if - Department of Statistics

..
•
GROUP SEQUENTIAL METHODS FOR
BIVARIATE SURVIVAL DATA IN CLINICAL TRIALS:
A PROPOSED ANALYTIC METHOD
by
Sergio R. Munoz
Department of Biostatistics
University of North Carolina
•
Institute of Statistics
Mimeo Series No. 2140T
December 1994
•
GROUP SEQUENTIAL METHODS FOR BIVARIATE SURVIVAL DATA IN CLINICAL TRIALS
A PROPOSED ANALYTIC METHOD.
Sergio R. Muiloz
A dissertation submitted to the faculty of the UniYersity of North
Carolina in partial fulfillment of the requirements for the degree
of Doctor of Philosophy in the Department of Biostatistics.
Chapel Hill
1994
ApproYed by:
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ABSTRACT
Sergio R. Munoz. Group Sequential Methods for Bivariate Survival Data in Clinical
Trials: A Proposed Analytic Method. (Under the direction of Shrikant I. Bangdiwala
and Pranab K. Sen)
Interim analysis on accumulating data in clinical trials is usually done to assess
whether there are significant differences between the experimental treatment under study
and the control treatment in order to decide whether or not to stop the trial prematurely.
Among many reasons for doing interim analysis are the possible early evidence of
treatment differences and also the ethical considerations that subjects should not be
exposed to an unsafe, inferior or ineffective treatment. Group sequential designs
(Fleming & DeMets, 1993) are suited for doing interim analysis as they allow correction
of the type I error which is known to increase as a consequence of repeated testing on
accumulated data. In many studies, more than one observation is obtained from the same
individual, e.g. studies on women where both breasts are the units of analysis,
odontological studies where several teeth are considered, ophthalmological studies
where both eyes are analyzed. In such situations it is necessary to account for the
correlated structure in the data in the context of group sequential analyses.
A model reflecting the correlated nature of a bivariate parametric survival
distribution is developed for the case of the bivariate exponential distribution of Sarkar
(1987). The uncensored and censored type of models are presented. The model
incorporates information from each of the individual organs, which is more efficient than
using information from subjects based on a single outcome summarizing the failure of
the related organs. The developed model is presented and justified on statistical and
biological grounds. The methodology of group sequential testing is developed after
parameter estimation and construction of suitable test statistics. Numerical simulations
are used to illustrate the application of the methodology developed.
.
ACKNOWLEDGEMENTS
I would like to thank my committee members, Drs. Shrikant I. Bangdiwala,
Pranab K. Sen, Harry A. Guess, Clarence E. Davis and Jianwen Cai for their increasing
support and thoughtful comments and suggestions.
In addition, I would like to
recognize Dr. Bangdiwala (Kant) for being exponentially biased to friendly and judicial
advice. To all of them "Many Thanks".
I also gratefully acknowledge the International Clinical Epidemiology Network
(INCLEN)
and the "Study Grant Program of the W.K. Kellogg Foundation" for
allowed me to complete my doctoral studies and research.
Last but not least, I want to recognize the enormous amount of love,
understanding and sacrifice my family have done in my favor. My degree is a result of
their faith in me, and I want to dedicate it to them and to the memory of my brother
Fernando.
11
TABLE OF CONTENTS
Page
v
LIST OF TABLES
LIST OF FIGURES
VI
Chapter
1.
INTRODUCTION
1
1.1
Motivation
1
1.2
Brief Summary of work done in this research
2
2
LITERATURE REVIEW AND MODEL PRESENTATION
5
2.1
Group Sequential Analysis
5
2.1.1
Introduction
5
2.1.2
Haybittle-Peto Boundaries
9
2.1.3
Pocock Boundaries
11
2.1.4
O'Brien-Fleming Boundaries
13
2.1.5
Lan and DeMets Boundaries
20
2.1.6
Bayesian Boundaries
22
2.2
Group Sequential Analysis for Survival Data
25
2.2.1
Introduction
25
2.2.2
Log-Rank Test and the Efficient Score Test for the Proportional Hazard
Model
25
2.2.3
Boundaries for Group Sequential Tests
31
2.3
Analysis of Clustered Data
33
3
MODEL FOR UNCENSORED
DISTRIBUTION OF SARKAR
BIVARIATE
EXPONENTIAL
38
3.1
Introduction
38
3.2
Vector of Bivariate Hazard Rate
38
III
.
•
3.2.1
Definition of Vector of Multivariate Hazard Rate
38
3.3
Sarkar's exponetial Distribution
41
3.3.1
Introduction
41
3.3.2
Definition and Properties of the
Distribution
45
3.3.2.1 Definition
45
3.3.2.2 Properties
46
3.3.3
Marginal MLE' s of A. and y
47
3.3.4
Joint MLE's of A. and y
48
3.3.5
Estimating Equations and Information Matrix
50
3.3.5.1 Estimating Equations
50
3.3.5.2 Observed Information Matrix
51
4
•
Sarkar Bivariate Exponential
MODEL FOR CENSORED
DISTRIBUTION OF SARKAR
BIVARIATE
EXPONENTIAL
54
4.1
Censoring Schemes
54
4.2
Model Specification and Maximum likelihood Estimation
55
4.3
Estimating Equations and Information Matrix
60
4.3.1
Estimating Equations
60
4.3.2
Observed Information Matrix
62
5
GROUP SEQUENTIAL TEST
66
5.1
Introduction
66
5.2
Bivariate Group Sequential Test
67
5.3
Concluding Remarks
71
6
NUMERICAL RESULTS
72
6.1
Introduction
72
6.2
Single analysis results st the End of the study
76
6.2.1
Ignoring the bivariate nature of the data
76
6.2.2
Incorporating the bivatiate nature of the data
79
6.3
Group Sequential boundaries
81
7
FUTURE RESEARCH
83
References
87
IV
LIST OF TABLES
Page
Table
8
2.1
Decisional Procedure in Group Sequential Test
2.2
Pocok Boundaries for a Two-Sided Group Sequential Test with Type I
error a
12
O'Brien-Fleming Boundaries for a Two-Sided Group Sequential Test with
Type I error a
14
Haybittle-Peto, Pocock and O'Brien-Fleming Boundaries for a Two-Sided
Group Sequential Test with Type I error a=0.05 and number of tests K=4
16
2.3
2.4
2.5
Group Sequential Boundaries for Haybittle-Peto, Pocock, O'Brien-Fleming
and Bayesian Approaches for 200 subjects
2.6
Number of Events at Tiem Tk for subjects at risk, by treatment group
6.1
Simulation Scenarios and values for sample of size 500: Bivariate
Exponential distribution of Sarkar
23
27
74
6.2
Summary statistics for scenarios A, B, C
78
6.3
Test statistic for the biavariate exponential distribution of Sarkar
80
lr
v
LIST OF FIGURES
Figure
,
2.1
2.2
Page
Habittle-Peto, Pocock and O'Brien-Fleming Critical Values for a Four
Group Sequential Test at 0.05 Significance Level
Habittle-Peto, Pocock and O'Brien-Fleming Critical Values for a Nine
Group Sequential Test at 0.05 Significance Level
18
19
6.1
Simulated Bivariate Exponential Distribution of Sarkar
75
6.2
Kaplan-Meier survival curves for scenario A
77
VI
CHAPTER 1
Introduction
·,
1.1.-Motivation
In analyzing clinical trials, there are many statistical aspects to be considered.
The type of endpoint to be analyzed, the sampling design and repeated looks at the data
for monitoring purposes, are, among others, considerations in doing statistical analysis.
Regarding these three considerations, our mam interest
repeated looks at correlated bivariate time to-event-data.
It
IS
focused in doing
IS
well known that
observations being taken from the same subject tend to be more alike than measurements
being gathered from independent observations. Ignoring the correlation structure and
•
treating correlated data as if they were independent can result in misleading analyses
depending on the magnitude of the intracluster correlation. There are several examples
of this type of data, including studies on women where both breasts are the units of
analysis, odontological studies where several teeth are considered, ophthalmologic
studies where both eyes are analyzed, and other situations that convey to naturally
clustered data. Community studies where a sample of schools, villages, or workplaces
are gathered are also a very important source of clustered data. In all these situations,
for practical or feasibility considerations, the investigator intervenes on the cluster
(person, school, community), but would prefer to evaluate the endpoint in each of the
individual members of the cluster. Aside from power considerations, this may also be
the logical choice for unit of analysis.
On the other hand, clinical trials are designed with ethical considerations for the
subject and usually thus involve interim analysis of data prior to the scheduled end of the
data collection period. Interim analysis on accumulating data in clinical trials is usually
done to assess whether there are significant differences between the experimental
,
treatment under study and the control treatment in order to essentially decide whether or
not to stop the trial prematurely. Among the many reasons for doing interim analysis are
the possible early evidence of treatment differences and also the ethical considerations
that subjects should not be exposed to an unsafe, inferior or ineffective treatment
(Jennison and Turnbull, 1991).
Group sequential designs (DeMets, 1987) are suited for doing interim analysis as
they allow correction of the type I error which is known to increase as a consequence of
repeated testing on cumulated data. Several procedures have been proposed in the last
fifteen years and some of them are going to be discussed in a later chapter.
.
Most of the procedures for doing interim analyses are suited to analyze
independent observations; therefore, adjustment for clustered data would need to be
done. The purpose of this dissertation is to propose a methodology that allows us to
analyze clustered bivariate survival data under the group sequential framework.
1.2.- Brief summary of work done in this research.
This dissertation intends to provide a solution for solving the problem of doing
interim analysis when parametric dependent bivariate survival data is gathered from each
subject, taking into account the correlated nature of the data and the needed adjustment
for a fixed overall type I error probability.
2
The specific model to be studied deals with subjects contributing two organs to
the analysis such as two eyes, two kidneys or two breasts. A real life example of this
type of data is taken from the "Sorbinil retinopathy trials research, 1990" . They
.
evaluated the drug Sorbinil, an aldose reluctance inhibitor, for ocular diabetic
retinopathy on a sample of 497 patients with insulin dependent diabetes mellitus.
Patients were randomly assigned to take either oral Sorbinil or placebo and the
occurrence of severe visual loss was one of the outcomes being analyzed. In this case,
patients are contributing two failure time observations to the analysis. These failure
times may be correlated and possibly censored.
The goal of the proposed research is to obtain adjusted interim analysis
boundaries for clusters of size two when K looks at the data are planned. The model
considers the joint parametric bivariate exponential distribution of Sarkar (1987). Each
component of the vector is based on each of the individual organs of a single patient,
which is more efficient than using a hazard based on a single outcome summarizing the
failure of the related organs of the patient. This joint distribution is assumed to be
symmetric, so that the corresponding marginal hazard function of each organ is the
same. Furthermore, the model considers the hazard function to be altered for the
remaining organ once the failure of one of them occurs. In other words, it is assumed
that patients have the same marginal hazard function for each organ while no failure has
occurred, but after one of the two organs fails, the hazard function of the remaining
organ is now different from the original marginal hazard function for that particular
organ. Thus, the remaining organ's hazard function is a new hazard conditional on
several characteristics determined by the failure of the first organ.
