Incompleteness is a major feature of time-to-event data. As one type of incompleteness, truncation refers to the unobservability of the time-to-event variable because it is smaller (or greater) than the truncation variable. A truncated sample always involves left and right truncation. Left truncation has been studied extensively while right truncation has not received the same level of attention. In one of the earliest studies on right truncation, Lagakos et al. (1988) proposed to transform a right truncated variable to a left truncated variable and then apply existing methods to the transformed variable. The reverse-time hazard function is introduced through transformation. However, this quantity does not have a natural interpretation. There exist gaps in the inferences for the regular forward-time hazard function with right truncated data. This dissertation discusses variance estimation of the cumulative hazard estimator, one-sample log-rank test, and comparison of hazard rate functions among finite independent samples under the context of right truncation. First, the relation between the reverse- and forward-time cumulative hazard functions is clarified. This relation leads to the nonparametric inference for the cumulative hazard function. Jiang (2010) recently conducted a research on this direction and proposed two variance estimators of the cumulative hazard estimator. Some revision to the variance estimators is suggested in this dissertation and evaluated in a Monte-Carlo study. Second, this dissertation studies the hypothesis testing for right truncated data. A series of tests is developed with the hazard rate function as the target quantity. A one-sample log-rank test is first discussed, followed by a family of weighted tests for comparison between finite $K$-samples. Particular weight functions lead to log-rank, Gehan, Tarone-Ware tests and these three tests are evaluated in a Monte-Carlo study. Finally, this dissertation studies the nonparametric inference for the hazard rate function for the right truncated data. The kernel smoothing technique is utilized in estimating the hazard rate function. A Monte-Carlo study investigates the uniform kernel smoothed estimator and its variance estimator. The uniform, Epanechnikov and biweight kernel estimators are implemented in the example of blood transfusion infected AIDS data.

Incompleteness is a major feature of time-to-event data. As one type of incompleteness, truncation refers to the unobservability of the time-to-event variable because it is smaller (or greater) than the truncation variable. A truncated sample always involves left and right truncation. Left truncation has been studied extensively while right truncation has not received the same level of attention. In one of the earliest studies on right truncation, Lagakos et al. (1988) proposed to transform a right truncated variable to a left truncated variable and then apply existing methods to the transformed variable. The reverse-time hazard function is introduced through transformation. However, this quantity does not have a natural interpretation. There exist gaps in the inferences for the regular forward-time hazard function with right truncated data. This dissertation discusses variance estimation of the cumulative hazard estimator, one-sample log-rank test, and comparison of hazard rate functions among finite independent samples under the context of right truncation. First, the relation between the reverse- and forward-time cumulative hazard functions is clarified. This relation leads to the nonparametric inference for the cumulative hazard function. Jiang (2010) recently conducted a research on this direction and proposed two variance estimators of the cumulative hazard estimator. Some revision to the variance estimators is suggested in this dissertation and evaluated in a Monte-Carlo study. Second, this dissertation studies the hypothesis testing for right truncated data. A series of tests is developed with the hazard rate function as the target quantity. A one-sample log-rank test is first discussed, followed by a family of weighted tests for comparison between finite $K$-samples. Particular weight functions lead to log-rank, Gehan, Tarone-Ware tests and these three tests are evaluated in a Monte-Carlo study. Finally, this dissertation studies the nonparametric inference for the hazard rate function for the right truncated data. The kernel smoothing technique is utilized in estimating the hazard rate function. A Monte-Carlo study investigates the uniform kernel smoothed estimator and its variance estimator. The uniform, Epanechnikov and biweight kernel estimators are implemented in the example of blood transfusion infected AIDS data.