ABSTRACT
In some analyses of failure-time data, it has been observed that a group of subjects may not react to treatments. A mixture model that incorporates different latency distributions for different groups seems to be appropriate. In this chapter we consider a mixture model where covariates may influence both the incidence probabilities and the conditional latency distributions. Let x denote the covariate vector (1 x q) associated with a subject of lifetime T. Then the mixture model assumes that the density ofT is
f(tix,6) = LPi(x,p)/j(tix,¢j), (1) j=l
where L,f=1 Pi(x, p) = 1. Let Y be an index variable for the subpopulations. We use Pi(x,p) to denote the mixing probability, P(Y = ilx,p), also called the incidence probability for the ph subpopulation. We use /j (tix, tPj) to denote /j (t!Y = j, x, tPj ), the conditional probability density (conditional latency density) function of the failure time for the ph subpopulation. We assume /j is a continuous density and indexed by an unknown parameter tPj that can be a vector. Moreover, we use 6 = ( ¢ 1 , ... , ¢ J, p) to denote the collection of all unknown parameters. The covariate x may include 1, the dosage (or log-dose) level, and other explanatory variables. In this study we only deal with time-independent explanatory variables. Logistic regression links can be chosen for the incidence probabilities. For example, we can
(2)
where ej = xpJ' with PI to be the transpose of Pj (1 X q) and p = (Pl' ... 'PJ ). In addition to the logistic link, other models such as the normalized probit link or the normalized complementary log-log link (McCullagh and Neider, 1989) can be considered for the incidence probabilities.
