Bayesian Analysis of Multi-Factorial Experimental Designs Using SEM.

Latent repeated measures ANOVA (L-RM-ANOVA) has recently been proposed as an alternative to traditional repeated measures ANOVA. L-RM-ANOVA builds upon structural equation modeling and enables researchers to investigate interindividual differences in main/interaction effects, examine custom contrasts, incorporate a measurement model, and account for missing data. However, L-RM-ANOVA uses maximum likelihood and thus cannot incorporate prior information and can have poor statistical properties in small samples. We show how L-RM-ANOVA can be used with Bayesian estimation to resolve the aforementioned issues. We demonstrate how to place informative priors on model parameters that constitute main and interaction effects. We further show how to place weakly informative priors on standardized parameters which can be used when no prior information is available. We conclude that Bayesian estimation can lower Type 1 error and bias, and increase power and efficiency when priors are chosen adequately. We demonstrate the approach using a real empirical example and guide the readers through specification of the model. We argue that ANOVA tables and incomplete descriptive statistics are not sufficient information to specify informative priors, and we identify which parameter estimates should be reported in future research; thereby promoting cumulative research.

A tutorial on using the paired t test for power calculations in repeated measures ANOVA with interactions.

The a priori calculation of statistical power has become common practice in behavioral and social sciences to calculate the necessary sample size for detecting an expected effect size with a certain probability (i.e., power). In multi-factorial repeated measures ANOVA, these calculations can sometimes be cumbersome, especially for higher-order interactions. For designs that only involve factors with two levels each, the paired t test can be used for power calculations, but some pitfalls need to be avoided. In this tutorial, we provide practical advice on how to express main and interaction effects in repeated measures ANOVA as single difference variables. In particular, we demonstrate how to calculate the effect size Cohen's d of this difference variable either based on means, variances, and covariances of conditions or by transforming [Formula: see text] or [Formula: see text] from the ANOVA framework into d. With the effect size correctly specified, we then show how to use the t test for sample size considerations by means of an empirical example. The relevant R code is provided in an online repository for all example calculations covered in this article.

Subgroup discovery in structural equation models.

Structural equation modeling is one of the most popular statistical frameworks in the social and behavioral sciences. Often, detection of groups with distinct sets of parameters in structural equation models (SEM) are of key importance for applied researchers, for example, when investigating differential item functioning for a mental ability test or examining children with exceptional educational trajectories. In the present article, we present a new approach combining subgroup discovery-a well-established toolkit of supervised learning algorithms and techniques from the field of computer science-with structural equation models termed SubgroupSEM. We provide an overview and comparison of three approaches to modeling and detecting heterogeneous groups in structural equation models, namely, finite mixture models, SEM trees, and SubgroupSEM. We provide a step-by-step guide to applying subgroup discovery techniques for structural equation models, followed by a detailed and illustrated presentation of pruning strategies and four subgroup discovery algorithms. Finally, the SubgroupSEM approach will be illustrated on two real data examples, examining measurement invariance of a mental ability test and investigating interesting subgroups for the mediated relationship between predictors of educational outcomes and the trajectories of math competencies in 5th grade children. The illustrative examples are accompanied by examples of the R package subgroupsem, which is a viable implementation of our approach for applied researchers. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Estimating and Testing Causal Mediation Effects in Single-Case Experimental Designs Using State-Space Modeling.

In this article, we present single-case causal mediation analysis as the application of causal mediation analysis to data collected within a single-case experiment. This method combines the focus on the individual with the focus on mechanisms of change, rendering it a promising approach for both mediation and single-case researchers. For this purpose, we propose a new method based on time-discrete state-space modeling to estimate the direct and indirect treatment effects. We demonstrate how to estimate the model for a single-case experiment on stress and craving in a routine alcohol consumer before and after an imposed period of abstinence. Furthermore, we present a simulation study that examines the estimation and testing of the standardized indirect effect. All parameters used to generate the data were recovered with acceptable precision. We use maximum likelihood and permutation procedures to calculate p-values and standard errors of the parameters estimates. The new method is promising for testing mediated effects in single-case experimental designs. We further discuss limitations of the new method with respect to causal inference, as well as more technical concerns, such as the choice of the time lags between the measurements.

Repeated Measures ANOVA with Latent Variables to Analyze Interindividual Differences in Contrasts.

Repeated measures analysis of variance (RM-ANOVA) is a broadly used statistical method to analyze data from experimental designs. RM-ANOVA aims at investigating effects of experimental conditions (i.e., factors) and predictors that affect the outcome of interest. It mainly considers contrasts that test standard main and interaction effects, even though more complex contrasts can in principle be used. Analyses, however, only focus on drawing conclusions about average effects and do not take into consideration interindividual differences in these effects. We propose an alternative approach to RM-ANOVA for analyzing repeated measures data, termed latent repeated measures analysis of variance (L-RM-ANOVA). The new approach is based on structural equation modeling and extends the latent growth components approach. L-RM-ANOVA enables the researcher to not only consider mean differences between different experimental conditions (i.e., average effects), but also to investigate interindividual differences in effects. Such interindividual differences are considered with regard to standard main and interactions effects and also with regard to customized contrasts that allow for testing specific hypotheses of interest. Furthermore, L-RM-ANOVA can include a measurement model for latent variables and can be used for the analysis of complex multi-factorial repeated measures designs. We conclude the presentation by demonstrating L-RM-ANOVA using a minimal repeated measures example.