Log-Linear Analysis - Notes and Technical Information Algorithms

The fitting of models (marginal tables) to observed frequency tables is accomplished via iterative proportional fitting (see Deming & Stephan, 1940; Brown, 1959; Ireland & Kullback, 1968; Haberman, 1972, 1974). The computations of the Pearson Chi-square and maximum likelihood ratio Chi-square statistics are described in detail by Bishop, Fienberg, and Holland (1975) and Fienberg (1978). The logic of the automatic selection algorithm is described in Goodman (1971).

Residual Statistics

The following residual statistics are available from the Results dialog (note: Fijk.. denotes the fitted or expected cell frequency for cell i,j,k,...; fijk.. denotes the observed frequency):

Raw residuals. Raw residuals (rijk..) are computed as:

rijk..=fijk..-Fijk..

Standardized residuals. Standardized residuals (sijk..) are computed as:

sijk..=(fijk..-Fijk..)/(Fijk..)

Components of maximum likelihood. These are the contributions of each cell to the overall maximum likelihood ratio Chi-square goodness-of-fit statistic:

cijk..=2*fijk..*ln(fijk../Fijk..)

In this equation, ln denotes the natural logarithm.

Freeman-Tukey deviates. Freeman-Tukey deviates (frijk..) represent a normalizing transformation that is appropriate when the frequencies in the table come from a Poisson distribution:

frijk..=fijk..+(fijk..+1)-(4*Fijk..+1)