We have no definitive explanation for the approximately constant proportion bias of -10 percent, which is observed across all proportions when heterogeneity is large, and the true summary proportion is different than 0.50.
 A plausible conjecture is based on the observation that the random effects discrete likelihood methods assume a logit-normal distribution of the true proportions across studies. A range of e.g., 0.10 in the proportion scale corresponds to vastly different ranges in the logit scale, depending on its location. For example the interval [0.01, 0.11] in the proportion scale corresponds to a length of [-4.60, -2.09] in the logit scale, but the interval [0.45, 0.55] corresponds to the much narrower [-0.20, 0.20] in the logit scale. The further away a given interval is located from 0.50 in the proportion scale, the more it expands in the logit scale. “Averaging” in the logit scale will therefore result in a negative bias in the proportion scale when the true mean is less than 0.50; no bias if the true mean is 0.50, and (by symmetry) a positive bias if the true mean is more than 0.50.
 The finding that for large expected counts (scenarios L4 and L5) the logit-transformed bias approaches that of the discrete likelihood method supports this conjecture.

From: Results

Cover of Simulation-Based Comparison of Methods for Meta-Analysis of Proportions and Rates
Simulation-Based Comparison of Methods for Meta-Analysis of Proportions and Rates [Internet].
Trikalinos TA, Trow P, Schmid CH.

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