Olive Jean Dunn

Discriminant Functions When Covariance Matrices are Unequal

Abstract This study compares by Monte Carlo methods the performance of three discriminant functions in classifying individuals into two multivariate normally distributed populations when covariance matrices are unequal—the quadratic, best linear and Fishers linear discriminant function. The comparison is carried out both asymptotically and using samples. The expected value of the probabilities is used as the measure of performance. Parameters that are varied in the study include the distance between the populations, covariance matrices, number of dimensions, samples size and a priori probabilities of origin from the populations.

Comparison of Tests of the Equality of Dependent Correlation Coefficients

Abstract When two correlation coefficients are calculated from a single sample, rather than from two samples, they are not statistically independent, and the usual methods for testing equality of the population correlation coefficients no longer apply. This article considers tests to be made using a sample from a multivariate normal distribution. Small sample level of significance and power are obtained using Monte Carlo methods for Hotellings test of , Williamss modification of Hotellings test, and for two tests of , based on Fishers z transformation.

The Robustness of Hotelling's T 2

Abstract Monte Carlo methods are used to study the effect of inequality of covariance matrices on the distribution of Hotellings T 2 statistic. Samples are drawn from multivariate normal distributions and from them the distribution of T 2 is approximated. One minus the significance level is calculated for various departures from equality, for tests with a supposed significance level of .05 and .01. The power of the tests is also given. The number of variates considered is 1, 2, 3, 5, 7, and 10. The effect of sample size on level of significance and power is studied; both equal and unequal sample sizes are used. Let ∑1 and ∑2 be the two covariance matrices. Graphs are presented for one minus the significance level versus d, for the case where all eigenvalues of the matrix ∑2∑1 −1 are equal to d. One minus the significance level is also plotted against the sample size for the case of equal sample size and equal eigenvalues of ∑2∑1 −1. Power curves are also included for several ratios of sample size. In gen...

Note on the use of stepwise regression in detecting outliers

Abstract Cases can be considered as possible outliers in a regression structure if deletion results in a sufficiently large reduction of the residual sum of squares. Calculations for selection of outlier cases on this basis can be accomplished by use of stepwise regression programs. Numerical examples are included.

LONGITUDINAL STUDIES OF INTELLECTUAL AND BEHAVIORAL DEVELOPMENT IN CHILDREN WITH CONGENITAL HEART DISEASE

This is the final paper in a series (5, 6, 9) which reports the results of a five year developmental study comparing the intelligence and adjustment of children with cyanotic congenital heart disease, those children with acyanotic cardiac disease and normal children. Previous papers have not dealt with changes during the five year study period. Possible effects of intervening cardiac operation upon the development of children with congenital heart disease are of primary interest. More than 40% of the children in the cardiac group underwent operation during the course of the investigation, and so we were able to compare the developmen; of operated and nonoperated children. We studied changes in physical capacity, intelligence and adjustment of the child and also changes in the mothers attitude toward her child, the extent of her anxiety and her need to protect and pamper him.

Alternative Approaches to Missing Values in Discriminant Analysis

Abstract This paper compares by simulations several methods of handling missing observations in discrimination. In an earlier paper, several methods were compared in discriminating by the usual linear discriminant function between two multivariate normal populations in which all pairs of variables are equally correlated. In the present study, a variety of population matrices was used and two additional methods were introduced: the first, a simpler regression technique and the second, a modified technique based on the first principal component. The new regression technique was found to give relatively high probabilities of correct classification.

The Treatment of Missing Values in Discriminant Analysis—I. The Sampling Experiment

Abstract Probabilities of correct classification under several commonly used methods of handling missing values are studied by Monte Carlo methods. The methods include use of only complete observation vectors; use of all observations with no replacement; substitution of means for missing observations; Bucks regression method; and, Dears principal component method. Discriminant functions were formed using independent random samples from two multivariate normal distributions with equal covariance matrices. Missing values occur randomly in each variable and independently of missing values in other variables. The mean substitution method and principal component method are, in general, superior to the other methods for cases considered.

Some Expected Values for Probabilities of Correct Classification in Discriminant Analysis

Monte Carlo estimates have been obtained for two quantities of interest in a discriminant analysis involving the usual linear discriminant function. The first is the unconditional probability of correct classification; the second is the expected value of its estimate based on the calculated Mahalanobis distance. These two quantities are shown in tables and graphs versus the population Mahalanobis distance. Equal sample sizes of 25, 50, and 100 have been used in forming the discriminant functions; 2, 6, 10, 15, 20, and 30 variates have been used. A comparison is made between the Monte Carlo estimates of the unconditional probability of correct classification and an approximation suggested by Lachenbruch [41].

A Note on the Asymptotic Aspect of the Treatment of Missing Values in Discriminant Analysis

Abstract In an earlier article the authors studied five methods of handling missing values in discriminant analysis for small samples; this note points out the asymptotic behavior of these methods when the variables are equally correlated. The methods using either complete observation vectors or all sample values always attain the maximum probability of correct classification. Differences of the asymptotic probability of correct classification from its maximum are found to be small for all methods. Hence, comparison of the asymptotic bias seems to be of little practical importance and missing values in discriminant analysis must be studied for small samples.

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.