3
Chapter 2 contains the literature review of group sequential tests for non
censored and censored data; it also contains a review of methodology on analysis of
clustered data. In chapter 3, the bivariate exponential distribution of Sarkar is examined
under no censoring and maximum likelihood estimators and variances are derived.
Censoring is incorporated in chapter 4. Chapter 5 contains the proposed group
sequential test statistics. Numerical results are presented in chapter 6 to illustrate the
methodology. Concluding remarks and future research are presented in chapter 7.
4
.
CHAPTER 2
Literature review and Model Presentation
2.1.- Group sequential analysis
2.1.1.- Introduction
Interim analysis is defined as any assessment of data done during either
patient enrollment or follow-up stages of the trial for the main purpose (among
others) of assessing treatment effects. Sometimes interim analysis conveys the
decision of stopping the trial. If the trial is stopped early for concern that the
..
experimental treatment may increase the incidence of death, then there is no
more data considered, and there are no subsequent statistical inferences
considered beyond estimation of the incidence rate. The complications caused
by interim analyses upon the estimation problem are not being considered in this
dissertation. However, when the trial is not stopped early, standard hypothesis
testing and confidence intervals of the treatment effects do require adjustment for
the previous analyses and are the topic of this dissertation. Table 1 shows
schematically what the decision procedure is in group sequential trials.
Traditional methods (fixed sample size) for the calculation of significance
levels and confidence intervals are not valid after repeated interim analyses,
because they do not achieve the error rates required. Multiple significance tests
can greatly increase the chance of false positive findings. The best known
statistical instrument for interim analyses in controlled clinical trials is group
sequential methodology. These methods depend crucially on the knowledge of
the joint distribution of the test statistic at the monitoring time.
To illustrate the general procedure of group sequential analysis, let us
assume that an experimental treatment is being compared against a control
treatment and a planned total of N patients is divided into K groups of 2n
patients each (2nK=N). Assume now that a group of 2n patients are randomly
allocated to each treatment so that n patients are assigned to the experimental
treatment and n are assigned to the control treatment. The decision to stop the
trial or continue is based on repeated significance tests of the accumulated data
after each group is evaluated. In real life problems, this fact is very restrictive
since data is continuously gathered. Another restriction is that it is also assumed
that response to treatments being compared is immediate.
Let Zi represent the test statistic for the i-th group and ci the critical
value associated with Zi ' i ::; K. Starting at the first interim analysis, in testing
the hypothesis of no treatment difference, the decision is stop the trial if IZ 11 > c1
or take another group of 2n patients and randomly allocate them into the
treatments in the same way described above. If the trial was not stopped at the
previous stage, compute Z2 based on the 4n patients and stop the trial if IZ21 >
c2 or take a new group of 2n patients to be allocated into the treatment being
tested. Keep doing the same procedure until either you stop the trial at any of the
stages or take a new group of patients up to the last group. The values of ci are
chosen such that
Pr{ZI>CJ or Z2 >c2 or ....or ZK>cK I HO}=a.,
6
where a is the pre-specified significance level to test the null hypothesis of no
treatment effect difference. One point to notice so far is that the number of
groups has to be specified in advance. Table 1 illustrates the procedure.
.
7
Table 2.1: Decisional Procedure in Group Sequential Trials.
Condition
Gro
up
Sample
size
Decision
1
2n
ar
Stop the trial
IZll > cl
br
Take a 2nd group
IZII:S; cl
a2:
Stop the trial
IZ21 > c2
b2:
Take a 3-th group
IZll:S; cl; IZ21:S; c2
ar
Stop the trial
IZil > ci
bi:
Take a i-th group
IZll:S; cl ·.. ·... IZi_ll:S; ci-l" IZil:S; ci
aK
RejeetHo
IZKI>CK
hi(
Do not reject Ho
IZll:S; cl ····.·.IZK-ll:S; CK-l··IZKI:s; CK
"
2
i
K
4n
2in
2nK=N
"
•
8
Several authors have proposed a variety of criteria and of methods for
•
choosing the critical values ci, to address the issue of requiring a total probability
of type I error of a pre-specified. Some of the most common strategies are
presented below.
2.1.2.- Haybittle-Peto Boundaries.
Haybittle (1971) and later Peto (1976) defined ad-hoc boundaries for
group sequential tests so that CJ
=
c2
= ....=
cK-1 were chosen in such a way
that the total probability of crossing these boundaries were almost negligible and
finally adjusting cK, so that the overall type I error probability was a. Although
..
it may appear pointless to set up a sequential test in which the probability of early
stopping is almost negligible under the null hypothesis, the approach does
provide reasonable power to detect the kind of major differences in efficacy
which would make early stopping an ethical necessity.
Consider the particular case of comparing the mean of an experimental
treatment against a control treatment and assume that XE and Xc represent the
experimental treatment and the control treatment responses, respectively. Further
•
assume that both XE and Xc are normally distributed with known common
variance 0 2 and mean J..lE and J..lC, respectively. Assume that the null hypothesis
is that of no treatment difference against the alternative hypothesis of treatment
difference and that a total probability of type I error equal to a is desired.
9
The goal is to compare K consecutive groups of 2n individuals where n
of them are randomly assigned to each treatment. Denote by X Ek and XCk the
observed mean responses of treatment and control group at the k-th interim
If
analysis (k=1,2, .... ,K).
l t(X xc)
Define the statistic d k = k
j=1
E J
for k=1,2, ... K. Then dk is
J
normally distributed with mean equal to IlE - IlC and variance equal to 2cr2/nk.
Notice that the increment dk-dk-l is statistically independent of dk-l and
it is normally distributed with mean IlE - IlC and variance equal to 2cr2/n.
To test the null hypothesis of no treatment effect difference versus the
two-sided alternative that experimental treatment effect is different from the
•
control one at a significance level of ct, the statistic to be used is
dk.Jnk
Zk =
~ N(O,l).
.
.J2cr
The Haybittle -Peto procedure is to reject the hypothesis of no treatment
difference if Idkl
~c
H_ P k
.J~
, vnk
for k=l, 2, ...., (K-l) and where cH_P k is a fixed
'
constant for every k less or equal to K-l, and finally for k=K, cH-PK
, is chosen
such that the total probability of Type I error is equal to ct. Equivalently, the
rejection region can be written as IZkl
~
cH_P,k
In practice, taking for example K=4, and assuming that the statistic to be
used to test the null hypothesis of no treatment difference corresponds to a test
10
derived from a standard normal distribution, then c1 = c2 = c3=3.0 and C4=1.98
for a desired overall level a=O.05 .
•
2.1.3.- Pocock Boundaries.
The name of group sequential design was formally introduced by Pocock
(1977). He suggested a constant adjustment to the K fixed critical values
associated with the K repeated tests coming from those K planned interim
analyses.
The decision of rejecting the null hypothesis of no treatment difference is
•
based on the same statistic Zk as defined above. The critical values ck are chosen
such that the null hypothesis is rejected after the k-th group if IZkl > cp(k)=cp,
..
independent of k, for k= 1, 2, ..... , K, where cp is a constant value chosen such
that the overall type I error probability is a. Notice that cp values are constant
for every k=1, 2, ...., K; they are different from the constant fixed values
proposed by Haybittle and Peto for the K-l first groups.
As an illustration of the Pocock procedure, consider the .two mean
comparison introduced in the previous section. The procedure is to reject the
hypothesis of no treatment difference if Idkl
•
~ c .J~
vnk
p
for k=1, 2, ....., K, and
where cp is a constant value. Pocock provided cp values for different values of
K and levels of significance. Table 2 reproduces some of those values.
11
Table 2.2 * Pocock cp Boundaries for a
two sided Group Sequential Tests with
type I error u.
K
u=0.05
u=0.10
2
2.178
1.875
3
2.289
1.992
4
2.361
2.067
5
2.413
2.122
2.555
• Extracted from Pocock ,1971.
•
2.270
10
..
12
In the case of K=4 and cx=0.05, the decision of rejecting the null
hypothesis is based on the fixed critical value 2.361 for every interim analysis.
This compares with the Haybittle-Peto critical value of 3.0 for k=1,2, and 3 and
1.98 for the final analysis (K=4).
2.1.4.- O'Brien - Fleming Boundaries.
The O'Brien - Fleming procedure is also based on the assumption that K,
the number of interim analyses, is predetermined and that groups of 2n
individuals are randomly assigned to the experimental and control groups
respectively. The critical values proposed by O'Brien - Fleming (1979)
..
monotonically decrease with k. This is intuitively accepted since one does not
wish to stop in the early stages of the trial unless differences are substantial
between the groups being compared.
For the comparison of two means presented earlier, the rejection region
of the no treatment difference null hypothesis is given by values of dk such that
I
Co-F
Id k ~.Jk
* ~2d'
.. k
.Jnk' whic. h'IS decreasmg
m .
O'Brien - Fleming provided simulated cO-F values for different values of
•
K, the number of interim analyses, and significance level cx. Table 3 reproduces
some of those values.
13
Table 2.3: * O'Brien - Fleming cO-F
Boundaries for a two sided Group
Sequential Tests with type I error u.
K
u=0.05
u=0.10
2
2.797
2.373
3
3.471
2.962
4
4.048
3.466
5
4.562
3.915
6.597
10
5.595
, Extracted from O'Bnen - FIeIDlng, 1979
14
Table 4 and Figure 1 compare the Haybittle - Peto, Pocock and O'Brien Fleming procedures of group sequential test critical values with K=4 pre-planned
interim analyses and a=O.05. Figure 2 shows a similar comparison for K=9.
Figures present only the upper region since the lower one is a symmetric copy of
the upper one.
.
15
Table 2.4: Haybittle-Peto, Pocock and O'Brien - Fleming Boundaries for
a two sided Group Sequential Tests with type I error a=0.05, number of
tests K=4.
Critical Value
Haybittle-Peto
Pocock
O'Brien-Fleming
cI
3.0
2.36
4.05
c2
3.0
2.36
2.86
c3
3.0
2.36
2.34
c4
1.98
2.36
2.02
16
From figure 2 we can see that the O'Brien-Fleming's method is more
conservative at the beginning of the trial when it is compared with both
Haybittle-Peto and Pocock boundaries. In doing either four or nine interim
analyses, the probability of rejecting the null hypothesis at earlier stages of the
trial is smaller for O'Brien-Fleming method, but the situation is reversed at latest
stages of the trial.
...
17
Figure 1.
Haybittle-Peto, Pocock and O'Brien-Fleming absolute critical values
for a four group Sequential tets at 0.05 significance level.
