An Extended GFfit Statistic Defined on Orthogonal Components of Pearson's Chi-Square

Area of Science:

• Statistics
• Multivariate Analysis
• Statistical Modeling

Background:

• Pearson and likelihood ratio statistics are standard for multinomial goodness-of-fit tests.
• Global tests offer limited insight into the source of poor model fit.
• Existing diagnostics like GFfit focus on lower-order marginals to address sparseness.

Purpose of the Study:

• To introduce an extended GFfit statistic for improved goodness-of-fit testing in cross-classified tables.
• To enhance diagnostic capabilities for identifying sources of model misspecification.
• To address limitations of traditional global tests in sparse, high-dimensional data.

Main Methods:

• Decomposing the Pearson statistic into orthogonal components based on marginal distributions.
• Defining a new GFfit statistic as a partial sum of these orthogonal components.
• Extending the GFfit statistic to higher-order tables, ensuring summation to the full Pearson statistic.

Main Results:

• The proposed GFfit statistics demonstrate higher power for detecting lack of fit compared to existing methods.
• These statistics maintain good Type I error control even with sparse joint frequencies.
• Theoretical results confirm known degrees of freedom and asymptotic independence, facilitating FDR/FWER control.

Conclusions:

• The extended GFfit statistic provides a powerful and computationally stable diagnostic for multinomial goodness-of-fit.
• It effectively addresses sparseness issues in high-dimensional tables.
• This method offers improved model misspecification diagnostics, applicable to various models including latent variable models.