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.
Highlights
The intraclass correlation plays a central role in the statistical analysis of hierarchical data
To aid Bayesian estimation and testing a class of stretched beta priors is proposed for the intraclass correlations
Our interest is in the default relative evidence based on the generalized fractional Bayes factor while varying the posterior modes of the intraclass correlations under five different hypotheses in and the trajectory of the estimated intraclass correlations (ρ1, ρ2) = (0, 0.5), to (0.5, 0)
Summary
The intraclass correlation plays a central role in the statistical analysis of hierarchical data. To aid Bayesian estimation and testing a class of stretched beta priors is proposed for the intraclass correlations This class of priors has positive support for negative intraclass correlations under the marginal model. Note that frequentist matching priors play an important role in objective Bayesian analysis (Welch and Peers, 1963; Severini et al, 2002; Berger and Sun , 2008) Another consequence of the marginal modeling approach is that significance type tests of whether an intraclass correlation equals zero can be performed using credible intervals with accurate error rates. Two prior classes are discussed, where a stretched beta distribution and a shifted F distribution is introduced to describe the distribution of the intraclass correlation and the between-groups variance, respectively, while taking account of restrictions on the parameter space to ensure that the covariance matrix is positive definite. A discussion is given and some generalizations are presented
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