Meta-analysis is frequently used in the analysis of safety data. In dealing with rare events, the commonly used risk measures, such as the odds ratio, or risk difference, or their variance, can become undefined when no events are observed in studies. The use of an arcsine transformation and arcsine difference (AD) as treatment effect were shown to have desirable statistical properties (Rucker, 2009). However, the interpretation of the AD remains challenging and this may hamper its utility. To convert the AD to a risk measure similar to the risk difference, two previously proposed linear approximation methods, along with new linear and non-linear methods were discussed and evaluated. The existing approximation methods generally provide satisfactory approximation when the event proportions are between 0.15 and 0.85. We propose a new linear approximation method, the modified rationalized arcsine unit (MRAU) which improves the approximation when proportions fall outside the range from 0.15 to 0.85. However, the MRAU can still lead to under- or over-estimation depending on the underlying proportion. We then proposed a non-linear approximation method, based on a Taylor series expansion (TRAUD), which shows the best approximation across the full range of risk levels. However, the variance for TRAUD is less easily estimated and requires bootstrap estimation. Results from simulation studies confirm these findings under a wide array of scenarios. In the second section, heterogeneity in meta-analysis is discussed along with current methods that address the issue. To provide an exploration of the nature of heterogeneity, finite mixture model methods (FMM) were presented, and their application in meta-analysis discussed. The estimates derived from the components in FMM indicate that even with a pre-specified protocol, the studies included in a meta-analysis may come from different distributions that can cause heterogeneity. The estimated number of components may suggest the existence of multiple sub-populations that a simple overall effect estimate will neglect. We propose that in the analysis of safety data, the estimates of the number of components and their respective means can provide valuable information for better patient care. In the final section, the application of the approximation methods and the use of FMM are demonstrated in the analysis of two published meta-analysis examples from the medical literature.

The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first few flips.