Bayesian mediation analysis for studies with small samples
Mediation analysis helps identify and understand the underlying mechanism of a phenomenon. As an example, a simple mediated effect occurs when an intervention changes a mediator (i.e., the α path) and that mediator changes the outcome (i.e., the β path). The mediated effect is then the product of α and β paths, αβ, which estimates the part of the total program effect transmitted through the mediator.
The analysis of mediated effect in prevention programs can be sometimes problematic if the sample size is small. The sampling distribution of the mediated effect estimate can have a nonnormal distribution due to low sample size which leads to biased estimates. An approach to this problem is to use a Bayesian perspective to estimate the mediated effect since Bayesian analysis does not require normality assumption on sampling distributions of estimates which results in exact estimates with finite samples (Yuan & MacKinnon, 2009). Another advantage of Bayesian approach is the statistical incorporation of prior research results into the statistical analysis which enables researchers to make use of the previous data. In Bayesian analysis, the unknown parameters are treated as random variables with a distribution. The posterior mean of the mediated effect is updated by using the prior knowledge of estimators which are assumed to have a probability distribution called the prior distribution:
Posterior ≈ Prior * Likelihood
Another advantage of Bayesian estimation is the interpretation of confidence intervals. Bayesian credible intervals and frequentist confidence intervals are different from each other. A 95% Bayesian credible interval means that there is a 95% chance that the credible interval contains the true value of the estimate based on the observed data; whereas a 95% frequentist confidence interval means that if we repeatedly sample from a population and calculate the confidence interval for each sample, on average 95% of those intervals contain the true value of the estimate. In this sense, Bayesian credible intervals are more intuitive than conventional confidence intervals since credible intervals rely on a more meaningful probability distribution rather than an idealistic assumption of repeated sampling under identical conditions (Yuan & MacKinnon, 2009).
Kisbu-Sakarya, Y., MacKinnon, D., & Tofighi, D. (June, 2011). Bayesian mediation analysis for studies with small samples. Presented at the Annual Meeting of Society of Prevention Research, Denver, Co.