Tests For Multivariate Normality Research Articles

A joint rank test for assessing multivariate normality in hydrologic data

Applications of stochastic hydrology to the synthesis of multisite data have commonly assumed that the historic data or some suitable transform thereof may be treated as a sample from a multivariate normal distribution. This assumption, generally made for mathematical tractability, is rarely subject to verification. A new test for multivariate normality, based on the properties of joint ranks, is presented. This test is of particular value in the synthesis of drought at multiple sites.

Some empirical distribution function tests for multivariate normality

This paper presents finite sample percentage points of the Anderson-Darling (AD) statistic for testing the composite hypothesis of Gaussianity (normality) for dimensionality 1≦p≦5 when the parameters are estimated from data. This paper also presents asymptotic percentage points for both the Anderson-Darling and the Cramervon Mises(CM) statistics for testing the composite hypothesis of Gaussianity for dimensionality 1≦p≦25 when the parameters are estimated from data. The AD test is developed from the fact that the quadratic form of the Gaussian density is distributed as chi-squared on p degrees of freedom. A small power study contrasts the finite sample performance of the AD and CM statistics. Several examples are discussed in the context of chi-squared probability plotting.

On assessing multivariate normality based on shapiro-wilk W statistic

Shapiro and Wilk's (1965) W statistic has been found to be the best omnibus test for detecting departures from univariate normality. Royston (1983) extends the application of W to testing multivariate normality but the procedure involves a certain approximation which needs to be justified. The procedures proposed in the present paper do not need such an approximation. The asymptotic null distributions are also given. Finally, a numerical example is used to illustrate the procedures.

Testing multivariate normality by simulation

The purpose of this paper is to propose a method of testing multivariate normality which guarantees the asymptotic size of the test. The basic idea is simulation of the actual percentage points of the distribution of a test statistic over a grid in a confidence region. Other tools which are called into service include non-parametric density estimation, likelihood ratio statistics, and spline interpolation. The procedure is compared with three well-known tests in a small simulation experiment.

An Improved Bonferroni Procedure for Multiple Tests of Significance

A modification of the Bonferroni procedure for testing multiple hypotheses is presented. The method, based on the ordered p-values of the individual tests, is less conservative than the classical Bonferroni procedure but is still simple to apply. A simulation study shows that the probability of a type I error of the procedure does not exceed the nominal significance level, α, for a variety of multivariate normal and multivariate gamma test statistics. For independent tests the procedure has type I error probability equal to α. The method appears particularly advantageous over the classical Bonferroni procedure when several highly-correlated test statistics are involved.

The greatest invariance-group of multivariate models

A useful theorem to determine the group of all affine transformations leaving a multivariate normal test problem invariant is presented. For the MANOVA and GMANOVA test problems the well-known invariance groups are shown to be the greatest invariance-groups.

An improved Bonferroni procedure for multiple tests of significance

SUMMARY A modification of the Bonferroni procedure for testing multiple hypotheses is presented. The method, based on the ordered p-values of the individual tests, is less conservative than the classical Bonferroni procedure but is still simple to apply. A simulation study shows that the probability of a type I error of the procedure does not exceed the nominal significance level, a, for a variety of multivariate normal and multivariate gamma test statistics. For independent tests the procedure has type I error probability equal to a. The method appears particularly advantageous over the classical Bonferroni procedure when several highly-correlated test statistics are involved.

Remarks on the use of Shapiro-Wilk statistic for testing multivariate normality

Al fine di verificare l'ipotesi di normalita multivariata Malkovich ed Afifi (4) hanno proposto una generalizzazione della statistica di Shapiro-Wilk ottenuta tramite l'applicazione del principio di unione-intersezione di Roy. Nel presente lavoro si considera una modifica di tale procedura che viene confrontata con quella originale attraverso il metodo di Monte Carlo.

