The metafor Package

A Meta-Analysis Package for R

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faq [2022/08/03 11:14] – Wolfgang Viechtbauerfaq [2022/09/25 11:23] – Wolfgang Viechtbauer
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 ??? For mixed-effects models, how is the $R^2$ statistic computed by the rma() function? ??? For mixed-effects models, how is the $R^2$ statistic computed by the rma() function?
  
-!!! The pseudo $R^2$ statistic (Raudenbush, 2009) is computed with $$R^2 = \frac{\hat{\tau}_{RE}^2 - \hat{\tau}_{ME}^2}{\hat{\tau}_{RE}^2},$$ where $\hat{\tau}_{RE}^2$ denotes the estimated value of $\tau^2$ based on the random-effects model (i.e., the total amount of heterogeneity) and $\hat{\tau}_{ME}^2$ denotes the estimated value of $\tau^2$ based on the mixed-effects model (i.e., the residual amount of heterogeneity). It can happen that $\hat{\tau}_{RE}^2 < \hat{\tau}_{ME}^2$, in which case $R^2$ is set to zero. Again, the value of $R^2$ will change depending on the estimator of $\tau^2$ used. Also note that this statistic is only computed when the mixed-effects model includes an intercept (so that the random-effects model is clearly nested within the mixed-effects model). You can also use the ''[[https://wviechtb.github.io/metafor/reference/anova.rma.html|anova()]]'' function to compute $R^2$ for any two models that are known to be nested.+!!! The pseudo $R^2$ statistic (Raudenbush, 2009) is computed with $$R^2 = \frac{\hat{\tau}_{RE}^2 - \hat{\tau}_{ME}^2}{\hat{\tau}_{RE}^2} = 1 - \frac{\hat{\tau}_{ME}^2}{\hat{\tau}_{RE}^2},$$ where $\hat{\tau}_{RE}^2$ denotes the estimated value of $\tau^2$ based on the random-effects model (i.e., the total amount of heterogeneity) and $\hat{\tau}_{ME}^2$ denotes the estimated value of $\tau^2$ based on the mixed-effects model (i.e., the residual amount of heterogeneity). It can happen that $\hat{\tau}_{RE}^2 < \hat{\tau}_{ME}^2$, in which case $R^2$ is set to zero. Again, the value of $R^2$ will change depending on the estimator of $\tau^2$ used. Also note that this statistic is only computed when the mixed-effects model includes an intercept (so that the random-effects model is clearly nested within the mixed-effects model). You can also use the ''[[https://wviechtb.github.io/metafor/reference/anova.rma.html|anova()]]'' function to compute $R^2$ for any two models that are known to be nested.
  
 ??? For random-effects models fitted with the rma() function, how is the prediction interval computed by the predict() function? ??? For random-effects models fitted with the rma() function, how is the prediction interval computed by the predict() function?
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 !!! By default, the interval is computed with $$\hat{\mu} \pm z_{1-\alpha/2} \sqrt{\mbox{SE}[\hat{\mu}]^2 + \hat{\tau}^2},$$ where $\hat{\mu}$ is the estimated average true outcome, $z_{1-\alpha/2}$ is the $100 \times (1-\alpha/2)$th percentile of a standard normal distribution (e.g., $1.96$ for $\alpha = .05$), $\mbox{SE}[\hat{\mu}]$ is the standard error of $\hat{\mu}$, and $\hat{\tau}^2$ is the estimated amount of heterogeneity (i.e., the variance in the true outcomes across studies). If the model was fitted with the Knapp and Hartung (2003) method (i.e., with ''test="knha"'' in ''[[https://wviechtb.github.io/metafor/reference/rma.uni.html|rma()]]''), then instead of $z_{1-\alpha/2}$, the $100 \times (1-\alpha/2)$th percentile of a t-distribution with $k-1$ degrees of freedom is used. !!! By default, the interval is computed with $$\hat{\mu} \pm z_{1-\alpha/2} \sqrt{\mbox{SE}[\hat{\mu}]^2 + \hat{\tau}^2},$$ where $\hat{\mu}$ is the estimated average true outcome, $z_{1-\alpha/2}$ is the $100 \times (1-\alpha/2)$th percentile of a standard normal distribution (e.g., $1.96$ for $\alpha = .05$), $\mbox{SE}[\hat{\mu}]$ is the standard error of $\hat{\mu}$, and $\hat{\tau}^2$ is the estimated amount of heterogeneity (i.e., the variance in the true outcomes across studies). If the model was fitted with the Knapp and Hartung (2003) method (i.e., with ''test="knha"'' in ''[[https://wviechtb.github.io/metafor/reference/rma.uni.html|rma()]]''), then instead of $z_{1-\alpha/2}$, the $100 \times (1-\alpha/2)$th percentile of a t-distribution with $k-1$ degrees of freedom is used.
  
