A bivariate parametric model for survival and intermediate event times

In diseases such as acquired immune deficiency syndrome (AIDS), there is great interest in describing the frequency of secondary diagnoses that occur during the course of the disease and their effect on survival. Casting this situation in a more general framework, one distinguishes a terminal event (TE) and an intermediate event (IE) that may or may not occur. In epidemiologic applications the TE is usually death. Earlier studies of IE and TE times have used the latter to censor the IE time for individuals who do not present it. For such cases, we argue that more appropriately the TE removes the individual from the risk set for the IE. With this view, one distinguishes observations of four types, each with a different formula for its likelihood contribution. We propose an approach that uses separate parametric models for the marginal distribution of the survival time D and for the conditional distribution of the time R to the IE given D = d and R ≤ D. A central quantity is the probability of presenting the IE given the occurrence of the TE at time d. This function of d can reveal important connections between the two events. We suggest a model derived from Weibull distributions where the parameters control the shape of this function. One can obtain inferences about other probabilities of interest such as the proportion of individuals who present the IE, P(R ≤ D), the marginal distribution of R among the if cases, P(R > r/R ≤ D) and the residual survival after the IE occurs, P(D - R > v/R ≤ D, R = r). We apply the model to the analysis of time to secondary Kaposi's sarcoma (KS) diagnosis and time to death in a large cohort study of homosexual men infected with the human immunodeficiency virus type 1 (HIV) and who had an initial non-KS AIDS diagnosis.