4.5 . , . . - - - - - - - - - - - - - - - - - - - - - - - ,
4 A~ ••
3.5
3
~ ~ •• - •• - : .. ~.t
2.5 ••
2
-. .----- .-.... ...... .....
. .. .. ..
- . . . _ _ -·.::..:..~. . :-;:-~ ........- :-II
•••• - """0
- •
- Haybittle-Peto
- . -Pocock
1.5
• ••
• O'Brien-fleming
1
0.5
o+--------+--..;..-----+-------~
1
3
2
number of interim analyses (K).
.
18
Figure 2.
Haybittle-Peto, Pocock and O'Brien-Fleming critical values for a nine
group Sequential tests ata 0.05 significance level.
..
7-r-----------------------,
5
4
- . -Pocock
- • • • O'Brien Fleming
- . . . . Hayblttle-Peto
1
O+--~I----+--_+_--+_-_t--_+--+_---f
1
2
3
4
5
6
7
number of interim analyses (K)•
..
19
8
9
2.1.5.- Lan and DeMets Boundaries.
Two main requirements of the group sequential procedures proposed by
Haybittle-Peto, Pocock and O'Brien-Fleming are (a) previous specification of the
number of interim analyses K to be performed and (b) equal increments of
information, say 2n observations in each group. This requirement does not
conform with the actual practical conduct of clinical trials. Interim analyses are
performed at predetermined time periods, and the number of observations in
each group at the k-th look is thus a random variable.
Lan and DeMets (1983) proposed a procedure to address the two
constraints above specified allowing for a total Type I error probability equal to
a. They proposed a flexible discrete boundary, called cO_F(k), for the sequence
of statistics Zk-, k=1, 2, ..., K. The procedure is based on the choice of a
function aCt), called the "spending function", which characterizes the rate at
which the Type I error level a is spent. Assuming that completion of the trial by
time T is scaled arbitrarily such that T=1, then the function a(t) is built so that a
(0)=0 and a( 1)=a. This function gives cumulative probabilities of Type I error
and it allocates the amount of Type I error one can "spend" at each analysis.
If t represents the first boundary crossing of a standardized test statistics
Zk, then aCt) can be written as: aCt) = P{ t
~
t} , o~ t
~
1.
Both Pocock and O'Brien-Fleming boundaries can be approximated
(DeMets, 1987) by functions as defined above:
(a) al (t)=alog{1+(e-l )t}approximates the Pocock boundaries, and
20
..
(b)a2(t)=2-2<I>(1.96/"t) approximates the O'Brien-fleming boundaries for a
=0.05, where <I> denotes the standard normal cumulative distribution function.
For K groups, set arbitrary points, 0 < t1 < t2 <.......<tK:::; 1, such that tk
is the scaled time for the k-th analysis (k=l, 2, ... K). The Lan and DeMets
boundary point cL_o(k) is chosen such that under the null hypothesis of no
treatment difference and given cL-O(l), cL_O(2), ........, cL_o(K-1) the probability
P{I Z 11::;cL-O(l), IZ 21::;cL_O(2), ..... IZ k-11::;cL_O(k-1) , IZkl>cL-O(k)}= a(tk)-a(tk_
1), where Zk represents the standardized test statistic at t=tk, k=l, 2, .... ,.K.
The increment a(tk)-a(tk-1) represents the additional amount of
significance level that one uses at time tk-
This procedure implicitly incorporates the concept of "information time".
On the scaled [0-1] interval, the information time t represents the fraction of
patients randomized up to that time. Having the knowledge of the target sample
size, calendar time can be transformed into information time having the
knowledge of the target sample size. This issue allows for analysis not having
equal sized groups constituting an advantage over the previous methods before
described.
21
2.1.6.- Bayesian Boundaries.
Bayesian monitoring of clinical trials has been addressed by several
authors. (Berry, 1985, 1989; Freedman and Spiegelhalter ,1989; Jennison and
Turnbull, 1990) among others have presented the Bayesian approach to group
sequential tests as an alternative to the previous methodologies presented above.
Consider a randomized clinical trial comparing two treatments. Let 8
denote the true treatment difference, with a pre-trial prior distribution denoted by
p(8). Suppose further that n pairs of subjects have been entered into the study
and one subject of the pair has been assigned to the experimental treatment and
A
the other one to the control treatment. Let 5 denotes the estimated treatment
difference. Assume that (i) the prior distribution of 8 is given by p(8)
2/no) and (ii) the conditional distribution of the estimate
A
8"
~
N(80 ' (j
A
given 8 is p( 8" 15) ~
N(8, (j2/n ), where (j2 is the variance of an individual pairwise difference, 80 .is
the pre-trial expectation of the treatment difference, and
no reflects the precision
of the prior information about the treatment difference.
A
~
The posterior distribution p(8 18 ) is given by: p(8 18 )
i
~
n8+no8o c? ]
----=-.::....;
. For a two sided test, assume that 8 > L\T and 8 < L\c
n+n o n+ no
reflect values where the experimental treatment is considered superior to the
control and values where the control treatment is considered to be superior to
the experimental one, respectively.
The decision whether or not to stop the trial is based on numbers Pc and
PE such that one should stop the trial if either:
22
r
8
Pc
=
p(018)do < E or PE
-~
=
r
p(018)do < E
.
8£
In words, the trial is stopped if either there is a very little chance that the control
treatment is superior (experimental treatment is preferred) or the same is true for
.
the experimental treatment (control treatment is better).
Since the posterior distribution of 0 is normal, rejection conditions can be
rewritten
as
follows:
(i)
"8 > c?
n
i<
c?[~E _oono _<1>-I(E)Jl
lJ
n
~
c?
,
l' ~c~ - oono
- <1>-1 (E) lJ
c?
and
(ii)
op
where <1>- 1 (8) is the inverse of the standard
op
normal cumulative distribution at point E, and o~
2
=
0
n+no
is the variance of the
posterior distribution of o.
As one can notice, Bayesian boundaries are dependent on the choice of
the prior and mainly on the choice of the prior variance.
Table 5 shows three Bayesian boundaries, assuming
0 2 =0.05,
00=0 and
E
=0.025, and their comparison with Haybittle-Peto, Pocock and O'Brien-Fleming
boundaries on the basis of 200 subjects for testing the null hypothesis of 0=0.
The three Bayesian rules correspond to three different standard deviations (o/-J
no) of the prior distribution, namely 0.025, 0.075, and 0.15. .
23
Table 2.5. Group sequential boundaries· for Haybittle-Peto, Pocock, O'BrienFleming and Bayesian approaches for 200 subjects.
..
!:i.
Non - Bayesian
Bayesian
cr/..}no
Interim
Analysis
# of pairs
of patients
H-P
Pocock
O-F
1
20
0.38
0.72
2
40
0.27
3
60
4
5
0.025
0.075
0.15
0.52
0.72
0.45
0.36
0.37
0.39
0.27
0.22
0.24
0.30
0.68
0.28
0.21
80
0.19
0.18
0.26
0.51
0.23
0.18
100
0.17
0.14
0.14
0.42
0.19
0.15
* Extracted from Friedman and Spiegelhalter (1989)
24
2.2.- Group sequential analysis for survival data.
2.2.1.- Introduction
One of the most frequently used test statistics in the analysis of survival
data in clinical trials is the log-rank test, where patients entering the study are
assigned randomly to the treatments. Most of these trials are designed so that the
log-rank test is evaluated at the end of the study. The log-rank test for group
sequential analysis was proposed by Tsiatis (1981). His derivation is based on the
efficient score test for the proportional hazard model, and shows that the logrank test is the special case when treatment assignment is the only covariate in
the model. The asymptotic joint distribution of the efficient score test for the
proportional hazard model at different points in time was established to be used
for repeated significance tests.
2.2.• 2.- Log-rank test and the efficient score test for the proportional hazard
model.
The development of the log rank test, Mantel, (1966) and Peto (1972); to
compare two treatments uses a conditioning argument based on the number of
subjects at risk of having the event just prior to each observed event time.
Let T1 < T2 <
< TL denote the ordered observed distinct event time
points in the sample formed by combining the two treatment groups to be
compared, and let Eik and Aik , k=1, 2, .....L, denote the number of observed
events and the number of subjects at risk, respectively, in group i at time point
25
Tk Let Ek and
Ak
denote the corresponding values in the combined sample.
The data at time Tk can be represented as in table 6.
26
Table 2.6. Number of events and no events at time Tk from those subjects at
risk, by treatment group.
Treatment Group
Event
Experimental
Control
Total
Yes
Elk
E2k
Ek
No
Alk - Elk
A2k - E2k
Ak-Ek
Total
Alk
A2k
Ak
27
Given Aik' the Eik have a binomial distribution with number of trials Aik
and, under the null hypothesis of a common event rate 'A in the two treatment
groups, approximate event probability
'A(Tk)~t.
Fisher's exact test for equal
binomial parameters (experimental and control treatment groups) is based on
conditioning further on Ek. Given the above stated conditions, Elk has a
hypergeometric distribution with mean and variance given by:
Alk
[]
A lkA 2k Ak- E k
-2
Elk = E [Elk ] = Ek --- ; ulk = V Elk = E k
Ak
A
Ak- 1
k
Given the margins in each of the L tables at the observed event times, the
vector {Ell - Ell> E 12 - E12' ....... , ElL - Eld is a vector of observed minus
conditionally expected number of event across observed event times. Further,
assuming independence among the elements of this vector, then the statistic
defined by:
known as the standardized two sample log rank test, has approximately standard
normal distribution.
Let us now introduce the Cox proportional hazard model (Cox, 1972).
Let 'A(t;x) represents the hazard function at time t for a subject with the set of
covariates x=(xl> x2, ...., xp)'. The Cox proportional hazard model specifies that:
'A(t;x) = 'Ao(t)ex~,
where "-o(t) is an arbitrary unspecified base-line hazard function (all covariates
are set equal to zero)
and
J3'=(~l> ~2,
parameters.
28
.....,
~p)
is a vector of regression
..
The partial likelihood for P is formed taking the product over all points ti,
i=l, 2, ...., k, to produce:
L
L(~) = nJ exp(xi~)
•
i=l
l LexP(x1P)J
Dei,
i=l
leA(t i )
where 8 i represents the conditional probability that subject i has the event at
time ti given that the total number of subjects at risk at that time is A(ti) and that
exactly one event occurs at ti.
When comparing the survival time subject to censoring among two
groups without any other covariate, and x denoting an indicator function with
value one if experimental treatment and zero if control, the corresponding hazard
function and partial likelihood related to the Cox proportional hazard model are
expressed as follows:
A(t;X)=AO(t)e
x
~ and L(P)=D J Lexp(xtP)
exp(x.~) } =D8
k
k
l
j
o
leA(tj)
The log likelihood is:
l =
Ln(L(~» ~ Xi~ - {~~P(XI~»)} ~Lnei .
