# A simplification and implementation of random-effects meta-analyses based on the exact distribution of Cochran's Q

*Methods of Information in Medicine 2014, 53 (1): 54-61*

BACKGROUND: The random-effects (RE) model is the standard choice for meta-analysis in the presence of heterogeneity, and the standard RE method is the DerSimonian and Laird (DSL) approach, where the degree of heterogeneity is estimated using a moment-estimator. The DSL approach does not take into account the variability of the estimated heterogeneity variance in the estimation of Cochran's Q. Biggerstaff and Jackson derived the exact cumulative distribution function (CDF) of Q to account for the variability of τ^².

OBJECTIVES: The first objective is to show that the explicit numerical computation of the density function of Cochran's Q is not required. The second objective is to develop an R package with the possibility to easily calculate the classical RE method and the new exact RE method.

METHODS: The novel approach was validated in extensive simulation studies. The different approaches used in the simulation studies, including the exact weights RE meta-analysis, the I² and τ² estimates together with their confidence intervals were implemented in the R package metaxa.

RESULTS: The comparison with the classical DSL method showed that the exact weights RE meta-analysis kept the nominal type I error level better and that it had greater power in case of many small studies and a single large study. The Hedges RE approach had inflated type I error levels. Another advantage of the exact weights RE meta-analysis is that an exact confidence interval for τ² is readily available. The exact weights RE approach had greater power in case of few studies, while the restricted maximum likelihood (REML) approach was superior in case of a large number of studies. Differences between the exact weights RE meta-analysis and the DSL approach were observed in the re-analysis of real data sets. Application of the exact weights RE meta-analysis, REML, and the DSL approach to real data sets showed that conclusions between these methods differed.

CONCLUSIONS: The simplification does not require the calculation of the density of Cochran's Q, but only the calculation of the cumulative distribution function, while the previous approach required the computation of both the density and the cumulative distribution function. It thus reduces computation time, improves numerical stability, and reduces the approximation error in meta-analysis. The different approaches, including the exact weights RE meta-analysis, the I² and τ² estimates together with their confidence intervals are available in the R package metaxa, which can be used in applications.

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