Semiparametric inference for transformation models via empirical likelihood

AMS 2000 subject classifications: 62N02 62G20 Keywords: Kaplan–Meier estimator Martingale Proportional hazards model Proportional odds model Right censoring U-statistic a b s t r a c t Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations. It is well known that the Cox [1] regression model is the most popular model used in survival analysis. The Cox model is semi-parametric, and its large sample inference properties have been demonstrated using martingale theory [2]. Moreover, practitioners have easy access to statistical software for this model. In practice, however, the proportional hazards assumption is often too restrictive, even for randomized clinical trials. In recent years, the transformation model has received a lot attention and provides a useful alternative to the Cox regression model in analyzing survival observations. Its simple structure and ease of interpretation make it an attractive method. The transformation model is becoming a valuable model for the analysis of survival data. Let T be the failure time, i.e. the response variable, and Z a corresponding covariate vector. Suppose that we are interested in making inferences about the effect of Z on the response variable. If there are censored observations in the data, one usually uses the Cox model to examine the covariate effect. Let S Z (·) be the survival function of T given Z. Suppose that h(t) is a completely unspecified strictly increasing function, which maps the positive half-line onto the whole real line. Thus, a natural generalization of the Cox regression model is g{S Z (t)} = h(t) + Z T β, where g(·) is a known decreasing function and β is a p × 1 vector of unknown …