1 Dependence in Meta-Analysis

Typical methods to conduct meta-analysis or meta-regression work under the assumption that the effect size estimates are independent. However, primary studies often report multiple, statistically dependent estimates of effect sizes. Dependent effect sizes can occur through correlated effects structure, a hierarchical effects structure, or a combination of the two (Hedges, Tipton, & Johnson, 2010; Pustejovsky & Tipton, 2021). A correlated effects data structure typically occurs due to multiple correlated measures of an outcome, repeated measures of the outcome data, or comparison of multiple treatment groups to the same control group (Hedges et al., 2010). A hierarchical effects structure typically occurs when the meta-analysis includes multiple primary studies conducted by the same researcher, by the same lab, or in the same region (Hedges et al., 2010).

Researchers may be inclined to ignore dependence and use methods that assume that each effect size estimate is independent. However, doing so can result in inaccurate standard errors and, consequently, hypothesis tests with incorrect Type I error rates and confidence intervals with incorrect coverage levels (Becker, 2000). Ad-hoc methods for handling dependence include averaging effect sizes by study or selecting a single effect size per study. These methods result in loss of information and are not suitable for studying within-study variation in effect sizes (Hedges et al., 2010). A method called shifting the unit-of-analysis involves running meta-analytic models for different subsets of the data (Cooper, 1998). However, this strategy is not useful if a researcher wants to summarize effects across the subsets or study differential effects (Becker, 2000).

The ideal solution for handling dependence would be to use a multivariate model (Becker, 2000; Hedges et al., 2010). This approach explicitly models dependence among the effect sizes (Becker, 2000; Hedges et al., 2010). However, multivariate meta-analysis requires knowledge of correlations or covariances between pairs of effect size estimates within each study, which are often difficult to obtain from primary studies.

To handle dependence without knowing the covariance structure between effect size estimates, Hedges et al. (2010) proposed the use of robust variance estimation (RVE). RVE involves estimating the variances for the meta-regression model’s coefficients using sandwich estimators that are valid even when the covariance structure is unknown or mis-specified (Hedges et al., 2010). However, the performance characteristics of RVE are asymptotic in that a large number of clusters or studies is required to provide accurate standard error estimates (Hedges et al., 2010). If the number of studies in a meta-analysis is small, RVE, as originally proposed by Hedges et al. (2010), can result in downwardly biased standard errors and inflation of Type I error rates (Hedges et al., 2010; Tipton, 2015).

Tipton (2015) and Tipton & Pustejovsky (2015) examined several small sample correction methods to improve the performance of RVE. Tipton (2015) recommended CR2 type correction for RVE as well as the use of Satterthwaite degrees of freedom for single coefficient tests. Tipton & Pustejovsky (2015) examined corrections for multiple-contrast hypothesis tests. Tipton & Pustejovsky (2015) found that the HTZ test, which is an extension of the CR2 correction method with the Satterthwaite degrees of freedom, controlled Type 1 error rate adequately even when the number of studies was small. However, Joshi, Pustejovsky, & Beretvas (2022) showed, through simulations, that the HTZ test can be conservative. The authors 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 the simulations from Joshi et al. (2022) showed that CWB adequately controlled for Type 1 error rate and provided higher power than the HTZ test, especially for multiple-contrast hypothesis tests.

2 Bootstrapping

General bootstrapping can be used to estimate measures of uncertainty, like standard errors, p-values and confidence intervals, even when other methods fail (Boos et al., 2003). Bootstrapping involves re-sampling from the original data many times to create an empirical distribution which is used in place of the distribution of an estimate or test statistic (Boos et al., 2003).

Several bootstrapping data generating processes are available. The most common one is pair bootstrapping, which involves re-sampling with replacement the set of outcome and covariate data for each case (Freedman, 1981, 1984). For data involving clusters, the entire cluster is re-sampled (Cameron, Gelbach, & Miller, 2008). In meta-analytic studies with small number of clusters, pairs bootstrapping can result in lack of variance in the distribution of covariates rendering estimation of coefficients infeasible (Cameron et al., 2008).

Another type of bootstrapping involves re-sampling residuals (Cameron et al., 2008). In case with clusters, entire vector of residuals for each cluster is re-sampled (Cameron et al., 2008; MacKinnon, 2009). Such a procedure requires clusters to be of equal size and has an underlying assumption that the errors are independently and identically distributed (MacKinnon, 2009).

