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Timevarying covariates are especially tractable within the Cox
model. Inspection of Table 8.4 shows how the partial likelihood involves contributions for the conditional probability of failure. These
conditional probabilities involv
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I
1
i;t \
lailbda0.5
. lamwa=l.o
I
I
lam2.0
I
0
2
I
6
4
8
10
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0
2
6
4
8
10
Time
Fig. 8.2. Survivor functions for exponential and Weibull distributions.
(This is a nonparametric estimator since no particula
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Table 8.3. Results from Fitting Accelerated Failure
Time Model to Survival Times from the Lung Cancer Trial.
Exponential
Estimate
SE
Estimate/SE
2.59
0.22
0.01
0.75
0.20
0.01
3.45
1.11
0.03
0.31
0.00
0.05
0.0
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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
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Table 8.7. Estimated Baseline Survivor Function at Selected Intervals for Lung Cancer Trial Data.
~
~
~
Survival time
Survival Prob.
Standard Treatment
Survival Prob.
Standard and Chemotherapy
1
125
249
373
497
6
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of interest (death, relapse, etc.). Consequently, their exact survival
times are not known. All that is known is that the survival times
are greater than the amount of time the patient has been in the
study. This is rightcensoring.
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Design and Analysis of Clinical Trials
conditional on past responses y Z j  ] ~ ,for k 2 1. The assumptions
of a crosssectional GLM would now be replaced by:
yij
are the expectation and variance of y i j conwhere now pfj and
ditional on all x j  k
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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
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of explanatory variables; for example, if dlj = [l,tij],then the elements of ~i correspond to the intercept and slope of a subject specific
timetrend in the mean response. We use p* here rather than p, to
emphas
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clearly departs from the expected y = z regression line but both the
compound symmetry and unstructured models lead to acceptable
plots.
6.5.2.
PostSurgical Recovery in Young
Children
In a comparison of
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241
of such an approach in which firstly, the groups to be compared represent not just a partitioning of the covariate space (patient groups) but
also of the time axis, and secondly, the expected deaths are derived
from the fitted Cox mo
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179
as we shall see later. The three principal approaches to introducing correlations are marginal, random eflects and transition models.
Each type will now be considered relatively briefly; full
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197
Table 7.9. Results from Random Effects Logistic Regression Model Fitted to Respiratory Status Data.
Covariate
Treatment
Visit
Sex
Estimate
2.14
0.12
0.32
Age/ 100
Baseline
Centre
2.45
SE
0
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treatment is consistent with treatment leading to a slightly quicker
recovery, but the effect is not significant and the confidence intervals
are wide. Thus the substantial differences in the point estimates
aris
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(1) an independence working model with robust standard errors,
(2) an exchangeable correlation model,
(3) as in (2) but with the scale parameter estimated by the Pearson chisquare/residual divided by the degr
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Table 6.10. Results from DiggleKenward Model for Dropouts
Oestrogen Patch Data (assuming informative d r o p o u t ) .
Maximised likelihood:
[l] 1291.848
Mean Parameters:
PARAMETER
STD. ERROR
(Intercept)
Group
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CHAPTER7
Models for NonNormal
Longitudinal Data
from Clinical Trials
7.1.
INTRODUCTION
In Chapter 4 we gave a brief description of generalised linear models and showed how they have unified regression analysis for discrete
and continuous independent resp
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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
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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
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Table 7.4. Results from Fitting Logistic Regression Model to
Respiratory Status Data under Various Assumptions about Correlational Structure.
Estimate
Classical
SE
ztest
Robust
SE
Covariate
Model
Treatment
Exch
1
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informativeness of the dropout and concludes that external information about the dropout mechanism be sought and used. A further
possible problem identified by Troxel et al. (1998) is that bias that
resu
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Multivariate Normal Regression Models for
Dose 20mg
Dose 15mg
w 
T
165
T
In
* m 
N
w

 
LL.I
A
0
I
0
Dose 30mg
Dose 25rng
m
m 
N

7

Fig. 6.4. Boxplots of surgicalrecovery data.
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It is important to keep in mind that what is being estimated by
the fitted model is the crosssectional relationship between variables.
Table 7.2 shows the sample frequencies over the joint distribution
of the ba
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visits. Two treatment groups were involved and in addition two further covariates, sex and age, were available together with a binary
indicator for centre. One possible logistic marginal model is given
by:
0
Corr(
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Survival Analysis
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I
II
,
,
,
lambda0.5
lambda1.0
I
I
II
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'.
II
:'
I
. I,
.,
%,
I
0
2
6
4
8
10
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Fig. 8.1. Weibull and exponential density functions.
Where there are no censored observations in the sample of survival times, a nonparametric
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Log(base1ine seizure count+l)
Fig. 7.1. Plot of log of total number of seizures (plus o