  Most classification methods provide either a prediction of class membership
or an assessment of class membership probability. In the case of two-group
classification the predicted probability can be described as "risk" of
belonging to a "special" class . When the required output is a set of
ordinal-risk groups, a discretization of the continuous risk prediction is
achieved by two common methods: by constructing a set of models that describe
the conditional risk function at specific points (quantile regression) or by
dividing the output of an "optimal" classification model into adjacent
intervals that correspond to the desired risk groups. By defining a new error
measure for the distribution of risk onto intervals we are able to identify
lower bounds on the accuracy of these methods, showing sub-optimality both in
their distribution of risk and in the efficiency of their resulting partition
into intervals. By adding a new form of constraint to the existing maximum
likelihood optimization framework and by introducing a penalty function to
avoid degenerate solutions, we show how existing methods can be augmented to
solve the ordinal risk-group classification problem. We implement our method
for logistic regression (LR) and show a numeric example.
