Using single variables to represent interactions

This paper suggests a way of using residuals to represent interaction terms to enable a stepwise (forward or backward) comparison of the interactions can be performed in SPSS. This overcomes the problem of having to have the main effects and any lower order interactions in a model, at all times, in a model with higher order interactions. For example the need for all models to contain main effects A and B when assessing the A x B interaction.

The rationale for the above method uses the fact that an interaction is the extra information in a product over and above that given by the combination of terms comprising that interaction e.g. a Downs Syndrome (Yes/No) group x age interaction represents the differences in an outcome at a given age over and above that expected assuming age has the same relationship with outcome in both groups and a constant difference in outcome between the two groups (tested by fitting the interaction in a model already containing the main effects of age and group).

Reference

Burrill, D. (1998). Modelling and interpreting interactions in multiple regression. [On-line]. Also available at: http://www.minitab.com/uploadedFiles/Shared_Resources/Documents/Articles/interactions_in_multiple_regression.pdf.