In longitudinal studies, the exact timing of an event often cannot be observed, and is usually detected at a subsequent visit, which is called interval censoring. Spacing of the visits is important when designing study with interval censored data. In a typical longitudinal study, the spacing of visits is usually the same across all subjects (balanced design). In this dissertation, I propose an unbalanced design: subjects at baseline are divided into a high risk group and a low risk group based on a risk factor, and the subjects in the high risk group are followed more frequently than those in the low risk group. Using a simple setting of a single binary exposure of interest (covariate) and exponentially distributed survival times, I derive the explicit formula for the asymptotic sampling variance of the estimate for the covariate effect. It shows that the asymptotic sampling variance can be simply reduced by increasing the number of examinations in the high risk group. The relative reduction tends to be greater when the baseline hazard rate in the high risk group is much higher than that in the low risk group and tends to be larger when the frequency of assessments in the low risk group is relatively sparse. Numeric simulations are also used to verify the asymptotic results in small samples and evaluate the efficiency of the unbalanced design in more complicated settings. Beyond comparing the asymptotic sampling variances, I further evaluate the power and empirical Type I error from unbalanced design and compare against the traditional balanced design. Data from a randomized clinical trial for type 1 diabetes are further used to test the performance of the proposed unbalanced design, and the parametric analyses of these data confirmed the findings from the theoretical and numerical studies.

This essay covers a designed experiment on paperboard where the quality under study is tear strength alongside and across. The objective is to examine what consequences the loss of balance in a designed experiment has on the explanatory power of the proposed empirical model. As did happen, the trial plan didn’t go as planned when the first run caused a disruption of the paperboard in the machine. Decision from the company was to raise the low level of one of the design factors to prevent this from happening again. The consequence of this is an alteration of the design during ongoing experimentation. This in turn affects what analysis approaches are appropriate for the problem. Three different approaches for analyzing the data are presented, each with different propositions on how to deal with the complication that occurred. The answer to the research question is that the ability of the empirical model to discover significant effects is moderately weakened by the loss of one run (out of eight total). The price payed for retrieving less information from the experiment is that the empirical model, for tear strength across, doesn’t deem the effects significant at the same level as for the candidate model with eight runs. Instead of concluding that the main effect of  and the interaction effect  is significant at the 2%- and 4%-level, respectively, we must now settle with deeming them significant at the 6%- and 7%-level.