 0.24 0.29 0.32 0.33 0.33 0.32 0.31 0.28 0.25
(0.319 (0.349% (0.366 (0.372 (0.369% (0.358 (0.342 (0.322) (0.301
0.9 0.22 0.27 0.30 0.32 0.32 0.31 0.29 0.264 0.21
(0.330) (0.359) (0.374) (0.378) (0.371) (0.355) (0.331) (0.300) (0.2661

 

Table 5.12. SLR method, interval length of p A — pg

:1: Numbers in the parentheses are Agresti results.

59

PA

 

 

 

 

p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.938 0.948 0. 947 0.952 0.954 0.954 0.939 0.937 0.939
(0.955 (0.960 (0. .950) (0.952 (0.956 (0.954 (0.943 (0.944 (0.959
0.2 0.93 0.96 0 94 0. 95 0.95 0.94 0.94 0. 93 0.93
(0.958 (0.963 (0. 949 (0. 952 (0.954 (0.948 (0.952 (0. 946 (0.958
0.3 0.93 0.95 0. 94 0. 94 0. 95 0.94 0.94 0. 94 0.93
(0.963 (0.961 (0. 957 (0. 951 (0. 951) (0.943 (0.951 (0. 947 (0.962
0.4 0.93 0.95 0. 94 0.94 0. 954 0.93 0. 94 0. 94 0.94
(0.968) (0.964) (0. 948 (0.954 (0. 955 (0.941 (0. 951 (0. 945 (0.957
0.5 0.931 0 94 0. 94 0.94 0. 95 0. 94 0. 94 0.92 0.93
(0.962) (0. 955 (0. 942 (0.950 (0. 954 (0. 946) (0. 949 (0.938 (0.960)
0.6 0.934 0. 94 0. 93 0.94 0.95 0. 954 0. 95 0.92 0.934
(0.968 (0. 954) (0. 946 (0.951 (0.958) (0. 953 (0. 953) (0.937) (0.965
0.7 0.92 0. 944 0. 94 0.94 0.954 0. 94 0. 954 0 93 0.93
(0.971 (0. 958 (0. 950 (0.950 (0.955 (0. 951 (0. 960 (0. 934 (0.960
0.8 0.99 0. 93 0. 93 0.93 0.95 0. 95 0.94 0. 93 0.93
(0.967 (0. .957) (0 .946 (0.946 (0.954) (0. 953 (0.955 (0. 942 (0. 961
0.9 0.88 0.93 0. 93 0.94 0.954 0.94 0.94 0.94 0. 92
(0.970) (0.964) (0. 954) (0.949) (0.951) (0.949) (0.960) (0.948) (0. 955)
Table 5.13. IID Bootstrap method, coverage probability of p A
:1: Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.110 0.145 0.163 0.170 0.168 ' 0.160 0.145 0.122 0.088
(0.121 (0.152 (0.167 (0 173 (0.171) (0.163 (0.148 (0.126) (0.093
0.2 0.11 0.14 0.16 0.17 0.17 0.16 0.14 0 124 0.08
(0.125 (0.156 (0.171) (0.177 (0.175 (0.166 (0.151 (0. 128 (0.094
0.3 0.11 0.15 0.17 0.17 0.17 0.16 0.15 0.12 0. 0
(0.129) (0.161 (0.176 (0.182 (0.179) (0.170) (0.154) (0.130 (0.096
0.4 0.12 0.15 0.17 0.18 0.18 0.17 0. 154 0.12 0. 09
(0.135 (0.167 (0.183) (0.188) (0.185 (0.175 (0.158 (0.133 (0.098)
0.5 0.12 0.16 0. 184 0.19 0.18 0.17 0.15 0.13 0. 094
(0.143) (0.175) (0.191) (0.196 (0.192 (0.181 (0.163% (0.137 (0.100
0.6 0.13 0.174 0.194 0.20 0.19 0.18 0 16 0.13 0. 09
(0.152) (0.185 (0.201 (0.206 (0.201 (0.189 (0.170) (0.142 (0.103
0.7 0.14 0.18 0.20 0. 21 0.20 0.19 0.174 0.14 0.10
(0.166) (0.200 (0.216) (0. 220 (0.214) (0. 200 (0.179 (0.149 (0.108
0.8 0.154 0.20 0.224 0. 23 0.224 0.21 0.18 0.15 0. 10
(0.188 (0.221 (0.236 (0. 239 (0.232 (0.216 (0.192) (0.160 (0.115)
0.9 0.17 0.22 0.25 0.25 0.25 0.23 0.204 0.16 0.114
(0.226) (0.258) (0.271) (0.271) (0.261) (0.241) (0.213) (0.176) (0.125)

