Diagnosing misspecification of the random-effects distribution in mixed models

Area of Science:

• Statistics
• Biostatistics

Background:

• Mixed models commonly assume multivariate normal distributions for random effects.
• This assumption can be violated in practice, leading to unreliable statistical inferences.

Purpose of the Study:

• To introduce a novel diagnostic test for assessing the random-effects distribution in mixed models.
• To provide a method for validating the distributional assumptions of random effects across various mixed model types.

Main Methods:

• Development of a diagnostic test based on the gradient function.
• Establishment of asymptotic properties, showing convergence to a weighted sum of chi-squared variables.
• Implementation of a parametric bootstrap algorithm for small sample sizes.

Main Results:

• The proposed test statistic's asymptotic properties were established.
• Weights for the chi-squared distribution are derived from matrix eigenvalues.
• Simulations and a real-world study demonstrated the test's utility.

Conclusions:

• The novel diagnostic test effectively assesses random-effects distributions in mixed models.
• The method is applicable to linear, generalized linear, and non-linear mixed models.
• This approach enhances the reliability of statistical analyses in diverse research settings.