Abstract
We study bias arising as a result of nonlinear transformations of random variables in random or mixed effects models and its effect on inference in group‐level studies or in meta‐analysis. The findings are illustrated on the example of overdispersed binomial distributions, where we demonstrate considerable biases arising from standard log‐odds and arcsine transformations of the estimated probability p^, both for single‐group studies and in combining results from several groups or studies in meta‐analysis. Our simulations confirm that these biases are linear in ρ, for small values of ρ, the intracluster correlation coefficient. These biases do not depend on the sample sizes or the number of studies K in a meta‐analysis and result in abysmal coverage of the combined effect for large K. We also propose bias‐correction for the arcsine transformation. Our simulations demonstrate that this bias‐correction works well for small values of the intraclass correlation. The methods are applied to two examples of meta‐analyses of prevalence.
Highlights
The main focus of this paper is the bias that arises as a result of transformations of random variables in random or mixed effects models and the deleterious effects of these biases on inference in group-level studies, such as ecological studies in epidemiology or cluster-randomized trials, and in meta-analyses
We have studied by simulation the bias and coverage of the parameter 2arcsin(√p) when the data is generated from an overdispersed binomial distribution with intracluster correlation coefficient ρ ≤ 0.1
We have investigated bias arising in the estimation of transformed probabilities under the assumptions of random or mixed effects models, and its deleterious effects on inference in a meta-analysis
Summary
The main focus of this paper is the bias that arises as a result of transformations of random variables in random or mixed effects models and the deleterious effects of these biases on inference in group-level studies, such as ecological studies in epidemiology or cluster-randomized trials, and in meta-analyses. If the overdispersion is small and undetectable in the data, it may still severely affect the inference on transformed effects in a group-level study or in a meta-analysis. We illustrate our findings with the comparatively simple example of overdispersed binomial data, where overdispersion arises as a result of an intracluster correlation ρ between Bernoulli random variables in cluster-randomized trials or within studies in meta-analyses. We illustrate our general findings in examples of biases from arcsine and logit (log-odds) transformations in single studies and in meta-analysis, concentrating on small values of the ICC, viz. Additional material, including the methods for generating Bernoulli variables and detailed simulation results, are provided in Web Appendix
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