Goodness-of-fit testing using components based on marginal frequencies of multinomial data

Area of Science:

• Statistics
• Statistical Modeling
• Data Analysis

Background:

• Pearson's chi-squared goodness-of-fit test is an omnibus test, often lacking specificity in identifying the source of poor fit.
• Focused or directional tests can outperform omnibus tests in detecting specific deviations from the null hypothesis.

Purpose of the Study:

• To develop a more informative goodness-of-fit test for models on cross-classified data.
• To present orthogonal components of the Pearson-Fisher statistic based on marginal frequencies.
• To enhance the power and interpretability of goodness-of-fit tests.

Main Methods:

• A score statistic is presented for overlapping cells corresponding to marginal frequencies of dichotomous variables.
• Orthogonal components of the Pearson-Fisher statistic are defined on marginal frequencies.
• A log-linear item response model is used to analyze test components and their projections.

Main Results:

• The proposed orthogonal components can form effective test statistics.
• These components demonstrate advantages in statistical power and in detecting the specific source of poor model fit.
• Using components based on marginal frequencies mitigates issues related to data sparseness.

Conclusions:

• The developed test statistics based on orthogonal components offer superior power and better localization of poor fit compared to traditional omnibus tests.
• These methods provide a more reliable asymptotic chi-squared distribution, particularly beneficial when dealing with sparse expected frequencies in joint distributions.