Correlation Analysis of Extreme Observations from a Multivariate Normal Distribution

Article excerpt

In measuring visual acuity, the extremes of a set of normally distributed measures are of concern, together with one or more covariates. This leads to a model in which (X, [Y.sub.1], [Y.sub.2]) are jointly normally distributed with [Y.sub.1], [Y.sub.2] exchangeable and (X, [X.sub.i]) having a common correlation. Inferential procedures are developed for correlations and linear regressions among X and the ordered Y values. This requires determination of the covariance matrix of X, [Y.sub.(1)] = min{[Y.sub.1], [Y.sub.2]} and [Y.sub.(2)] = max{[Y.sub.1], [Y.sub.2]}. The inadequacy of certain estimates that ignore the nonnormality of {X, [Y.sub.(1)], [Y.sub.(2)] is also discussed. Although the bivariate case is emphasized because of the context of the visual acuity model, many results are given for the more general multivariate case.

KEY WORDS: Elliptically contoured distributions; Order statistics; Regression with order statistics.

1. INTRODUCTION

Visual acuity is measured by a (Snellen) eye chart and expressed in log units of the minimum angle of resolution, or log MAR. Smaller values of log MAR correspond to better visual acuity. Normally, a single measure of visual acuity is made in each eye (say [Y.sub.1], [Y.sub.2]), together with one or more covariates X, such as the subject's age, physical condition, etc. But because visual acuities generally are unequal in the presence of certain macular lesions, of interest are not the measures [Y.sub.1], [Y.sub.2] but rather the extreme visual acuities: the "best" acuity, [Y.sub.(1)] = min {[Y.sub.1], [Y.sub.2]} and the "worst" acuity, [Y.sub.(2)] = max{[Y.sub.1], [Y.sub.2]}. A common example is the current criterion for an unrestricted driver's license, which in most states is based on the visual acuity of the best eye (see Fishman et al. 1993 and Szlyk, Fishman, Sovering, Alexander, and Viana 1993). Other applications include the assessment of defective hearing in mentally retarded adults based on the ear with best hearing (Parving and Christensen 1990), the predictive value of the worst vision following a filtration operation in the aphakic eyes of elderly patients with glaucoma (Frenkel and Shin 1986), the comparison between best vision achieved following cataract surgery and two methods of secondary posterior capsulotomy (Knolle 1985), sports injury data on the reduction of best vision in damaged eyes (Aburn 1990), and the analysis of worst vision in patients treated for macular edema (Rehak and Vymazal 1989).

Although we can obtain maximum likelihood estimates (MLE's) or Bayes estimates of the parameters of the underlying distribution of (X, [Y.sub.1], [Y.sub.2]), our current interest is with the parameters of the distribution of (X, [Y.sub.(1)], [Y.sub.(2)]). More specifically, we are interested in making inferences on the covariance structure defined by

[phi] = Cov ( X, [Y.sub.(10], [Y.sub.(2)],

with particular interest in assessing the correlations [[eta].sub.i] between X and [Y.sub.(i)], the correlation [theta] between [Y.sub.(1)] and [Y.sub.(2)], and the parameters defining the best linear predictors between X and the ordered Y values. Because the assumption of a multivariate normal distribution for (X, [Y.sub.1], [Y.sub.2]) does not imply a multivariate normal distribution for (X, [Y.sub.(1)], [Y.sub.(2)]), we need to determine how well the correlation parameters [[eta].sub.i] and [theta] reflect the actual dependence between the corresponding variables.

Although the model of interest relates to two Y values, the results obtained in the following section are generalizable to p variables [Y.sub.1], [Y.sub.2],..., [Y.sub.p] and to the order statistics [Y.sub.(1)] [less than or equal to] [Y.sub.(2)] [less than or equal to] [Y.sub.(p)]. Also, the normality assumption can be weakened to that of an elliptically contoured distribution. In some instances we state our results under these more general conditions.

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