Calculates various between-study heterogeneity measures in meta-analysis, including the conventional measures (e.g., \(I^2\)) and the alternative measures (e.g., \(I_r^2\)) which are robust to outlying studies; p-values of various tests are also calculated.
metahet(y, s2, data, n.resam = 1000)This function returns a list containing p-values of various heterogeneity tests and various heterogeneity measures with 95% confidence intervals. Specifically, the components include:
p-value of the \(Q\) statistic (using the resampling method).
p-value of the \(Q\) statistic using the \(Q\)'s theoretical chi-squared distribution.
p-value of the \(Q_r\) statistic (using the resampling method).
p-value of the \(Q_m\) statistic (using the resampling method).
the \(Q\) statistic.
95% CI of the \(Q\) statistic.
DerSimonian--Laird estimate of the between-study variance.
95% CI of the between-study variance based on the DerSimonian--Laird method.
the \(H\) statistic.
95% CI of the \(H\) statistic.
the \(I^2\) statistic.
95% CI of the \(I^2\) statistic.
the \(Q_r\) statistic.
95% CI of the \(Q_r\) statistic.
the between-study variance estimate based on the \(Q_r\) statistic.
95% CI of the between-study variance based on the \(Q_r\) statistic.
the \(H_r\) statistic.
95% CI of the \(H_r\) statistic.
the \(I_r^2\) statistic.
95% CI of the \(I_r^2\) statistic.
the \(Q_m\) statistic.
95% CI of the \(Q_m\) statistic.
the between-study variance estimate based on the \(Q_m\) statistic.
95% CI of the between-study variance based on the \(Q_m\) statistic
the \(H_m\) statistic.
95% CI of the \(H_m\) statistic.
the \(I_m^2\) statistic.
95% CI of the \(I_m^2\) statistic.
a numeric vector specifying the observed effect sizes in the collected studies; they are assumed to be normally distributed.
a numeric vector specifying the within-study variances.
an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, y and s2, should be specified as their corresponding column names in data.
a positive integer specifying the number of resampling iterations for calculating p-values of test statistics and 95% confidence interval of heterogeneity measures.
Suppose that a meta-analysis collects \(n\) studies. The observed effect size in study \(i\) is \(y_i\) and its within-study variance is \(s^{2}_{i}\). Also, the inverse-variance weight is \(w_i = 1 / s^{2}_{i}\). The fixed-effect estimate of overall effect size is \(\bar{\mu} = \sum_{i = 1}^{n} w_i y_i / \sum_{i = 1}^{n} w_i\). The conventional test statistic for heterogeneity is $$Q = \sum_{i = 1}^{n} w_i (y_{i} - \bar{\mu})^2.$$ Based on the \(Q\) statistic, the method-of-moments estimate of the between-study variance \(\hat{\tau}_{DL}^2\) is (DerSimonian and Laird, 1986) $$\hat{\tau}^2_{DL} = \max \left\{ 0, \frac{Q - (n - 1)}{\sum_{i = 1}^{n} w_{i} - \sum_{i = 1}^{n} w_{i}^{2} / \sum_{i = 1}^{n} w_{i}} \right\}.$$ Also, the \(H\) and \(I^2\) statistics (Higgins and Thompson, 2002; Higgins et al., 2003) are widely used in practice because they do not depend on the number of collected studies \(n\) and the effect size scale; these two statistics are defined as $$H = \sqrt{Q/(n - 1)};$$ $$I^{2} = \frac{Q - (n - 1)}{Q}.$$ Specifically, the \(H\) statistic reflects the ratio of the standard deviation of the underlying mean from a random-effects meta-analysis compared to the standard deviation from a fixed-effect meta-analysis; the \(I^2\) statistic describes the proportion of total variance across studies that is due to heterogeneity rather than sampling error.
Outliers are frequently present in meta-analyses, and they may have great impact on the above heterogeneity measures. Alternatively, to be more robust to outliers, the test statistic may be modified as (Lin et al., 2017): $$Q_{r} = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \bar{\mu}|.$$ Based on the \(Q_r\) statistic, the method-of-moments estimate of the between-study variance \(\hat{\tau}_r^2\) is defined as the solution to $$Q_r \sqrt{\frac{\pi}{2}} = \sum_{i = 1}^{n} \left\{1 - \frac{w_{i}}{\sum_{j = 1}^{n} w_{j}} + \tau^{2} \left[ w_{i} - \frac{2 w_{i}^{2}}{\sum_{j = 1}^{n} w_{j}} + \frac{w_{i} \sum_{j = 1}^{n} w_{j}^{2}}{(\sum_{j = 1}^{n} w_{j})^2} \right]\right\}.$$ If no positive solution exists to the equation above, set \(\hat{\tau}_{r}^{2} = 0\). The counterparts of the \(H\) and \(I^2\) statistics are defined as $$H_{r} = Q_r \sqrt{\pi/[2 n (n - 1)]};$$ $$I_{r}^{2} = \frac{Q_{r}^{2} - 2 n (n - 1) / \pi}{Q_{r}^{2}}.$$
To further improve the robustness of heterogeneity assessment, the weighted mean in the \(Q_r\) statistic may be replaced by the weighted median \(\hat{\mu}_m\), which is the solution to \(\sum_{i = 1}^{n} w_i [I (\theta \geq y_i) - 0.5] = 0\) with respect to \(\theta\). The new test statistic is $$Q_m = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \hat{\mu}_m|.$$ Based on \(Q_m\), the new estimator of the between-study variance \(\hat{\tau}_m^2\) is the solution to $$Q_m \sqrt{\pi/2} = \sum_{i = 1}^{n} \sqrt{(s_i^2 + \tau^2)/s_i^2}.$$ The counterparts of the \(H\) and \(I^2\) statistics are $$H_m = \frac{Q_m}{n} \sqrt{\pi/2};$$ $$I_m^2 = \frac{Q_m^2 - 2 n^2/\pi}{Q_m^2}.$$
DerSimonian R, Laird N (1986). "Meta-analysis in clinical trials." Controlled Clinical Trials, 7(3), 177--188. <tools:::Rd_expr_doi("10.1016/0197-2456(86)90046-2")>
Higgins JPT, Thompson SG (2002). "Quantifying heterogeneity in a meta-analysis." Statistics in Medicine, 21(11), 1539--1558. <tools:::Rd_expr_doi("10.1002/sim.1186")>
Higgins JPT, Thompson SG, Deeks JJ, Altman DG (2003). "Measuring inconsistency in meta-analyses." BMJ, 327(7414), 557--560. <tools:::Rd_expr_doi("10.1136/bmj.327.7414.557")>
Lin L, Chu H, Hodges JS (2017). "Alternative measures of between-study heterogeneity in meta-analysis: reducing the impact of outlying studies." Biometrics, 73(1), 156--166. <tools:::Rd_expr_doi("10.1111/biom.12543")>
data("dat.aex")
set.seed(1234)
metahet(y, s2, dat.aex, 100)
metahet(y, s2, dat.aex, 1000)
data("dat.hipfrac")
set.seed(1234)
metahet(y, s2, dat.hipfrac, 100)
metahet(y, s2, dat.hipfrac, 1000)
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