Bayesian Estimation of the Indirect Effect in a Mediation Model With Cross-Classified Data

• In Event: Cross-Classified and Multiple Membership Random Effects Models: Demonstrations, Evaluations, and Extensions

Abstract

This study introduces a two-level cross-classified mediation model for handling two patterns of cross-classification in multisite 1->1->1 mediation data. We demonstrate the model using real data fitting the two patterns that are distinguished by the maximum number and timing of cluster changes for a participant. In the first pattern with two cross-classified factors, some students changed schools between treatment onset and mediator measurement but not between measurement of the mediator and distal outcomes. In the second pattern (three cross-classified factors) some students attended three schools, changing schools twice (from treatment to mediator to outcome). Derivations for the mediated effect formulas are presented.

Theoretical Framework

Multilevel mediation models extend the single-level mediation model to allow handling of clustered data. However, if data are not purely nested, ignoring the structure and using a conventional multilevel mediation model may lead to biased parameter and SE estimates. In the simplest cross-classified data example, lower level units are purely nested within two higher level clustering units although neither higher level cluster is nested within the other (e.g., Goldstein, 2010; Rasbash & Brown, 2001).

A mediation model for cross-classified data has not yet been developed. Given how frequently cross-classified data are encountered, it is important for researchers to appropriately handle this kind of data structure when testing mediation hypotheses. The current study introduces a bivariate multilevel model for testing mediation for cross-classified multisite 1->1->1 data.

Model for Two Cross-Classified Factors

A single, combined model for the mediator, M, and outcome, Y, variables is introduced that uses dummy-coding variables. The model allows outcome-specific (i.e., for the mediator and distal outcome) level-1 residuals’ variances. At level-2, there are five parameters including M’s intercept (M-Beta_0_j1j2), the effect of the treatment on M (a_j1j2), the intercept for Y (Y-Beta_0_j1j2), the effect of M on Y (b_j1j2), and the direct effect (c-prime_j1j2). Each parameter is modeled as randomly varying across each cross-classified factor. Matching convention, covariances between residuals for different cross-classified factors are set to zero. The covariances between residuals for the same cross-classified factor are estimated. The mediated effect’s expected value was derived to equal the product of a and b fixed effects plus the covariance between a_j1j2’s and b_j1j2’s residuals for cross-classified factor j1 plus the covariance between a_j1j2’s and b_j1j2’s residuals for cross-classified factor j2.

Model for Three Cross-Classified Factors

When some students are associated with a different cluster for treatment onset, mediator and distal outcome measurements, a third random effect is added to some of the previously mentioned five parameters’ equations resulting in: M-Beta_0_j1j2, a_j1j2, Y-Beta_0_j1j2j3, b_j1j2j3, and c-prime_j1j2j3. The same expected value results for the mediated effect.

Method, Results and Discussion
MCMC estimation through RStan was used to estimate the two models using ECLS-K data (where X=full- versus part-time kindergarten, M=1st grade reading IRT and Y=5th grade math IRT). Tables 7 and 8 list parameter and SE estimates for the first pattern. The final paper includes a fuller discussion of both models’ results and their justification, derivations of the mediated effects and guidelines for applied researchers.

Authors

• Anita Israni, The University of Texas - Austin

• Susan Natasha Beretvas, The University of Texas - Austin