Marginal Likelihood Estimation for Proportional Odds Models with Right Censored Data

One major aspect in medical research is to relate the survival times of patients with the relevant covariates or explanatory variables. The proportional hazards model has been used extensively in the past decades with the assumption that the covariate effects act multiplicatively on the hazard function, independent of time. If the patients become more homogeneous over time, say the treatment effects decrease with time or fade out eventually, then a proportional odds model may be more appropriate. In the proportional odds model, the odds ratio between patients can be expressed as a function of their corresponding covariate vectors, in which, the hazard ratio between individuals converges to unity in the long run. In this paper, we consider the estimation of the regression parameter for a semiparametric proportional odds model at which the baseline odds function is an arbitrary, non-decreasing function but is left unspecified. Instead of using the exact survival times, only the rank order information among patients is used. A Monte Carlo method is used to approximate the marginal likelihood function of the rank invariant transformation of the survival times which preserves the information about the regression parameter. The method can be applied to other transformation models with censored data such as the proportional hazards model, the generalized probit model or others. The proposed method is applied to the Veteran's Administration lung cancer trial data.