Ordinal Logic Regression: A classifier for discovering combinations of binary markers for ordinal outcomes.

In medicine, it is often useful to stratify patients according to disease risk, severity, or response to therapy. Since many diseases arise from complex gene-gene and gene-environment interactions, patient strata may be defined by combinations of genetic and environmental factors. Traditional statistical methods require specifying interactions a priori making it difficult to identify high order interactions. Alternatively, machine learning methods can model complex interactions, however these models are often difficult to interpret in a clinical setting. Logic regression (LR) enables modeling a binary outcome using logical combinations of binary predictors yielding easily interpretable models. However LR, as currently available, cannot model ordinal responses. This paper extends LR to model an ordinal response and the resulting method is called Ordinal Logic Regression (OLR). Several simulations comparing OLR and Classification and Regression Trees (CART) demonstrate that OLR is superior to CART for identifying variable interactions associated with an ordinal response. OLR is applied to data from a study to determine associations between genetic and health factors with severity of adult periodontitis.