This vignette demonstrates how to use the clubSandwich
package to conduct a meta-analysis of dependent effect sizes with robust
variance estimation. Tests of meta-regression coefficients and F-tests
of multiple-coefficient hypotheses are calculated using small-sample
corrections proposed by Tipton (2015) and Tipton and Pustejovsky (2015).
The example uses a dataset of effect sizes from a Campbell Collaboration
systematic review of dropout prevention programs, conducted by Sandra Jo
Wilson and colleagues (2011).
The original analysis included a meta-regression with covariates that
capture methodological, participant, and program characteristics. The
regression specification used here is similar to Model III from Wilson
et al. (2011), but treats the evaluator_independence and
implementation_quality variables as categorical rather than
interval-level. Also, the original analysis clustered at the level of
the sample (some studies reported results from multiple samples),
whereas here we cluster at the study level. The meta-regression can be
fit in several different ways. We first demonstrate using the
robumeta package (Fisher & Tipton, 2015) and then using
the metafor package (Viechtbauer, 2010).
library(clubSandwich)
library(robumeta)## Warning: package 'robumeta' was built under R version 4.1.3data(dropoutPrevention)
# clean formatting
names(dropoutPrevention)[7:8] <- c("eval","implement")
levels(dropoutPrevention$eval) <- c("independent","indirect","planning","delivery")
levels(dropoutPrevention$implement) <- c("low","medium","high")
levels(dropoutPrevention$program_site) <- c("community","mixed","classroom","school")
levels(dropoutPrevention$study_design) <- c("matched","unmatched","RCT")
levels(dropoutPrevention$adjusted) <- c("no","yes")
m3_robu <- robu(LOR1 ~ study_design + attrition + group_equivalence + adjusted
+ outcome + eval + male_pct + white_pct + average_age
+ implement + program_site + duration + service_hrs,
data = dropoutPrevention, studynum = studyID, var.eff.size = varLOR,
modelweights = "HIER")
print(m3_robu)## RVE: Hierarchical Effects Model with Small-Sample Corrections
##
## Model: LOR1 ~ study_design + attrition + group_equivalence + adjusted + outcome + eval + male_pct + white_pct + average_age + implement + program_site + duration + service_hrs
##
## Number of clusters = 152
## Number of outcomes = 385 (min = 1 , mean = 2.53 , median = 1 , max = 30 )
## Omega.sq = 0.24907
## Tau.sq = 0.1024663
##
## Estimate StdErr t-value dfs P(|t|>) 95% CI.L 95% CI.U Sig
## 1 X.Intercept. 0.016899 0.615399 0.0275 16.9 0.97841541 -1.28228 1.31608
## 2 study_designunmatched -0.002626 0.185142 -0.0142 40.5 0.98875129 -0.37667 0.37141
## 3 study_designRCT -0.086872 0.140044 -0.6203 38.6 0.53869676 -0.37024 0.19650
## 4 attrition 0.118889 0.247228 0.4809 15.5 0.63732597 -0.40666 0.64444
## 5 group_equivalence 0.502463 0.195838 2.5657 28.7 0.01579282 0.10174 0.90318 **
## 6 adjustedyes -0.322480 0.125413 -2.5713 33.8 0.01470796 -0.57741 -0.06755 **
## 7 outcomeenrolled 0.097059 0.139842 0.6941 16.5 0.49727848 -0.19862 0.39274
## 8 outcomegraduation 0.147643 0.134938 1.0942 30.2 0.28253825 -0.12786 0.42315
## 9 outcomegraduation.ged 0.258034 0.169134 1.5256 16.3 0.14632629 -0.10006 0.61613
## 10 evalindirect -0.765085 0.399109 -1.9170 6.2 0.10212896 -1.73406 0.20389
## 11 evalplanning -0.920874 0.346536 -2.6574 5.6 0.04027061 -1.78381 -0.05794 **
## 12 evaldelivery -0.916673 0.304303 -3.0124 4.7 0.03212299 -1.71432 -0.11903 **
## 13 male_pct 0.167965 0.181538 0.9252 16.4 0.36824526 -0.21609 0.55202
## 14 white_pct 0.022915 0.149394 0.1534 21.8 0.87950385 -0.28704 0.33287
## 15 average_age 0.037102 0.027053 1.3715 21.2 0.18458247 -0.01913 0.09333
## 16 implementmedium 0.411779 0.128898 3.1946 26.7 0.00358205 0.14714 0.67642 ***
## 17 implementhigh 0.658570 0.123874 5.3164 34.6 0.00000635 0.40699 0.91015 ***
## 18 program_sitemixed 0.444384 0.172635 2.5741 28.6 0.01550504 0.09109 0.79768 **
## 19 program_siteclassroom 0.426658 0.159773 2.6704 37.4 0.01115192 0.10303 0.75028 **
## 20 program_siteschool 0.262517 0.160519 1.6354 30.1 0.11236814 -0.06525 0.59028
## 21 duration 0.000427 0.000873 0.4895 36.7 0.62736846 -0.00134 0.00220
## 22 service_hrs -0.003434 0.005012 -0.6852 36.7 0.49752503 -0.01359 0.00672
## ---
## Signif. codes: < .01 *** < .05 ** < .10 *
## ---
## Note: If df < 4, do not trust the resultsNote that robumeta produces small-sample corrected
standard errors and t-tests, and so there is no need to repeat those
calculations with clubSandwich. The eval
variable has four levels, and it might be of interest to test whether
the average program effects differ by the degree of evaluator
independence. The null hypothesis in this case is that the 10th, 11th,
and 12th regression coefficients are all equal to zero. A small-sample
adjusted F-test for this hypothesis can be obtained as follows. The
vcov = "CR2" option means that the standard errors will be
corrected using the bias-reduced linearization estimator described in
Tipton and Pustejovsky (2015).
