Abstract

Tianyuan Guan, Department of Epidemiology & Biostatistics, College of Public Health, Kent State University, Kent, OH 44242, USA
Suyang Gao, Shesh Rai, and M.B. Rao
Department of Biostatistics, Medical Informatics, and Data Science, College of Medicine, University of Cincinnati, Cincinnati, OH 45267, USA

Abstract
A fundamental question when pursuing Meta-Analysis in assessing the effectiveness of a treatment, for example, is how many studies are to be included. Is there a way to calculate the power of meta-analysis based on the number of studies included? We will answer this question in this presentation.
Power calculations in the environment of testing hypotheses are well hashed out. To unleash these calculations, we need 1. A family of models to explain the underlying random phenomenon in the population of interest; 2. A subset of the family exemplifying the null hypothesis; 3. The alternative hypothesis, typically composite; 4. Data from the underlying population; 5. A test statistic T, which is a function of the data; 6. The distribution of T under the null hypothesis; 7. The distribution of T under any specified alternative; 8. A level of significance; 9. A test embodying when to reject the null hypothesis. Then we are ready to calculate the power of the test at any chosen alternative. Power calculations serve two purposes. 1. If we have two competing tests for the same testing problem, power lets us choose one. 2. For the chosen test, calculations help us to determine the sample size to achieve a specified power.
We bring the entire modus operandi in the environment of testing hypotheses into the realm of Meta-Analysis. Let m be the number of studies chosen for synthesis. Information on the studies is collected in two ways. 1. Relevant summary statistics from each study. 2. P-values from each study. There are studies which give only p-values. These studies are the focus of this presentation. There are scores of tests proposed and used in literature on synthesis. Tippett’s test, Fisher’s test, Pearson’s test are some examples. We initiate power calculations as a function m of the number of studies on these tests. The data for Meta-Analysis are the p-values coming from the individual studies. Theoretically, the p-values are independently identically distributed with common uniform distribution under the null hypothesis. A test statistic is chosen to combine the p-values. One needs to determine the distribution of the test statistic under any specified alternative. We show then how power calculations can be done. We compare the power functions of the Tippett, Fisher, and Pearson tests.