=
{
=
Since Xi is either zero or one, the log-likelihood can be rewritten as:
k
f oc P~dli -
k
(
'1
~1'te~~xP(Pdlil))
where d Ii is the number of events at time ti in the experimental group and d Iii is
the number of subjects at risk of experimental group at time ti.
29
The efficient score statistic and the information for f3 are gIven
respectively by:
where Aj(f3) and Vj(f3) denote respectively the mean and the variance of d iiI
under weighted sampling without replacement from the risk set at time tj.
To test the hypothesis of no treatment difference, f3=0, the score test can
be applied. The score test requires the computation of S(O) and 1(0). Under the
null hypothesis, the sampling scheme defining
Alf3) and Vj(f3) reduces to simple
random sample without replacement, so that Aj(O) and Vj(O) are just the mean
and variance of an hypergeometric random variable and their expressions are:
Aj (0) = di rIj
rj
Vj(O) = dj rOjr~i(ri -d i )
ri (rj -1)
where rOj and rIi are the risk set sizes in the control and experimental group
respectively.
The score test, which compares U(O) to its estimated vanance 1(0)
corresponds to the log rank test previously described here. The name comes
from the relation to the exponential ordered scores of the test (Cox and Oakes,
1984). This test can also be obtained by setting up a separate 2x2 contingency
table at each event time and carrying out the combined Mantel-Haenszel test
(piece-wise exponential fitting).
30
•
2.2.3.- Boundaries for Group Sequential Tests.
Group sequential boundaries with time to event data are based on the fact
that the log-rank test behaves like a partial sum of independent normal random
variables with variance proportional to the
number of observed events.
Therefore, all the group sequential methods described before are applicable,
using the standardized two sample log-rank test Q defined previously.
Suppose that at calendar time Tc' there are e distinct event times t 1> t2'
....., te, where tj represents the time from entering to the study to the occurrence
of the event. Then the Log-rank tests is to be computed a scheduled maximum of
N calendar times and at each time Tn,
the log-rank test is computed and
compared to a boundary cn' n=l, 2, ..., N.
The Haybittle-Peto, Pocock and O'Brien-Fleming boundaries can
be
used and they required the number of looks to be predetermined and to have
equal number of events between consecutive looks. This is an equivalent
requi~ement
for normal and binomial data of equally spaced analyses (equal sized
groups).
The Lan and DeMets boundaries can be computed using the "estimated
information fraction" (Lan, Lachin; 1990). That is estimating the fraction of
patients having the event at the time of the analysis. The actual value of this
fraction is generally unknown since the total number of events at the end of the
trial is unknown. Lan, Reboussin and DeMets (1993) defined what they called
31
the "natural estimate of information" as a function of the cumulative distribution
function of the time to event random variable.
•
.
32
2.3. Analysis of Clustered Data.
.
Some studies involve randomization of clusters of units rather than units
themselves in the allocation of treatment groups. Examples of cluster allocation
ranges from studies where both eyes of an individual are allocated to a systemic
treatment regimen in ophthalmologic studies; to the evaluation of educational
strategies, where either schools or classrooms are randomly allocated to different
teaching methods; and to evaluation of health care procedures where medical
practices or even entire communities are randomly assigned to either intervention
or control group.
Application of methods for this kind of data can be found widely in the
literature. Donner (1989) and Rosner (1984) proposed statistical methods for
analysis of clustered data in ophtalmological studies; DeRouen et al (1991) and
Hujoel et al (1991) presented methods applied to dental studies. In both
ophtalmological and dental studies, the patient represents the cluster and the eyes
or teeth, respectively, represent the units of analysis within clusters.
Since units within clusters are generally not statistically independent,
randomization of clusters are in most cases less efficient than studies in which the
randomization has been carried out on units themselves. Despite this
disadvantage, cluster randomization often conveys economical, ethical or
administrative practical advantages. In some cases, it is the only feasible
allocation procedure available to the investigator.
A key issue in cluster randomization is the fact that the variability among
clusters tends to be higher than the variability within clusters and the larger the
33
ratio between these two sources of variation, the larger the degree of the within
cluster dependency (Donner at aI, 1990).
As an example, consider the simple case of sampling one cluster of size
two from an infinite population of such clusters. Suppose additionally that a
variable y is being measured with expected value Il, variance 0 2 and correlation p
between units within clusters. Let Yl and Y2 be the two observations within the
cluster. Then, an unbiased estimator of Il is ~ = Y1 + Y2 .
2
If we ignore the fact that Yl and Y2 are correlated, aSSUmIng
independence among the observations, the variance of the above proposed
unbiased estimator is Vi [~]
the
dependence
2
= ~.
2
between the
On the other hand, the actual variance when
observations
is taken
into
account
is
2
Vc[~] = ~(1 + p). Therefore, the larger the dependence is among units, the less
2
.
efficient is the estimator.
The ratio of Vc[~] to Vi[~]is the so called "Kish design effect", denoted
by
deff(~) = Vc[[~]]. In this example, if p=0.6, then deff(~)=1.6, implying that
Vi Il
the actual variance of the estimator is 60% larger that what would have being
obtained from a sample of the same number of independent observations.
Adjustments for dependency have been proposed by several authors and
the type of adjustment varies according to the type of data being analyzed. For
univariate analysis, Kupper and Haseman (1978) proposed the "Correlated
Binomial Model" when the dependency is on dichotomous variables. The model
34
is based on a "correction factor" proposed initially by Bahadur (1961) based on
the correction factor applied to the Binomial distribution. They take the second
order approximation as the correction factor to the Binomial probability density
function ending with the following pdf:
P(X
~ x;) =(::";(1-8)n,-,,{ 1+Z8(;-8) [(x; - no)' +X;(Z8-1)-n;8'j}
An application of this method to the evaluation of a diagnostic test i n the
presence of dependent data can be found in DeRouen et al (1991).
Donner (1989) proposed a method he called the "adjusted chi-square
test" for comparing proportions coming from sampling clusters of size two. The
method is based on an adjustment made to the standard chi-square test for the
•
equality of two proportions. He considers the fact that the variance of a
Bernoulli random variable, with mean 8, is not larger than 8(1-8); therefore, the
variance can be approximated, when p:t:O, by the quantity p8(1-8), where p is the
intraclass correlation which it is assumed ranges from zero to one. If p=O, then
the standard chi-square test is applicable (no dependence among units within the
cluster). When p>O, the standard chi-square test statistic can no longer be
approximated by a chi-square distribution, but it can be adjusted so that a
modified version of the test statistic follows a chi-square distribution. The
procedure implies the estimation of p, which can be estimated by the standard
analysis of variance estimator of an intraclass correlation for a stratified cluster
design. Donald and Donner (1987) proposed adjustment to the Mantel-Haenszel
chi-square statistic and to the variance of the Mantel-Haenszel estimate of the
common odds ratio in sets of 2x2 contingency tables. They used the same
35
technique described above of approximating the vanance of the binomial
distribution by the quantity p8( 1-8).
In the case of multivariate analysis, the vanance component model
.
strategy can be used to account for the dependency of the data incorporating
cluster specific covariates as random effects into the model. Sterne et al (1988)
applied a variance component model to the study of periodontal disease in which
the response is measured at five different sites (teeth) in the same subject. The
model allow for components of variation both between and within subjects and
therefore avoids the assumption of independence between different sites in the
same subject. The only one assumption about the structure of correlations is that
the within subject error variance is the same for all sites and subjects.
A more general class of models, allowing for a more general correlation
structure, is the mixed model, and a recent discussion can be found in Helms
(1992). In the previous example, one might allow for a different site-level
variance for each type of site; or more generally, a coefficient at either the
subject or site-level may be assumed to be random.
The generalized estimating equation (GEE) approach (Liang and Zeger,
1986; Zeger and Liang, 1986) is a method which offers the opportunity to apply
-
regression models for continuous and categorical responses to situations in
which observations have dependencies of varying forms. DeRouen et al (1991)
applied GEE models to periodontal diseases, where the patient is the cluster and
teeth site are the units of analysis within clusters.
36
•
For censored survival data, Lee et al (1992) showed that for a large
number of small groups of correlated failure time observations, the standard
.
maximum likelihood estimate of the regression coefficients are still consistent
and asymptotically normally distributed; but their variance covariance matrix may
no longer be valid due to the dependency among groups. They finally proposed a
"correct" variance-covariance estimate which takes account of the intra group
correlation.
All these methods take into account the dependency among observations
in terms of p. What we intend to do for solving our problem is similar, using the
vector of bivariate hazard rate, which is described in the next section.
•
37
CHAPTER 3
Model for Uncensored Bivariate Exponential Distribution of Sarkar
...
3.1.- Introduction
In this dissertation, the problem of accounting for clustered correlated time to
failure data is tackled in a parametric setup for the bivariate situation. In this chapter,
we initially introduce the concept of bivariate hazard rate and then we present
specifically the bivariate exponential distribution of Sarkar. After this, we develop the
simple case when the data is complete, that is when there is no censoring. Maximum
likelihood estimators and estimated varainces of the estimators are presented under this
framework.
3.2.- Vector of Bivariate Hazard Rate.
3.2.1.- Definition of Vector of Multivariate Hazard Rate.
Johnson, N.L. and Kotz, S. (1975) defined what they called the joint multivariate
hazard rate (JMHR) of m absolutely continuous random variables X J, X2, ....., Xm as
the vector:
•
where Gx(x)=P{Xi>xi, i=l, 2, .... , m}. Thej-th element of the JMHR will be denoted as
hx(x)j
( a 'I
.
=iaxJLnGx(X), J = 1,2, .... ,m.
From now on, we will focus on the case of m=2, the joint bivariate hazard rate
(JBHR).
Some of the properties of the JBHR are:
(i).- IfXJ, X2 are mutually independent then hx(x)j
= h Xj (Xj) where the left hand side
is the j-th component (j=1,2) of a bivariate hazard rate and the right hand side is a
univariate hazard rate.
(ii). - If the bivariate hazard rate is constant, so that h X (x) = c, which implies that
a
'I
j
ax LnGx(x) = -c j 0=1,2) , then
j
(2
G x (x)
'I
= exp(-cjx)gj(X") (j:;tj'; j=l, 2), therefore G x (x) ex: exol- LCjXj)'
• \. j=l
J
Then the X's are mutually independent exponential random variables if and only if the
bivariate hazard rate is constant.
(iii).- Noting that
Gx(x) = p{X i >
Xi;
i = 1,2}
= p{X 2 > X 2 } *p{X I > XII X 2 > X 2 }
=GX2,(X2)*GXIIX2(Xllx2)
we can see that:
39
A similar expression can be derived for the 2nd component of the vector. Thus,
the components of the vector of bivariate hazard rate are in fact univariate hazard rates
of conditional distributions of each variable, given certain inequalities on the remainder.
