8 Meta-regression
# fit mixed-effects meta-regression model with percent female as predictor
res <- rma(yi, vi, mods = ~ Perc_Female, data=dat)
res##
## Mixed-Effects Model (k = 51; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity): 0.0175 (SE = 0.0050)
## tau (square root of estimated tau^2 value): 0.1325
## I^2 (residual heterogeneity / unaccounted variability): 77.84%
## H^2 (unaccounted variability / sampling variability): 4.51
## R^2 (amount of heterogeneity accounted for): 2.04%
##
## Test for Residual Heterogeneity:
## QE(df = 49) = 185.0467, p-val < .0001
##
## Test of Moderators (coefficient 2):
## QM(df = 1) = 2.2554, p-val = 0.1331
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## intrcpt 0.1995 0.0713 2.7997 0.0051 0.0598 0.3391 **
## Perc_Female -0.0017 0.0011 -1.5018 0.1331 -0.0040 0.0005
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# use percentage of females and year as predictors
res <- rma(yi, vi, mods = ~ Perc_Female, data=dat)
res##
## Mixed-Effects Model (k = 51; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity): 0.0175 (SE = 0.0050)
## tau (square root of estimated tau^2 value): 0.1325
## I^2 (residual heterogeneity / unaccounted variability): 77.84%
## H^2 (unaccounted variability / sampling variability): 4.51
## R^2 (amount of heterogeneity accounted for): 2.04%
##
## Test for Residual Heterogeneity:
## QE(df = 49) = 185.0467, p-val < .0001
##
## Test of Moderators (coefficient 2):
## QM(df = 1) = 2.2554, p-val = 0.1331
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## intrcpt 0.1995 0.0713 2.7997 0.0051 0.0598 0.3391 **
## Perc_Female -0.0017 0.0011 -1.5018 0.1331 -0.0040 0.0005
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# fit model with branch as moderator
res <- rma(yi, vi, mods = ~ Sample, data=dat)
res##
## Mixed-Effects Model (k = 51; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity): 0.0157 (SE = 0.0047)
## tau (square root of estimated tau^2 value): 0.1253
## I^2 (residual heterogeneity / unaccounted variability): 75.69%
## H^2 (unaccounted variability / sampling variability): 4.11
## R^2 (amount of heterogeneity accounted for): 12.33%
##
## Test for Residual Heterogeneity:
## QE(df = 48) = 163.9977, p-val < .0001
##
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 5.5984, p-val = 0.0609
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## intrcpt 0.0425 0.0395 1.0753 0.2822 -0.0350 0.1200
## SampleHigh School 0.0636 0.0478 1.3300 0.1835 -0.0301 0.1573
## SampleWorkplace 0.2018 0.0872 2.3127 0.0207 0.0308 0.3728 *
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# intercept = estimated average corrected correlation for studies using a community sample
# SampleHigh School = estimated difference in the average corrected correlation between studies using
# a high school sample compared to studies using a community sample
# SampleWorkplace = estimated difference in the average corrected correlation between studies using
# a workplace sample compared to studies using a community sample# fit model with Sample as moderator removing the intercept
res <- rma(yi, vi, mods = ~ Sample - 1, data=dat)
res##
## Mixed-Effects Model (k = 51; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity): 0.0157 (SE = 0.0047)
## tau (square root of estimated tau^2 value): 0.1253
## I^2 (residual heterogeneity / unaccounted variability): 75.69%
## H^2 (unaccounted variability / sampling variability): 4.11
##
## Test for Residual Heterogeneity:
## QE(df = 48) = 163.9977, p-val < .0001
##
## Test of Moderators (coefficients 1:3):
## QM(df = 3) = 26.5936, p-val < .0001
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## SampleCommunity 0.0425 0.0395 1.0753 0.2822 -0.0350 0.1200
## SampleHigh School 0.1061 0.0269 3.9458 <.0001 0.0534 0.1588 ***
## SampleWorkplace 0.2443 0.0778 3.1413 0.0017 0.0919 0.3967 **
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# SampleCommunity = estimated average corrected correlation for studies using a community sample
# SampleHigh School = estimated average corrected correlation for studies using a high school sample
# SampleWorkplace = estimated average corrected correlation for studies using a workplace sample8.1 Comparing meta-regression and sub groups
rma(yi, vi, data=dat, subset=Sample=="Community")##
## Random-Effects Model (k = 15; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0.0119 (SE = 0.0072)
## tau (square root of estimated tau^2 value): 0.1092
## I^2 (total heterogeneity / total variability): 70.09%
## H^2 (total variability / sampling variability): 3.34
##
## Test for Heterogeneity:
## Q(df = 14) = 42.0802, p-val = 0.0001
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## 0.0434 0.0360 1.2065 0.2276 -0.0271 0.1140
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1rma(yi, vi, data=dat, subset=Sample=="Workplace")##
## Random-Effects Model (k = 4; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0.0504 (SE = 0.0484)
## tau (square root of estimated tau^2 value): 0.2245
## I^2 (total heterogeneity / total variability): 86.31%
## H^2 (total variability / sampling variability): 7.30
##
## Test for Heterogeneity:
## Q(df = 3) = 24.0253, p-val < .0001
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## 0.2284 0.1217 1.8767 0.0606 -0.0101 0.4669 .