 

Table 5.14. IID Bootstrap method, interval length of p A

=1: Numbers in the parentheses are Agresti results.

60

PA

 

 

 

 

193 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.957 0.951 0.947 0.946 0.939 0.937 0.937 0.921 0.862
(0.978 (0.962 (0.961) (0.957 (0.955 (0.952 (0.962 (0.961 (0.957)
0.2 0.94 0.95 0.954 0.95 0.95 0.95 0.94 0.93 0.92
(0.949 (0.968 (0.964 (0.968 (0.961 (0.960 (0.956) (0.951 (0.949
0.3 0.93 0.95 0.95 0.95 0.94 0.95 0.95 0.94 0.93
(0.954 (0.960 (0.957) (0.964 (0.954 (0.960% (0.955) (0.958 (0.949
0.4 0.94 0.95 0.95 0.95 0.95 0.95 0.94 0.93 0.93
(0.957 (0.954 (0.953) (0.954 (0.957 (0.958) (0.953 (0.945 (0.945
0.5 0.92 0.94 0.95 0.94 0.93 0.944 0.94 0.93 0.94
(0.951 (0.957 (0.958) (0.957 (0.944 (0.950 (0.946 (0.938 (0.947
0.6 0.93 0.93 0.94 0.94 0.94 0.95 0.94 0.93 0.93
(0.958 (0.950) (0.952 (0.949) (0.953 (0.962) (0.947 (0.941) (0.941)
0.7 0.92 0.934 0.93 0.94 0.94 0.944 0.93 0.934 0.92
(0.954 (0.945 (0.946 (0.950 (0.956 (0.954 (0.945) (0.941 (0.936
0.8 0.89 0.92 0.93 0.93 0.94 0.93 0.94 0.92 0.94
(0.948) (0.951 (0.943 (0.941) (0.949 (0.946 (0.950 (0.945 (0.960
0.9 0.854 0.92 0.92 0.93 0.93 0.94 0.93 0.93 0.92
(0.948) (0.953) (0.948) (0.942) (0.945) (0.949) (0.952) (0.958) (0.961)
Table 5.15. IID Bootstrap method, coverage probability of p A — p B
* Numbers in the parentheses are Agresti results.
PA
p3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.157 0.186 0.201 0.209 0.211 0.209 0.203 0.198 0.197
(0.172 (0.197) (0.212 (0.220 (0.223 (0.223 (0.223 (0.227 (0.246
0.2 0.18 0.21 0.22 0.23 0.23 0.23 0.23 0.23 0.24
(0.197) (0.221 (0.235) (0.244) (0.247 (0.249% (0.250)) (0.256 (0.275
0.3 0.20 0.22 0.24 0.25 0.25 0.25 0.25 0.25 0.26
(0.212) (0.235 (0.250) (0.258 (0.262 (0.264 (0.265) (0.271 (0.288
0.4 0.28 0.31 0.334 0.34 0.35 0.34 0.34 0.32 0.31
(0.220) (0.244 (0.2585), (0.266 (0.270 (0.270) (0.271 (0.275 (0.290
0.5 0.21 0.23 0.25 0.26 0.26 0.264 0.26 0.26 0.26
(0.223 (0.248 (0.262 (0.270 (0.272 (0.271 (0.269 (0.270) (0.281
0.6 0.20 0.23 0.25 0.26 0.26 0.26 0.25 0.25 0.25
(0.224) (0.249 (0.264)) (0.270 (0.271 (0.268 (0.263 (0.259 (0.264
0.7 0.204 0.23 0.25 0.26 0.26 0.25 0.24 0.23 0.22
(0.223 (0.251 (0.265 (0.271) (0.269) (0.263 (0.254 (0.244 (0.240
0.8 0.19 0.23 0.25 0.264 0.26 0.25 0.23 0.21 0.19
(0.227 (0.256 (0.270% (0.274) (0.270 (0.260 (0.244 (0.227 (0.212
0.9 0.19 0.24 0.26 0.274 0.26 0.25 0.22 0.20 0.16
(0.245) (0.275) (0.288) (0.289) (0.280) (0.264) (0.240) 40.212) (0.181)

