Co-Ordinate Transformations to Normality and the Power of Normal Tests for Independence

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY This paper considers properties of the normal theory test for independence applied to observations from bivariate distributions constrained to have normal marginals. Coordinate transformations to normality are applied to a variety of non-normal distributions (X, Y) and it is found that the normal theory test for independence is more powerful when applied to the transformed vector than when (X, Y) is used. The general transformation of an arbitrary continuous variate to standard normality is discussed and a method for obtaining an accurate estimator of this transformation is presented. The method is applied to several sets of real data and improvement is demonstrated in terms of agreement with previously established theory.

Efficiency of test for independence after Box–Cox transformation

“…These effects are why one may consider a transformation to normality. For example, Kowalski [10] and Kowalski and Tarter [12] assume that normal correlation analysis is robust with respect to the kinds of non-normality possible when the marginals are constrained to be normal. Kowalski [10] notes that for a wide range of bivariate distributions, transformed correlation agrees more closely with the normal theory distribution of the sample correlation coefficient for a wide range of values of the correlation.…”

“…Pitman [18] shows that r x , the sample correlation between X 1 and X 2 , has zero expectation and variance 1/(n − 1) when X 1 and X 2 are independent. Kowalski and Tarter [12] study the use of normalizing transformations as a prelude to applying normal-theory techniques. Given non-normal bivariate random variables (X 1 , X 2 ), the method consists of making co-ordinate transformations Y i = −1 (F i (X i )) for i = 1, 2 where −1 is the quantile function of the standard normal distribution andF i are the Fourier estimators of the marginal distribution functions.…”

“…. , n. Using Monte Carlo simulation, Kowalski and Tarter [12] show that the normal-theory test of independence is generally more powerful if they are based on r y than r x .…”

The OC and ASN Functions of Some SPRT's for the Correlation Coefficient

“…While it must be admitted that marginal normality is not sufficient for bivariate normality, Kowalski (1970) has discussed this problem and argued that normal correlation analyses are robust with respect to the kinds of departures from bivariate normality possible when the margins are constrained to be normal. Kowalski and Tarter (1969) showed that the use of the transformed observations results in a more powerful test for independence than that based on the original observations for a wide class of non-normal (X, Y) distributions.…”

Non-Normal Bivariate Distributions with Normal Marginals

“…Non-Normal Bivariate Distributions with Normal Marginals CHARLES J. KOWALSKI* Examples of non-normal bivariate (multivariate) distributions with normal marginals [15,16,18] can be used in the classroom to (a) contrast the correlation/ regression structures of these distributions with that of the corresponding bivariate normal model [9,10,11,12], (b) investigate the feasibility of employing coordinate transformations to normality as a prelude to analyses based on the assumption of joint normality [2,13,15,16,17,18,27] and (c) motivate the need for the development of multidimensional goodness-of-fit tests [2,13,16]. In addition, since most of the examples presented here are concerned with ways to generate distributions with arbitrary marginals, they can also be used to obtain a wide variety of non-normal bivariate (multivariate) distributions for these, and other, purposes [2,11,12,13].…”