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1rma(yi, vi, data=dat, subset=Sample=="High School")##
## Random-Effects Model (k = 32; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0.0137 (SE = 0.0052)
## tau (square root of estimated tau^2 value): 0.1171
## I^2 (total heterogeneity / total variability): 73.88%
## H^2 (total variability / sampling variability): 3.83
##
## Test for Heterogeneity:
## Q(df = 31) = 97.8922, p-val < .0001
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## 0.1061 0.0257 4.1345 <.0001 0.0558 0.1563 ***
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Note, the results of the subgroups analysis are different to the meta-regression approach. Note that the two estimates of because they are based on different amounts of (residual) heterogeneity.
A discussion/comparison of these two approaches (i.e., assuming a single tau square value or allowing tau sqaure to differ across subsets) can be found in the following article:
Rubio-Aparicio, M., López-López, J. A., Viechtbauer, W., Marín-Martínez, F., Botella, J., & Sánchez-Meca, J. (2020). Testing categorical moderators in mixed-effects meta-analysis in the presence of heteroscedasticity. Journal of Experimental Education, 88(2), 288-310. https://doi.org/10.1080/00220973.2018.1561404
The results suggest that using a pooled estimate of tau sqare (through meta-regression) across categories is the best option in most conditions, although using separate estimates of tau square (subgroups analysis) is preferable if the residual heterogeneity variances are heteroscedastic i.e. the levels of heterogeneity vary a lot between groups (this kind of makes sense, averaging things that are very different is, in general, a bad idea )
8.2 Changing Reference Category
# change the reference level for Sample
res <- rma(yi, vi, mods = ~ relevel(factor(Sample), ref="High School"), data=dat)
res##
## Mixed-Effects Model (k = 51; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity): 0.0157 (SE = 0.0047)
## tau (square root of estimated tau^2 value): 0.1253
## I^2 (residual heterogeneity / unaccounted variability): 75.69%
## H^2 (unaccounted variability / sampling variability): 4.11
## R^2 (amount of heterogeneity accounted for): 12.33%
##
## Test for Residual Heterogeneity:
## QE(df = 48) = 163.9977, p-val < .0001
##
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 5.5984, p-val = 0.0609
##
## Model Results:
##
## estimate se
## intrcpt 0.1061 0.0269
## relevel(factor(Sample), ref = "High School")Community -0.0636 0.0478
## relevel(factor(Sample), ref = "High School")Workplace 0.1382 0.0823
## zval pval ci.lb
## intrcpt 3.9458 <.0001 0.0534
## relevel(factor(Sample), ref = "High School")Community -1.3300 0.1835 -0.1573
## relevel(factor(Sample), ref = "High School")Workplace 1.6791 0.0931 -0.0231
## ci.ub
## intrcpt 0.1588 ***
## relevel(factor(Sample), ref = "High School")Community 0.0301
## relevel(factor(Sample), ref = "High School")Workplace 0.2995 .
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 18.3 Pseudo R-square
Just an FYI, here’s how you can calculate a pseudo R-sqaure
# show how the pseudo R^2 value is computed
res0 <- rma(yi, vi, data=dat)
res1 <- rma(yi, vi, mods = ~ Perc_Female, data=dat)
round(100 * (res0$tau2 - res1$tau2) / res0$tau2, digits=2)## [1] 2.04