Power in Bayesian Mediation Analysis for Small Sample Research

Bayesian methods have the potential for increasing power in mediation analysis (Koopman, Howe, Hollenbeck, & Sin, 2015; Yuan & MacKinnon, 2009). This article 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

Small Sample Size Solutions: A Guide for Applied Researchers and Practitioners

Researchers often have difficulties collecting enough data to test their hypotheses, either because target groups are small or hard to access, or because data collection entails prohibitive costs. Such obstacles may result in data sets that are too small for the complexity of the statistical model needed to answer the research question. This unique book provides guidelines and tools for implementing solutions to issues that arise in small sample research. Each chapter illustrates statistical methods that allow researchers to apply the optimal statistical model for their research question when the sample is too small. This essential book will enable social and behavioral science researchers to test their hypotheses even when the statistical model required for answering their research question is too complex for the sample sizes they can collect. The statistical models in the book range from the estimation of a population mean to models with latent variables and nested observations, and solutions include both classical and Bayesian methods. All proposed solutions are described in steps researchers can implement with their own data and are accompanied with annotated syntax in R. The methods described in this book will be useful for researchers across the social and behavioral sciences, ranging from medical sciences and epidemiology to psychology, marketing, and economics

Statistical properties of four effect-size measures for mediation models

This project examined the performance of classical and Bayesian estimators of four effect size measures for the indirect effect in a single-mediator model and a two-mediator model. Compared to the proportion and ratio mediation effect sizes, standardized mediation effect-size measures were relatively unbiased and efficient in the single-mediator model and the two-mediator model. Percentile and bias-corrected bootstrap interval estimates of ab/s Y , and ab(s X )/s Y in the single-mediator model outperformed interval estimates of the proportion and ratio effect sizes in terms of power, Type I error rate, coverage, imbalance, and interval width. For the two-mediator model, standardized effect-size measures were superior to the proportion and ratio effect-size measures. Furthermore, it was found that Bayesian point and interval summaries of posterior distributions of standardized effect-size measures reduced excessive relative bias for certain parameter combinations. The standardized effect-size measures are the best effect-size measures for quantifying mediated effects

Statistical properties of four effect-size measures for mediation models

This project examined the performance of classical and Bayesian estimators of four effect size measures for the indirect effect in a single-mediator model and a two-mediator model. Compared to the proportion and ratio mediation effect sizes, standardized mediation effect-size measures were relatively unbiased and efficient in the single-mediator model and the two-mediator model. Percentile and bias-corrected bootstrap interval estimates of ab/s Y , and ab(s X )/s Y in the single-mediator model outperformed interval estimates of the proportion and ratio effect sizes in terms of power, Type I error rate, coverage, imbalance, and interval width. For the two-mediator model, standardized effect-size measures were superior to the proportion and ratio effect-size measures. Furthermore, it was found that Bayesian point and interval summaries of posterior distributions of standardized effect-size measures reduced excessive relative bias for certain parameter combinations. The standardized effect-size measures are the best effect-size measures for quantifying mediated effects

Going multivariate in clinical trial studies: A Bayesian framework for multiple binary outcomes

In an era where medicine is increasingly personalized, clinical trials often suffer from small samples. As a consequence, treatment comparison based on the data of these trials may result in inconclusive decisions. Efficient decision-making strategies are highly needed so decisions can be made with smaller samples without increasing the risk of errors. The current chapter centers around one such strategy: Including information from multiple outcomes in the decision, thereby focusing on data from binary outcomes. Key elements of the approach are (1) criteria for treatment comparison that are suitable for two outcomes, and (2) a multivariate Bayesian technique to analyze multiple binary outcomes simultaneously. The conceptual discussion of these elements is complemented with software to implement the approach. To further facilitate trials with small samples, the chapter also outlines how interim analyses may result in more efficient decisions compared to the traditional sample size estimation before data collection

Testing Mediators of Youth Intervention Outcomes Using Single-Case Experimental Designs