-Note that this differs slightly from Riley et al. (2001), who suggest to use a t-distribution with $k-2$ degrees of freedom for constructing the interval. Neither a normal, nor a t-distribution with $k-1$ or $k-2$ degrees of freedom is correct; all of these are approximations. The computations in metafor are done in the way described above, so that the prediction interval is identical to the confidence interval for $\mu$ when $\hat{\tau}^2 = 0$, which could be argued is the logical thing that should happen. If the prediction interval should be computed exactly as described by Riley et al. (2001), one can use argument ''pi.type="riley"'' in ''predict()''.+Note that this differs slightly from Riley et al. (2011), who suggest to use a t-distribution with $k-2$ degrees of freedom for constructing the interval. Neither a normal, nor a t-distribution with $k-1$ or $k-2$ degrees of freedom is correct; all of these are approximations. The computations in metafor are done in the way described above, so that the prediction interval is identical to the confidence interval for $\mu$ when $\hat{\tau}^2 = 0$, which could be argued is the logical thing that should happen. If the prediction interval should be computed exactly as described by Riley et al. (2011), one can use argument ''pi.type="riley"'' in ''predict()''.
  
 ??? How is the Freeman-Tukey transformation of proportions and incidence rates computed? ??? How is the Freeman-Tukey transformation of proportions and incidence rates computed?
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 ==== References ==== ==== References ====
  
-Freeman, M. F., & Tukey, J. W. (1950). Transformations related to the angular and the square root. //Annals of Mathematical Statistics, 21//(4), 607--611.+Freeman, M. F., & Tukey, J. W. (1950). Transformations related to the angular and the square root. //Annals of Mathematical Statistics, 21//(4), 607--611. https://doi.org/10.1214/aoms/1177729756
  
-Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. //Statistics in Medicine, 21//(11), 1539--1558.+Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. //Statistics in Medicine, 21//(11), 1539--1558. https://doi.org/10.1002/sim.1186
  
-van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. //Statistics in Medicine, 21//(4), 589--624.+van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. //Statistics in Medicine, 21//(4), 589--624. https://doi.org/10.1002/sim.1040
  
 Lipsey, M. W., & Wilson, D. B. (2001). //Practical meta-Analysis.// Sage, Thousand Oaks, CA. Lipsey, M. W., & Wilson, D. B. (2001). //Practical meta-Analysis.// Sage, Thousand Oaks, CA.
  
 Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), //The handbook of research synthesis and meta-analysis// (2nd ed., pp. 295--315). New York: Russell Sage Foundation. Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), //The handbook of research synthesis and meta-analysis// (2nd ed., pp. 295--315). New York: Russell Sage Foundation.
 +
 +Riley, R. D., Higgins, J. P. T. & Deeks, J. J. (2011). Interpretation of random effects meta-analyses. //British Medical Journal, 342//, d549. https://doi.org/10.1136/bmj.d549 
  
 Sterne, J. A. C. (Ed.) (2009). //Meta-analysis in Stata: An updated collection from the Stata Journal.// Stata Press, College Station, TX. Sterne, J. A. C. (Ed.) (2009). //Meta-analysis in Stata: An updated collection from the Stata Journal.// Stata Press, College Station, TX.
  
faq.txt · Last modified: 2024/06/18 19:26 by Wolfgang Viechtbauer