The application of bias to discriminant analysis

When classification rules are constructed using sample estimatest it is known that the probability of misclassification is not minimized. This article introduces a biased minimum X² rule to classify items from a multivariate normal population. Using the principle of variance reduction, the probability of misclassification is reduced when the biased procedure is employed. Results of sampling experiments over a broad range of conditions are provided to demonstrate this improvement.     

A three-population constrained discrimination procedure

ABSTRACT

Classification rules with a reserve judgment option provide a way to satisfy constraints on the misclassification probabilities when there is a high degree of overlap among the populations. Constructing rules which maximize the probability of correct classification while satisfying such constraints is a difficult optimization problem. This paper uses the form of the optimal solution to develop a relatively simple and computationally fast method for three populations which has a non parametric quality in controlling the misclassification probabilities. Simulations demonstrate that this procedure performs well.     

Known Mean,Unknown Maxima? Testing the Maximum Knowing Only the Mean

In the quantitative group testing problem, the use of the group mean to identify if the group maximum is greater than a prefixed threshold (infected group) is analyzed, using n independent and identically distributed individuals. Under these conditions, it is shown that the information of the mean is sufficient to classify each group as infected or healthy with low probability of misclassification when the underline distribution is a unilateral heavy-tailed distribution.     

The influence of observations on misclassification probability in multiple discriminant analysis

The influence of observations in estimating the misclassification probability in multiple discriminant analysis is studied using the common omission approach. An empirical influence function for the misclassification probability is also derived, It can give a very good approximation to the omission approach, but the computational load is much reduced, Various extensions of the measures are suggested. The proposed measures are applied to the famous Iris data set. The same three observations are identified as having the most influence under different measures.     

Maximum Likelihood Estimators in Regression Models for Error‐prone Group Testing Data

Since Dorfman's seminal work on the subject, group testing has been widely adopted in epidemiological studies. In Dorfman's context of detecting syphilis, group testing entails pooling blood samples and testing the pools, as opposed to testing individual samples. A negative pool indicates all individuals in the pool free of syphilis antigen, whereas a positive pool suggests one or more individuals carry the antigen. With covariate information collected, researchers have considered regression models that allow one to estimate covariate‐adjusted disease probability. We study maximum likelihood estimators of covariate effects in these regression models when the group testing response is prone to error. We show that, when compared with inference drawn from individual testing data, inference based on group testing data can be more resilient to response misclassification in terms of bias and efficiency. We provide valuable guidance on designing the group composition to alleviate adverse effects of misclassification on statistical inference.     

Posterior probability estimators in classification simulations

Class specific stratified posterior probability estimators of misclassification probabilities in discriminant analysis simulations are introduced. These estimators afford a significant variance reduction over the usual count estimators. Sufficient conditions for a variance reduction are given. The stratified posterior probability estimator is generalized to other class specific expectations.     

Systematically misclassified binary dependent variables

ABSTRACT

When a binary dependent variable is misclassified, that is, recorded in the category other than where it really belongs, probit and logit estimates are biased and inconsistent. In some cases, the probability of misclassification may vary systematically with covariates, and thus be endogenous. In this paper, we develop an estimation approach that corrects for endogenous misclassification, validate our approach using a simulation study, and apply it to the analysis of a treatment program designed to improve family dynamics. Our results show that endogenous misclassification could lead to potentially incorrect conclusions unless corrected using an appropriate technique.     

An investigation of the effects of inspection errors on the run-length control charts

For continuous inspection schemes in an automated manufacturing environment, a useful alternative to the traditional p or np chart is the Run-Length control chart, which is based on plotting the run lengths (the number of conforming items) between successive nonconforming items. However, its establishment relies on the error-free inspection assumption, which can seldom be met in practice. In this paper, the effects of inspection errors on the Run-Length chart are investigated based on that these errors are assumed known. The actual false alarm probability and the average number inspected (ANI) in the presence of inspection errors are studied. This paper also presents the adjusted control limits for the Run-Length chart, which can provide much closer ANI curves to the ones obtained under error-free inspection.     

Simple computation of a bivariate normal integral arising from a problem of misclassification with applications to the diagnosis of hypertension

The joint distribution of the true and observed values of a variable that is subject to measurement error is bivariate normal.An important special case occurs when we want the joint probability of the true value being below a cutoff point and the observed value above it.In that case the required integral can be simply evaluated using a Gaussian quadrature formula, which can easily be evaluated using a calculator.This formula is used to estimate the probabilities of misclassification of participants in screening programs for hypertension.It shows that basing a diagnosis on a single visit, at which a single measurement was made leads to a very high risk of misclassification.The probability of a subject having a blood pressure below the cutoff point, given that the observed pressure is above it, would be 0.45.Increasing the number of visits to three, and measuring the blood pressure twice at each visit, as advocated by Rosner and Polk (1979), would bring the probability down to 0.29.     

Testing for Mutual Exclusivity

A test for two events being mutually exclusive is presented for the case in which there are known rates of misclassification of the events. The test can be utilized in other situations, such as to test whether a set is a subset of another set. In the test, the null value of the probability of the intersection is replaced by the expected value of the number determined to be in the intersection by the imperfect diagnostic tools. The test statistic is the number in a sample that is judged to be in the intersection. Medical testing applications are emphasized.     

