Analysis of Sets of Two-Way Contingency Tables Using Association Models

Abstract A class of models is introduced for the analysis of group differences in the association between two discrete variables. The RC(M) association model for two-way tables is reviewed, and alternative weighting systems for identifying interaction parameters are presented. This model is generalized for the setting where a two-way contingency table is available for two or more groups. Various restricted models can be used to examine possible sources of intergroup heterogeneity in the association. These sources pertain to heterogeneity in the intrinsic association and/or in the scores for the row and column variables. The importance of weights used to identify the row and column scores is emphasized. A classical set of data previously analyzed by many authors is used to illustrate the advantages of the models and methods developed here.

[1]  R. Fisher.  THE PRECISION OF DISCRIMINANT FUNCTIONS , 1940 .

[2]  K. Maung,et al.  MEASUREMENT OF ASSOCIATION IN A CONTINGENCY TABLE WITH SPECIAL REFERENCE TO THE PIGMENTATION OF HAIR AND EYE COLOURS OF SCOTTISH SCHOOL CHILDREN , 1941 .

[3]  A. W. Edwards.  The Measure of Association in a 2 × 2 Table , 1963 .

[4]  M. W. Birch.  Maximum Likelihood in Three-Way Contingency Tables , 1963 .

[5]  Frederick Mosteller,et al.  Association and Estimation in Contingency Tables , 1968 .

[6]  J. Lingoes.  The Multivariate Analysis Of Qualitative Data , 1968 .

[7]  H. O. Lancaster.  The chi-squared distribution , 1971 .

[8]  L. A. Goodman.  The Multivariate Analysis of Qualitative Data: Interactions among Multiple Classifications , 1970 .

[9]  P. Altham.  The Measurement of Association in a Contingency Table: Three Extensions of the Cross‐Ratios and Metrics Methods , 1970 .

[10]  K. Gabriel,et al.  The biplot graphic display of matrices with application to principal component analysis , 1971 .

[11]  Leo A. Goodman,et al.  Guided and Unguided Methods for the Selection of Models for a Set of T Multidimensional Contingency Tables , 1973 .

[12]  Robin Plackett.  The analysis of categorical data , 1974 .

[13]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[14]  L. A. Goodman.  Simple Models for the Analysis of Association in Cross-Classifications Having Ordered Categories , 1979 .

[15]  Erling B. Andersen,et al.  Discrete Statistical Models with Social Science Applications. , 1980 .

[16]  Leo A. Goodman,et al.  Association Models and Canonical Correlation in the Analysis of Cross-Classifications Having Ordered Categories , 1981 .

[17]  Clifford C. Clogg,et al.  Some Models for the Analysis of Association in Multiway Cross-Classifications Having Ordered Categories , 1982 .

[18]  Clifford C. Clogg,et al.  The Analysis of Categorical Data (2nd Ed.). , 1983 .

[19]  R. Clarke,et al.  Theory and Applications of Correspondence Analysis , 1985 .

[20]  L. A. Goodman,et al.  Latent Structure Analysis of a Set of Multidimensional Contingency Tables , 1984 .

[21]  New methods for the analysis of two-way contigency tables: An alternative to Diaconis and Efron , 1985 .

[22]  Discussion: Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic , 1985 .

[23]  D. Gheva,et al.  Methods for diagnosing multiplicative-interaction models for two-way contingency tables , 1985 .

[24]  L. A. Goodman.  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 , 1985 .

[25]  S. Haberman,et al.  Canonical Analysis of Contingency Tables by Maximum Likelihood , 1986 .

[26]  Leo A. Goodman,et al.  Some Useful Extensions of the Usual Correspondence Analysis Approach and the Usual Log-Linear Models Approach in the Analysis of Contingency Tables , 1986 .

[27]  A. Krieger,et al.  Ordinal Association in Contingency Tables: Some Interpretive Aspects , 1988 .

[28]  Shelby J. Haberman,et al.  The analysis of multivariate contingency tables by restricted canonical and restricted association models , 1988 .