Note that, the first component of the JBHR is:
h x X (xJ, x 2h = -~LnGx X (xJ, x 2), which can be rewritten as a function of the
I'
Oxl
2
I'
2
joint cumulative distribution function and the joint survival function as follows:
h x102
x (X1,X2 )1 =- G
* :...
1
UA:
Xl.X2 (xl.x2)
=- G
1
X!.X2 (Xl ,x2)
1
{l-Fx1 (x1)-Fx2 (x 2 )+Fx 102
x (X 1,X2 )}
{-fx,(XI
)+:
!Fx,,x,(xl ,x2 )1}
1
The second component of the vector can be easily obtained by interchanging the
indexes in the above expression.
Now, given these relationships, we will focus our methodological development
on the joint bivariate density function rather than in the joint bivariate hazard rate.
40
..
3.3.- Sarkar's Bivariate Exponential Distribution (1987)
3.3.1.- Introduction
Several bivariate exponential distributions were previously examined in order to
detennine the "best" according to both their underlying assumptions and their
applicability to real life data. The bivariate exponential distributions we considered
initially were
the
Morgenstern-Gumbel-Farlie
Distribution,
Gumbel's
Bivariate
Exponential Distribution and Bivariate Exponential Distributions of Marshall and Olkin.
A brief representation of these distributions are given below.
a.- The Morgenstern-Gumbel-Farlie Distribution (Johnson, N.L., Kotz, S. 1975)
The joint survival and the first element of the vector of hazard function are given
respectively by:
GX) 'X 2 (x"x2) = Gx) (xd G x 2 (X2)[ 1+a.FX1 (xd Fx 2 (X2)]
h X1 'X 2 (x"x2h = -
lal < 1
~l {LnG x ) (xI)+ LnG X2 (x2)+ Ln[l +a.Fx ) (xI)FX2 (X2)])
=hx
a(I-G x2 (X2»)fX1 (xl)
(Xl) - ------::;---"-)
1+a.Fx) (XI)FX2 (x2)
where lal<l, and Xl,X2 >0.
This distribution has the two special cases:
(i).- When X}, X2 each have exponential distributions so that the corresponding hazards
functions are constants, i.e., hj(xj)=hj j=I,2. Then the first component of the vector of
bivariate hazard is:
41
h X I' x 2 (x}'x2h
with
J3
=[1-{J3exP(hlXl)-I}-l]hl
= 1+[a{1-exp(-h2X2)-1}r
1
This particular distribution is a reasonable distribution if we consider that each of the
marginal distributions is exponentially distributed and therefore has a constant hazard
rate. The association between the components of the vector depends on the parameter
a. We do not consider this distribution as the distribution of interest mainly because the
parameter a does not have a direct interpretation.
(ii).- When X 1, X2 each have Weibull distributions, then the marginal survival and the
first component of the vector of hazard rates are respectively:
G Xj (Xj) = exp ( -x~j )
Cj
> 0,
hx"x, (x]'x2h = [1- {pexp(x~,)
where
J3 = 1+[a{1- exp(
Xj > 0;
j = 1,2. Then
-It}IXI
-X~2 ) -1}]-l
This distribution was not considered any longer because our maIO focus
distributions with constant hazard rates.
42
IS
on
b.- Gumbel's Bivariate Exponential Distribution (Johnson, N.L., Kotz, S. 1975).
The corresponding joint cumulative distribution, joint survival and the first component
of the vector of bivariate hazard function are:
Each marginal component has a hazard rate equal to one, and thus this distributional was
not considered further.
c.- Bivariate Exponential Distributions of Marshall and Olkin (Johnson, N.L., Kotz, S.
1975).
The joint survival function has the form:
G x l' X 2 (XJ, X2) = exp{-AIXI- A2 X2 -A12 max (Xlo x 2)}
Al > 0,
1.. 2 > 0, 1.. 12 > 0; xl > 0,
X2 >
°
This distribution has the following nice properties such as: (i) its marginal distributions
are exponentially distributed, so that their hazard are constant, (ii) it allows for different
hazard rates for each of the components of the vector, (iii) the correlation between the
43
components is a function of 1. 12 , one of its parameters. The main reason for not using
this bivariate distribution is that this is not an absolutely continuous bivariate distribution
..
since the probability of the components being equal is greater than zero.
We finally ended in using the bivariate exponential distribution of Sarkar because of its
properties to be discussed in the next section.
Ryu in 1993, extended the Marshall & Olkin's Bivariate exponential distribution such
that it is absolutely continuous and need not to be memoryless. In addition, the new
marginal distribution has an increasing hazard rate, and the joint distribution exhibits an
aging pattern. This distribution offers an attractive alternative to the extention provided
by Sarkar to the case of non constant hazard rates.
In summary, several parametric bivariate failure time distributions, among these
the family of the bivariate exponential distributions, were considered for application to
this problem. Specifically, the Marshall-Olkin bivariate exponential distribution (Johnson
& Kotz, 1975) was initially considered since its marginal distributions are univariate
exponential, which is a common univariate distribution for the type of data we are
interested in. However, it assigns a positive probability to the event of simultaneous
occurrence of the failure of the two components of the vector, which is rarely the case in
actual practice. In addition, this distribution also poses a probabilistic problem given the
fact that it is not an absolutely continuous distribution, since it assigns a positive
probability to the event {X=Y}. In many practical situations, like in rare diseases, an
absolutely continuous bivariate exponential distribution is desirable.
The bivariate
exponential distribution proposed by Sarkar retains the property of exponential
marginals distributions and it is absolutely continuous.
44
..
3.3.2.- Definition and properties of the Sarkar's Bivariate Exponential
Distribution:
3.3.2.1.- Definition.
We will focus on the bivariate exponential distribution of Sarkar due to the
properties this distribution has as going to showalter in this section.
The random vector (X,Y) has the bivariate exponential distribution of Sarkar if
the joint bivariate survival distribution is given by:
P~X ~ x, Y. ~ y) = exp{-(A 2 -
y
A12 h}{I-[ A(Aly)r [ A(Alx)f+Y} *I[x<y]
+
y
exp{-(A I - 1.. 12 )X}{l- [A(A2 x)r [A( A2Y)] I+Y} *I[x>y]
where x>O, y>O, 1..1>0, 1..2>0,
A12~O,
y = '1
1.. 12
J\,I
'1
+ J\,2
and A(t)=l-exp(-t) for t>O.
The joint bivariate density of (X,Y) is given by:
fx,y(x,y)
1..1..*
I )2 exp{-AI x - (1..2 + A12h} *{( Al + 1..2)( 1..2 + 1..\2) - 1..2Aexp( -AlY)} *
Al +1.. 2
=(
[A(A,Ix)r[ A(A,Iy)r(I+Y) *I[x<y]
(
+
A 1..*
2 )2 exp{-A,2Y-(A,1 +A,I2)X}*{(A I +A,J(A 2 +A,12)-A IAexp(-A,2 x)}*
Al +1.. 2
[A( A,2Y)] Y[A( A,2 X)]-( I+Y) *I[x>y]
45
3.3.2.2.- Properties.
ii) The marginal distributions of X and Yare exponential
X - exp(A I + AIJ
Y - exp( A2 + A12 )
ii)
If A12=O, then X and Yare independent exponential distributions with
parameters Al and A2 respectively.
iii) The distribution of the minimum of X and Y is also exponential
min(X, Y) - exp( A*
= Al + A2 + A12 )
iv) min(X, Y) is independent of g(X,Y) for any g
ill = {g( x, y): g(x, x)
E
ill where
= 0; g( x, y) is strictly increasing (decreasing) in x(y) for fixed
y(x) }
v) The correlation between X and Y is given by:
PX,Y
AI2 AI2 ~
j!
{ j A( *
)-1
j A( *
)-I}
=-*
+ -* L... * (
). Ad 1 A + kA I + A211 A + kA 2
k=I
k=I
A
A j=I A + Al + A J
2
From this last property we can see that when AI2=0 then px,y=O. Thus, AI2 can be
interpreted as a measure of association between X and Y.
The above properties make the Bivariate exponential distribution of Sarkar
applicable to many practical situations such as eye trials or similar.
In what follows we will assume that X and Y have the same marginal
distribution, i.e. that AI=AI=A. This is a very reasonable assumption because it implies
that the marginal failure time distributions are the same for both organs.
46
3.3.3.- Marginal MLE's of Aand y.
Assuming that A1=A2=A, then y = ~~
~
Al2
= 2Ay, and therefore:
(a).- The joint bivariate exponential distribution of Sarkar is reduced to:
fx,y(x,y)
=
A2 (1 + y) exp{-Ax - A(I + 2y)y}{1 + 2y - (I + y) exp(-Ay)}[A(AX)r [A(Ay)t1+Y\x<y)
A2 (I + y) exp{-AY - A(I + 2y)x}{ 1+ 2y - (I + y) exp(-AX)}[A(AY)]Y [A(AX)]-(l+Y) Ilx>y)
(b).- We can re-parameterize the marginal distributions of X and Y in terms ofS 1 as
follows:
x-
exp(Sl
= A(I + 2y))
Y - exp(Sl
=A(I + 2y))
(c).- And the distribution ofthe minimum can be written in terms of S2
min(X, Y) - exp(S2
=2A(1 +y))
From (b) and (c) above, we can obtain the marginal MLE's ofS 1 and S2 as follows:
-
Since Sl can be estimated by either SI
= X1
-
or SI
1
= y;
we average both estimators and
- ="'1(1
2 X + Y1)
we can estimate Sl by Sl
-
From the distribution of the minimum of X and Y we estimate S2 by S2
= X1
(1)
Now, solving for the original parameters Aand y, we have:
_ 1 I( 1 1)
1 1) 1
(
_ 1 X+y -~
!C
r
A = XII) - 2 X + Y , and y = 2 1 _
+1
X(1) 2 X Y
47
.
3.3.4.- Joint MLE's of A and y.
We can rewrite the joint density defined in (a) above as:
fX,Y (x,y) =
fjJ,~ (x, y)I,x<Y) + fjZ~ (x, y)I,x,n ' and defining (; = {~
if X<Y
if X > Y ,
•
(1)
]O[ fx,Y(x,y)
(2)
]1-0 5 = 0, .
this
becomes fx,Y(x,y) = [ fx,Y(x,y)
The log likelihood for a single observation is given by:
'- i = 10g(fx,Y (Xi, yJ) = 510g[ f*~~ (Xi, Yi)] + (1- 5) log[ f*:~ (Xi,y i)]
Now,
log[f*l,~ (x, y)] = 10g(1 + y) + 10g(A) + {-AX - A.( 1+ 2Y)Y} + log{ 1+ 2y - (1 + y)e- AY } +
y 10g[A(Ax)] - (1 + y) log[A(AY)]
and
log[f*:~ (x, y)] = 10g(I + y) + 10g(A) + {-AY - A.( 1+ 2y)x} + log{ 1+ 2y - (1 + y)e- Ax } +
ylog[A(AY)] - (1 + y) 10g[A(Ax)]
Now, '-i can be rewritten as:
2
,-.I =
J
j=l
t 1
4
It. +5."
