analyses:berkey1998
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analyses:berkey1998 [2022/08/03 16:52] – Wolfgang Viechtbauer | analyses:berkey1998 [2023/06/22 11:42] (current) – Wolfgang Viechtbauer | ||
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Berkey et al. (1998) describe a meta-analytic multivariate model for the analysis of multiple correlated outcomes. The use of the model is illustrated with results from 5 trials comparing surgical and non-surgical treatments for medium-severity periodontal disease. Reported outcomes include the change in probing depth (PD) and attachment level (AL) one year after the treatment. The effect size measure used for this meta-analysis was the (raw) mean difference, calculated in such a way that positive values indicate that surgery was more effective than non-surgical treatment in decreasing the probing depth and increasing the attachment level. The data are provided in Table I in the article and are stored in the dataset '' | Berkey et al. (1998) describe a meta-analytic multivariate model for the analysis of multiple correlated outcomes. The use of the model is illustrated with results from 5 trials comparing surgical and non-surgical treatments for medium-severity periodontal disease. Reported outcomes include the change in probing depth (PD) and attachment level (AL) one year after the treatment. The effect size measure used for this meta-analysis was the (raw) mean difference, calculated in such a way that positive values indicate that surgery was more effective than non-surgical treatment in decreasing the probing depth and increasing the attachment level. The data are provided in Table I in the article and are stored in the dataset '' | ||
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<code rsplus> | <code rsplus> | ||
library(metafor) | library(metafor) | ||
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dat | dat | ||
</ | </ | ||
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(I copy the dataset into '' | (I copy the dataset into '' | ||
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<code output> | <code output> | ||
| | ||
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10 | 10 | ||
</ | </ | ||
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So, the results from the various trials indicate that surgery is preferable for reducing the probing depth, while non-surgical treatment is preferable for increasing the attachment level. | So, the results from the various trials indicate that surgery is preferable for reducing the probing depth, while non-surgical treatment is preferable for increasing the attachment level. | ||
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Before we can proceed with the model fitting, we need to construct the full (block-diagonal) variance-covariance for all studies from these two variables. We can do this using the '' | Before we can proceed with the model fitting, we need to construct the full (block-diagonal) variance-covariance for all studies from these two variables. We can do this using the '' | ||
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<code rsplus> | <code rsplus> | ||
V <- vcalc(vi=1, cluster=author, | V <- vcalc(vi=1, cluster=author, | ||
</ | </ | ||
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The '' | The '' | ||
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<code output> | <code output> | ||
[,1] | [,1] | ||
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==== Multivariate Random-Effects Model ==== | ==== Multivariate Random-Effects Model ==== | ||
- | A multivariate random-effects model can now be used to meta-analyze the two outcomes simultaneously. | + | A multivariate random-effects model can now be used to meta-analyze the two outcomes simultaneously. |
+ | | ||
+ | | ||
+ | \end{array} \right] | ||
+ | = | ||
+ | \left[ \begin{array}{c} | ||
+ | | ||
+ | | ||
+ | \end{array} \right] | ||
+ | + | ||
+ | \left[ \begin{array}{c} | ||
+ | | ||
+ | | ||
+ | \end{array} \right] | ||
+ | + | ||
+ | \left[ \begin{array}{c} | ||
+ | | ||
+ | | ||
+ | \end{array} \right],$$ where $$G = \mbox{Var} | ||
+ | \left[ \begin{array}{c} | ||
+ | | ||
+ | | ||
+ | \end{array} \right] | ||
+ | = | ||
+ | \left[ \begin{array}{cc} | ||
+ | | ||
+ | \rho \tau_{PD} \tau_{AL} & \tau^2_{AL} | ||
+ | \end{array} \right]$$ and $$V_i = \mbox{Var} | ||
+ | \left[ \begin{array}{c} | ||
+ | | ||
+ | | ||
+ | \end{array} \right] | ||
+ | = | ||
+ | \left[ \begin{array}{cc} | ||
+ | | ||
+ | | ||
+ | \end{array} \right],$$ where the elements in this second variance-covariance matrix are given by the $2 \times 2$ blocks along the diagonal in the '' | ||
+ | V_1 & & \\ | ||
+ | & \ddots & \\ | ||
+ | & | ||
+ | \end{array} \right].$$ | ||
