  Random-effects meta-analysis requires an estimate of the between-study
variance, $\tau^2$. We study methods of estimation of $\tau^2$ and its
confidence interval in meta-analysis of odds ratio, and also the performance of
related estimators of the overall effect. We provide results of extensive
simulations on five point estimators of $\tau^2$ (the popular methods of
DerSimonian-Laird, restricted maximum likelihood, and Mandel and Paule; the
less-familiar method of Jackson; and the new method (KD) based on the improved
approximation to the distribution of the Q statistic by Kulinskaya and
Dollinger (2015)); five interval estimators for $\tau^2$ (profile likelihood,
Q-profile, Biggerstaff and Jackson, Jackson, and KD), six point estimators of
the overall effect (the five inverse-variance estimators related to the point
estimators of $\tau^2$ and an estimator (SSW) whose weights use only
study-level sample sizes), and eight interval estimators for the overall effect
(five based on the point estimators for $\tau^2$; the
Hartung-Knapp-Sidik-Jonkman (HKSJ) interval; a KD-based modification of HKSJ;
and an interval based on the sample-size-weighted estimator). Results of our
simulations show that none of the point estimators of $\tau^2$ can be
recommended, however the new KD estimator provides a reliable coverage of
$\tau^2$. Inverse-variance estimators of the overall effect are substantially
biased. The SSW estimator of the overall effect and the related confidence
interval provide the reliable point and interval estimation of log-odds-ratio.
