The two-way mixed model: a long and winding controversy.

Area of Science:

• Statistics
• Statistical Modeling

Background:

• The testing of statistical significance for random factors in two-way mixed models is a long-standing controversy in statistics.
• Discrepancies exist between classical ANOVA texts and professional statistical software regarding the significance of random effects.

Purpose of the Study:

• To analyze the controversy surrounding mixed models and the choice between restrictive and non-restrictive models.
• To address key questions that extend beyond the ongoing debate in statistical significance testing.

Main Methods:

• Detailed analysis of the mixed model controversy.
• Examination of the non-restrictive versus restrictive model options.
• Evaluation of the linear mixed approach and REML estimation.

Main Results:

• The equivalence of two classical models is questioned.
• The marginality principle's limitation in testing main effects with significant interactions is highlighted.
• The relevance of the linear mixed approach for models with fixed and random effects is discussed.

Conclusions:

• The classical linear approach is deemed inapplicable in this context.
• The study proposes the mixed linear approach with REML estimation as a practical solution.
• This approach offers a more robust method for analyzing mixed models.