Primary studies in education and social sciences often contain multiple effect sizes (Hedges, Tipton & Johnson, 2010). Presence of multiple effect sizes leads to dependence as the estimates within each study are likely correlated (e.g., because the same participants provide multiple outcome scores). The increasingly popular method to handle such dependence, robust variance estimation (RVE), results in inflated Type 1 error rate when the number of studies is small (Hedges, Tipton & Johnson, 2010; Tipton, 2015).

Tipton (2015) and Tipton and Pustejovsky (2015) recommended a small sample correction, the HTZ test (CR2 correction with Satterthwaite degrees of freedom). The HTZ test has been shown to control Type 1 error rate adequately even when the number of studies is small (Tipton, 2015; Tipton & Pustejovsky, 2015). Through simulations that I ran for my dissertation, I showed the the HTZ test can be conservative. I examined another method, cluster wild bootstrapping (CWB), that has been studied in the econometrics literature but not in the meta-analytic context. The results of my dissertation simulations showed that CWB adequately controls for Type 1 error rate and has more power than the HTZ test.

The goal of this package is to provide applied meta-analytic researchers a function with which they can conduct single coefficient tests or multiple-contrast hypothesis tests using cluster wild bootstrapping.


You can install the development version from GitHub with:

# install.packages("devtools")


The following example uses the SATCoaching dataset from the clubSandwich package (Pustejovksy, 2020), originally from DerSimonian and Laird (1983). The standardized mean differences represent the effects of SAT coaching on SAT verbal (SATV) and/or SAT math (SATM) scores. The data contains the study_type variable indicating whether groups compared in primary studies were matched, randomized, or non-equivalent. The code below runs cluster wild bootstrapping to test the multiple-contrast hypothesis that the effect of coaching does not differ based on study type.



full <- robu(d ~ study_type,
             studynum = study,
             var.eff.size = V,
             small = FALSE,
             data = SATcoaching)

cwb(full_model = full,
    indices = 2:3,
    R = 99)
#>   test working_model     p_val
#> 1  CWB            CE 0.5252525


Canty A. & Ripley B. (2020). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-25.

DerSimonian, R., & Laird, N. (1983). Evaluating the effect of coaching on SAT scores: A meta-analysis. Harvard Educational Review, 53(1), 1-1

Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39–65.

Graham N., Arai M., & Hagströmer, B (2016). multiwayvcov: Multi-Way Standard Error Clustering. R package version 1.2.3.

Heyman, M. (2019). lmboot: Bootstrap in Linear Models. R package version 0.0.1.

Pustejovsky, J. E. (2020). clubSandwich: Cluster-robust (sandwich) variance estimators with small-sample corrections Rpackageversion0.4.2. R package version 0.4.2.

Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375–393.

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 (Vol. 40).