Introduction

This is the Supplemental Material of the article “Evaluating informative hypotheses with equality and inequality constraints: A tutorial using the Bayes factor via the encompassing prior approach” available at https://psyarxiv.com/6kc5u. The paper aims to provide a clear and detailed description of the Bayes Factor with the encompassing prior approach considering an applied example.

Study Summary

When conducting a study, researchers usually have expectations based on hypotheses or theoretical perspectives they want to evaluate. Equality and inequality constraints on the model parameters are used to formalize researchers’ expectations or theoretical perspectives into the so-called informative hypotheses. However, traditional statistical approaches, such as the Null Hypothesis Significance Testing (NHST) or the model comparison using information criteria (e.g., AIC and BIC), are unsuitable for testing complex informative hypotheses. An alternative approach is to use the Bayes factor. In particular, the Bayes factor based on the encompassing prior approach allows researchers to easily evaluate complex informative hypotheses in a wide range of statistical models (e.g., generalized linear). This paper provides a detailed introduction to the Bayes factor with encompassing prior. First, all steps and elements involved in the formalization of informative hypotheses and the computation of the Bayes factor with encompassing prior are described. Next, we apply this method to a real case scenario, considering the attachment theory. Specifically, we analyzed the relative influence of maternal and paternal attachment on children’s social-emotional development by comparing the various theoretical perspectives debated in the literature.

Supplemental Material Structure

In the supplemental material, we provide the complete statistical analyses of the case study regarding attachment. For more information regarding the Bayes factor with encompassing prior approach or the formalization of theoretical perspectives regarding the role of mother attachment and father attachment on children socio-emotional development, see the main article available at https://psyarxiv.com/6kc5u. The supplemental material is structured into three parts:

1) Presentation

In this section, we describe the aim of the study and the sample included in the study.

Chapter 1 - Case Study. We describe the sample included in the study and present the results of the cluster analysis to classify the different attachment patterns.

2) Externalizing Problems

In this section, we present the analyses evaluating the different roles of mother attachment and father attachment on children’s externalizing problems. We follow three different approaches.

Chapter 2 - Models Family Choice. We discuss the appropriate models’ family to take into account data characteristics.
Chapter 3 - NHST. Analysis is conducted following the traditional Null Hypothesis Significance Testing (NHST).
Chapter 4 - Model Comparison. Analysis is conducted according to the Model Comparison approach using the AIC and BIC criteria.
Chapter 5 - Bayes Factor. Analysis is conducted following the Bayes Factor with the encompassing prior approach.
Chapter 6 - Conclusions. Results of the three approaches are discussed.

3) Internalizing Problems

In this section, we present the analyses evaluating the different roles of mother attachment and father attachment on children’s internalizing problems. We follow the three different approaches like in the analysis of the externalizing problems. This section is shorter than the previous one as we focus only on the results without repeating all the descriptions of the methods.

Chapter 7 - Models Family Choice. We discuss the appropriate models’ family to take into account data characteristics.
Chapter 8 - NHST. Analysis is conducted following the traditional Null Hypothesis Significance Testing (NHST).
Chapter 9 - Model Comparison. Analysis is conducted according to the Model Comparison approach using the AIC and BIC criteria.
Chapter 10 - Bayes Factor. Analysis is conducted following the Bayes Factor with the encompassing prior approach.
Chapter 11 - Conclusions. Results of the three approaches are discussed.

GitHub

All the materials are available at the GitHub repository https://github.com/ClaudioZandonella/Attachment.

Refer to the README file for further information.

Evaluating Informative Hypotheses with Equality and Inequality Constraints: A Tutorial Using the Bayes Factor via the Encompassing Prior Approach