Multiplicative random effect model of meta-analysis

For a random-effect model meta-analysis we usually assume that the random effects are additive. In a multiplicative random-effects model, the variant-specific estimates are assumed to be normally distributed with variance \(\phi^2v_i\) where \(\phi^2\) is an overdispersion parameter.

The appropriate value of \(\phi\) can be established by running a linear regression of the observed effect sizes against a constant, with weights \(w_i = 1/v_i\) and extracting the mean squared error. Noted, Higgins’s \(H^2 =Q/k- 1\) is equivalent to this quantity, which is also referred to as Birge’s ratio.\(^{1,2}\)

Ref:

1.Mawdsley, D., Higgins, J. P., Sutton, A. J., & Abrams, K. R. (2017). Accounting for heterogeneity in meta‐analysis using a multiplicative model—an empirical study. Research Synthesis Methods, 8(1), 43-52

2.Thompson, Simon G., and Stephen J. Sharp. “Explaining heterogeneity in meta‐analysis: a comparison of methods.” Statistics in medicine 18.20 (1999): 2693-2708.