Multilevel models account for clustered data by incorporating group-specific random coefficients. Testing whether a random coefficient should be included in a multilevel model involves the test of whether the variance of that random coefficient is equal to 0. This is problematic because the null hypothesis lies on the boundary of the parameter space. Such issues are addressed in the literature in the context of linear mixed models, but there is relatively little research in the context of multilevel models. We extend an approach for the linear mixed model to multilevel models by scaling the random coefficients to the residual variance and introducing parameters that control the relative contribution of the random coefficients. After integrating over the random coefficients and variance components, the resulting integrals needed to calculate the Bayes factor can be efficiently approximated with Laplace’s method. The method incorporates default prior distributions that are shown to have good frequentist properties and large sample consistency. A major contribution of our method is the ability to test several variance components for multiple factors simultaneously, and to do so for nested, non-nested, or cross-nested multilevel designs. We illustrate our method using a study of infant birth weights in New York City.
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