Regression models with parametrically weighted explanatory variables

This paper describes linear regression models with parametrically weighted explanatory variables and related logistic regression models that estimate parameters characterizing (1) the effects of weighted variables on the dependent variable and (2) weights for the components of weighted variables. The models also characterize parsimoniously the interaction effects between weighted variables and covariates on the dependent variable by the use of various constraints on parameters. In particular, the models are concerned with testing the significance of variation with covariates in the weights of weighted variables separately from the significance of variation with those covariates in the effects of weighted variables. The usefulness of these models in sociological research is demonstrated by an illustrative analysis of the class identifications of married working women using education, occupational prestige, and income as three variables weighted between own and spousal attributes, and using year, age, race, part-time-full-time distinction, and employment status as covariates.