title: Generalized t-statistic for two-group classification.
abstract: In the classic discriminant model of two multivariate normal distributions with equal variance matrices, the linear discriminant function is optimal both in terms of the log likelihood ratio and in terms of maximizing the standardized difference (the t-statistic) between the means of the two distributions. In a typical case-control study, normality may be sensible for the control sample but heterogeneity and uncertainty in diagnosis may suggest that a more flexible model is needed for the cases. We generalize the t-statistic approach by finding the linear function which maximizes a standardized difference but with data from one of the groups (the cases) filtered by a possibly nonlinear function U. We study conditions for consistency of the method and find the function U which is optimal in the sense of asymptotic efficiency. Optimality may also extend to other measures of discriminatory efficiency such as the area under the receiver operating characteristic curve. The optimal function U depends on a scalar probability density function which can be estimated non-parametrically using a standard numerical algorithm. A lasso-like version for variable selection is implemented by adding L1-regularization to the generalized t-statistic. Two microarray data sets in the study of asthma and various cancers are used as motivating examples.
authors.affiliation: Institute of Statistical Mathematics, Midori-cho, Tachikawa, Tokyo 190-8562, Japan.
authors.forename: Osamu
authors.initials: O
authors.lastname: Komori
authors.affiliation: Institute of Statistical Mathematics, Midori-cho, Tachikawa, Tokyo 190-8562, Japan.
authors.forename: Shinto
authors.initials: S
authors.lastname: Eguchi
authors.affiliation: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
authors.forename: John B
authors.initials: JB
authors.lastname: Copas
mesh_terms: D000485:Allergens; D001249:Asthma / Q000276:immunology; D001699:Biometry; D016022:Case-Control Studies; D016002:Discriminant Analysis; D006801:Humans; D016013:Likelihood Functions; D016014:Linear Models; D015233:Models, Statistical; D015999:Multivariate Analysis; D012372:ROC Curve; D018709:Statistics, Nonparametric
chemical_list: D000485:Allergens
keywords: Area under the ROC curve; Asymptotic variance; Fisher linear discriminant function; Kullback-Leibler divergence; Lasso; t-Statistic