A Bayesian model for combining standardized mean differences and odds ratios in the same meta-analysis

In meta-analysis practice, researchers frequently face studies that report the same outcome differently, such as a continuous variable (e.g., scores for rating depression) or a binary variable (e.g., counts of patients with depression dichotomized by certain latent and unreported depression scores). For combining these two types of studies in the same analysis, a simple conversion method has been widely used to handle standardized mean differences (SMDs) and odds ratios (ORs). This conventional method uses a linear function connecting the SMD and log OR; it assumes logistic distributions for (latent) continuous measures. However, the normality assumption is more commonly used for continuous measures, and the conventional method may be inaccurate when effect sizes are large or cutoff values for dichotomizing binary events are extreme (leading to rare events). This article proposes a Bayesian hierarchical model to synthesize SMDs and ORs without using the conventional conversion method. This model assumes exact likelihoods for continuous and binary outcome measures, which account for full uncertainties in the synthesized results. We performed simulation studies to compare the performance of the conventional and Bayesian methods in various settings. The Bayesian method generally produced less biased results with smaller mean squared errors and higher coverage probabilities than the conventional method in most cases. Nevertheless, this superior performance depended on the normality assumption for continuous measures; the Bayesian method could lead to nonignorable biases for non-normal data. In addition, we used two case studies to illustrate the proposed Bayesian method in real-world settings.