Goodman Association Model
Also known as: RC association model, row-column association model, log-multiplicative model, RC(M) model
Goodman's association models, especially the row-column (RC) model, analyze the association in a two-way contingency table by representing it as a product of estimated scores for the row categories and scores for the column categories, scaled by an intrinsic association parameter. Introduced by Leo Goodman in 1979, they are log-multiplicative rather than purely log-linear, allowing ordered categories to be assigned data-driven scores and the strength of association to be summarized in a single, interpretable coefficient.
Key highlights
- Summarizes association in one (or a few) interpretable parameters rather than a full set of cell interactions.
- Estimates category scores from the data, revealing the empirical ordering and spacing of ordinal categories.
- The unidiff/log-multiplicative layer model compares association strength across many tables parsimoniously.
- Connects log-linear modeling, correspondence analysis, and latent-structure analysis within one framework.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use a Goodman association model when you have a contingency table — especially with ordered row and column categories — and want to summarize the association compactly, recover data-driven category scores, or compare association strength across tables (the unidiff design). It is well suited to mobility tables, attitude-by-attitude tables, and education-by-occupation tables. It is less appropriate when categories are purely nominal with no latent ordering (though RC can still be applied), when the association is genuinely cell-specific and not low-dimensional, or when cells are too sparse for stable score estimation. Identifiability constraints and dimension selection require care.
Strengths & limitations
- Summarizes association in one (or a few) interpretable parameters rather than a full set of cell interactions.
- Estimates category scores from the data, revealing the empirical ordering and spacing of ordinal categories.
- The unidiff/log-multiplicative layer model compares association strength across many tables parsimoniously.
- Connects log-linear modeling, correspondence analysis, and latent-structure analysis within one framework.
- Estimation is nonlinear and requires identifying constraints; scores are determined only up to those normalizations.
- Choosing the number of RC(M) dimensions is a model-selection problem that can be ambiguous.
- Assumes the association has low-dimensional multiplicative structure; genuinely irregular cell associations are missed.
- Sparse cells destabilize the estimated scores and the association parameter.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
How does the RC model differ from a saturated log-linear model?
A saturated log-linear model fits a separate interaction parameter for every cell and so reproduces the table exactly but offers no parsimony or interpretation. The RC model replaces those many interaction terms with a single multiplicative product of estimated row and column scores times an association parameter, capturing the bulk of the association with far fewer parameters and yielding interpretable category scores and a summary measure of strength.
What is the unidiff (uniform difference) model?
The unidiff, or log-multiplicative layer-effect, model assumes that several tables (e.g., different countries or cohorts) share the same pattern of association but differ in its overall strength by a single multiplicative factor per table. It thus summarizes whether social fluidity is stronger or weaker in each table with one parameter, making cross-table comparison both parsimonious and interpretable.
Do the categories need to be ordered to use these models?
The models are most natural and interpretable for ordered categories, where estimated scores recover the latent ordering. They can also be applied to nominal categories, where the RC scores simply reflect the dimensions of association rather than a known order. Uniform-association models with fixed integer scores, however, presuppose ordinal categories with meaningful spacing.
Sources
- 1.Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74(367), 537–552.
- 2.Goodman, L. A. (1985). The analysis of cross-classified data having ordered and/or unordered categories: association models, correlation models, and asymmetry models for contingency tables with or without missing entries. The Annals of Statistics, 13(1), 10–69.
You have read it. What now?
Cite this page
ScholarGate. (2026, June 22). Goodman Association Model. ScholarGate. https://scholargate.app/sociology/goodman-association-model