Wald_test(m3_robu, constraints = constrain_zero(10:12), vcov = "CR2")## test Fstat df_num df_denom p_val sig
## HTZ 2.78 3 16.8 0.0732 .By default, the Wald_test function provides an F-type
test with degrees of freedom estimated using the approximate Hotelling’s
\(T^2_Z\) method. The test has less
than 17 degrees of freedom, even though there are 152 independent
studies in the data, and has a p-value that is not quite significant at
conventional levels. The low degrees of freedom are a consequence of the
fact that one of the levels of evaluator independence has
only a few effect sizes in it:
table(dropoutPrevention$eval)##
## independent indirect planning delivery
## 6 33 43 303clubSandwich also works with models fit using the
metafor package. Here we re-fit the same regression
specification, but use REML to estimate the variance components
(robumeta uses a method-of-moments estimator), as well as a
somewhat different weighting scheme than that used in
robumeta.
library(metafor)## Warning: package 'metafor' was built under R version 4.1.3m3_metafor <- rma.mv(LOR1 ~ study_design + attrition + group_equivalence + adjusted
+ outcome + eval
+ male_pct + white_pct + average_age
+ implement + program_site + duration + service_hrs,
V = varLOR, random = list(~ 1 | studyID, ~ 1 | studySample),
data = dropoutPrevention)
