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We propose a general approach based on the concept of overdispersion for specification of a random effects model (REM) in meta-analysis. This approach is similar to that used in generalized linear models, and includes the traditional REM as a particular case. A key feature of the model is the interpretation of the multiplicative factor as an intra-class correlation parameter. We provide several motivating examples, discuss statistical inference, and compare the new and standard methods on two examples of published meta-analyses. Estimation of the overdispersion parameter in the proposed model is compared in simulations to that of the traditional between-studies variance in the case of normal means. For small values of heterogeneity, the coverage of the confidence intervals for the overdispersion parameter is more stable.