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tips:i2_multilevel_multivariate

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tips:i2_multilevel_multivariate [2021/01/06 10:58] – [General Equation for I^2] Wolfgang Viechtbauertips:i2_multilevel_multivariate [2022/03/18 09:48] – Wolfgang Viechtbauer
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-===== 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. 
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 <code rsplus> <code rsplus>
 k <- res$k k <- res$k
-wi <- 1/dat$vi+wi <- 1/res$vi
 vt <- (k-1) * sum(wi) / (sum(wi)^2 - sum(wi^2)) vt <- (k-1) * sum(wi) / (sum(wi)^2 - sum(wi^2))
 100 * res$tau2 / (res$tau2 + vt) 100 * res$tau2 / (res$tau2 + vt)
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 </code> </code>
  
-==== General Equation for $\boldsymbol{I}^2$ ====+==== General Equation for $\boldsymbol{I^2}$ ====
  
 Before we continue with more complex models, it is useful to point out a more general equation for computing $I^2,$ which also applies to models involving moderator variables (i.e., mixed-effects meta-regression models). This will also become important when dealing with models where sampling errors are no longer independent. So, let us define $$\mathbf{P} = \mathbf{W} - \mathbf{W} \mathbf{X} (\mathbf{X}' \mathbf{W} \mathbf{X})^{-1} \mathbf{X}' \mathbf{W},$$ where $\mathbf{W}$ is (for now) a diagonal matrix with the inverse sampling variances (i.e., $1/v_i$) along the diagonal and $\mathbf{X}$ is the model matrix. In the random-effects model, $\mathbf{X}$ is just a column vector with 1's, but in meta-regression models, it will contain additional columns with the values of the moderator variables. Then we define $$I^2 = 100\% \times \frac{\hat{\tau}^2}{\hat{\tau}^2 + \frac{k-p}{\mathrm{tr}[\mathbf{P}]}},$$ where $\mathrm{tr}[\mathbf{P}]$ denotes the trace of the $\mathbf{P}$ matrix (i.e., the sum of the diagonal elements) and $p$ denotes the number of columns in $\mathbf{X}$. Before we continue with more complex models, it is useful to point out a more general equation for computing $I^2,$ which also applies to models involving moderator variables (i.e., mixed-effects meta-regression models). This will also become important when dealing with models where sampling errors are no longer independent. So, let us define $$\mathbf{P} = \mathbf{W} - \mathbf{W} \mathbf{X} (\mathbf{X}' \mathbf{W} \mathbf{X})^{-1} \mathbf{X}' \mathbf{W},$$ where $\mathbf{W}$ is (for now) a diagonal matrix with the inverse sampling variances (i.e., $1/v_i$) along the diagonal and $\mathbf{X}$ is the model matrix. In the random-effects model, $\mathbf{X}$ is just a column vector with 1's, but in meta-regression models, it will contain additional columns with the values of the moderator variables. Then we define $$I^2 = 100\% \times \frac{\hat{\tau}^2}{\hat{\tau}^2 + \frac{k-p}{\mathrm{tr}[\mathbf{P}]}},$$ where $\mathrm{tr}[\mathbf{P}]$ denotes the trace of the $\mathbf{P}$ matrix (i.e., the sum of the diagonal elements) and $p$ denotes the number of columns in $\mathbf{X}$.
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 Let's try this out for the example above: Let's try this out for the example above:
 <code rsplus> <code rsplus>
-W <- diag(1/dat$vi)+W <- diag(1/res$vi)
 X <- model.matrix(res) X <- model.matrix(res)
 P <- W - W %*% X %*% solve(t(X) %*% W %*% X) %*% t(X) %*% W P <- W - W %*% X %*% solve(t(X) %*% W %*% X) %*% t(X) %*% W
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 Based on the discussion above, it is now very easy to generalize the concept of $I^2$ to such a model (see also Nakagawa & Santos, 2012). That is, we can first compute: Based on the discussion above, it is now very easy to generalize the concept of $I^2$ to such a model (see also Nakagawa & Santos, 2012). That is, we can first compute:
 <code rsplus> <code rsplus>
-W <- diag(1/dat$vi)+W <- diag(1/res$vi)
 X <- model.matrix(res) X <- model.matrix(res)
 P <- W - W %*% X %*% solve(t(X) %*% W %*% X) %*% t(X) %*% W P <- W - W %*% X %*% solve(t(X) %*% W %*% X) %*% t(X) %*% W
tips/i2_multilevel_multivariate.txt · Last modified: 2022/10/24 10:10 by Wolfgang Viechtbauer