metahet: Meta-Analysis Heterogeneity Measures
In altmeta: Alternative Meta-Analysis Methods
metahet R Documentation
Meta-Analysis Heterogeneity Measures
Description
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.
Usage
metahet(y, s2, data, n.resam = 1000)
Arguments
y
a numeric vector specifying the observed effect sizes in the collected studies; they are assumed to be normally distributed.
s2
a numeric vector specifying the within-study variances.
data
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.
n.resam
a positive integer specifying the number of resampling iterations for calculating p-values of test statistics and 95% confidence interval of heterogeneity measures.
Details
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}.
Value
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.Q
p-value of the Q statistic (using the resampling method).
p.Q.theo
p-value of the Q statistic using the Q's theoretical chi-squared distribution.
p.Qr
p-value of the Q_r statistic (using the resampling method).
p.Qm
p-value of the Q_m statistic (using the resampling method).
Q
the Q statistic.
ci.Q
95% CI of the Q statistic.
tau2.DL
DerSimonian–Laird estimate of the between-study variance.
ci.tau2.DL
95% CI of the between-study variance based on the DerSimonian–Laird method.
H
the H statistic.
ci.H
95% CI of the H statistic.
I2
the I^2 statistic.
ci.I2
95% CI of the I^2 statistic.
Qr
the Q_r statistic.
ci.Qr
95% CI of the Q_r statistic.
tau2.r
the between-study variance estimate based on the Q_r statistic.
ci.tau2.r
95% CI of the between-study variance based on the Q_r statistic.
Hr
the H_r statistic.
ci.Hr
95% CI of the H_r statistic.
Ir2
the I_r^2 statistic.
ci.Ir2
95% CI of the I_r^2 statistic.
Qm
the Q_m statistic.
ci.Qm
95% CI of the Q_m statistic.
tau2.m
the between-study variance estimate based on the Q_m statistic.
ci.tau2.m
95% CI of the between-study variance based on the Q_m statistic
Hm
the H_m statistic.
ci.Hm
95% CI of the H_m statistic.
Im2
the I_m^2 statistic.
ci.Im2
95% CI of the I_m^2 statistic.
References
DerSimonian R, Laird N (1986). "Meta-analysis in clinical trials." Controlled Clinical Trials, 7(3), 177–188. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/biom.12543")}>
See Also
metahet.hybrid
Examples
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)
altmeta documentation built on April 30, 2026, 9:08 a.m.
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| metahet | R Documentation |
Meta-Analysis Heterogeneity Measures
Description
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.
Usage
metahet(y, s2, data, n.resam = 1000)
Arguments
y |
a numeric vector specifying the observed effect sizes in the collected studies; they are assumed to be normally distributed. |
s2 |
a numeric vector specifying the within-study variances. |
data |
an optional data frame containing the meta-analysis dataset. If |
n.resam |
a positive integer specifying the number of resampling iterations for calculating p-values of test statistics and 95% confidence interval of heterogeneity measures. |
Details
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}.
Value
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.Q |
p-value of the |
p.Q.theo |
p-value of the |
p.Qr |
p-value of the |
p.Qm |
p-value of the |
Q |
the |
ci.Q |
95% CI of the |
tau2.DL |
DerSimonian–Laird estimate of the between-study variance. |
ci.tau2.DL |
95% CI of the between-study variance based on the DerSimonian–Laird method. |
H |
the |
ci.H |
95% CI of the |
I2 |
the |
ci.I2 |
95% CI of the |
Qr |
the |
ci.Qr |
95% CI of the |
tau2.r |
the between-study variance estimate based on the |
ci.tau2.r |
95% CI of the between-study variance based on the |
Hr |
the |
ci.Hr |
95% CI of the |
Ir2 |
the |
ci.Ir2 |
95% CI of the |
Qm |
the |
ci.Qm |
95% CI of the |
tau2.m |
the between-study variance estimate based on the |
ci.tau2.m |
95% CI of the between-study variance based on the |
Hm |
the |
ci.Hm |
95% CI of the |
Im2 |
the |
ci.Im2 |
95% CI of the |
References
DerSimonian R, Laird N (1986). "Meta-analysis in clinical trials." Controlled Clinical Trials, 7(3), 177–188. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{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. <\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/biom.12543")}>
See Also
metahet.hybrid
Examples
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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