Power in Bayesian Mediation Analysis for Small Sample Research

Abstract

It was suggested that Bayesian methods have potential for increasing power in mediation analysis (Koopman, Howe, Hollenbeck, & Sin, 2015; Yuan & MacKinnon, 2009). This paper compares the power of Bayesian credibility intervals for the mediated effect to the power of normal theory, distribution of the product, percentile, and bias-corrected bootstrap confidence intervals at N≤ 200. Bayesian methods with diffuse priors have power comparable to the distribution of the product and bootstrap methods, and Bayesian methods with informative priors had the most power. Varying degrees of precision of prior distributions were also examined. Increased precision led to greater power only when N≥ 100 and the effects were small, N < 60 and the effects were large, and N < 200 and the effects were medium. An empirical example from psychology illustrated a Bayesian analysis of the single mediator model from prior selection to interpreting results.

Bayesian Versus Frequentist Estimation for Structural Equation Models in Small Sample Contexts: A Systematic Review

“…Although one of these 14 studies reported that both frequentist and BayesN methods lead to minimal bias in the parameter estimates (Yuan & MacKinnon, 2009), 6 of 14 studies reported that both methods resulted in poor parameter estimates (Browne & Draper, 2000, 2006Depaoli, 2013; 2 simulation studies in McNeish, 2016a;van de Schoot et al, 2015). The remaining studies show that the conclusion depends on: the choice of the naive prior distribution (McNeish, 2016b;McNeish & Stapleton, 2016;e.g., McNeish and Stapleton (2016) show that BayesN with Inverse Gamma or half-Cauchy prior distributions for the variance components in a multilevel model perform better in comparison to the other BayesN option with a uniform prior distribution); the choice of the frequentist estimation method to which the BayesN is compared (Koopman, Howe, Hollenbeck, & Sin, 2015;McNeish, 2016b;Miočević et al, 2017;e.g., McNeish (2016b) concludes that REML with Kenward-Roger correction performs better than ML and BayesN); or that the conclusions depend on the interest in either point estimates or interval estimates (2 simulation studies in Chen, Choi, Weiss, & Stapleton, 2014).…”

“…In 7 out of 22 studies that examined BayesN and frequentist methods, no severely biased estimates were reported when using BayesN. However, 6 of these studies focused on mediation or multilevel mediation models and did not evaluate the variance parameters (2 simulation studies in Chen et al, 2014;Hox et al, 2014;Koopman et al, 2015;Miočević et al, 2017;Yuan & MacKinnon, 2009). As shown in Figure 5, the variance parameters are more often problematic in terms of bias than the structural parameters.…”

“…From the 18 studies that compared BayesT to frequentist methods, 9 studies concluded that BayesT performed better than the frequentist methods (Depaoli & Clifton, 2015;Natesan, 2015;Price, 2012;Serang et al, 2015;van de Schoot et al, 2015;Yuan & MacKinnon, 2009; 2 simulation studies in Zondervan-Zwijnenburg et al, 2018;Miočević et al, 2017). The other 9 studies did not report a clear preference for one of the two methods, either because BayesT and the frequentist methods performed equally well (Farrell & Ludwig, 2008) or because the superiority of one of the two estimation methods depended on the amount or accuracy of information incorporated in the prior distributions (2 simulation studies in Depaoli, 2012;Depaoli, 2013; 2 simulation studies in McNeish, 2016b;Holtmann et al, 2016), the choice of the prior distributions (Baldwin & Fellingham, 2013), or the evaluation criteria and parameters of interest (Chen et al, 2014).…”

“…Only when BayesT was used, and thus prior information was included, power of 0.80 was reached in a substantial portion (48.71%) of the reported cases. In studies in which power levels of 0.80 or higher were reported when using BayesT, it is explained that power increased when the variance hyperparameter of the prior distribution became smaller, that is, when specific prior information is included (Miočević, MacKinnon, & Levy, 2017;Price, 2012;van de Schoot et al, 2015;Zondervan-Zwijnenburg et al, 2018). Thus, using BayesT increased chances of reaching a power level of 0.80 or higher.…”

“…These priors were obtained by varying the level of informativeness via adjusting the variance hyperparameter of the prior distribution (see e.g., 2 simulation studies in Depaoli, 2012;Depaoli & Clifton, 2015;Depaoli, 2013;van de Schoot et al, 2015; 2 simulation studies in Zondervan-Zwijnenburg et al, 2018;Holtmann et al, 2016), or by adjusting both hyperparameters (2 simulation studies in McNeish, 2016b; Natesan, 2015). Other variations of thoughtful priors are obtained by varying the parameters for which a thoughtful prior was specified or by adjusting the accurateness of the prior information included in the distributions (e.g., Depaoli, 2013;Miočević et al, 2017), or finally, by varying the distribution that is specified (see e.g., Baldwin & Fellingham, 2013;Yuan & MacKinnon, 2009).…”