Page 125 - Contributed Paper Session (CPS) - Volume 7

CPS2044 Mohd Asrul Affendi A. et al.
the event time T can be modelled via the scale parameter b as,
′
= [ ] (4)
′
=
The form of parameterization in (4) is when the fixed time covariate or
proportional assumption is satisfied. But when time‐varying covariate z(t) exist,
(4) can be modified as;
′
= [ + ()] (5)
Where and represent the parameters of the fixed and time varying
covariate.

For the HIV‐TB mortality model, () is a binary vector representing TB
infection at varying interval of time before the end of study. The matrix
represents the fixed covariate such as gender, age, marital status etc.
substituting (5) in (1), we can obtain the density function for Weibull with fixed
and time varying covariate effects as;

(|, , ()) = ( ) ( ) −1 [− ( ) ](6)
′
[ +()] [ +()] [ +()]
′
′
Now assuming z(t) to be a piece‐wise function as in the case of TB infection
occurring with HIV infection, z(t) can be define as;
0, <
() = {
1, ≥

Where t_c is the time at which the covariate z(t) changes. This implies that
f(t│a,x,z(t)) will also be a piece‐wise function. Therefore f(t│a,x,z(t)) can be
define as;
(|, ), <
(|, , ()) = {
(|, , ()), ≥

(|, , ())
−1
( ) ( ) [− ( ) ] , <
[ ] [ ] [ ]
′
′
′
=
−1
( ) ( ) [− ( ) ] , ≥
′
{ [ + ()] [ + ()] [ + ()]
′
′

Consequently, the hazard function h(t│a,x,z(t))associated with T can be
obtained as;

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