Models for non-additive interaction effects

Ordinary least squares is typically used to model continuous outcomes in risk adjustment analysis, even when some of the model assumptions are not met or the population is highly comorbid. Attempts at adding interactions into a model are made but the model quickly becomes hard to explain and analyze. We found that when predicting cost, a main effects model on square-root transformed cost performs best followed by a generalized linear model assuming a gamma distribution and a square-root link. These two models implicitly introduce 2-way interactions into the modeling process. Motivated by the need for interactions and with the goal that effect estimates should be in a form where they can be easily understood, we propose a series of models that account for interactions in data but with reduced complexity and that have parameters that are easier to interpret than the models with square-root transformed outcome or link.;The first model proposed adds one multiplicative parameter to a main effects model, which attempts to smooth all possible interactions present in the data by assuming that interactions have a similar effect. To remove this constraint, other models are developed that allow the use of multiple parameters for disjoint sets of combinations of the main effects. The effect of interactions is expressed as a value greater than zero. It represents the weight given to the sum of the main effects based on the independent variable indicators a particular subject has. For example, an estimate of 1.2 means that a person with two or more diseases is predicted to cost 20% higher than the sum of the main effects for those diseases.;Simulation results show that the new models perform better than the main effects model when the data is generated with some form of interaction included. When applied to real data, there are cases where the new models perform well achieving better results than the main effects model. In the situations where the new models perform as well as the main effects model, the new model has the added benefit of grouping interactions with similar effects aiding the understanding of the data.