Elastic Net Constrained Stereotype Logit Model for Ordered Categorical Data

In biomedical studies the outcome of interest may be an ordinal, rather than a dichotomous, class label such as progression of disease. An example of an ordinal variable is drug toxicity levels evaluated as mild, moderate or severe. Another example is the Breast Imaging Reporting and Data System (BIRADS) [1] classification system. After a mammogram is read, a subjective score is assigned based on the condition of the breast tissue. These categories are: Category 0 Incomplete; Category 1Negative; Category 2 Benign; Category 3 -Probably Benign; Category 4 Suspicious Abnormality; Category 5 Highly Suspicious of Malignancy; and Category 6 Known Biopsy Proven Malignancy. The ordinality of these categories is evident. As another example, when cancer treatments are applied there is usually an interest in how patients respond. A typical way to measure this response is called the Revised Response Evaluation Criteria in Solid Tumors (RECIST) [2]. Based on a wide variety of tools, as well as defined rules for classification, Revised RECIST defines the responses as: Complete Response, Partial Response, Stable Disease, and Progressive Disease. The types of models used to model ordinal data include the multinomial, adjacent category logit, continuation ratio logit, proportional odds logit, stereotype logit, and cumulative link models [3]. These models have the assumption, among others, that there are considerably more observations than variables. However, there are many types of data for which there are more variables than observations. When using microarray based, or other high throughout technologies, due to the expense of obtaining samples, there may be few observations but thousands of variables. The afore-mentioned models are not estimable by traditional means or without additional assumptions. Although there are data dimensionality reduction techniques, such as principle component analysis, due to the severely unbalanced nature of the data it may still be impossible to satisfactorily reduce the subset of variables to be less than the number of observational units without a significant loss of information in the data. This paper is concerned with the development of an ordinal classification model using the Least Absolute Shrinkage and Selection Operator (LASSO) and ridge penalizations to accommodate the case where there are considerably more variables than observations. This described procedure uses the stereotype logit model [4] with the applied penalty in an attempt to overcome said problems. The proposed method is applied to simulated data. An algorithm is presented in which the above penalized likelihood is utilized to model high dimensional data with an ordinal outcome; the algorithm is applied to an actual data set with promising results.