Coding invariance in factorial linear models and a new tool for assessing combinatorial equivalence of factorial designs

This paper provides new insights into coding invariance for linear models with qualitative factors, including a coding invariant way of denoting the model coefficients. On this basis, “interaction contributions” (ICs) are proposed for decomposing generalized word counts for factorial designs into contributions that neither depend on level allocation nor on the coding of factors. Combinatorially equivalent designs yield the same ICs, so that ICs are suitable for classifying factorial designs with qualitative factors. ICs are based on singular value decomposition and have an interpretation in terms of bias contributions of an interaction on the estimation of the overall mean. The paper introduces ICs and their tabulations in interaction contribution frequency tables and illustrates their behavior in various examples. ICs are compared to several other tools for assessing combinatorial equivalence of general factorial designs, and they are found to provide a useful complement to existing methods.