In a typical comparative clinical trial, participants are randomized to receive either an experimental or a control treatment, and the response variable of interest (e.g., bone density) is measured at baseline and at pre-specified post-baseline time points.  The validity of the resulting data analysis depends on whether the key underlying assumptions are true.  For example, if the treatments are compared using a standard linear mixed effects model for the longitudinal data, the analysis assumes that the response vector follows a multivariate normal distribution and missing data (e.g., due to dropouts) are "missing at random".  Similarly, in a stratified trial with a time-to-event endpoint (e.g., endpoint = heart attack), a key assumption in the standard stratified Cox model analysis is that the treatment hazard ratio (HR) is constant across strata.  In both these examples, even moderate violations of the key assumptions can result in misleading conclusions.  In this two-part presentation, we will demonstrate how some standard analyses can be made stronger by weakening assumptions!  In part I, we will illustrate the utility of using multiple imputation to tackle missing values in conjunction with M-estimation (to avoid the normality assumption) for the analysis of longitudinal clinical trials.  In part II, we will present an efficient alternative to the stratified Cox model approach in which first the [log] HR is estimated separately for each stratum (to avoid the stratum-invariant HR assumption), and then the stratum-specific estimates are combined for overall estimation and inference using either sample size or "minimum risk" stratum weights.  Real examples and computer simulations will be used to compare the standard and new approaches, and to reinforce the key points.