Noncentrality Parameters in Chi-Squared Goodness-of-Fit Analyses with an Application to Log-Linear Procedures

Abstract

Limiting noncentral chi-squared distributions are obtained for statistics commonly used in sequential stepwise testing to select from a general class of functions the correct model for T multinomial cell frequencies in an m-way cross-classification table. The simple quadratic form expressions are used to establish the asymptotic independence of tests of nested models used in stepwise selection. In an application of these results, the quadratic form expression for the noncentrality parameter of the asymptotic distribution of chi-squared statistics testing H ₀ versus H ₁, when H ₂ is true is used to compare the powers of hierarchical log-linear model-selection procedures. Tables and charts of approximate powers of stepwise testing procedures to select the correct log-linear model for 2 × 2 × 2 and 2 × 3 × 3 cross-classification tables are presented for a range of sample sizes and magnitudes of higher-order effect parameters. The powers of backward elimination and forward selection procedures are compared.

Key Words:

• Noncentral chi-square
• Asymptotic independence
• Numerical powers of sequential procedures
• Log-linear model selection
• Backward elimination
• Forward selection