metahet {altmeta}R Documentation

Meta-Analysis Heterogeneity Measures

Description

Calculates various between-study heterogeneity measures in meta-analysis, including the conventional measures (e.g., I2I^2) and the alternative measures (e.g., Ir2I_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 nn studies. The observed effect size in study ii is yiy_i and its within-study variance is si2s^{2}_{i}. Also, the inverse-variance weight is wi=1/si2w_i = 1 / s^{2}_{i}. The fixed-effect estimate of overall effect size is μˉ=i=1nwiyi/i=1nwi\bar{\mu} = \sum_{i = 1}^{n} w_i y_i / \sum_{i = 1}^{n} w_i. The conventional test statistic for heterogeneity is

Q=i=1nwi(yiμˉ)2.Q = \sum_{i = 1}^{n} w_i (y_{i} - \bar{\mu})^2.

Based on the QQ statistic, the method-of-moments estimate of the between-study variance τ^DL2\hat{\tau}_{DL}^2 is (DerSimonian and Laird, 1986)

τ^DL2=max{0,Q(n1)i=1nwii=1nwi2/i=1nwi}.\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 HH and I2I^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 nn and the effect size scale; these two statistics are defined as

H=Q/(n1);H = \sqrt{Q/(n - 1)};

I2=Q(n1)Q.I^{2} = \frac{Q - (n - 1)}{Q}.

Specifically, the HH 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 I2I^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):

Qr=i=1nwiyiμˉ.Q_{r} = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \bar{\mu}|.

Based on the QrQ_r statistic, the method-of-moments estimate of the between-study variance τ^r2\hat{\tau}_r^2 is defined as the solution to

Qrπ2=i=1n{1wij=1nwj+τ2[wi2wi2j=1nwj+wij=1nwj2(j=1nwj)2]}.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 τ^r2=0\hat{\tau}_{r}^{2} = 0. The counterparts of the HH and I2I^2 statistics are defined as

Hr=Qrπ/[2n(n1)];H_{r} = Q_r \sqrt{\pi/[2 n (n - 1)]};

Ir2=Qr22n(n1)/πQr2.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 QrQ_r statistic may be replaced by the weighted median μ^m\hat{\mu}_m, which is the solution to i=1nwi[I(θyi)0.5]=0\sum_{i = 1}^{n} w_i [I (\theta \geq y_i) - 0.5] = 0 with respect to θ\theta. The new test statistic is

Qm=i=1nwiyiμ^m.Q_m = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \hat{\mu}_m|.

Based on QmQ_m, the new estimator of the between-study variance τ^m2\hat{\tau}_m^2 is the solution to

Qmπ/2=i=1n(si2+τ2)/si2.Q_m \sqrt{\pi/2} = \sum_{i = 1}^{n} \sqrt{(s_i^2 + \tau^2)/s_i^2}.

The counterparts of the HH and I2I^2 statistics are

Hm=Qmnπ/2;H_m = \frac{Q_m}{n} \sqrt{\pi/2};

Im2=Qm22n2/πQm2.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 QQ statistic (using the resampling method).

p.Q.theo

p-value of the QQ statistic using the QQ's theoretical chi-squared distribution.

p.Qr

p-value of the QrQ_r statistic (using the resampling method).

p.Qm

p-value of the QmQ_m statistic (using the resampling method).

Q

the QQ statistic.

ci.Q

95% CI of the QQ 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 HH statistic.

ci.H

95% CI of the HH statistic.

I2

the I2I^2 statistic.

ci.I2

95% CI of the I2I^2 statistic.

Qr

the QrQ_r statistic.

ci.Qr

95% CI of the QrQ_r statistic.

tau2.r

the between-study variance estimate based on the QrQ_r statistic.

ci.tau2.r

95% CI of the between-study variance based on the QrQ_r statistic.

Hr

the HrH_r statistic.

ci.Hr

95% CI of the HrH_r statistic.

Ir2

the Ir2I_r^2 statistic.

ci.Ir2

95% CI of the Ir2I_r^2 statistic.

Qm

the QmQ_m statistic.

ci.Qm

95% CI of the QmQ_m statistic.

tau2.m

the between-study variance estimate based on the QmQ_m statistic.

ci.tau2.m

95% CI of the between-study variance based on the QmQ_m statistic

Hm

the HmH_m statistic.

ci.Hm

95% CI of the HmH_m statistic.

Im2

the Im2I_m^2 statistic.

ci.Im2

95% CI of the Im2I_m^2 statistic.

References

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>

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)

[Package altmeta version 4.1 Index]