Introduction
Classic receiver operator characteristic (ROC) curve analysis addresses the relation of continuous measurements to binary outcomes \cite{Agresti2014}, including identification of a cutpoint or threshold on the continuous measurement scale discriminating the outcome levels. From its origins in signal detection theory \cite{Egan1966} and application in early radio detection and ranging systems, the technique has been used in fields as diverse as clinical chemistry \cite{Hanley2005}, radiology \cite{Pepe2004}, psychology \cite{Swets_1973}, and machine learning \cite{Hernandez-Orallo2012,Majnik2013,Prati2011a}.
Extension beyond binary outcomes would be desirable for the increased scope of applications. One readily implemented approach is to group multinomial outcome levels into binomial levels and run classic ROC curve analysis, but this loses information and biases test accuracy \cite{Obuchowski2004a}. There have been other, more sophisticated proposals spanning a range of theoretical approaches \cite{Nakas2010,Nakas2013,Inacio2011,Li2012,Kijewski1989,Mossman1999,Dreiseitl2000,Edwards2004}, but the complexity of these noteworthy proposals has limited their application. Additional methods have been implemented and some enjoy broad use \cite{Lopez-Raton2014,Chang2017,Budczies2012a,Camp2004}, yet theoretical justification may be sparse.
This paper proposes a two-stage, semiparametric approach combining conventional cumulative logit regression with a cumulative extension of ROC curve analysis to discriminate ordinal outcome levels. The performance of this approach is evaluated under simulation, with comparison of several criteria used with classic ROC curves to select cutpoints. Results from these criteria are compared to cutpoints computed from maximum likelihood estimates (MLEs) of the regression parameters. The procedure is also demonstrated with publicly available data.