Inference for the difference in the area under the ROC curve derived from nested binary regression models.

The area under the curve (AUC) statistic is a common measure of model performance in a binary regression model. Nested models are used to ascertain whether the AUC statistic increases when new factors enter the model. The regression coefficient estimates used in the AUC statistics are computed using the maximum rank correlation methodology. Typically, inference for the difference in AUC statistics from nested models is derived under asymptotic normality. In this work, it is demonstrated that the asymptotic normality is true only when at least one of the new factors is associated with the binary outcome. When none of the new factors are associated with the binary outcome, the asymptotic distribution for the difference in AUC statistics is a linear combination of chi-square random variables. Further, when at least one new factor is associated with the outcome and the population difference is small, a variance stabilizing reparameterization improves the asymptotic normality of the AUC difference statistic. A confidence interval using this reparameterization is developed and simulations are generated to determine their coverage properties. The derived confidence interval provides information on the magnitude of the added value of new factors and enables investigators to weigh the size of the improvement against potential costs associated with the new factors. A pancreatic cancer data example is used to illustrate this approach.

General single-index survival regression models for incident and prevalent covariate data and prevalent data without follow-up.

This article mainly focuses on analyzing covariate data from incident and prevalent cohort studies and a prevalent sample with only baseline covariates of interest and truncation times. Our major task in both research streams is to identify the effects of covariates on a failure time through very general single-index survival regression models without observing survival outcomes. With a strict increase of the survival function in the linear predictor, the ratio of incident and prevalent covariate densities is shown to be a non-degenerate and monotonic function of the linear predictor under covariate-independent truncation. Without such a structural assumption, the conditional density of a truncation time in a prevalent cohort is ensured to be a non-degenerate function of the linear predictor. In light of these features, some innovative approaches, which are based on the maximum rank correlation estimation or the pseudo least integrated squares estimation, are developed to estimate the coefficients of covariates up to a scale factor. Existing theoretical results are further used to establish the n -consistency and asymptotic normality of the proposed estimators. Moreover, extensive simulations are conducted to assess and compare the finite-sample performance of various estimators. To illustrate the methodological ideas, we also analyze data from the Worcester Heart Attack Study and the National Comorbidity Survey Replication.

Order-constrained linear optimization.

Despite the fact that data and theories in the social, behavioural, and health sciences are often represented on an ordinal scale, there has been relatively little emphasis on modelling ordinal properties. The most common analytic framework used in psychological science is the general linear model, whose variants include ANOVA, MANOVA, and ordinary linear regression. While these methods are designed to provide the best fit to the metric properties of the data, they are not designed to maximally model ordinal properties. In this paper, we develop an order-constrained linear least-squares (OCLO) optimization algorithm that maximizes the linear least-squares fit to the data conditional on maximizing the ordinal fit based on Kendall's τ. The algorithm builds on the maximum rank correlation estimator (Han, 1987, Journal of Econometrics, 35, 303) and the general monotone model (Dougherty & Thomas, 2012, Psychological Review, 119, 321). Analyses of simulated data indicate that when modelling data that adhere to the assumptions of ordinary least squares, OCLO shows minimal bias, little increase in variance, and almost no loss in out-of-sample predictive accuracy. In contrast, under conditions in which data include a small number of extreme scores (fat-tailed distributions), OCLO shows less bias and variance, and substantially better out-of-sample predictive accuracy, even when the outliers are removed. We show that the advantages of OCLO over ordinary least squares in predicting new observations hold across a variety of scenarios in which researchers must decide to retain or eliminate extreme scores when fitting data.