A method for negative exponential modeling of survival times relates to the Poisson loglinear model for rates (Aitkin and Clayton 1980). Let T denote the time to some event, with pdf f and cdf F. For subject i, let wi = 1 for death and 0 for censoring, and let t = ∑

i ti and w = ∑

i wi.

a. Explain why the survival-time log-likelihood for n independent observations is10 L(????) = ∑

i wi log[f(ti)] +

∑

i

(1 − wi) log[1 − F(ti)].

Assuming f(t) = ???? exp(−????t), show that ????̂ = w∕t. Conditional on t, explain why w has a Poisson distribution with mean t????. Using the Poisson likelihood, show that ????̂ = w∕t.

b. With ???? replaced by ???? exp(x????) and with ????i = ti???? exp(xi????), show that L simplifies to L(????, ????) = ∑
i wi log ????i −
∑
i ????i −
∑
i wi log ti.
Explain why maximizing L(????, ????) is equivalent to maximizing the likelihood for the Poisson loglinear model log ????i − log ti = log ???? + xi????
with offset log(ti), using “observations” {wi}.

c. When we sum terms in L for subjects having a common value of x, explain why the observed data are the numbers of deaths (∑
i wi) at each setting of x, and the offset is log(∑
i ti) at each setting.