Unplanned missing data commonly arise in longitudinal trials. When the mechanism driving the missing data process is related to the outcome under investigation, traditional methods of analysis may yield seriously biased parameter estimates. Motivated by data from two clinical trials, this thesis explores various approaches to dealing with data incompleteness. In the first part, a Monte Carlo EM algorithm is developed and used to fit so called random-co efficient-based dropout models; these models relate the probability of a patient's dropout in follow-up studies to some subject-specific characteristics such as their deviation from the average rate of progression of the disease over time. The approach is used to model incomplete data from a 5-year study of patients with Parkinson's disease. The validity of the results obtained using these methods however, depends in general on distributional and modelling assumptions about the missing data that are inherently untestable as no data were collected. For this reason, many have advocated the need for a sensitivity analysis aimed at assessing the robustness of the conclusions from an analysis that ignores the missing data mechanism. In the second part of the thesis we address these issues. In particular, we present results from sensitivity analyses based on local influence and sampling-based methods used in conjunction with the random-coefficient-based dropout model described in the first part. Recently, a more formal approach to sensitivity analysis for missing data problems has been proposed whereby traditional point estimates are replaced by intervals encoding our lack of knowledge due to incompleteness of the data. In the third part of the thesis, we extend these methods to longitudinal ordinal data. Also, for cross-sectional discrete data having distribution belonging to the exponential family, we propose using the proportion of possible estimates of a parameter of interest, over all solutions corresponding to all sample completions, as a measure of ignorance. We develop a computationally efficient algorithm to calculate this proportion and illustrate our methods using data from a dental pain trial.