A default Bayes factor for testing null hypotheses about the fixed effects of linear two-level models.

Abstract

Testing null hypotheses of the form "β = 0," by the use of various Null Hypothesis Significance Tests (rendering a dichotomous reject/not reject decision), is considered standard practice when evaluating the individual parameters of statistical models. Bayes factors for testing these (and other) hypotheses allow users to quantify the evidence in the data that is in favor of a hypothesis. Unfortunately, when testing equality-contained hypotheses, the Bayes factors are sensitive to the specification of prior distributions, which may be hard to specify by applied researchers. The paper proposes a default Bayes factor with clear operating characteristics when used for testing whether the fixed parameters of linear two-level models are equal to zero. This is achieved by generalizing an already existing approach for linear regression. The generalization requires: (a) the sample size for which a new estimator for the effective sample size in two-level models containing random slopes is proposed; (b) the effect size for the fixed effects for which the so-called marginal R² for the fixed effects is used. Implementing the aforementioned requirements in a small simulation study shows that the Bayes factor yields clear operating characteristics regardless of the value for sample size and the estimation method. The paper gives practical examples and access to an easy-to-use wrapper function to calculate Bayes factors for hypotheses with respect to the fixed coefficients of linear two-level models by using the R package bain. (PsycInfo Database Record (c) 2025 APA, all rights reserved).