The notion of frailty, also called random effect, provides a convenient way to introduce association and unobserved heterogeneity into the models in survival analysis. The semiparametric transformation models with random effects or frailties are useful in analyzing dependent data, for example, recurrent data and clustered data. With the error and random effect distributions specified, Zeng and Lin (2007a) proved the nonparametric maximum likelihood estimators (NPMLEs) are semiparametric efficient.

In the first part of this thesis, we focus on the Cox model with gamma frailty, which can be also written as the transformation model with gamma frailty whose error term follows extreme value distribution. We propose an empirical Bayes estimator with closed form expression for the es...[ Read more ]

The notion of frailty, also called random effect, provides a convenient way to introduce association and unobserved heterogeneity into the models in survival analysis. The semiparametric transformation models with random effects or frailties are useful in analyzing dependent data, for example, recurrent data and clustered data. With the error and random effect distributions specified, Zeng and Lin (2007a) proved the nonparametric maximum likelihood estimators (NPMLEs) are semiparametric efficient.

In the first part of this thesis, we focus on the Cox model with gamma frailty, which can be also written as the transformation model with gamma frailty whose error term follows extreme value distribution. We propose an empirical Bayes estimator with closed form expression for the estimation of unknown variables in the absence of censoring as well as for censored data, through the comparison with other two estimates in the simulation studies, we show the superiority of the empirical Bayes estimation in terms of the root-mean-square error (RMSE) criterion.

In the second part of this thesis, we consider a more general class of transformation models with random effects, under which an unknown monotonic transformation of the response is linearly related to the covariates and the random effects with unspecified error and random effect distributions. This class of models is broad enough to include many popular models and allows various random effect distributions. We propose an estimator based on the maximum rank correlation, which does not reply on any further model assumption except the symmetry of the random effect distribution. The consistency and asymptotic normality of the proposed estimator are established. A random weighting resampling scheme is employed for inference. Moreover, the proposed method can be easily extended to handle censored data and clustered data. Numerical studies demonstrate that the proposed method performs well in practical situations. Applications are illustrated with an AIDS clinical trial study and the Framingham cholesterol data set.