A comparison of statistical methods for allocating disease costs in the presence of interactions

We consider the non-trivial problem of estimating a health cost repartition among diseases from patients' hospital stays' global costs in the presence of multimorbidity, that is, when the patients may suffer from more than one disease. The problem is even harder in the presence of interactions among the disease costs, that is, when the costs of having, for example, two diseases simultaneously do not match the sum of the basic costs of having each disease alone, generating an extra cost which might be either positive or negative. In such a situation, there might be no "true solution" and the choice of the method to be used to solve the problem will depend on how one wishes to allocate the extra costs among the diseases. In this article, we study mathematically how different methods proceed in this regard, namely ordinary least squares (OLS), generalized linear models (GLM), and an iterative proportional repartition (IPR) algorithm, in a simple case with only two diseases. It turned out that only IPR allowed to retrieve the total costs and the unambiguous solution that one would have in a setting without interaction, that is, when no extra cost has to be allocated, while OLS and GLM may produce some negative health costs. Also, contrary to OLS, IPR is taking into account the basic costs of the diseases for the allocation of the extra cost. We conclude that IPR seems to be the most natural method to solve the problem, at least among those considered.