In the analysis of bivariate correlated failure time data, it is important to measure the strength of association among the correlated failure times. One commonly used measure is the cross-ratio. In the literature, the functional form of cross-ratio is rather restrictive, often assumed to be constant, piece-wise constant or completely determined by specific model assumption for the joint survival function, e.g. copula model. In this dissertation, we focus on estimating the cross-ratio as a smooth function of bivariate times in various settings without imposing any model assumption for the joint survival function.

Motivated by Cox's partial likelihood idea, we propose in the first chapter a novel parametric estimator for the cross-ratio that is a flexible polynomial function of both survival times. We show that the estimates of cross-ratio regression coefficients are consistent and asymptotically normal. The performance of the proposed technique in finite samples is examined using simulation studies. The proposed method is applied to the Australian twin data for the estimation of dependence of the risk for appendicitis between twin pairs.

In the second chapter, we extend our model to accommodate covariates. Motivated by the Tremin study, we propose a multiplicative model for covariates effect. When the covariate is discrete, we modify our estimator in chapter one by grouping subjects with the same covariate value into strata. When the covariate is continuous, we propose using kernel smoothing applied to the estimating equations. The estimates of regression coefficients are shown to be consistent and asymptotically normal. Numerical studies are conducted for both discrete and continuous covariates.

In observational follow up studies, delayed entry observations are common. In an AIDS incubation cohort study, for example, lag time between HIV infection and death is left-truncated by lag time between HIV infection and the beginning of the study if the patient was infected before the beginning of the study. Ignoring left truncation yields biased estimates. We adjust our model by modifying the risk set and relevant indicators to handle left truncations in the third chapter. We show that the estimates of cross-ratio regression coefficients are consistent and asymptotically normal. Numerical studies are conducted.