Optimal Scaling of Interaction Effects in Generalized Linear Models

Multiplicative interaction models, such as Goodman's (1981) RC(M) association models, can be a useful tool for analyzing the content of interaction effects. However, most models for interaction effects are suitable only for data sets with two or three predictor variables. Here, we discuss an optimal scaling model for analyzing the content of interaction effects in generalized linear models with any number of categorical predictor variables. This model, which we call the optimal scaling of interactions model, is a parsimonious, one-dimensional multiplicative interaction model. We discuss how the model can be used to visually interpret the interaction effects. Several extensions of the one-dimensional model are also explored. Finally, two data sets are used to show how the results of the model can be applied and interpreted. The first data set is based on the Student/Teacher Achievement Ratio project and is used to investigate the effects of class size on the performance of primary school students. The second data set comprises four questions from the 1994 General Social Survey (Davis & Smith, 1996) on attitudes of the labor roles of women.

[1]  L. Guttman,et al.  The Quantification of a class of attributes : A theory and method of scale construction , 1941 .

[2]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[3]  J. Kruskal.  Analysis of Factorial Experiments by Estimating Monotone Transformations of the Data , 1965 .

[4]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[5]  Fred A. van Eeuwijk,et al.  MULTIPLICATIVE INTERACTION IN GENERALIZED LINEAR MODELS , 1995 .

[6]  M. W. Richardson,et al.  Making a rating scale that measures. , 1933 .

[7]  Jeroen K. Vermunt,et al.  3. Log-Multiplicative Association Models as Latent Variable Models for Nominal and/or Ordinal Data , 2000 .

[8]  A. Gifi,et al.  NONLINEAR MULTIVARIATE ANALYSIS , 1990 .

[9]  A. Mooijaart,et al.  Three-factor association models for three-way contingency tables , 1997 .

[10]  B. Francis,et al.  Algorithmic approaches for fitting bilinear models. , 1992 .

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

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

[13]  F. A. van Eeuwijk,et al.  Between and beyond additivity and non-additivity : the statistical modelling of genotype by environment interaction in plant breeding , 1996 .

[14]  Jacqueline J Meulman,et al.  Nonlinear principal components analysis: introduction and application. , 2007, Psychological methods.

[15]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[16]  Forrest W. Young.  Quantitative analysis of qualitative data , 1981 .

[17]  Vartan Choulakian,et al.  Generalized bilinear models , 1996 .

[18]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[19]  R. Wong.  Multidimensional Association Models , 2001 .

[20]  P. Groenen,et al.  A New Model for Visualizing Interactions in Analysis of Variance , 2004 .

[21]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation: Rejoinder , 1985 .

[22]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[23]  Forrest W. Young,et al.  Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features , 1976 .

[24]  Carolyn J. Anderson.  The analysis of three-way contingency tables by three-mode association models , 1996 .

[25]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[26]  K. Gabriel,et al.  Generalised bilinear regression , 1998 .

[27]  C. Borror.  Generalized Linear Models and Extensions, Second Edition , 2008 .

[28]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

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

[30]  J. P. Benzécri,et al.  Histoire et préhistoire de l'analyse des données , 1976 .

[31]  Forrest W. Young,et al.  Nonmetric Common Factor Analysis: An Alternating Least Squares Method with Optimal Scaling Features , 1979 .

[32]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[33]  Pieter M. Kroonenberg,et al.  Multiplicative Models for Interaction in Three-Way ANOVA, with Applications to Plant Breeding , 1998 .

[34]  Forrest W. Young,et al.  Additive structure in qualitative data: An alternating least squares method with optimal scaling features , 1976 .

[35]  M. Hill,et al.  Nonlinear Multivariate Analysis. , 1990 .