Transforming the Model T: random effects meta‐analysis with stable weights

Abstract

Standard meta-analytic theory assumes that study outcomes are normally distributed with known variances. However, methods derived from this theory are often applied to effect sizes having skewed distributions with estimated variances. Both shortcomings can be largely overcome by first applying a variance stabilizing transformation. Here we concentrate on study outcomes with Student t-distributions and show that we can better estimate parameters of fixed or random effects models with confidence intervals using stable weights or with profile approximate likelihood intervals following stabilization. We achieve even better coverage with a finite sample bias correction. Further, a simple t-interval provides very good coverage of an overall effect size without estimation of the inter-study variance. We illustrate the methodology on two meta-analytic studies from the medical literature, the effect of salt reduction on systolic blood pressure and the effect of opioids for the relief of breathlessness. Substantial simulation studies compare traditional methods with those newly proposed. We can apply the theoretical results to other study outcomes for which an effective variance stabilizer is available.

Combining Statistical Evidence

“…Further, both traditional and new methods for the FEM are shown to be robust to small τ 2 > 0, so that if K is too small to estimate τ 2 , a practical solution is to revert to the FEM, even though one might prefer the REM. These results should be able to be extended to the standardised mean difference, see Malloy et al (, Section 1.4).…”

“…Yet, another possibility is the profile likelihood (Hardy and Thompson, ; Malloy et al . ). All these methods require numerical maximisation.…”

“…Another approach by Malloy et al . () for random effects having the Student‐ t distribution uses variance stabilisation before finding maximum likelihood and profile estimates of both θ and τ 2 ; this approach yields explicit formulae for the standard errors of each estimator in terms of the number of studies K . In addition, these authors show that the best performing confidence interval for δ is a simple t ‐interval, which does not require an estimate of τ 2 .…”

“…Instead, the MLE of 2 or the restricted MLE (REML) can be used. Yet, another possibility is the profile likelihood (Hardy & Thompson, 1996;Malloy et al, 2013). All these methods require numerical maximisation.…”

Inference for the Standardized Median

“…As seen earlier in this paper, the Key plays an important role in estimation by confidence intervals. Another advantage of variance stabilized statistics is that they can be readily combined in a meta-analysis of effects from multiple studies, as shown in (Kulinskaya et al 2008;Malloy et al 2013;and Morgenthaler and Staudte 2012).…”