## Bayesian Analysis

### Bayes Factor Testing of Multiple Intraclass Correlations

#### Abstract

The intraclass correlation plays a central role in modeling hierarchically structured data, such as educational data, panel data, or group-randomized trial data. It represents relevant information concerning the between-group and within-group variation. Methods for Bayesian hypothesis tests concerning the intraclass correlation are proposed to improve decision making in hierarchical data analysis and to assess the grouping effect across different group categories. Estimation and testing methods for the intraclass correlation coefficient are proposed under a marginal modeling framework where the random effects are integrated out. A class of stretched beta priors is proposed on the intraclass correlations, which is equivalent to shifted $F$ priors for the between groups variances. Through a parameter expansion it is shown that this prior is conditionally conjugate under the marginal model yielding efficient posterior computation. A special improper case results in accurate coverage rates of the credible intervals even for minimal sample size and when the true intraclass correlation equals zero. Bayes factor tests are proposed for testing multiple precise and order hypotheses on intraclass correlations. These tests can be used when prior information about the intraclass correlations is available or absent. For the noninformative case, a generalized fractional Bayes approach is developed. The method enables testing the presence and strength of grouped data structures without introducing random effects. The methodology is applied to a large-scale survey study on international mathematics achievement at fourth grade to test the heterogeneity in the clustering of students in schools across countries and assessment cycles.

#### Article information

Source
Bayesian Anal., Volume 14, Number 2 (2019), 521-552.

Dates
First available in Project Euclid: 10 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ba/1533866668

Digital Object Identifier
doi:10.1214/18-BA1115

Mathematical Reviews number (MathSciNet)
MR3934096

Zentralblatt MATH identifier
07045441

#### Citation

Mulder, Joris; Fox, Jean-Paul. Bayes Factor Testing of Multiple Intraclass Correlations. Bayesian Anal. 14 (2019), no. 2, 521--552. doi:10.1214/18-BA1115. https://projecteuclid.org/euclid.ba/1533866668

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#### Supplemental materials

• The supplementary material for “Bayes Factor Testing of Multiple Intraclass Correlations”. The supplementary material for “Bayes factor testing of multiple intraclass correlations” contains the proof of Lemma 1, the proof of Lemma 2, the conditional posterior distributions for the Gibbs sampler, the proof of Lemma 3, the Gibbs sampler under a constrained model, the analytic expression of the marginal likelihood (with derivation) for a standard random intercept model using fractional Bayes methodology, and a simulation study when testing interval hypotheses.