The major aim of this manuscript is to bring together two important topics that have recently received much attention in child and adolescent research, albeit separately from each other: single-case experimental designs and statistical mediation analysis. Single-case experimental designs (SCEDs) are increasingly recognized as a valuable alternative for Randomized Controlled Trials (RCTs) to test intervention effects in youth populations. Statistical mediation analysis helps provide understanding about the most potent mechanisms of change underlying youth intervention outcomes. In this manuscript we: (i) describe the conceptual framework and outline desiderata for methods for mediation analysis in SCEDs; (ii) describe the main aspects of several data-analytic techniques potentially useful to test mediation in SCEDs; (iii) apply these methods to a single-case treatment data set from one clinically anxious client; and (iv) discuss pros and cons of these methods for testing mediation in SCEDs, and provide future directions

Relations between mathematics achievement and motivation in students of diverse achievement levels

Due to the diverse achievement experiences of students with diverse achievement levels in heterogeneous primary school classrooms, the relations between motivation and achievement may develop differentially depending upon achievement level. This study investigated the relations between several core aspects of motivation for mathematics – self-efficacy, self-concept, task value, and mathematics anxiety – and achievement with particular attention for potential differences between students of diverse achievement levels. Participants (N = 4306 students of grade 2–6) completed a standardised mathematics achievement test at T1 and T3 and a mathematics motivation questionnaire at T2. T1 achievement positively predicted perceived competence (self-efficacy and self-concept combined) and task value and negatively predicted mathematics anxiety. Only perceived competence had a significant effect on T3 achievement after controlling for T1 achievement and working memory, and partially mediated the relation between previous and subsequent achievement. This pattern of effects was largely similar across the full range of achievement levels, although the effect of previous achievement on perceived competence was stronger within the subsample of average-achieving students. These findings highlight the unique contribution of perceived competence in predicting subsequent achievement across students of diverse achievement levels in primary school

Relations between mathematics achievement and motivation in students of diverse achievement levels

Due to the diverse achievement experiences of students with diverse achievement levels in heterogeneous primary school classrooms, the relations between motivation and achievement may develop differentially depending upon achievement level. This study investigated the relations between several core aspects of motivation for mathematics – self-efficacy, self-concept, task value, and mathematics anxiety – and achievement with particular attention for potential differences between students of diverse achievement levels. Participants (N = 4306 students of grade 2–6) completed a standardised mathematics achievement test at T1 and T3 and a mathematics motivation questionnaire at T2. T1 achievement positively predicted perceived competence (self-efficacy and self-concept combined) and task value and negatively predicted mathematics anxiety. Only perceived competence had a significant effect on T3 achievement after controlling for T1 achievement and working memory, and partially mediated the relation between previous and subsequent achievement. This pattern of effects was largely similar across the full range of achievement levels, although the effect of previous achievement on perceived competence was stronger within the subsample of average-achieving students. These findings highlight the unique contribution of perceived competence in predicting subsequent achievement across students of diverse achievement levels in primary school

Relations between mathematics achievement and motivation in students of diverse achievement levels

Due to the diverse achievement experiences of students with diverse achievement levels in heterogeneous primary school classrooms, the relations between motivation and achievement may develop differentially depending upon achievement level. This study investigated the relations between several core aspects of motivation for mathematics – self-efficacy, self-concept, task value, and mathematics anxiety – and achievement with particular attention for potential differences between students of diverse achievement levels. Participants (N = 4306 students of grade 2–6) completed a standardised mathematics achievement test at T1 and T3 and a mathematics motivation questionnaire at T2. T1 achievement positively predicted perceived competence (self-efficacy and self-concept combined) and task value and negatively predicted mathematics anxiety. Only perceived competence had a significant effect on T3 achievement after controlling for T1 achievement and working memory, and partially mediated the relation between previous and subsequent achievement. This pattern of effects was largely similar across the full range of achievement levels, although the effect of previous achievement on perceived competence was stronger within the subsample of average-achieving students. These findings highlight the unique contribution of perceived competence in predicting subsequent achievement across students of diverse achievement levels in primary school