Artificial Mixture Methods for Correlated Nominal Responses and Discrete Failure Time.

Abstract

Multinomial logit model with random effects is a common choice for modeling correlated nominal responses. But due to the presence of random effects and the complex form of the multinomial probabilities, the computation is often costly. We generalize the artificial mixture method for independent nominal response to correlated nominal responses. Our method transforms the complex multinomial likelihood to Poisson-type likelihoods and hence allows for the estimates to be obtained iteratively solving a set of independent low-dimensional problems. The methodology is applied to real data and studied by simulations. For discrete failure time data in large data sets, there are often many ties and a large number of distinct event time points. This poses a challenge of a high-dimensional model. We explore two ideas with the discrete proportional odds (PO) model due to its methodological and computational convenience. The log-likelihood function of discrete PO model is the difference of two convex functions; hence difference convex algorithm (DCA) carries over and brings computational efficiency. An alternative method proposed is a recursive procedure. As a result of simulation studies, these two methods work better than Quasi-Newton method in terms of both accuracy and computational time. The results from the research on the discrete PO model motivate us to develop artificial mixture methods to discrete failure time data. We consider a general discrete transformation model and mediate the high-dimensional optimization problem by changing the model form at the “complete-data” level (conditional on artificial variables). Two complete data representations are studied: proportional hazards (PH) and PO mixture frameworks. In the PH mixture framework, we reduce the high-dimensional optimization problem to many one-dimensional problems. In the PO mixture framework, both recursive solution and DCA can be synthesized into the M-step of EM algorithm leading to simplification in the optimization. PO mixture method is recommended due to its simplicity. It is applied to real data sets to fit a discrete PH and PHPH models. Simulation study fitting discrete PH model shows that the advocated PO mixture method outperforms Quasi-Newton method in terms of both accuracy and speed.