I adapted a method designed by Ioannidis and Trikalinos, which compares the observed number of positive studies in a meta-analysis with the expected number, if the summary measure of effect, averaged over the individual studies, were assumed true. Excess in the observed number of positive studies, compared to the expected, is taken as evidence of publication bias. The observed number of positive studies, at a given level for statistical significance, is calculated by applying Fisher's exact test to the reported 2x2 table data of each constituent study, doubling the Fisher one-sided P-value to make a two-sided test. The corresponding expected number of positive studies was obtained by summing the statistical powers of each study. The statistical power depended on a given measure of effect which, here, was the pooled odds ratio of the meta-analysis was used. By simulating each constituent study, with the given odds ratio, and the same number of treated and non-treated as in the real study, the power of the study is estimated as the proportion of simulated studies that are positive, again by a Fisher's exact test. The simulated number of events in the treated and untreated groups was done with binomial sampling. In the untreated group, the binomial proportion was the percentage of actual events reported in the study and, in the treated group, the binomial sampling proportion was the untreated percentage multiplied by the risk ratio which was derived from the assumed common odds ratio. The statistical significance for judging a positive study may be varied and large differences between expected and observed number of positive studies around the level of 0.05 significance constitutes evidence of publication bias. The difference between the observed and expected is tested by chi-square. A chi-square test P-value for the difference below 0.05 is suggestive of publication bias, however, a less stringent level of 0.1 is often used in studies of publication bias as the number of published studies is usually small.
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