 

Table 5.16. IID Bootstrap method, interval length of p A — pg

:1: Numbers in the parentheses are Agresti results.

61

Figures

62

Coverage Probability of pA

Coverage Probability

 

Figure 5.1. NBB method, RPW(1,0,1) rule, coverage probability of p A-
:1: The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA

Interval Length

 

Figure 5.2. NBB method, RPW(1,0,1) rule, interval length of pA.
a: The dark surface indicates resampling method,
the light surface indicates Agresti method.

63

Coverage Probability of pA—pB

Coverage Probability

 

Figure 5.3. NBB method, RPW(1,0,1) rule, coverage probability of p A — pg.
:1: The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA-pB

Interval Length

 

Figure 5.4. NBB method, RPW(1,0,1) rule, interval length of 1),; — p3.
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

64

Coverage Probability of pA

Coverage Probability

 

Figure 5.5. MBB method, RPW(1,0,1) rule, coverage probability of pA.
=1: The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA

 

Figure 5.6. MBB method, RPW(1,0,1) rule, interval length of pA.
:1: The dark surface indicates resampling method,
the light surface indicates Agresti method.

65

Coverage Probalility of pA-pB

9,099.0
889.88

Coverage Proballlty

 

Figure 5.7. MBB method, RPW(1,0,1) rule, coverage probability of p A -— p3.
at The dark surface indicatw resampling method,
the light surface indicates Agresti method.

Interval Length of pA—pB

Interval Length

 

Figure 5.8. MBB method, RPW(1,0,1) rule, interval length of pA — p3.
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

66

Coverage Probability of pA

Coverage Probability

 

Figure 5.9. SLR method, RPW(1,0,1) rule, coverage probability of 11A.
:0: The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA

Interval Length

 

Figure 5.10. SLR method, RPW(1,0,1) rule, interval length of mg.
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

67

Coverage Probability

Coverage Probability of pA—pB

 

Figure 5.11. SLR method, RPW( 1,0,1) rule, coverage probability of p A — p3.

Interval Length

* The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA-pB

     
  

  
 
  
     

ogzzzzstia
5 ’ooe‘ g3
‘II.;¢:.$‘

Figure 5.12. SLR method, RPW(1,0,1) rule, interval length of pA —— p3.

at The dark surface indicates resampling method,
the light surface indicates Agresti method.

68

Coverage Probability of pA

Coverage Probability

 

Figure 5.13. IID Bootstrap method, RPW(1,0,1) rule, coverage probability of p A-
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA

Interval Length

 

Figure 5.14. IID Bootstrap method, RPW( 1,0,1) rule, interval length of pA.
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

69

Coverage Probability of pA-pB

Coverage Probability

 

Figure 5.15. IID Bootstrap method, RPW(1,0,1) rule, coverage probability of p A ‘PB-
so: The dark surface indicates resampling method,
the light surface indicates Agresti method.

Interval Length of pA—pB

Interval Length

 

Figure 5.16. IID Bootstrap method, RPW(1,0,1) rule, interval length of pA - PB-
* The dark surface indicates resampling method,
the light surface indicates Agresti method.

