Scale of interest versus scale of estimation: comparing alternative estimators for the incremental costs of a comorbidity

We investigate how the scale of estimation in risk-adjustment models for health-care costs affects the covariate effect, where the scale of interest for the covariate effect may be different from the scale of estimation. As an illustrative example, we use claims data to estimate the incremental costs associated with heart failure within one year subsequent to myocardial infarction. Here, the scale of interest for the effect of heart failure on costs is additive. However, traditional methods for modeling costs use predetermined scale of estimation - for example, ordinary least squares (OLS) regression assumes an additive scale while log-transformed OLS and generalized linear models with log-link assume a multiplicative scale of estimation. We compare these models with a new flexible model that lets the data determine the appropriate scale of estimation. We use a variety of goodness-of-fit measures along with a modified Copas test to assess robustness, lack of fit, and over-fitting properties of the alternative estimators. Biases up to 19% in the scale of interest are observed due to the misrepresentation of the scale of estimation. The new flexible model is found to appropriately represent the scale of estimation and less susceptible to over-fitting despite estimating additional parameters in the link and the variance functions.