Page 55 - Contributed Paper Session (CPS) - Volume 7

CPS2028 Ayon M.
defined as the set of individuals still at risk at time , where is the
ℎ

ordered event time. Let the hazard for the individual, with covariate vector
ℎ

, be h(| ) = h(t| = 0) where h(t| = 0) denotes the baseline

hazard function.This calculates the hazard rate when all the covariate values
for a patient is set to zero. Throughout this paper it is assumed that the survival
responses follow a continuous time model, so that only one event occurs at
any one time. Therefore, the conditional probability is given by;

Thus the baseline hazard cancels out from the expression. This is the
essence of the analyis: to evaluate the conditional probability the hazard at
the event times ti only needs to be considered. The product of these
conditional probabilities over all the ordered event times is termed the

partial likelihood, where j(i) is the index of the individual who dies at the
ℎ
time :

It can be seen from equation (4) above that the individual times ti do not
appear in the expression of partial likelihood. This can be justified by the
argument that in the absence of a parametric form for the hazard, there is no
information about its value between successive : it could quite possibly be

zero. It follows that the partial likelihood is a function of only the ranks of the
times and it would be unchanged if the time scale were transformed by any
monotonic transformation.

3. The Proposed CARA Designs
Let and be the population characteristics representing the treatment

effects of A and B, respectively. During the initial phase of the trial, one uses
some restricted randomization procedure to allocate the initial 2 patients
0
equally among treatments A and B, where is a positive integer. This ensures
0
that at least patients are allocated to each treatment, and is chosen so
0
0
that estimates of the parameters ( , ) can be obtained from this initial

sample. At stage , one computes the partial likelihood estimates
( ̂ , ̂ ) based on the responses of the first patients, eliminating the
th
effects of the prognostic factors. When the ( + 1) patient enters the
clinical trial with covariate vector + 1 , this patient is randomized to

treatment A with probability ( ̂ , ̂ , + 1) where 0 ≤ (. ) ≤ 1 is an

allocation function which bridges the past allocation pattern, response
th
histories and the covariate vector of the patients to the ( + 1)
allocation with the covariate vector + 1. This allocation is chosen with the

intention of skewing the treatment allocation probability in favour of the
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