Assume that we have right-censored survival data and that all covariates are time-fixed. Let
be the vector of martingale residuals for the additive regression model (Section 4.2.4). Prove that
Assume that the counting processes
have intensity processes of the form
where the Yi(t) are at risk indicators and the
are fixed covariates. In Chapter 5 we show that the log likelihood takes the form [cf. (5.5)]
Where
is the study time interval and
a) Show that
The log-likelihood in question (a) may be made arbitrarily large by letting α0(t) be zero except from close to the observed event times where we let it peak higher and higher. However, if we consider an extended model, where the cumulative baseline hazard A0(t) may be any nonnegative, non decreasing function, the log-likelihood achieves a maximum. For such an extended model, the log-likelihood is maximized if A0(t) is a step function with jumps at the observed event times T1
2
<>
Where
is the increment of the cumulative baseline hazard at Tj.
b) Show that for a given value of β , the baseline hazard increments that maximize the log-likelihood (4.108) are given by
c) If we insert (4.109) in (4.108), we get a profile log-like hood for β . Show that this profile log-likelihood may be written
where L(β ) is the partial likelihood (4.7). Thus an alternative interpretation of (4.7) is as a profile likelihood.
d) Explain that the results in questions (b) and (c) imply that the maximum partial likelihood estimator
and the Breslow estimator (4.13) are maximum likelihood estimators (in the extended model).