Estimation and Testing in a Two-Sample Generalized Odds-Rate Model

Abstract

The generalized odds ratio for a survival variable T is defined as Λ _T (t | c) = c ^-1[1 - S^c (t)]/S^c (t) for c > 0 and -log S(t) for c = 0, where S(t) = Pr(T > t). This ratio coincides with the integrated hazard for c = 0 and the odds ratio for c = 1. When the distribution of T depends on a covariate vector, we assume that conditionally on the covariates, log Λ _T (t | c) is linear in the covariates. This model is a generalization of the proportional hazard model (PHM), which has an interpretation both as a PHM with random nuisance effects (Clayton and Cuzick 1986) and as a proportional odds-rate model with the odds rate defined from the response times of series systems. Harrington and Fleming (1982) and Bickel (1986a) considered rank tests for this semiparametric model, Clayton and Cuzick (1986) considered estimation. We use the odds-rate representation to define a class of estimates of the proportionality parameter in the two-sample case. We show that the estimates are consistent and asymptotically normal, and that they lose little efficiency when compared with the optimal parametric estimates. Moreover, when testing for equality of distributions, the test statistics based on the semiparametric estimates are fully efficient when compared with best parametric tests.

Key Words:

• Proportional hazard models
• Proportional odds-rate models
• Rank statistics
• Semiparametric models