Semiparametric AUC regression for ordered treatment effects.

We investigated distribution free methods for testing covariate adjusted treatment effects when the researchers believe that these effects are ordered. Dodd and Pepe (2003) proposed a semi-parametric logistic regression model for the area under the ROC curve (AUC). Their approach was motivated by the observation that the Mann-Whitney statistic is a non-parametric estimate of the AUC. Their results allow one to test hypotheses using distribution free methods when the covariates are discrete, however, the standard errors generated using standard GLM software are not correct since the Bernoulli data generated by the Mann-Whitney statistic are correlated. They used the bootstrap method to estimate the standard errors for the AUC regression parameters. Zhang (2008) and Zhang et. al (2011) considered an analytical method for estimating the standard errors based on a modification of a method by DeLong et. al (1988), as an alternative to the bootstrap procedure. In Chapter Two, we compare the DeLong method to two alternative analytical methods for estimating the standard errors. In Chapter Three, we extend the AUC regression model, with and without discrete covariates, to the situation where there are k >2 ordered treatment levels as the alternative hypothesis. This approach extends the Jonckheere-Terpstra statistic (Jonckheere (1954) and Terpstra (1952)) to allow for covariates. In Chapter Four, we introduce a multiple comparison method for the Jonckheere-Terpstra statistic. Chapter Five gives a summary of the results and describes future work.