metahet | R Documentation |
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)
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. |
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{μ} = ∑_{i = 1}^{n} w_i y_i / ∑_{i = 1}^{n} w_i. The conventional test statistic for heterogeneity is
Q = ∑_{i = 1}^{n} w_i (y_{i} - \bar{μ})^2.
Based on the Q statistic, the method-of-moments estimate of the between-study variance \hat{τ}_{DL}^2 is (DerSimonian and Laird, 1986)
\hat{τ}^2_{DL} = \max ≤ft\{ 0, \frac{Q - (n - 1)}{∑_{i = 1}^{n} w_{i} - ∑_{i = 1}^{n} w_{i}^{2} / ∑_{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 = √{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} = ∑_{i = 1}^{n} √{w_i} |y_{i} - \bar{μ}|.
Based on the Q_r statistic, the method-of-moments estimate of the between-study variance \hat{τ}_r^2 is defined as the solution to
Q_r √{\frac{π}{2}} = ∑_{i = 1}^{n} ≤ft\{1 - \frac{w_{i}}{∑_{j = 1}^{n} w_{j}} + τ^{2} ≤ft[ w_{i} - \frac{2 w_{i}^{2}}{∑_{j = 1}^{n} w_{j}} + \frac{w_{i} ∑_{j = 1}^{n} w_{j}^{2}}{(∑_{j = 1}^{n} w_{j})^2} \right]\right\}.
If no positive solution exists to the equation above, set \hat{τ}_{r}^{2} = 0. The counterparts of the H and I^2 statistics are defined as
H_{r} = Q_r √{π/[2 n (n - 1)]};
I_{r}^{2} = \frac{Q_{r}^{2} - 2 n (n - 1) / π}{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{μ}_m, which is the solution to ∑_{i = 1}^{n} w_i [I (θ ≥q y_i) - 0.5] = 0 with respect to θ. The new test statistic is
Q_m = ∑_{i = 1}^{n} √{w_i} |y_{i} - \hat{μ}_m|.
Based on Q_m, the new estimator of the between-study variance \hat{τ}_m^2 is the solution to
Q_m √{π/2} = ∑_{i = 1}^{n} √{(s_i^2 + τ^2)/s_i^2}.
The counterparts of the H and I^2 statistics are
H_m = \frac{Q_m}{n} √{π/2};
I_m^2 = \frac{Q_m^2 - 2 n^2/π}{Q_m^2}.
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. |
DerSimonian R, Laird N (1986). "Meta-analysis in clinical trials." Controlled Clinical Trials, 7(3), 177–188. <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. <doi: 10.1002/sim.1186>
Higgins JPT, Thompson SG, Deeks JJ, Altman DG (2003). "Measuring inconsistency in meta-analyses." BMJ, 327(7414), 557–560. <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. <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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