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analyses:konstantopoulos2011 [2020/05/01 14:01] – Wolfgang Viechtbaueranalyses:konstantopoulos2011 [2021/10/20 19:03] – Wolfgang Viechtbauer
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 We can fit the same model using a multivariate parameterization, which directly provides us with the ICC. For this, we use the syntax: We can fit the same model using a multivariate parameterization, which directly provides us with the ICC. For this, we use the syntax:
 <code rsplus> <code rsplus>
-res.mv <- rma.mv(yi, vi, random = ~ factor(study) | district, data=dat)+res.mv <- rma.mv(yi, vi, random = ~ study | district, data=dat)
 print(res.mv, digits=3) print(res.mv, digits=3)
 </code> </code>
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 Variance Components: Variance Components:
  
-outer factor: district      (nlvls = 11) +outer factor: district (nlvls = 11)                                                                                    
-inner factor: factor(study) (nlvls = 56)+inner factor: study    (nlvls = 56)
  
            estim   sqrt  fixed             estim   sqrt  fixed 
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 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 </code> </code>
-The ''random = ~ factor(study) | district'' argument adds correlated random effects for the different studies within districts to the model, where the variance-covariance matrix of the random effects takes on a compound symmetric structure (''struct="CS"'' is the default). Note that the estimate of $\rho$ that is obtained is exactly the same as the ICC value we computed earlier based on the multilevel model. Also, the estimate of $\tau^2$ obtained from the multivariate parameterization is the same as the total amount of heterogeneity computed earlier based on the multilevel model. Note that ''random = ~ factor(school) | district'' would again yield the same results.+The ''random = ~ study | district'' argument adds correlated random effects for the different studies within districts to the model, where the variance-covariance matrix of the random effects takes on a compound symmetric structure (''struct="CS"'' is the default). Note that the estimate of $\rho$ that is obtained is exactly the same as the ICC value we computed earlier based on the multilevel model. Also, the estimate of $\tau^2$ obtained from the multivariate parameterization is the same as the total amount of heterogeneity computed earlier based on the multilevel model. Note that ''random = ~ school | district'' would again yield the same results.
  
 As long as $\rho$ is estimated to be positive, the multilevel and multivariate parametrizations are in essence identical. In fact, the log likelihoods of the two models should be identical, which we can confirm with: As long as $\rho$ is estimated to be positive, the multilevel and multivariate parametrizations are in essence identical. In fact, the log likelihoods of the two models should be identical, which we can confirm with:
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 Finally, for illustration purposes, it is instructive to examine what can happen when we fit an overparameterized model. For example, suppose we combine the multilevel and multivariate structures above in a single model: Finally, for illustration purposes, it is instructive to examine what can happen when we fit an overparameterized model. For example, suppose we combine the multilevel and multivariate structures above in a single model:
 <code rsplus> <code rsplus>
-res.op <- rma.mv(yi, vi, random = list(~ factor(study) | district, ~ 1 | district, ~ 1 | study), data=dat)+res.op <- rma.mv(yi, vi, random = list(~ study | district, ~ 1 | district, ~ 1 | study), data=dat)
 print(res.op, digits=3) print(res.op, digits=3)
 </code> </code>
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 sigma^2.2  0.026  0.162     56     no     study  sigma^2.2  0.026  0.162     56     no     study 
  
-outer factor: district      (nlvls = 11) +outer factor: district (nlvls = 11) 
-inner factor: factor(study) (nlvls = 56)+inner factor: study    (nlvls = 56)
  
            estim   sqrt  fixed             estim   sqrt  fixed 
analyses/konstantopoulos2011.txt · Last modified: 2022/08/22 16:00 by Wolfgang Viechtbauer