4
+ (1-5.I )"1=
1£.J'l"IJ
£.J~IJ.. , where
\II ..
j=l
= 10g(1 + y)
j=l
t
z =210g(A)
'IIi 1
=-AX i - A.( 1+ 2y)y i
'IIi2
= log{ 1+ 2y - (1 + y)e -AYi }
'IIi3
= ylog[ A(Axi)]
'IIi4
= -(1 +y) 10g[A(AyJ]
~l = -AY - A.( 1+ 2y)x i
~2 = 10g{I+2y-(I+y)e- AYi }
~3 = ylog[A(AYi)]
~4 = -(I+y)log[A(Ax i)]
48
In what follows we will denote by ~'
= [A y].
To obtain the MLE's of A and y, we need to solve the estimating equations given by:
Un(A)
=
tnat.a~ =
0 and
. nat·
Un{Y)
=
t ayl =
0, which are clearly non linear.
Therefore the Newton-Raphson method is to be applied to iteratively obtain the MLE's.
We will use as naive initial estimators those obtained as marginal MLE' s in the previous
section.
&t1
Defining e = ( A
at = (at
at)
y), then ae
aA' ay
aAay I
Eft
ay2
J'
then the iteration scheme to be used for each of the components of e is
.
49
3.3.5.- Estimating Equations and Information Matrix
In this section we develop the algebraic solution to the estimating equations and
for the information matrix.
3.3.5.1.- Estimating Equations.
By definition of f
n
of.
n
2
i'
the estimating equation for A. is given by
a
n
4
a
n
4
a
Un(A.)=~ o~ =~~OA. tj+~8i~OA. \IIij+~(l-8J~OA. ~ij
where
a
OA. t
l
=0
a
OA. t
2
2
= A.
a
Yi(l+y)e- AYi
= 1+ 2y - (1+ Y)e-AYi
OA. \IIi2
a
(1+Y)Yie-AYi
OA. \IIi4 = -
a
OA. ~2
[A(A.YJ]
Xi(l+y)e- Axi
= 1+2y-(1+y)e-Axj
~~ __ (1 +Y)Xie-Axi
OA.
4 -
[A(A.xJ]
The corresponding estimating equation for Yis:
.
n Of.
n 2 a
n 4 a
n
4 a
j
Un(Y) = ~ Oy. = ~~ Oy t + ~8i~ Oy \IIij + t(1-8i)~ Oy ~j
where
a
-t
Oy
I
1
=-l+y
a
-t
Oy
2
a
=0
a
2-e- AYi
= 1+ 2y - (1+ y)e-AYi
Oy \IIil =-2A.Yi
Oy \IIi2
~ \IIi3 = 10~A(A.xJ]
~ \IIi4 =-log[A(AyJ]
50
..
o
Cty 1;1
C
2 - e-)·x,
8y ~2 = 1+2y-(I+y)e-Ax,
=-2AX i
o
~ 1;3 = log[A(AyJ]
8y 1;4
= -lo~ A( AXJ]
3.3.5.2.- Observed Information Matrix.
The observed information matrix is given by:
iffl
OACty I
if f I, where each of the components of the matrix can be
J
8y2
obtained as:
&f
.
02
n
4 if
n
4 if
(a).- :l'I2 = ~ :l'I2 = ~~ :1,\2 t j + ~8i~ :1,\2 \!Iij + ~(1-8J~ :1,\2 ~ij
n
iff.
1
1=1 VI\,
VI\,
n
2
1=1 .F1 VI\,
02
oA2
-t
&
aA2 \!IiI
=0
&
1=1.F1 VI\,
-
2 -
-
2
oA
yx~e-Axi
[A(AxJ]
(l+y)y~e-2AY,
aA2 \!Ii4 = [1-e-Ay,t
&
if
(l+y)y~e-AYi
= - [1 +2y-(1 +y)e-AYi]2 - [1 + 2y -(1 +y)e- AYi ]
yx~e-2Axi
-2 1;1 = 0
01.:
2
(l+y)2y~e-2AYi
\!Ii2
J=1 VI\,
A2
if
oA2 'Vi3 =-[I-e-Axit
&
-
1=1
(l+y)y~e-AY,
+
[A(AYi))
(l+y)2x~e-2Axi
(l+y)x~e-Axi
-2 ~i2 = 2- [
]
01.:
[1+2y-(I+y)e-Axi]
1+2y-(I+y)e-Axi
yy~e-AYi
[A(AYJ]
51
..
82
fJy2
1
't 1
if
fJy2 \IIil
if
fJy2 \IIi3
if
fJy2 ~1
if
fJy2 ~3
82
=0
=- (1 +y)2
fJy2
=0
-2 \IIi2 =2
fJy
[1+2y-(1+y)e- AYi ]
=0
fJy2 \IIi4
't 2
if
if
(2_e- AYi
)2
=0
=0
=0
if
fJy8'A. 't 1 = 0
if
fJy8'A. \IIil = -2Yi
if
(1+y)(2-e- AYi hie-AYi
Yie-AYi
fJy8'A. \IIi2 =- [1+2y-(1+y)e-AYit + 1+2y-(1+y)e-AYi
if
fJyOA. \IIi3 =
xie- Axi
[A( 'A.xJ]
82
fJy8'A. \IIi4 =
52
Yie-AYi
[A( 'A.y J]
&
fJy8'A ~il
82
=- 2x
j
(1+y)(2-e- Ax ')x j e- Axi
xje- h
,
fJy8'A ~i2 =- [1+2y_(1+y)e-Axi]2 + 1+2y-(1+y)e- h
&
fJy8'A ~3
Yje- AYi
= [A('AYJ]
;j
fJy8'A ~j4
x·e- h
,
,
=- [~('AxJ]
Since the components of the observed information matrix are expressed as a sum
of independent random variables, by Khintchine's WLLN, the observed information
matrix converges to the expected information matrix (Sen & Singer, 1994, pp 206).
"
53
CHAPTER 4
..
Model for Censored Bivariate Exponential Distribution of Sarkar
We now consider the more realistic case when the failure times of any of
the two organs are censored.
4.1.- Censoring schemes
Let C be the censored random variable which we will assume
IS
exponentially distributed with parameter I..l, and independent of(X,Y).
Censoring can happen in three different ways:
i.- Total Censoring: Censoring comes before the smallest failure time, i.e. C <
~l)
ii.- Partial Censoring: The censored time is in between the smallest and the
largest failure time, i.e.
~l)
< C < ~2)
iii.- No censoring at all: the smallest and the largest failure time are observed,
C>~2).
The following two indicator variables are defined in order to identify the type of
censoring corresponding to a particular observation:
t,
t,
={~
if C <X(l)
if C > X(l)
={~
if
X(I) < C < X(2)
otherwise
54
•
Thus, for total censoring (11 ,12)=(1, 0), for partial censoring (11 ,12)=(0,
1), and for no censoring (11 ,12)=(0, 0).
•
4.2.- Model Specification and Maximum Likelihood Estimation.
When censoring is total, we only observe the censored time and therefore
the only available information we have can be represented by the observed event
E 1={ C=c ;
~1»c}.
When censoring is partial we observe the minimum failure
time and the censored time, the available information can be represented by the
observed event E2={~1) ~I)
;
C=c and ~2»C}. Similarly for the uncensored
case E3={~1)~1), ~2)~2); C>X(2) }.
Therefore, the i-th observation, i=l, 2, .... n,
can be represented as
1t~Ii1t~2i1t~-tli-t2i, where 1tj is defined as follows 0=1,2,3), for all i=l, 2, ....,n
.
1t 1=fc(Ci)*P{
Xi(1»cd, where fc is the density function of the censored
time which is known has exponential distribution with parameter fJ., and
1t 2
=
r
f xW ,x(2) (xi( I)' Y) *fc {cJdY , and
Ci
The likelihood for a sample of size n therefore can be expressed as:
55
Alternatively, we can write the likelihood as:
L:
n
where
!fC<c;)' Fx,,, (cJ!",
{I fxen,x", (X;iJ) ,y) •fc{c;)dY}'"
{fX(Ij,x(2) (Xi( J) ,X i(2))
* Fe (Xi(2))}
Fx
(Ij
and
Fe
!-tjJ-tj2
represent the survival function of ~!) and the censored
variable respectively.
Evaluating the likelihood piece by piece, we obtain:
(a).- f e (c.)
1
= lie
,....
-l!Cj
.
,
FX(I) (c.) = e -21..(
1
!+Y)Cj
However, by the definition of the joint density we have
•
(s=1,2) can be expressed as follows:
j f~~~ (xi(I) , y)dy = (1 + y)[ A( AXi(I))
Cj
r
Ae -J.xj(J)
*
After consecutive integration by parts of the integral on the right side above, we
have that
56
which can be expresed as e-2A.YCj [A(ACJr times a power series on A(ACi), called
S(A(ACi)). Therefore,
1Ci f~~~ (Xi(l) ,y)dy =(I + Y)[A( AXi(l))r Ae-A.xi(l) e-2A.YCi[A(ACi)r s( A(ACJ)
=(I + Y)[A( AXi(l) )A(ACi)r Ae -~xj(1l+2ycd~ A(AC j))
Finally, the last component of the likelihood is given by:
.
Therefore, the likelihood can be expressed as:
n
r
{~e-llCje-2A(1+Y)Ci i1 {(I + Y)[A( AXi(l) )A(A.cJr A.e -~Xj(l)+2YC;)S(A(ACJ) fi2
L=
{{ f~~~ (Xi' Y
J} {f~~~ (Xi' Yi)}
i
6
j
H
t
e-~(2) } l-til- i2
The loglikelihood is thus given by:
n
f
= L 'til {log~- ci(~+2A.(1 +y))} +
i=l
~ 't i2 IOg{( 1+Y)[A( Axi(l) )A(ACJr Ae-~Xi(I)+2YC;)S(A(ACJ)} +
t(1=1
1- 'til - 'ti2){Oi
logf~~~ (Xi' Yi) +( I-Oj) logf~~~ (Xi, Yj) - ~Xj(2)}
57
, where
~il
= 10gJ..l
~i3
= -J..lX H2 )
U1
= 10g{A(I+y)}
U2
U4
= 10g{S(A(AcJ)}
\IIj2
\IIi3
= ylog[A(AxJ]
\IIi4
= yIOg{A(Axj(I))A(AcJ}
= log{ 1+ 2y - (1 + y)e- AYi }
=-(I+y)log[A(AYi)]
1;1
=-AY-Ml+2y)x i
1;2
= log{ 1+ 2y - (1 + y)e -AYi }
1;3
= ylog[A(AyJ]
1;4
= -(I+y)log[A(AxJ]
In what follows we will denote by ~ = [A y J..ll
To obtain the MLE's of A, y and J..l, we need to solve the estimating equations
given by:
clearly non linear, therefore the Newton-Raphson method is to be applied to
obtain the MLE's.
