This module contains log-linear modeling procedures for multi-way frequency tables. The term log-linear derives from the fact that we can, through logarithmic transformations, restate the problem of analyzing multi-way frequency tables in terms that are very similar to ANOVA. Specifically, we may think of the multi-way frequency table to reflect various main effects and interaction effects that add together in a linear fashion to bring about the observed table of frequencies.

When analyzing four-way or higher tables, finding the best fitting model can become increasingly difficult. Therefore an option was included for automatic model fitting to facilitate the search for a good model.

The statistical significance of the goodness-of-fit of a particular model is evaluated via a Chi-square test. This module computes two types of Chi-squares:

- traditional Pearson Chi-square statistic
- maximum likelihood ratio Chi-square statistic

Both tests evaluate whether the expected cell frequencies under the respective model are significantly different from the observed cell frequencies.

The following residual statistics are available. Note: F_{ijk}.. denotes the fitted or expected cell frequency for cell i,j,k,... And f_{ijk}.. denotes the observed frequency.

Raw residuals (r_{ijk}..) are computed as:

r_{ijk..}= f_{ijk..} - F_{ijk..}

Standardized residuals (sijk..) are computed as:

s_{ijk..} = (f_{ijk..} - F_{ijk..}) / (F_{ijk..})^{½}

In the equation below, **ln** denotes the natural logarithm. These are the contributions of each cell to the overall maximum likelihood ratio Chi-square goodness-of-fit statistic:

c_{ijk..} = 2 * f_{ijk..} * ln(f_{ijk..} / F_{ijk..})

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

fr_{ijk..} = f_{ijk..}^{½} + (f_{ijk..} + 1)^{½ }- (4 * F_{ijk..} + 1)^{½}

For additional information on the computations of the Pearson Chi-square and maximum likelihood ratio Chi-square statistics, see Bishop, Fienberg, and Holland (1974) and Fienberg (1978).

For additional information on the logic of automatic model selection see Goodman (1971).

See Deming & Stephan, 1940; Brown, 1959; Ireland & Kullback, 1968; Haberman, 1972, 1974 for information on the fitting of models (marginal tables) to observed frequency tables via iterative proportional fitting.

Note: The General Linear Models module provides additional options for analyzing binomial and multinomial logit models with coded ANOVA/ANCOVA-like designs.

Historical Note: The term likelihood ratio for Chi-square was first introduced by Neyman and Pearson, in 1931. The term maximum likelihood was first used by Fisher, 1922a.

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