. 2018 Jan 9;9:98–107. doi: 10.1016/j.conctc.2017.11.012

Table 1.

Summary of heterogeneity estimators, including their equation, abbreviation and source.

Methods	Equation	Abbreviation	Source
DerSimonian Laird	${\hat{τ}}_{d l}^{2} = m a x (0, (Q_{F E} - (k - 1)) / c_{F E})$	dl	[15]
Positive DerSimonian Laird	${\hat{τ}}_{d l_{p}}^{2} = {\hat{τ}}_{d l}^{2}, {\hat{τ}}_{d l}^{2} > 0$ and ${\hat{τ}}_{d l}^{2} = 0.01, {\hat{τ}}_{d l}^{2} < = 0$	dlp	[17]
Two-step Der Simonian Laird	${\hat{τ}}_{d l 2}^{2} = m a x (0, Q_{R E} - (w_{i, R E}^{2} s_{i}^{2} - \frac{\sum_{i} w_{i, R E}^{2} s_{i}^{2}}{\sum_{i} w_{i, R E}}) / c_{R E})$	dl2	[16]
Hedges	${\hat{τ}}_{h e}^{2} = m a x (0, \frac{\sum_{i} {(Y_{i} - {\bar{Y}}_{F E})}^{2}}{k - 1} - \frac{\sum_{i} s_{i}^{2}}{k})$	he	[24]
Two step Hedges	Similar to DL2 using the Hedges estimator	he2	[16]
Positive Sidik-Jonkman	${\hat{τ}}_{s j}^{2} = m a x (\frac{\sum_{i} ({(Y_{i} - \bar{Y_{F E}})}^{2} / (r_{i} + 1))}{k - 1}, 0.01)$ , $r_{i} = s_{i}^{2} / {\hat{τ}}_{O}^{2}$	sj	[20]
Model error variance - vc	${\hat{τ}}_{m v v c}^{2} = \frac{\sum_{i} ({(Y_{i} - \bar{Y_{F E}})}^{2} / / r_{i}^{} + 1)}{k - 1}$ , $r_{i}^{} = s_{i}^{2} / {\hat{τ}}_{H E}^{2}$	mvvc	[20]
Paul-Mandel	$(τ_{p m}^{2})$ , $F (τ^{2}) = \sum_{i} w_{i, R E} {[Y_{i} - Y_{w} (τ^{2})]}^{2} - (k - 1)$	pm	[18]
Improved Paul-Mandel	$(τ_{i p m}^{2})$ , $F (τ^{2}) = \sum_{i} w_{i, R E}^{*} {[Y_{i} - Y_{w} (τ^{2})]}^{2} - (k - 1)$	Ipm	[19]
Hartung - Makambi	${\hat{τ}}_{h m}^{2} = \frac{Q_{F E}^{2}}{[2 (k - 1) + Q_{F E}] c_{F E}}$	hm	[22]
Hunter-Schmidt	${\hat{τ}}_{h s}^{2} = m a x (0, (Q_{F E} - k) / \sum_{i} w_{i, F E})$	hs	[23]
Maximum Likelihood	${\hat{τ}}_{m l}^{2} = m a x (0, \sum_{i} w_{i, R E}^{2} ({(Y_{i} - {\bar{Y}}_{M L})}^{2} - s_{i}^{2}) / \sum_{i} w_{i, R E}^{2})$	ml	–
Restricted Maximum likelihood	${\hat{τ}}_{r e m l}^{2} = m a x (0, \frac{\sum_{i} w_{i, R E}^{2} ({(Y_{i} - {\bar{Y}}_{M L})}^{2} - s_{i}^{2})}{\sum_{i} w_{i, R E}^{2}} + \frac{1}{\sum_{i} w_{i, R E}})$	reml	–
Rukhin Bayes zero estimator	${\hat{τ}}_{r b 0}^{2} = \frac{\sum_{i} {(Y_{i} - {\bar{Y}}_{F E})}^{2}}{k + 1} - \frac{\sum_{i} (n_{i} - k) (k - 1) \sum_{i} s_{i}^{2}}{k (k + 1) \sum_{i} (n_{i} - k + 2)}$	rb0	[21]
Rukhin Bayesian positive	${\hat{τ}}_{r b p}^{2} = \sum_{i} {(Y_{i} - {\bar{Y}}_{F E})}^{2} / (k + 1)$	rbp	[21]

$w_{i, R E} = \frac{1}{(s_{i}^{2} + τ^{2})}, w_{i, F E} = \frac{1}{s_{i}^{2}}$ , ${\bar{Y}}_{R E / F E} = \frac{\sum_{i} w_{i, R E / F E} Y_{i}}{\sum_{i} w_{i, R E / F E}}$ , $Q_{R E / F E} = \sum_{i} w_{i, R E / F E} {(Y_{i} - {\bar{Y}}_{R E / F E})}^{2}$ , $c_{R E / F E} = \sum_{i} w_{i, R E / F E} - \frac{\sum_{i} w_{i, R E / F E}^{2}}{\sum_{i} w_{i, R E / F E}}$ , $w_{i}^{*} = \frac{1}{(τ^{2} + v_{i, i p m}^{*})}, v_{i, i P M} = \frac{1}{n_{(T, i)} + 1} (e^{- P r c_{O} - \tilde{Y} + τ^{2} / 2} + 2 + e^{P r c_{O} + \tilde{Y} + τ^{2} / 2}) + \frac{1}{n_{(C i)} + 1} (e^{- P r c_{O}} + 2 + e^{P r c_{O}})$ $P r_{c, o}$ : Observed control event rate, $τ_{O}^{2} = \sum_{i} (Y_{i} - \bar{Y_{F E}}) / k$ . The pm, ipm, ml and reml are iterative estimators.