ABSTRACT
FIGURE 6.2: Marginal standardized residuals versus tted values for the longitudinal outcome for the AIDS dataset.
subject as
rtmi (t) = Ni(t) Z t 0
Ri(s)hi(s j M^i(s); ^) ds
= Ni(t) Z t 0
Ri(s)h^0(s) expf ^>wi + ^m^i(s)g ds; (6.5)
where Ni(t) is the counting process denoting the number of events for subject i by time t, Ri(t) is the left continuous at risk process with Ri(t) = 1 if sub-
ject i is at risk at time t, and Ri(t) = 0 otherwise, m^i(t) = x > i (t)^ + z
and h^0() denotes the estimated baseline risk function. The idea behind these residuals is based on the Doob-Meyer decomposition of a counting process to a compensator plus a martingale process, which can be seen analogous, in a broad sense, to the classical statistical decomposition in which the data are described by a model plus noise. Thus, the martingale process can been seen as the equivalent of the residual term in the standard statistical decomposition. In plain terms, the residual rtmi (t) can be viewed as the dierence between the observed number of events for the ith subject by time t, and the expected number of events by the same time based on the tted model. The theoretical framework behind the use of martingales to investigate the t of relative risk models has been provided by Barlow and Prentice (1988) and Therneau
model) for evaluating whether the appropriate functional form for a covariate interest has been used in the model.
