Classification of samples into two or more ordered populations with application to a cancer trial

Abstract

In many applications, especially in cancer treatment and diagnosis, investigators are interested in classifying patients into various diagnosis groups on the basis of molecular data such as gene expression or proteomic data. Often, some of the diagnosis groups are known to be related to higher or lower values of some of the predictors. The standard methods of classifying patients into various groups do not take into account the underlying order. This could potentially result in high misclassification rates, especially when the number of groups is larger than two. In this article, we develop classification procedures that exploit the underlying order among the mean values of the predictor variables and the diagnostic groups by using ideas from order-restricted inference. We generalize the existing methodology on discrimination under restrictions and provide empirical evidence to demonstrate that the proposed methodology improves over the existing unrestricted methodology. The proposed methodology is applied to a bladder cancer data set where the researchers are interested in classifying patients into various groups.

dawai: AnRPackage for Discriminant Analysis with Additional Information

“…These rules are based on the use of additional information to obtain alternative estimators of the vector means. The generalization to the k > 2 populations case appears in Conde et al (2012). These alternative estimators are defined via an iterative procedure whose convergence is shown in Fernández et al (2006) and that is described here for completeness.…”

“…For more details on these restricted linear rules and their properties the reader is referred to Fernández et al (2006) and Conde et al (2012).…”

“…This is the second theoretical novelty appearing in this paper. As happened when the restricted discriminant rule was extended to the more than 2 populations case in Conde et al (2012), the extension is not immediate as the appropriate parameter spaces and projection matrices have to be considered. Following the arguments already appearing in the original paper, Efron (1979), or in Boos (2003), the underlying idea in the definition of the new estimators of the true error rate is that the "bootstrap world" should mirror the "real world".…”

“…However, that paper provided limited results for the case of two populations with simple order restrictions and identity covariance matrices only. In a series of papers, the rules appearing in that initial paper have been improved, first to deal with more general types of information expressed in terms of cones of restrictions and general covariance matrices in Fernández et al (2006) and later to the case of more than two populations in Conde et al (2012). The robustness of the rules has also been studied in Salvador, Fernández, Martín, and Rueda (2008) and good estimators of the performance of the rules (which is an essential issue in discriminant analysis) have been provided in Conde et al (2013).…”

“…In Section 2 we describe the statistical problem and the methodology of Fernández et al (2006), Conde et al (2012) and Conde et al (2013), which we extend to the above mentioned situations. In Section 3 we introduce the dawai package and explain some details about the main user-lever functions it includes.…”