Random Effects for Mantel Haenszel I'm familiar with the standard Mantel Haenszel method for Odds ratios and Risk ratios. Is there also a random effects method? If so, can anyone provide the formulae and/or citations?
 A: Jonathan Deeks and Julian Higgins have a nice document showing all the calculations used in Review Manager. Scroll down to page 8  for DerSimonian and Laird random-effects models that can be used with the Mantel-Haenszel summary models.
A: A proper random-effects model extension to the standard Mantel-Haenszel procedure is described by van Houwelingen, Zwinderman, and Stijnen (1993). In essence, one can think of the M-H procedure as a model based on the (non-central) hypergeometric distribution (Mantel & Haenszel, 1959). So, using this as the starting point, van Houwelingen and colleagues extend the method by adding a random effect to the model where the 2x2 tables are modeled by non-central hypergeometric distributions. See also Stijnen, Hamza, and Ozdemir (2010). The resulting model can also be thought of as a conditional mixed-effects logistic regression model.
The equations get a bit messy, but we can write things pretty compactly if we let $L(\theta_i|a_i, b_i, c_i, d_i)$ denote the likelihood function of a non-central hypergeometric distribution for the $i$th study, where $a_i, b_i, c_i, d_i$ are the 2x2 table counts and $\theta_i$ is the true log odds ratio (see wikipedia for the pmf of the non-central hypergeometric distribution). Now let $f(\theta_i)$ denote the density of a normal distribution with mean $\mu$ and variance $\tau^2$. So the log-likelihood for the random-effects model is given by $$ll = \sum_{i=1}^k \ln \left[\int_{-\infty}^\infty L(\theta_i|a_i, b_i, c_i, d_i) f(\theta_i)d\theta_i\right].$$ There is no closed-form solution to the values of $\mu$ and $\tau^2$ that maximize $ll$, so those values must be obtained numerically.
References
van Houwelingen, H. C., Zwinderman, K. H., & Stijnen, T. (1993). A bivariate approach to meta-analysis. Statistics in Medicine, 12(24), 2273-2284.
Mantel, N., & Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22(4), 719-748.
Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29(29), 3046-3067.