70

Chapter 6

Conclusion and Future Work

6. 1 Conclusion

In this dissertation, we have investigated the resampling methods of adaptive design.
The dependence structure is accounted for by blocking or estimating of the data
generating process. The martingale structure of binomial difference was explored
extensively. We proved resampling consistency for Non-Overlapping Block Bootstrap
and Martingale Based Bootstrap. Confidence intervals based on resampling methods
are shown to outperform the traditional ones based on normal estimation for realistic
p A and p B in clinical trials. The results of this dissertation can be extended beyond

binary response under mild assumptions of the distributions.

71

6.2 Future Work

Phrther work based on this study could take several directions. First, as we mentioned
in Chapter 4, we will explore the theoretical properties of Sequential Likelihood Re-
sampling estimator.

Second, a common tool in exploring the merit of resampling methods is the Edge-
worth Expansion. The major finding is that the resampling estimator is second order
correct. Edgeworth Expansion has been developed for a long time for i.i.d. observa-
tions. Basic work can be found in Hall (1988). In the context of adaptive designs,
we need to go above and beyond that because the observations are dependent due to
adaptive allocation. The validity of using Edgeworth Expansion needs to be checked.

Third, there are other resampling methods that could be explored in the setting of
adaptive designs. The Sequential Likelihood Resampling method could be extended
from simple empirical likelihood estimates to the inversion of empirical likelihood
ratio test. The frame-work of empirical likelihood is natural and appealing. It is
a nonparametric method but has likelihood theoretic foundations. The maximum
empirical likelihood estimator is transition invariant, and a nonparametric analog
of Wilks’ theorem also holds: take the log-empirical likelihood ratio estimate by
—2, we obtain the empirical likelihood ratio statistic (ELR) that converges to a X2
distribution. This is an important point, since the ELR—based test achieve asymptotic
pivotalness without explicit studentization. Pivoting is theoretically important. when
applying bootstrap. It is often advantageous to select a pivotal statistic because the

distribution of a pivotal statistic is independent of all parameters. Implicit pivotalness

72

is very useful when estimating the variance of the studentized statistic is difficult.
Subsampling technique may be incorporated into the resampling. Subsampling is
another branch of resampling methods, where the resample size is smaller than the
original sample size. It is known that subsampling may achieve better coverage when
full resampling is not.

Fourth, we use the percentile confidence interval in our Simulation. There are many
other non-parametric intervals can be applied in adaptive design, such as percentilet
method and 300 method. In the BC“ method, the confidence interval incorporates
the biased correction derived from Edgeworth Expansion.

Fifth, as we observed from the simulation results, the interval length is similar
but not exactly the same for resampling methods vs. the Agresti Interval. We think
that if we fix the interval length and then observe the coverage probability, the merit
of resampling methods will be more apparent and persuasive.

Sixth, in the spirit of the empirical likelihood method, we may view some adaptive
design processes as Markov Chain with four possible states. The ergodic theorem may
be applied and a Monte Carlo Markov Chain (MCMC) can be used for transition
probabilities. Statistical inference can be conducted based on the limiting transition
probabilities.

Finally, as we mentioned in Chapter 1, in simulation, RPW(l, 0, 1) is popular
because of its simple implementation. The initial urn composition is an important
parameter whose effect could be explored by further simulations. We would expect
more stable results by having a few more balls of each color to start with. These will

be areas of further research.

73

Appendices

74

Appendices

Appendix A: Definition Index

Adaptive Biased Coin Design, 6
Adaptive Design, 2

Agresti Interval, 46

Allocation Adaptive Designs, 2
Biased Coin Design, 5

Bootstrap, 14

Complete Randomization, 5

Doubly Adaptive Biased Coin Design, 8
Empirical Likelihood Method, 40
Martingale Based Bootstrap, 27
Naive Parametric Resampling, 11
Non-overlapping Block Bootstrap, 17
Normal Approximation, 13
Play-the—winner Rule, 7

Profile Maximum Empirical Likelihood Estimators, 40
Randomized Adaptive Design, 8
Randomized Play-the—winner Rule, 7
Resampling Method, 38

Response Adaptive Designs, 2
Sequential Likelihood Resampling, 40
Strict Stationary, 16

Strong Mixing, 16

Weak Dependence, 15

75

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76

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81

   

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