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Design and Analysis of Clinical Trials
Within each of these two types of model, either the baseline
hazard in the proportional hazards model, or the baseline survivor
function in the acccelerated failure time model, can be assumed t o
be either fully
Design and Analysis of Clinical Dials
200
Table 7.11. Results from Fitting Transition Model
Logistic Regression to Respiratory Status Data.
~
~
Covariate
~
Estimate
Classical SE
Robust SE
0.321
0.314
0.461
0.380
0.384
0.568
~
Treatment
yt-1
Interact ion
0
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Design and Analysis of Clinical Trials
g ( t ) = t , g ( t ) = ln(t) and a step function such that g ( t ) = 0 if
t <T
and 1 if t 2 T . We describe and illustrate how such time-varying
variables can be included in Section 8.6. A likelihood ratio test
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Design and Analysis of Clinical Trials
Table 6.3. Results from Likelihood Analysis of Oestrogen Patch
Data (assuming compound symmetry for the covariance structure
of the repeated observations).
Parameter
constant
group
linear time
quadratic time
base
Design and Analysis of Clinical Trials
194
of explanatory variables; for example, if dlj = [l,tij],then the elements of ~i correspond to the intercept and slope of a subject specific
time-trend in the mean response. We use p* here rather than p, to
emphas
Design and Analysis of Clinical Trials
146
In many examples, the model for the covariances and variances of
the repeated measures will need to allow for non-stationarity, with
changes (most often increases) in variances across time being particularly comm
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Design and Analysis of Clinical Trials
Schoenfeld (1994) present a method that uses the time to the intermediate state as a covariate for subsequent survival. Their simulations
suggest that although improvements in precision of the estimates of
main i
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Design and Analysis of Clinical Trials
hold if the hypothetical complete-data, Y, can be written as:
where Yi,Xi and ~i are obtained by deleting rows from Yf,Xf and
5, respectively.
Columns of each design matrix correspond to terms in the model
under
184
Design and Analysis of Clinical Trials
viewed as a multivariate extension of the generalised linear model
and the quasi-likelihood method (see Chapter 4). The use of the
latter leads to consistent inferences about mean responses without
requiring spec
CHAPTER
8
Survival Analysis
8.1.
INTRODUCTION
In many clinical trials, the main response variable is often the time
to the occurrence of a particular event. In a cancer study, for example, surgery, radiation and chemotherapy might be compared with
respect
Design and Analysis of Clinical Dials
134
Power curves for three methods
of analysing repeated measure designs
0
20
40
60
Sample size
Fig. 5.9. Power curves for POST, CHANGE and ANCOVA.
The results of applying each of POST, CHANGE and ANCOVA
to the oestro
168
Design and Analysis of Clinical Trials
oestrogen patch data takes the form:
where yijk denotes the depression score of subject k in treatment
group i on the j t h visit, p represents an overall mean effect, cq, i =
1 , 2 represent the effect of treatm
210
Design and Analysis of Clinical Traals
Two probability distributions often used to introduce the analysis of survival data are the exponential distribution and the Weibull
distribution. The probability density function of the former is:
and of the lat
Design and Analysis of Clinical Tnals
252
Beta-Blocker Heart Attack Trial
r
Z*ue
I
1.88
0
Juna
1978
136
0
May
1979
Ocl. MMcI)
1979 1980
Ocl.
1980
%Y
Ocl.
1981
June
1482
Dab
Fig. 8.10. OBrien and Fleming boundary for log-rank test in a randomised
double bl
Design and Analysis of Clinical lhals
174
A novel approach to the analysis of longitudinal data is suggested by Tango (1998). Tango postulates that each treatment group
consists of a mixture of several distinct latent profiles, a situation
he models using
170
Design and Analysis of Clinical Dials
Table 6.8. Diggle and Kenward Model for Dropouts.
Let Y * represent the complete vector of intended measurements and
t = [ t l , t 2 , . . , t,] the corresponding set of times at which measurements
are taken.
Let
Design and Analysis of Clinical Trials
220
For the exponential distribution, the hazard function is simply A; for
the Weibull distribution, it is Xyt7-l. The Weibull can accomodate
increasing, decreasing and constant hazard functions.
The hazard function
226
Design and Analysis of Clinical T ? d s
individuals with covariates xi = [ X I ~212,
,
. . . ,3clp] and x; = [QI, 222,
. . . ,zzp]:does not vary with time t. This implies that, given a set of
covariates x, the hazard function can be written as:
where
242
Design and Analysis of Clinical Trials
little difference to the estimated treatment effect, but the hazard
ratio effect estimate for the log-Performance Index has moved closer
to the null value of one. The inclusion of the weights has also increased t
Design and Analysis of Clinical Trials
148
Table 6.1. Separability of Likelihood for Non-informative Missing
Values.
Let the complete set of measurements Y * be partioned into Y * =
[Y("),
Y'")],with Y(")representing the observed measurements and
Y(")the
Design and Analysis of Clinical Trials
140
Table 5.9 (Continued)
Patient
Treat.
Sex
Age
BL
V1
V2
V3
V4
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
P
P
P
P
A
P
P
P
A
P
A
P
A
A
A
A
A
M
M
M
M
M
M
M
M
M
F
M
11
14
15
66
34
43
33
48
20
39
28
38
43
39
68
Design and Analysis of Clinical Dials
218
Table 8.2 (Continued)
~
Calculating log-rank test
Time
T1
T2
Total
Dead
l(0.5)
O(0.5)
1
Alive
Total
2
3
3
3
5
6
6.1
15.2
Dead
Alive
Total
18.7
Dead
Alive
Total
T1
T2
Total
l(0.5)
1
2
O(0.5)
2
2
1
3
T1
T2
Total
O(0
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Design and Analysis of Clinical Trials
survival data is the presence of censored observations and this has
led to the development of a wide range of methodology for analysing
survival times. Of the available methods, the most widely used
is Coxs propo
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Design and Analysis of Clinical Rials
and Prentice (1973). In the particular case where there are no tied
survival times the estimated baseline hazard function at time t ( j ) is
given by:
(8.25)
i L O ( t ( 2 , ) = 1- i z
(8.26)
where x ( ~is
) the v
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Design and Analysis of Clinical Dials
some way - we shall use the second of the pre-randomisation measures of depression as a further covariate. The model to be considered
can be written as follows:
where group is a dummy variable coding treatment rec
196
Design and Analgsis of Clinical Trials
The likelihood of the data, expressed as a function of the unknown
parameters, is given by:
where a represents the parameters of the random effects distribution.
The likelihood is the integral over the unobserved
Design and Analysis of Clinical Trials
130
possibilities when the average response over time is the chosen summary measure:
0
0
0
POST - an analysis that ignores the pre-randomisation values
available and analyses only the mean of each subjects postrandom