As in the uncensored case, we will use as naive initial estimators, those obtained
as marginal MLE's in the uncensored case for A and y, and for J..l its
corresponding marginal MLE .
58
I ele
.
Defining
Oe
(oe Oe oe)
0).' (}y , Oil ;
oe =
I 0).2
I &e
&e
oeoe' =1
(}yo).
- - I &e
lOllo).
ele l
I
o).(}y o).ollj
a2 e &e I
fie
(}y2
02e
01l(}Y
(}yOIl I
&e I
0112
J
and the iteration scheme to be used is the same described in chapter 3.
•
59
4.3.-Estimating Equations and Information Matrix.
4.3.1.- Estimating Equations.
The estimating equation for A is given by:
nat.
n
2a
4a
n
Un(A) = ~ a'll = ~til~a'l ~jj + Ltj2~a'l Ujj +
1=1 I\, 1=1 .F1 I\,
1=1 )=1 I\,
t(
1- til - t i2 )Oj ±a~ \j!jj + I( 1- til - t i2 ){(1-0J± a~ ~jj +~i3}
1=1
.F1
1=1
.F1
I\,
I\,
where
a
aA ~il = 0
a
aA ~i3
=0
a
1
aA Uil = A
a
•
c ie -A.ci A( AX j(1)) + Xi( 1) e -AXj(I) A{ ACj)
aA Uj2 =
A( AxjO) )A( ACJ
: Uj3 = -(Xi(I) +2yC i )
: Ui4 = : logS(A(AC i ))
a
aA \j!i2
a
aA \j!j3
a
yxje- AXi
= [A{ AXJ]
aA 1;1 =-Yi -(1+2y)x j
Yi(l+y)e- AYi
= 1+2y-(1+y)e-AYi
a
aA \j!i4 = -
(1 + y)Yje- AYi
[A{ AyJ]
a
Xi(l+y)e- Axi
aA 1;2 = 1+2y -(1 +y)e- Axi
Ax
~I; __ (1 +y)xie- ;
aA 4 [A{AxJ]
60
.
where
a
a
r 02 = -2ACoI
Or~l
Or l;il = 0
a
Or l;i3 =0
a
1
-u - -
Or
il -
1+y
a =-2ACo
Oru13
0
1
a
2 - e-A.Yi
Or \IIi2 = 1+2y-(1+y)e-A.Yi
~ \IIi3 = log[A{AxJ]
Or \IIi4 =-log[A{Ay J]
a
Or ~l =-2AX i
a
Or ~3 = log[A{AyJ]
•
a
Or \IIil =- 2Ay i
a
Or ~2 = 1+2y-(1+y)e-A.xi
a
Or ~4 =-lo~ A{MJ]
where
a
alll;il
a
alll;i3
1
= 11
=-Xj(2)
61
o
-u··=o
Oil IJ
o
-~
oil
j=l, 2, 3, 4
.IJ =o
j=1,2,3,4
j=1,2,3,4
4.3.2.- Observed Information Matrix.
02 £
o'A.Oy
&£
Oy2
02 £
°IlOy
&£ l
-I
O'A.OIl
&£ I
OyOIl I ,where
1
&£
Oll2
1
J
(a).-
,
and
&
0'A.2 l;ij
=0
j=l, 2, 3
0'A.
&
il -
0'A.2 \j!ij
'A.2
2 -Ac
::12
2
0
1
-u
2 ---
-
u
o'A.2
.
u
- i2 -
Ci
[(
e
1
A 'A.c i )] 2
x 2 e-Ax;(l)
i(I)
- -----'-----'--:[(
A 'A.xi( 1) )] 2
02
and o'A.2 ~ij are the same as those derived in chapter 3 for j= 1, 2,
3,4.
62
(b).-
82
()y2 ~ij
=0
j=l, 2, 3
EP
()y2
if
()y2
Ui3
if
()y2
EP
1
U il
- 2 Ui2
=- {1+y)2
()y
02
=0
()y2
Ui4
=0
if
= ()y2
logS(A(AcJ)
if
\j!ij
and ()y2 ~ij are the same as those derived in chapter 3 for j=l, 2,
3,4.
and
02
if
0J.!2
Ujj
= 0 , 0J.!2
02
\j!ij
= 0 and 0J.!2 ~ij = 0 for j=l, 2, 3, 4.
63
02
(}yOA ~il
02
(}yOA Uil
~~
\j!ij
82
(}yOA ~i2
=0
82
=0
and
(}yOA U i2
~~A~ij
= -2c i
82
(}yOA ~i3
82
=0
(}yOA Ui3
=0
= -2c i
are the same as those derived in chapter 3.
02
~OA ~ij = 0 for j=l, 2, 3.
&
~aA Uij
02
o~(}y ~ij
&
0J.U7y Uij
02
=0
, o~oA \j!ij
=0
for j=l, 2, 3.
02
= 0 , o~(}y \j!ij
= 0, and
•
&
o~A ~j
=0
02
= 0 , and a~(}y ~ij
64
for j=l, 2, 3, 4.
.
=0
for j=l, 2, 3,4.
Large sample properties for the score statistics can be applied to make inferences
about the parameters involved in both the uncensored and the censored case.
•
65
CHAPTER 5
Group Sequential Test
5.1.- Introduction
In clinical trials in which more than one observation is obtained from the
same individual, such as ophtalmological studies where both eyes are analyzed,
or when two meaningful endpoints are measured, group sequential interim
analysis methodology must be applied. The methodology developed in chapters
3 & 4 of this dissertation for the bivariate exponential distribution of Sarkar is
used to develop group sequential interim analysis method. Our model
incorporates information from each of the individual organs, which is more
efficient in the sense that marginal analysis can also be performed in the
traditional way.
Group sequential test statistics based on the bivariate
exponential distribution of Sarkar are presented under the absence and presence
of censoring. In what follows, we first present the setup for repeated significance
test statistics and we show that the discrete sequence process generated by the
application of repeated test along the time converges to a Brownian motion
process. We use the above convergency property to obtain group sequential
boundaries for the bivariate exponential distribution of Sarkar.
•
5.2.- Bivariate Group Sequential Test
Let us consider a clinical trial where subjects are randomly allocated into
•
either an experimental or control treatment.
Let us assume further that an
unspecified number of interim analyses, K, are planned to be done during the
study period, and that the decision of stooping the trial is based on repeated
significance test statistics after evaluation of each group.
Let t denote the study time and that the interim analyses are performed at
the point times tl, h, ....., tK, where K is unspecified in advance, and 0<t1< t2<
.....< tK.
To make statistical inferences under the Group Sequential framework, it
is necessary to show that the discrete sequence of Score statistics at the different
time points converges to a Brownian Motion Process
At a particular monitoring time tk (k=l, 2, ....., K) we have 5 possible
situations:
{X < Y ~ tk}
Both X and Y failed before tk
{Y<X~td
{X < tk < Y}
One of then failed before tk
{Y <tk <X}
{X> tk ; Y> td
None of them failed before tk
Therefore for tk, the likelihood, for a single observation, is given by:
~~, tk
T
(
)
(j)
Jk
= 5 [ gx,y(x,y)
E
j
, where
•
Ok1=I{X < Y
~
td
(1) (
gx,y
x,y)
(1) (
= fx,y
x,y )
Ok2=I{Y < X
~
td
(2) (
gx,y
x,y)
(2) (
= fx,Y
x,y)
67
(3) (
gx,y
x,y )
=
OC!f
(1) (
fX,y
x,y ) dy
lk
(4) (
gx,y
x,y )
=
OC!f
(2) (
fX,y
x,y) dx
lk
Define the Score statistics and the information at time point t respectively
We have that E[U n (8;t)]
that Ie(t = 0) =
°
= 0,
and that Ie(t) is an increasing function in t, such
and Ie(t = 00) = Ie
•
By the Central Limit Theorem, for each tk we have that:
lun(e;tk)=wn(e;t k) ~ N(O,Ie(tk))
Let us now consider the time points tl and h (tl < h), and denote .3(tl)
the set of all possible events generated up to the time tl
-----1------1
L(tl)
h
L(h)
h
The conditional likelihood up to the time h given the history up to the time h is
L(h 1.3(tl» and it is represented by:
68
1
if 8 11 =1
•
8 12=1
f
y(t 2)[ (2)
r-Iy(t
gx,YI013=1
h X,Ylo13=l
[ (l)
[ (I)
gX,YI0 14 =1
or
f
X(t 2)[ (2)
r-IX(t
hX,Ylo I 4=1
Ell5 [gX,YlOls=l t2r
(r)
if 8 13=1
if 8 14=1
if 815=1
where ly(h)=I{Y < h} and Ix(h)=I{X < h} and 02r are defined as above
The expressions for the components of the likelihood are simply given by:
(1) (
(1)
fx,Y
x,y)
. gX,YlolS =1 = k....
.I.X,Y
(t b t)1
r<i~{x,y)
(2)
gX,YI81s=1 = k....
.I.x,Y
(t b t)1
00
-f f XY
(2) ( x,y) dx
(4)
t2'
gX,YI81s=1 = --=-F--"(-t-t-:-)X,Y b 1
69
r
\
Now, L(e; h)=L(e; tl)* L(t2 3(tl)) , then
1
log L(e; h)=logL(e; tl)+log L(h I 3(tl))
log L(h I 3(tl))= log L(e; t2)-logL(e; tl)
Thus, Un ( t
213{ t 1)) = Un ( t 2) - Un ( t 1)
Similarly for any pair tk-I and tk, the increments in the scores are
[U n( t
113( to)); Un ( t 21 3(t 1));
; Un ( t K 13{ t K-I))].
The
vector of components has independent elements (conditionally independent).
The sequence of conditional scores
{U n( t k 13( t k-1)):
k = 1,2,.... K}
is a zero mean martingale and it converges to a Brownian Motion Process (Weak
convergence of martingales)
As a consequence of the result above, we can use the Lan & DeMets
spending function to obtain the boundary crossing probabilities for the developed
model.
70
5.3.- Concluding Remarks
The proposed model is based on the Bivariate Exponential Distribution of
..
Sarkar. This bivariate distribution is absolutely continuous with marginal hazard
rates being constant and non-negatively correlated. These properties make this
distribution the appropriate choice for several real life problems such as eye
studies or any type of study comprised of pairs of organs.
The proposed method allows one to use the information coming from
each of the organs being analized, and therefore it allows for no loss of
information such as when one uses a summary measure of the participating
organs.
In a particular clinical trial application when the outcome of the two
organs can be assumed to follow the bivariate exponential distribution af Sarkar,
one may utilize the proposed method to obtain estimates of the parameter A.
accounting for the correlation between organs. Testing of the hypothesis for
treatment differences can also easily be constructed based on the A.' s of each
treatment group.
