Fisher's Linear Discriminant Function Analysis and its Potential Utility as a Tool for the Assessment of Health-and-Wellness Programs in Indigenous Communities.

Diabetes mellitus is a growing public health problem affecting persons in both developed and developing nations. The prevalence of type 2 diabetes mellitus (T2DM) is reported to be several times higher among Indigenous populations compared to their non-Indigenous counterparts. Discriminant function analysis (DFA) is a potential tool that can be used to quantitatively evaluate the effectiveness of Indigenous health-and-wellness programs (e.g., on-the-land programs, T2DM interventions), by creating a type of pre-and-post-program scoring system. As the communities of the Eeyou Istchee territory, subarctic Quebec, Canada, have varying degrees of isolation, we derived a DFA tool for point-of-contact evaluations to aid in monitoring and assessment of health-and-wellness programs in rural and remote locations. We developed several DFA models to discriminate between those with and without T2DM status using age, fasting blood glucose, body mass index, waist girth, systolic and diastolic blood pressure, high-density lipoprotein, triglycerides, and total cholesterol in participants from the Eeyou Istchee. The models showed a ~97% specificity (i.e., true positives for non-T2DM) in classification. This study highlights how varying risk factor models can be used to discriminate those without T2DM with high specificity among James Bay Cree communities in Canada.

Generalized t-statistic for two-group classification.

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.