A nonparametric allocation scheme for classification based on transvariation probabilities

In this paper, a nonparametric discriminant analysis procedure that is less sensitive than traditional procedures to deviations from the usual assumptions is proposed. The procedure uses the projection pursuit methodology where the projection index is the two-group transvariation probability. Montanari [A. Montanari, Linear discriminant analysis and transvariation, J. Classification 21 (2004), pp. 71–88] proposed and used this projection index to measure group separation but allocated the new observation using distances. Our procedure employs a method of allocation based on group–group transvariation probability to classify the new observation. A simulation study shows that the procedure proposed in this paper provides lower misclassification error rates than classical procedures like linear discriminant analysis and quadratic discriminant analysis and recent procedures like maximum depth and Montanari's transvariation-based classifiers, when the underlying distributions are skewed and/or the prior probabilities are unequal.     

A Classification Rule Whose Probability of Misclassification is Independent of the Variance

An observation ×_o is to be classified into one of two normal populations φ₁ and φ₂. A classification rule, the Two-stage sample Rule, R(TS), whose probability of misclassification, P[MC], is independent of the common but unknown variance is proposed. Some optimal properties of R(TS) are also discussed and some values of P[MC | R(TS)], the probability of misclassification given the rule R(TS), are tabulated.     

Robustness of the linear discriminant function to nonnormality: Edgeworth series distribution

The effects of applying the normal classificatory rule to a nonnormal population are studied here. These are assessed through the distribution of the misclassification errors in the case of the Edgeworth type distribution. Both theoretical and empirical results are presented. An examination of the latter shows that the effects of this type of nonnormality are marginal. The probability of misclassification of an observation from ∏₁, using the appropriate LR rule, is always larger than one using the normal approximation (μ₁<μ₂). Converse condition holds for the misclassification of an observation from ∏₂. Overall error rates are not affected by the skewness factor to any great extent.     

When a Constant Classifier is as Good as Any Linear Classifier

A classifier is constant if it classifies all examples into just one class. Call a training data set “(linearly) indiscriminate” if a constant classifier minimizes, among all linear classifiers, the misclassification rate on the training data set. General sufficient conditions are presented for the probability of getting an indiscriminate data set to be positive. Similarly, general sufficient conditions are also presented for the probability of getting an indiscriminate data set to be 0.

A small simulation study examines how our results are reflected in the behavior of logistic regression.     

A multiple testing protocol for exploratory data analysis and the local misclassification rate

A false discovery rate (FDR) procedure is often employed in exploratory data analysis to determine which among thousands or millions of attributes are worthy of follow-up analysis. However, these methods tend to discover the most statistically significant attributes, which need not be the most worthy of further exploration. This article provides a new FDR-controlling method that allows for the nature of the exploratory analysis to be considered when determining which attributes are discovered. To illustrate, a study in which the objective is to classify discoveries into one of several clusters is considered, and a new FDR method that minimizes the misclassification rate is developed. It is shown analytically and with simulation that the proposed method performs better than competing methods.     

Biased discriminant analysis: Evaluation of the optimum probability of misclassification

This article extends the work of DiPillo (1976) on the Biased Minimum x² Rule. The optimum value of k (the biasing factor) Is determined and the true probability of misclassification is found. The proportion improvements reported in the 1976 paper are shown to be conservative. Some suggestions for algorithms to determine the optimal value of k are presented.     

A note on optimum linear feature extraction for gaussian populations with equal covariances and equal a priori probabilities

We consider the linear feature selection problem of obtaining a nonzero 1 × n matrix B which minimizes the probability of misclassification based on the Bayes decision rule applied to the random variable Y = BX, where X is a random n-vector arising from one of m Gaussian populations with equal covariances and equal apriori probabilities. It is shown that the optimal B satisfies a fixed point equation B = F(B) which can be solved by successive substitution.     

Smooth Nonparametric Allocation of Classification

A nonparametric discriminant analysis procedure that is robust to deviations from the usual assumptions is proposed. The procedure uses the projection pursuit methodology where the projection index is the two-group transvariation probability. We use allocation based on the centrality of the new point measured using a smooth version of point-group transvariation. It is shown that the new procedure provides lower misclassification error rates than competing methods for data from skewed heavy-tailed and skewed distributions as well as unequal training data sizes.     

Factors causing occurrence of artificial dif: A simulation study for dichotomous data

ABSTRACT

In modern test theory, differential item functioning (DIF) appears where respondents from two different groups have the same ability but different probability to respond to an item correctly. If some items favouring one group lead to the appearance of DIF in any other item favouring the other group, this type of problem is called artificial differential item functioning (A-DIF). The purpose of this paper is to deal with the effect of different factors causing A-DIF under the Rasch theoretical model for dichotomous responses. A simulation study was conducted to explore how various factors affect real DIF and simultaneous A-DIF expected proportion including total sample size, percentage of individuals in focal and reference groups, percentage of items exhibiting real DIF and DIF magnitude for two item sets including 10 and 20 items. It is concluded that DIF magnitude is the most essential factor while evaluating A-DIF for each item set. This is followed by percent of items exhibiting real-DIF.     

Bayesian adjustment for unidirectional misclassification in ordinal covariates

In this paper, we study the identification of Bayesian regression models, when an ordinal covariate is subject to unidirectional misclassification. Xia and Gustafson [Bayesian regression models adjusting for unidirectional covariate misclassification. Can J Stat. 2016;44(2):198–218] obtained model identifiability for non-binary regression models, when there is a binary covariate subject to unidirectional misclassification. In the current paper, we establish the moment identifiability of regression models for misclassified ordinal covariates with more than two categories, based on forms of observable moments. Computational studies are conducted that confirm the theoretical results. We apply the method to two datasets, one from the Medical Expenditure Panel Survey (MEPS), and the other from Translational Research Investigating Underlying Disparities in Acute Myocardial infarction Patients Health Status (TRIUMPH).