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tips:i2_multilevel_multivariate [2022/03/18 09:48] – Wolfgang Viechtbauertips:i2_multilevel_multivariate [2022/10/24 10:10] (current) – Wolfgang Viechtbauer
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 ===== $\boldsymbol{I^2}$ for Multilevel and Multivariate Models ===== ===== $\boldsymbol{I^2}$ for Multilevel and Multivariate Models =====
  
-The $I^2$ statistic was introduced by Higgins and Thompson in their seminal 2002 paper and has become a rather popular statistic to report in meta-analyses, as it facilitates the interpretation of the amount of heterogeneity present in a given dataset. +The $I^2$ statistic was introduced by Higgins and Thompson in their seminal 2002 paper and has become a rather popular statistic to report in meta-analyses, as it facilitates the interpretation of the amount of heterogeneity present in a given dataset.
  
 For a standard random-effects models, the $I^2$ statistic is computed with $$I^2 = 100\% \times \frac{\hat{\tau}^2}{\hat{\tau}^2 + \tilde{v}},$$ where $\hat{\tau}^2$ is the estimated value of $\tau^2$ and $$\tilde{v} = \frac{(k-1) \sum w_i}{(\sum w_i)^2 - \sum w_i^2},$$ where $w_i = 1/v_i$ is the inverse of the sampling variance of the $i^{th}$ study. The equation for $\tilde{v}$ is equation 9 in Higgins & Thompson (2002) and can be regarded as the 'typical' within-study (or sampling) variance of the observed effect sizes or outcomes.((There are also other suggestions in the literature for defining the 'typical' or 'average' sampling variance of the estimates. For example, Takkouche et al. (1999, 2013) suggest to use the harmonic mean of the sampling variances in an analogous computation of an $I^2$-type statistic, but this never really caught on in practice.)) For a standard random-effects models, the $I^2$ statistic is computed with $$I^2 = 100\% \times \frac{\hat{\tau}^2}{\hat{\tau}^2 + \tilde{v}},$$ where $\hat{\tau}^2$ is the estimated value of $\tau^2$ and $$\tilde{v} = \frac{(k-1) \sum w_i}{(\sum w_i)^2 - \sum w_i^2},$$ where $w_i = 1/v_i$ is the inverse of the sampling variance of the $i^{th}$ study. The equation for $\tilde{v}$ is equation 9 in Higgins & Thompson (2002) and can be regarded as the 'typical' within-study (or sampling) variance of the observed effect sizes or outcomes.((There are also other suggestions in the literature for defining the 'typical' or 'average' sampling variance of the estimates. For example, Takkouche et al. (1999, 2013) suggest to use the harmonic mean of the sampling variances in an analogous computation of an $I^2$-type statistic, but this never really caught on in practice.))
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 Multivariate Meta-Analysis Model (k = 56; method: REML) Multivariate Meta-Analysis Model (k = 56; method: REML)
  
-Variance Components: +Variance Components:
  
             estim    sqrt  nlvls  fixed           factor             estim    sqrt  nlvls  fixed           factor
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 sigma^2.2  0.0327  0.1809     56     no  district/school sigma^2.2  0.0327  0.1809     56     no  district/school
  
-Test for Heterogeneity: +Test for Heterogeneity:
 Q(df = 55) = 578.8640, p-val < .0001 Q(df = 55) = 578.8640, p-val < .0001
  
 Model Results: Model Results:
  
-estimate       se     zval     pval    ci.lb    ci.ub           +estimate       se     zval     pval    ci.lb    ci.ub 
-  0.1847   0.0846   2.1845   0.0289   0.0190   0.3504        * +  0.1847   0.0846   2.1845   0.0289   0.0190   0.3504        *
  
 --- ---
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 [1] 63.32484 31.86248 [1] 63.32484 31.86248
 </code> </code>
-Therefore, about 63% of the total variance is estimated to be due to between-cluster heterogeneity, with the remaining 32% due to within-cluster heterogeneity. And the remaining 5% are sampling variance.+Therefore, about 63% of the total variance is estimated to be due to between-cluster heterogeneity, 32% due to within-cluster heterogeneity, and the remaining 5% due to sampling variance.
  
 ==== Multivariate Models ==== ==== Multivariate Models ====
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 Multivariate Meta-Analysis Model (k = 10; method: REML) Multivariate Meta-Analysis Model (k = 10; method: REML)
  
-Variance Components: +Variance Components:
  
 outer factor: trial   (nlvls = 5) outer factor: trial   (nlvls = 5)
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 PD  0.6088       1     5   - PD  0.6088       1     5   -
  
-Test for Residual Heterogeneity: +Test for Residual Heterogeneity:
 QE(df = 8) = 128.2267, p-val < .0001 QE(df = 8) = 128.2267, p-val < .0001
  
-Test of Moderators (coefficient(s) 1,2): +Test of Moderators (coefficient(s) 1,2):
 QM(df = 2) = 108.8616, p-val < .0001 QM(df = 2) = 108.8616, p-val < .0001
  
 Model Results: Model Results:
  
-           estimate      se     zval    pval    ci.lb    ci.ub     +           estimate      se     zval    pval    ci.lb    ci.ub
 outcomeAL   -0.3392  0.0879  -3.8589  0.0001  -0.5115  -0.1669  *** outcomeAL   -0.3392  0.0879  -3.8589  0.0001  -0.5115  -0.1669  ***
 outcomePD    0.3534  0.0588   6.0057  <.0001   0.2381   0.4688  *** outcomePD    0.3534  0.0588   6.0057  <.0001   0.2381   0.4688  ***
tips/i2_multilevel_multivariate.txt · Last modified: 2022/10/24 10:10 by Wolfgang Viechtbauer