analyses:raudenbush1985
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analyses:raudenbush1985 [2019/05/05 14:05] – external edit 127.0.0.1 | analyses:raudenbush1985 [2022/03/26 15:02] – Wolfgang Viechtbauer | ||
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==== The Methods and Data ==== | ==== The Methods and Data ==== | ||
- | Raudenbush and Bryk (1985) describe the meta-analytic random- and mixed-effects models and describe restricted maximum-likelihood estimation for the amount of (residual) heterogeneity (p. 80-82).((Raudenbush and Bryk (1985) do not explicitly mention that they are using restricted maximum-likelihood estimation, but equation (27) in their article corresponds to the restricted log likelihood. So don't let the title of the paper confuse you. They do not use the empirical Bayes estimator for the amount of (residual) heterogeneity (e.g., as used by [[analyses: | + | Raudenbush and Bryk (1985) describe the meta-analytic random- and mixed-effects models and describe restricted maximum likelihood estimation for the amount of (residual) heterogeneity (p. 80-82).((Raudenbush and Bryk (1985) do not explicitly mention that they are using restricted maximum likelihood estimation, but equation (27) in their article corresponds to the restricted log likelihood. So don't let the title of the paper confuse you. They do not use the empirical Bayes estimator for the amount of (residual) heterogeneity (e.g., as used by [[analyses: |
<code rsplus> | <code rsplus> | ||
library(metafor) | library(metafor) | ||
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H^2 (total variability / sampling variability): | H^2 (total variability / sampling variability): | ||
- | Test for Heterogeneity: | + | Test for Heterogeneity: |
Q(df = 18) = 35.830, p-val = 0.007 | Q(df = 18) = 35.830, p-val = 0.007 | ||
Model Results: | Model Results: | ||
- | estimate | + | estimate |
- | | + | |
--- | --- | ||
- | Signif. codes: | + | Signif. codes: |
</ | </ | ||
REML estimation is the default for the '' | REML estimation is the default for the '' | ||
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R^2 (amount of heterogeneity accounted for): 100.00% | R^2 (amount of heterogeneity accounted for): 100.00% | ||
- | Test for Residual Heterogeneity: | + | Test for Residual Heterogeneity: |
QE(df = 17) = 16.571, p-val = 0.484 | QE(df = 17) = 16.571, p-val = 0.484 | ||
- | Test of Moderators (coefficient(s) 2): | + | Test of Moderators (coefficient 2): |
QM(df = 1) = 19.258, p-val < .001 | QM(df = 1) = 19.258, p-val < .001 | ||
Model Results: | Model Results: | ||
- | | + | |
intrcpt | intrcpt | ||
weeks.c | weeks.c | ||
--- | --- | ||
- | Signif. codes: | + | Signif. codes: |
</ | </ | ||
These results again match the findings from Raudenbush and Bryk (1985). The residual amount of heterogeneity is now $\hat{\tau}^2 \approx 0$ (p. 90) and the test statistic for $H_0: \tau^2 = 0$ is $Q_E(df=17) = 16.57$ (p. 90). The estimated model is $E(d_i) = .407 - 0.157 x_i$, where $x_i$ is the number of prior contact weeks (p. 90). The standard errors of the model coefficients are $SE[b_0] = .087$ and $SE[b_1] = .036$, so that the test statistics are $z_0 = 4.68$ and $z_1 = -4.39$, respectively (p. 92). | These results again match the findings from Raudenbush and Bryk (1985). The residual amount of heterogeneity is now $\hat{\tau}^2 \approx 0$ (p. 90) and the test statistic for $H_0: \tau^2 = 0$ is $Q_E(df=17) = 16.57$ (p. 90). The estimated model is $E(d_i) = .407 - 0.157 x_i$, where $x_i$ is the number of prior contact weeks (p. 90). The standard errors of the model coefficients are $SE[b_0] = .087$ and $SE[b_1] = .036$, so that the test statistics are $z_0 = 4.68$ and $z_1 = -4.39$, respectively (p. 92). | ||
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Finally, we can draw a scatterplot of the observed standardized mean differences as a function of the weeks variable (with the radius of the points drawn proportional to the inverse standard errors and hence the area of the points drawn proportional to the inverse sampling variances) with: | Finally, we can draw a scatterplot of the observed standardized mean differences as a function of the weeks variable (with the radius of the points drawn proportional to the inverse standard errors and hence the area of the points drawn proportional to the inverse sampling variances) with: | ||
<code rsplus> | <code rsplus> | ||
- | preds <- predict(res, newmods=seq(0, | + | regplot(res, xlab="Weeks of Prior Contact" |
- | + | ||
- | size <- 1 / sqrt(dat$vi) | + | |
- | size <- 0.05 * size / max(size) | + | |
- | + | ||
- | plot(NA, NA, xlim=c(0, | + | |
- | xlab="Weeks of Prior Contact" | + | |
axis(side=1, | axis(side=1, | ||
- | |||
- | symbols(dat$weeks.c, | ||
- | |||
- | lines(seq(0, | ||
- | lines(seq(0, | ||
- | lines(seq(0, | ||
- | |||
- | abline(h=0, lty=" | ||
</ | </ | ||
The predicted (average) effect as a function of the weeks of prior contact (with 95% CI bounds) is also added to the plot. The resulting plot is shown below. | The predicted (average) effect as a function of the weeks of prior contact (with 95% CI bounds) is also added to the plot. The resulting plot is shown below. |
analyses/raudenbush1985.txt · Last modified: 2022/08/03 11:20 by Wolfgang Viechtbauer