Bayesian Analysis of Multi-Factorial Experimental Designs Using SEM

Abstract

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.

Keywords:

• Bayesian estimation
• ANOVA
• growth curves
• factorial designs
• Monte-Carlo simulation

Article information

Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant MA 7702/1-2 from the German Research Foundation.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgements: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions or the German Research Foundation is not intended and should not be inferred. This manuscript is based on the dissertation of the author Benedikt Langenberg.

Notes

1 The software code can be found in the accompanying online repository at https://osf.io/acz62/.

2 We call the variance liberal and not uncertain (as in Section “Study: Approach 1”) because the prior does not contain a location assumption. Furthermore, small variances pull the means and regression coefficients toward zero. Larger variances allow for larger values for means and regression coefficients—thus, more liberal.

3 The formula to calculate the intercept row is ${\mathbf{C}}_{1l}=\sqrt{\frac{{\sigma }_{Y_i}^2-{\sigma }_{Y_i{Y}_j}}{p{\sigma }_{Y_i}^2+2(p-1)p/2{\sigma }_{Y_i{Y}_j}}}$ where p is the number of dependent variables (in our case 6). Note that the equation is only valid if the contrast matrix is orthonormal, and the variances and covariances of all dependent variables are equal (i.e., compound symmetry).

4 The variance of the inverse Wishart is a function of the degrees of freedom ν and the entries of the scale matrix S. Different entries s_ij will have different variances because the degrees of freedom are equal for all entries s_ij.

5 Text books usually refer to the matrix of regression coefficients as B. We use ${\mathbf{B}}^{\text{GLM}}$ to refer to this matrix to avoid confusion with the B matrix from the structural equation of SEM (e.g., Bollen, 1989; Jöreskog, 1969) and ${\mathbf{B}}^{*}$ from L-RM-ANOVA (see Equation (3)).