An ideal way to bootstrap when the number of clusters is small is to use cluster wild bootstrapping, which involves sampling weights and multiplying residuals with the random weights (Cameron et al., 2008; MacKinnon, 2009). In contrast to the process of pair bootstrapping, the process of CWB does not involve re-sampling the distribution of predictor variables. Thus, the problem of lack of variance in covariates due to re-sampling does not occur with CWB (Cameron et al., 2008; MacKinnon, 2009). Further, in contrast to residual bootstrapping, CWB does not require clusters to have the same size and does not require the errors to be independently and identically distributed (Cameron et al., 2008; MacKinnon, 2009).

3 Example from wildmeta

This section presents examples of how to implement cluster wild bootstrapping using functions from our wildmeta package. The functions in our package work with models fit using robumeta::robu() (Fisher, Tipton, & Zhipeng, 2017) and metafor::rma.mv() (Viechtbauer, 2010).

library(wildmeta)
library(clubSandwich)
library(robumeta)

robu_model <- robu(d ~ 0 + study_type + hrs + test,
                   studynum = study,
                   var.eff.size = V,
                   small = FALSE,
                   data = SATcoaching)

Wald_test_cwb(full_model = robu_model,
              constraints = constrain_equal(1:3),
              R = 99,
              seed = 20201228)
#>   Test Adjustment CR_type Statistic  R     p_val
#> 1  CWB        CR0     CR0   Naive-F 99 0.3939394

Wald_test_cwb(full_model = robu_model,
              constraints = constrain_equal(1:3),
              R = 99,
              adjust = "CR2",
              seed = 20201229)
#>           Test Adjustment CR_type Statistic  R     p_val
#> 1 CWB Adjusted        CR2     CR0   Naive-F 99 0.3535354

library(metafor)

rma_model <- rma.mv(yi = d ~ 0 + study_type + hrs + test,
                    V = V,
                    random = ~ study_type | study,
                    data = SATcoaching,
                    subset = !is.na(hrs) & !is.na(test))

Wald_test_cwb(full_model = rma_model,
              constraints = constrain_equal(1:3),
              R = 19,
              seed = 20210314)
#>   Test Adjustment CR_type Statistic  R     p_val
#> 1  CWB        CR0     CR0   Naive-F 19 0.2631579

system.time(
  res <- Wald_test_cwb(full_model = robu_model,
                       constraints = constrain_equal(1:3),
                       R = 1999, 
                       seed = 20201229)
)
#>    user  system elapsed 
#>  22.182   0.578  26.052

library(future)

if (parallelly::supportsMulticore()) {
  plan(multicore) 
} else {
  plan(multisession)
}

nbrOfWorkers()
#> system 
#>      4

system.time(
  res <- Wald_test_cwb(full_model = robu_model,
                       constraints = constrain_equal(1:3),
                       R = 1999, 
                       seed = 20201229)
)
#>    user  system elapsed 
#>   2.316   0.144  30.790

plan(sequential)

reps
#> [1] 99

system.time(
  res_seq <- Wald_test_cwb(full_model = rma_model,
                           constraints = constrain_equal(1:3),
                           R = reps,
                           seed = 20210314)
)
#>    user  system elapsed 
#>  26.068   3.498  32.836

res_seq
#>   Test Adjustment CR_type Statistic  R     p_val
#> 1  CWB        CR0     CR0   Naive-F 99 0.2828283

library(future)

if (parallelly::supportsMulticore()) {
  plan(multicore) 
} else {
  plan(multisession)
}

nbrOfWorkers()
#> system 
#>      4

system.time(
  res_para <- Wald_test_cwb(full_model = rma_model,
                            constraints = constrain_equal(1:3),
                            R = reps,
                            seed = 20210314)
)
#>    user  system elapsed 
#>   2.692   0.281  29.845

plan(sequential)

res_para
#>   Test Adjustment CR_type Statistic  R     p_val
#> 1  CWB        CR0     CR0   Naive-F 99 0.2828283

4 Bootstrap Distribution Plot

The function plot() creates a ggplot figure representing the distribution of the bootstrap replicates based on the results from the Wald_test_cwb() function. The dashed line represents the \(F\) test statistic from the original full model. The bootstrapped p-value is the proportion of bootstrapped \(F\) test statistics that are greater than the original \(F\) statistic.

plot(res, fill = "darkred", alpha = 0.5)