## Bayesian covariance structure modeling: An __overview__ and new __developments__

** ****Presenter: Jean-Paul Fox (Online Seminar Department of Statistics UConn, 23 April 2021). **

**(papers and other information see www.Jean-PaulFox.com)**

There is large family of statistical models to understand clustered or hierarchical structures in the data (e.g., multilevel models, mixed effect models, random effect models). The general modeling technique is to use a latent variable (i.e., random effect, frailty parameter) to describe the covariance among clustered observations, where the strength of the covariance is represented by the latent variable variance. This approach has several disadvantages. It is only possible to describe positive within-cluster correlation (similarity), and not dissimilarity (Nielsen et al., 2021). Sample size restriction and model complexity is often implied by the number and type of latent variables. Furthermore, the latent variable variance is restricted to be positive, which leads to boundary issues at/around zero, and statistical issues in evaluating the data support in favor of a latent variable.

A new approach for modeling clustered data is presented, which is known as Bayesian covariance structure modeling (BCSM). This is a multivariate modeling approach, where the dependence structure is directly modeled through a structured covariance matrix. The BCSM is a general modeling approach which can also identify negative dependence structures and even dependence structures implied by non-identifiable random effects. BCSMs have been developed for different applications and for complex dependence structures (Fox et al., 2017, Klotzke and Fox, 2019a,2019b; Mulder and Fox, 2019).

In this presentation, an overview is given of the BCSM, in which several applications and new developments will be discussed. (1) The BCSM for measurement invariance testing is shown to test simultaneously whether items perform in the same way across groups. The BCSM measurement invariance procedure is applied to PISA data of the 2015 cycle, to illustrate the advantages of the method (Fox et al., 2020). (2) The BCSM for negative clustering effects is shown using an application about personalized treatment effects in counseling. It is argued that the identification of a negative within-cluster correlation under the BCSM can identify personalized (treatment) effects. (3) An application is shown of a BCSM (multivariate probit model) for interval-censored clustered event time data from a three-armed randomized clinical trial. Patients required a drug-eluting stent during a coronary intervention. A Bayes factor has been developed to test for a difference/equivalence in performance of the stents. Insight is given in the priors, the multiple hypothesis testing problem, and the required computational demands.

**References**

Fox, J.-P., Koops, J., Feskens, R., Beinhauer, L. (2020). Bayesian covariance structure modelling for measurement invariance testing. *Behaviormetrika 47*, 385–410. DOI 10.1007/s41237-020-00119-3.

Klotzke, K., Fox, J.-P. (2019a) Modeling dependence structures for response times in a Bayesian framework. *Psychometrika 84*, 649–672 (2019). DOI: 10.1007/s11336-019-09671-8

Klotzke, Konrad, Fox, J.-P. (2019b): Bayesian covariance structure modeling of responses and process data. *Frontiers.* Collection. DOI: 10.3389/fpsyg.2019.01675

J Mulder, J.-P. Fox (2019). Bayes factor testing of multiple intraclass correlations. *Bayesian Analysis 14 *(2), 521-552.

Nielsen, N.M., Smink, W.A.C. & Fox, J.-P. (2021). Small and negative correlations among clustered observations: limitations of the linear mixed effects model. *Behaviourmetrika, 48*, 51-77. DOI: 10.1007/s41237-020-00130-8.