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+ | We can fit this model with: | ||
<code rsplus> | <code rsplus> | ||
res <- rma.mv(yi, V, mods = ~ outcome - 1, | res <- rma.mv(yi, V, mods = ~ outcome - 1, | ||
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Signif. codes: | Signif. codes: | ||
</ | </ | ||
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This is what Berkey et al. (1998) call a multivariate maximum likelihood (MML) random-effects model.((Berkey et al. (1998) claim that this approach can only be used when all the trials in the meta-analysis provide results for all the outcomes considered. This does not apply to the way '' | This is what Berkey et al. (1998) call a multivariate maximum likelihood (MML) random-effects model.((Berkey et al. (1998) claim that this approach can only be used when all the trials in the meta-analysis provide results for all the outcomes considered. This does not apply to the way '' | ||
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The results given in Table II in the paper actually are based on a meta-regression model, using year of publication as a potential moderator. To replicate those analyses, we use: | The results given in Table II in the paper actually are based on a meta-regression model, using year of publication as a potential moderator. To replicate those analyses, we use: | ||
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<code rsplus> | <code rsplus> | ||
res <- rma.mv(yi, V, mods = ~ outcome + outcome: | res <- rma.mv(yi, V, mods = ~ outcome + outcome: | ||
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Signif. codes: | Signif. codes: | ||
</ | </ | ||
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Note that publication year was centered at 1983, as was done by the authors. These results correspond to those given in the rightmost column in Table II on page 2545 (column " | Note that publication year was centered at 1983, as was done by the authors. These results correspond to those given in the rightmost column in Table II on page 2545 (column " | ||
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<code rsplus> | <code rsplus> | ||
round(res$G[1, | round(res$G[1, | ||
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To test whether the slope of publication year actually differs for the two outcomes, we can fit the same model with: | To test whether the slope of publication year actually differs for the two outcomes, we can fit the same model with: | ||
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<code rsplus> | <code rsplus> | ||
res <- rma.mv(yi, V, mods = ~ outcome*I(year - 1983) - 1, | res <- rma.mv(yi, V, mods = ~ outcome*I(year - 1983) - 1, | ||
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print(res, digits=3) | print(res, digits=3) | ||
</ | </ | ||
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The output is identical, except for the last part, which is now equal to: | The output is identical, except for the last part, which is now equal to: | ||
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<code output> | <code output> | ||
Model Results: | Model Results: | ||
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Signif. codes: | Signif. codes: | ||
</ | </ | ||
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Therefore, the slope is actually not significantly different for the two outcomes ($p = .553$). In fact, it does not appear as if publication year is at all related to the two outcomes. | Therefore, the slope is actually not significantly different for the two outcomes ($p = .553$). In fact, it does not appear as if publication year is at all related to the two outcomes. | ||
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One could actually consider a simpler model for these data, which assumes a compound symmetry structure for the random effects (this would imply that the amount of heterogeneity is the same for the two outcomes). A formal comparison of the two models can be conducted using a likelihood ratio test: | One could actually consider a simpler model for these data, which assumes a compound symmetry structure for the random effects (this would imply that the amount of heterogeneity is the same for the two outcomes). A formal comparison of the two models can be conducted using a likelihood ratio test: | ||
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<code rsplus> | <code rsplus> | ||
res1 <- rma.mv(yi, V, mods = ~ outcome - 1, | res1 <- rma.mv(yi, V, mods = ~ outcome - 1, | ||
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Reduced | Reduced | ||
</ | </ | ||
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Since model '' | Since model '' | ||
analyses/berkey1998.txt · Last modified: 2023/06/22 11:42 by Wolfgang Viechtbauer