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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?

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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.

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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.

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    $\begingroup$ Note that this is some strange hybrid of the Mantel-Haenszel method and the 'normal-normal' model. I would not consider this a proper random-effects model extension of the M-H method. $\endgroup$
    – Wolfgang
    Mar 5 '17 at 12:00
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    $\begingroup$ I won't argue with you as I'm not a statistician, but it also makes we wonder why these methods haven't been challenged by statisticians in a more concerted effort since Cochrane reviews uses RevMan (and as an extension these formulas) as well as I would dare to guess a large number of published reviews. I also use Comprehensive Meta-analysis by Michael Borenstein, Higgins and others and I don't have their calculations with me right now assume that they use similar methods. To be honest, I've never been challenged by using these RE models in my published work including at JAMA, BMJ, CMAJ, etc $\endgroup$
    – abousetta
    Mar 5 '17 at 16:59
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    $\begingroup$ As far as I know, RevMan is the only software that implements this hybrid approach. This aside, I am not saying that the approach necessarily needs to be challenged -- it's just not a proper random-effects model generalization of the M-H method. In fact, it's basically just the standard normal-normal model, but with the amount of heterogeneity estimated using a DL-type estimator that uses the M-H estimate for computing the Q-statistic. $\endgroup$
    – Wolfgang
    Mar 5 '17 at 18:12
  • $\begingroup$ Thanks Wolfgang for the explanation. Hopefully new versions of RevMan will be more robust in their analytic options. $\endgroup$
    – abousetta
    Mar 5 '17 at 22:31

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