Since the test statistic can be calculated at any given tieme point, and the
discrete process generated converges to a Brownian motion, one can utilize the
Group Sequential boundaries for repeated testing on accumulated data.
71
CHAPTER 6
Numerical Results
6.1.- Introduction
We have chosen to simulate the behavior of the methodology in a given
hypothetical sample generated from the bivariate exponential distribution of Sarkar.
Thus, we are not examining the goodness of fit of the parametric model, but rather the
behavior of the procedure in a situation where it applies. In real applications, one could
examine the goodness of fit of the bivariate exponential distribution of Sarkar by using
its properties, i.e., examine if the minimum failure time of the two organs distribution has
a constant hazard.
We compare the methodology to the standard method of using the
rmmmum failure time of the two organs, under a variety of scenarios of different
correlations. These are outlined in table 6.1.
First, we simulate 500 observations from the bivariate exponential distribution
of Sarkar for both an treatment and a control group. The simulated distributions are
such that the corresponding mean response time for each of the participating organs is
20 months for the experimental group and 16 for the control group. This represents a
•
treatment difference in the survival of 4 months.
between the two organs are also considered.
Different degrees of correlation
Table 6.1 contains (a) the different
scenarios (true parameter values) used for simulations and (b) descriptive statistics of the
simulated sample.
.
73
Table 6.1. Simulation Scenarios and values for sample of size 500:
Bivariate Exponential Distribution of Sarkar
Experimental Group
Control Group
n=500,E[X]=E[Y]=16
n=500,E[X]=E[Y]=20
True values
A
Sample values
True values
Al =0.04
x=19.8
Al =0.0500
x=16.3
AI2 =0.01
Y=20.5
AI2 =0.0125
Y=14.5
r=0.19
B
r=0.13
Al =0.025
x=19.8
Al =0.0300
X=15.6
AI2 =0.025
Y=19.7
A12 =0.0325
Y=14.6
r=0.45
C
Sample values
r=0.49
Al =0.01
x=20.9
Al =0.0200
x=16.8
A12 =0.04
Y=20.0
A12 =0.0425
Y=15.9
r=0.82
r=0.65
74
~
As can be seen in figure 6.1 (two-way scatterplots and univariate box plot), the
marginal distributions of the simulated data are similar under two different scenarios
with different correlations. In addition, the bottom set of graphs with higher correlations
display a tighter cloud for the bivariate distribution, as expected.
75
Figure 6.1. Simulated Bivariate Exponential Distribution of Sarkar
Simulateo SSr'1r8r'S BI\lI!lr'l8tE- rlponl?ntldJ
n=500, mean=20
C'lstrlCu!ton
SllllulBIe>a Sar ... ars,
CQ/"r:O.17
n=500, llean=16
l -=-----
8l~arlalE'
• • !,
200
200
0
lSO
'50
e:.oonpnoal
•
D1StrlOlJt~on
cOI'"I"':0.l7
'!I'
II
J
0
0
'00
50
0
~
9rS't9
lOa
0
0
~
R£ct>
50
lOa
v
S~lIlulatec Sar-kar's Billilr1ate [):ponentlal
n=':500. IIIrlln:20 COI""'''Q 75
150
200
~
50
0
0
k
00
<5Z>
®
50
tOO
v
S1111ulateo Sarkar'g Bhartate E):DonenUal
D~stntlul1on
,>0
200
~
O~slrltJuUon
n=500. IIrl!!f'l=J6. co",."O.68
200
'00
--=-
0
ISO
'00
50
0
J;
lSO
0
1~ocB
50
_otto
0
laO
0
000
'00
v
lSO
200
~
76
50
0
00
fPJO~
50
0
,00
y
'50
200
~
6.2.- Single Analysis Results at the end of the Study.
..
6.2.1.- Ignoring the bivariate nature of the data.
As mentioned in chapter 2, most analyses of bivariate failure time data is reduced
to a univariate situation such as by studying the time to failure of the first organ. In our
setup, this is essentially studying the time of the random variable min(X,V).
bivariate exponential distribution of Sarkar min(X,Y) - exp(2A.+A.IZ).
In the
Figure 6.2
displays the Kaplan-Meier estimates of the survival curves for the dataset for the
experimental group and the control group under scenario A.
summary statistics for all three simulation scenarios.
77
Table 6.2 displays
Figure 6.2. Kaplan-Meier Survival Curves for Scenario A
Graue> 0
..
Graue: J
1.00 -
.
>-
0.75 -
D
fU
D
o
!l
0.50 -
fU
>
>
L
:J
U1
0.25 -
0.00 I
o
I
I
20
10
I
30
I
~o
I
50
tlme
Kaplan-Meler Survlval Curve
logrank test: '1. 2 = 21.33
p<O.OOOI
•
78
•
Scenario
. . fior Scenanos A , B ande
T abl e 62
.. Summary Statlstlcs
2
Events
Median
Group
Obs
Mean
X
Control
500
429
6.69
8.80
A
Experimental
500
406
7.97
12.09
Control
500
412
7.12
10.11
B
Experimental
500
406
9.68
13.15
Control
500
402
9.25
9.25
C
Experimental
500
387
12.46
.
•
79
p
12.46
21.33
0.0000
14.10
0.0002
20.07
0.0000
6.2.2. Incorporating the Bivariate Nature of the Data.
•
In this section section we present the analysis results comparing the experimental
to the control group with no interim analyses and complete data. We now consider the
test procedure using the bivariate information under the four scenarios presented in table
6.1. We use the score statistics to compare AE to
Ac , i.e., we compare the distributions
treating y as a nuisance parameter.
A program in
e++ was written to
obtain the estimators and standard errors by
using the Newton-Raphson procedure described earlier.
To test Ho: AE = Ac we will use the statistic
T=
In( Ln( ~E) - Ln( ~c)) - N( 0, ~)
or the equivalent
Z =.J2rl( Ln( ~E)
- Ln( ~c)) - N( 0,1).
The results for the three different scenarios are
shown in table 6.3.
80
Table 6.3. Test Statistics for the Bivariate Exponential Distribution of Sarkar
Scenario
Estimated Parameters
Z Statistic
p-value
Z=-11.38
P = 0.0000
Z = -7.93
P = 0.0000
Z = -26.14
P = 0.0000
A
A
AE = 0.0360
Ac =0.0516
A
B
AE = 0.0249
Ac = 0.0320
A
C
AE = 0.0091
A
Ac = 0.0208
81
6.3.- Group Sequential Boundaries
..
Interim analyses on the simulated censored bivariate distribution of Sarkar, are
based on the Z statistic Z = .J2r1( Ln( ~E)
analysis on the accumulated data.
- Ln( ~c)) - N( 0,1) ,.computed at each interim
The corresponding p-value for the k-th interim
analysis is computed by using as the accumulated level of significance
a..Jt:, where tk is
the proportion of elapsed time at to the point tk. Our numerical results are based on the
assumption of a total study time of 60 months, and interim analyses are going to be
performed at 12, 24, 36 and 48 months. Therefore tl=12/60=0.2, t2=0.4, t3=0.6, t4=0.8
and t5=1. By using a program written in e++, the estimated parameters and Z statistics
A
for the interim analysis at time point 12 months, for the scenario A, are respectively: AE
A
= 0.035, Ac = 0.052, Z=-12.52.
The
level
of
a..Jt: =0.05,J0.2=0.0224,
significance
of
5%
gIves
a
rejection
level
of
which implies a rejection of the null hypothesis of no
difference between the lambdas. Therefore an early termination of the trial must be
considered because of the observed treatment difference.
82
CHAPTER 7
..
Future Research
The methodology presented in this research allows the investigator to use the
bivariate nature of the data and therefore it is not necessary to restrict the analysis to a
single result summarizing the pair of outcomes by using either the minimum of the
failure of the organs or some other summary statistics.
In building the group sequential boundaries for censored bivariate survival data,
we have considered a series of parametric bivariate models with different functional
forms and underlying assumptions. Having in mind the applicability of these parametric
distributions in real life data, we focused our interest in a kind of model that fits in
situations in which both marginal distributions have constant and equal hazard rates such
as the family of bivariate exponential distributions. After discussing the applicability of
several bivariate exponential distributions, we used the bivariate exponential distribution
of Sarkar since it is an absolutely continuous random vector and fulfills both the
mathematical and the applicability requirements. We restricted_ the Sarkar's exponential
bivariate distribution to the situation in which both marginal distributions have constant
hazard rates.
This constraint was done in order to satisfy many real life bivariate
83
survival data, such as ophthalmological data where illnesses with constant hazard rates
in each eye is a reasonable assumption.
..
However, there are a series of situations in which this assumption may not work
and different hazard rates must be considered. As an example of this, let us consider a
study where our main outcome variable is the "survival" in breast cancer in women.
Kelsey & Hom-Ross (1993) reported a study on breast cancer in which there is an
excess of left-sided tumors, with ratios in the range of 1.05 to 1.20. Therefore, in a
clinical trial on breast cancer prevention, it would be more nearly correct to assume a
higher hazard rate for the left breast than for the right breast in the placebo group.
Another study on testicular cancer by Stone, Cruickshank, Sanderman & Matthews
..
(1991), reported that in men with unilateral cryptorchidism, there would be a much more
higher rate of testicular cancer on the cryptorchid side, even though rates on both sides
are above the rates in normal men. The above examples show that a bivariate model that
incorporates different hazard rates has actual applications in cancer epidemiology.
Therefore, future research must include the study of different hazard rates for
each of the individual organs of the patients. This assumption of non equal hazard rates
allows the investigator to compare pairs of organs at different stages of the disease and
therefore with different hazard rates. The bivariate exponential distribution of Sarkar
allows unequal but constant hazard rates in each of the organs (AI
84
:t.
A2). This extension
.
,
follows the same line of the development presented here and it adds some extra terms to
the equations used in the derivations and therefore it can be easily done.
..
We have assumed that data fit the bivariate exponential distribution of Sarkar.
However, departures from this assumption may invalidate the results we have obtained.
A possible extension of this research would be to investigate the robustness of the
results. One could examine the behavior of the test statistics for data with non-constant
marginal hazards. A non-constancy parameter could be introduced and its effects studied
with further simulations.
Another extension of the methodology presented here is to use a bivariate
•
parametric distribution which allows non constant hazard rates, such as the bivariate
exponential Weibull distribution. This distribution can be applied also to situations in
which both the non constant and the non equal hazard rates of the participating organs is
the most realistic situation.
Additional areas of research to be considered are sample-size estimation and
interval estimation after group sequential test. To develop these statistical areas would
improve the group sequential design and consequently their use in clinical trials when
bivariate survival data are generated.
85
Other adjustments to the methodology presented here are related to studies when
more than two observations are generated on the same individual such as dental studies,
or in studies when clusters of size bigger than two are the unit of analysis such as in
family or community studies.
"
.
86
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"