Modern Stochastics: Theory and Applications

The area under the receiver operating characteristic curve (AUC) is a suitable measure for the quality of classification algorithms. Here we use the theory of U-statistics in order to derive new confidence intervals for it. The new confidence intervals take into account that only the total sample size used to calculate the AUC can be controlled, while the number of members of the case group and the number of members of the control group are random. We show that the new confidence intervals can not only be used in order to evaluate the quality of the fitted model, but also to judge the quality of the classification algorithm itself. We would like to take this opportunity to show that two popular confidence intervals for the AUC, namely DeLong’s interval and the Mann–Whitney intervals due to Sen, coincide.

In clustering of high-dimensional data a variable selection is commonly applied to obtain an accurate grouping of the samples. For two-class problems this selection may be carried out by fitting a mixture distribution to each variable. We propose a hybrid method for estimating a parametric mixture of two symmetric densities. The estimator combines the method of moments with the minimum distance approach. An evaluation study including both extensive simulations and gene expression data from acute leukemia patients shows that the hybrid method outperforms a maximum-likelihood estimator in model-based clustering. The hybrid estimator is flexible and performs well also under imprecise model assumptions, suggesting that it is robust and suited for real problems.

In this paper we develop a general framework for quantifying how binary risk factors jointly influence a binary outcome. Our key result is an additive expansion of odds ratios as a sum of marginal effects and interaction terms of varying order. These odds ratio expansions are used for estimating the excess odds ratio, attributable proportion and synergy index for a case-control dataset by means of maximum likelihood from a logistic regression model. The confidence intervals associated with these estimates of joint effects and interaction of risk factors rely on the delta method. Our methodology is illustrated with a large Nordic meta dataset for multiple sclerosis. It combines four studies, with a total of 6265 cases and 8401 controls. It has three risk factors (smoking and two genetic factors) and a number of other confounding variables.