summary(m3_metafor)##
## Multivariate Meta-Analysis Model (k = 385; method: REML)
##
## logLik Deviance AIC BIC AICc <U+200B>
## -489.0357 978.0714 1026.0714 1119.5371 1029.6217
##
## Variance Components:
##
## estim sqrt nlvls fixed factor
## sigma^2.1 0.2274 0.4769 152 no studyID
## sigma^2.2 0.1145 0.3384 317 no studySample
##
## Test for Residual Heterogeneity:
## QE(df = 363) = 1588.4397, p-val < .0001
##
## Test of Moderators (coefficients 2:22):
## QM(df = 21) = 293.8694, p-val < .0001
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub <U+200B>
## intrcpt 0.5296 0.7250 0.7304 0.4651 -0.8915 1.9506
## study_designunmatched -0.0494 0.1722 -0.2871 0.7741 -0.3870 0.2881
## study_designRCT 0.0653 0.1628 0.4010 0.6884 -0.2538 0.3843
## attrition -0.1366 0.2429 -0.5623 0.5739 -0.6126 0.3395
## group_equivalence 0.4071 0.1573 2.5877 0.0097 0.0988 0.7155 **
## adjustedyes -0.3581 0.1532 -2.3371 0.0194 -0.6585 -0.0578 *
## outcomeenrolled -0.2831 0.0771 -3.6709 0.0002 -0.4343 -0.1320 ***
## outcomegraduation -0.0913 0.0657 -1.3896 0.1646 -0.2201 0.0375
## outcomegraduation/ged 0.6983 0.0805 8.6750 <.0001 0.5406 0.8561 ***
## evalindirect -0.7530 0.4949 -1.5214 0.1282 -1.7230 0.2171
## evalplanning -0.7700 0.4869 -1.5814 0.1138 -1.7242 0.1843
## evaldelivery -1.0016 0.4600 -2.1774 0.0294 -1.9033 -0.1000 *
## male_pct 0.1021 0.1715 0.5951 0.5518 -0.2341 0.4382
## white_pct 0.1223 0.1804 0.6777 0.4979 -0.2313 0.4758
## average_age 0.0061 0.0291 0.2091 0.8344 -0.0509 0.0631
## implementmedium 0.4738 0.1609 2.9445 0.0032 0.1584 0.7892 **
## implementhigh 0.6318 0.1471 4.2965 <.0001 0.3436 0.9201 ***
## program_sitemixed 0.3289 0.2413 1.3631 0.1729 -0.1440 0.8019
## program_siteclassroom 0.2920 0.1736 1.6821 0.0926 -0.0482 0.6321 .
## program_siteschool 0.1616 0.1898 0.8515 0.3945 -0.2104 0.5337
## duration 0.0013 0.0009 1.3423 0.1795 -0.0006 0.0031
## service_hrs -0.0003 0.0047 -0.0654 0.9478 -0.0096 0.0090
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1metafor produces model-based standard errors, t-tests,
and confidence intervals. The coef_test function from
clubSandwich will calculate robust standard errors and
robust t-tests for each of the coefficients:
coef_test(m3_metafor, vcov = "CR2")## Coef. Estimate SE t-stat d.f. (Satt) p-val (Satt) Sig.
## intrcpt 0.529569 0.724851 0.7306 20.08 0.47347
## study_designunmatched -0.049434 0.204152 -0.2421 58.42 0.80952
## study_designRCT 0.065272 0.149146 0.4376 53.17 0.66342
## attrition -0.136575 0.306429 -0.4457 10.52 0.66485
## group_equivalence 0.407108 0.210917 1.9302 23.10 0.06595 .
## adjustedyes -0.358124 0.136132 -2.6307 43.20 0.01176 *
## outcomeenrolled -0.283124 0.237199 -1.1936 7.08 0.27108
## outcomegraduation -0.091295 0.091465 -0.9981 9.95 0.34188
## outcomegraduation/ged 0.698328 0.364882 1.9138 8.02 0.09188 .
## evalindirect -0.752994 0.447670 -1.6820 6.56 0.13929
## evalplanning -0.769968 0.403898 -1.9063 6.10 0.10446
## evaldelivery -1.001648 0.355989 -2.8137 4.89 0.03834 *
## male_pct 0.102055 0.148410 0.6877 9.68 0.50782
## white_pct 0.122255 0.141470 0.8642 16.88 0.39961
## average_age 0.006084 0.033387 0.1822 15.79 0.85772
## implementmedium 0.473789 0.148660 3.1871 22.44 0.00419 **
## implementhigh 0.631842 0.138073 4.5761 28.68 < 0.001 ***
## program_sitemixed 0.328941 0.196848 1.6710 27.47 0.10607
## program_siteclassroom 0.291952 0.146014 1.9995 42.70 0.05195 .
## program_siteschool 0.161640 0.171700 0.9414 29.27 0.35420
## duration 0.001270 0.000978 1.2988 31.96 0.20332
## service_hrs -0.000309 0.004828 -0.0641 49.63 0.94915Note that coef_test assumed that it should cluster based
on studyID, which is the outer-most random effect in the
metafor model. This can be specified explicitly by including the option
cluster = dropoutPrevention$studyID in the call.
The F-test for degree of evaluator independence uses the same syntax as before:
Wald_test(m3_metafor, constraints = constrain_zero(10:12), vcov = "CR2")## test Fstat df_num df_denom p_val sig
## HTZ 2.71 3 18.3 0.0753 .Despite some differences in weighting schemes, the p-value is very
close to the result obtained using robumeta.
Fisher, Z., & Tipton, E. (2015). robumeta: An R-package for robust variance estimation in meta-analysis. arXiv:1503.02220
Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375-393. https://doi.org/10.1037/met0000011
Tipton, E., & Pustejovsky, J. E. (2015). Small-sample adjustments for tests of moderators and model fit using robust variance estimation in meta-regression. Journal of Educational and Behavioral Statistics, 40(6), 604-634. https://doi.org/10.3102/1076998615606099
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1-48. URL: https://doi.org/10.18637/jss.v036.i03
Wilson, S. J., Lipsey, M. W., Tanner-Smith, E., Huang, C. H., & Steinka-Fry, K. T. (2011). Dropout prevention and intervention programs: Effects on school completion and dropout Among school-aged children and youth: A systematic review. Campbell Systematic Reviews, 7(1), 1-61. https://doi.org/10.4073/csr.2011.8