Methodological Issues in Absenteeism Research: Multiple Absence Measures and Multivariate Normality

Four methods for inducing or testing for multivariate normality when using multiple measures of absenteeism are described. They are the elimination of multivariate outliers, the transformation of absence measures to approximate multivariate normality, testing for univariate normality, and testing for multivariate normality. Evidence indicating that application of these methods can enhance the validity of absenteeism studies is presented.

Univariate and multivariate distributions of the job descriptive index's measures of job satisfaction

This study examines the univariate and multivariate distributions of the Job Descriptive Index's (JDI) scales for job satisfaction with work, supervision, pay, promotion, and co-workers. The results indicate that the sample distributions of four of the five scales differ significantly from normality. The exception is satisfaction with work. After normalizing square-root transformations were applied to the nonnormal scales, the results of a test for multivariate normality indicated that the transformed sample data for the satisfaction scales reasonably approximate multivariate normality. The implication for organizational researchers is that the use of univariate statistical procedures that are sensitive to nonnormality is questionable in analyses involving the JDI scales. When it is appropriate to analyze the scales jointly, normalizing transformations are recommended to induce approximate multivariate normality for the joint distribution.

DENBRAN: A basic program for a significance test for multivariate normality of clusters from branching patterns in dendrograms

A BASIC program is presented for significance tests to determine whether a dendrogram is derived from clustering of points that belong to a single multivariate normal distribution. The significance tests are based on statistics of the Kolmogorov—Smirnov type, obtained by comparing the observed cumulative graph of branch levels with a graph for the hypothesis of multivariate normality. The program also permits testing whether the dendrogram could be from a cluster of lower dimensionality due to character correlations. The program makes provision for three similarity coefficients, (1) Euclidean distances, (2) squared Euclidean distances, and (3) Simple Matching Coefficients, and for five cluster methods (1) WPGMA, (2) UPGMA, (3) Single Linkage (or Minimum Spanning Trees), (4) Complete Linkage, and (5) Ward's Increase in Sums of Squares. The program is entitled DENBRAN.

A measure of skewness and kurtosis and a graphical method for assessing multivariate normality

Using principal components, a measure of skewness and kurtosis is developed for multivariate populations. The sample analogues of these measures are proposed as tests of multivariate normality. Also, a graphical method is presented for assessing multivariate normality.

Two Statistics for Testing for Multivariate Normality

Malkovich & Afifi (1973) proposed measures of multivariate skewness and kurtosis. In the present paper, distributional properties of the two statistics are examined. A method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8, for dimensions 2, 3 and 4, and, in principle, for all higher dimensions. The percentage points are derived from appropriate transformations of the asymptotic distributions. A simulation study shows the approximations to be good for sample sizes of 25 and above.

A Bivariate Normal Test for Elliptical Home-Range Models: Biological Implications and Recommendations

I propose a goodness-of-fit test to evaluate compliance of the spatial distribution of activity with bivariate normality. Home-range data for 2 Columbian white-tailed deer fawns (Odocoileus virginianus leucurus) were subjected to this test. The distribution for 1 fawn departed significantly from the expected; the distribution of the other fawn did not. J. WILDL. MANAGE. 47(3):613-619 The study of home ranges and territories (Burt 1943) has become an increasingly prominent approach for elucidating intraand interspecific ecological phenomena among terrestrial vertebrates, notably rodents (Randolph 1977, O'Farrell 1978), insectivores (Hawes 1977), cervids (Robinette 1966, Martinka 1969, Miller 1970, Phillips et al. 1973, Inglis et al. 1979), canids (Ables 1968, Fuller 1978, Andelt and Gipson 1979, Fuller and Keith 1980), felids (Hornocker 1969, Bailey 1975), ursids (Pearson 1975, Landers et al. 1979), mustelids (Mitchell 1961, Shirer and Fitch 1970, Storm 1972), procyonids (Shirer and Fitch 1970, Urban 1970), didelphids (Verts 1963, Shirer and Fitch 1970), leporids (Rongstad and Tester 1971, Trent and Rongstad 1974), fringillids (Wasserman 1980), parulids (Zach and Falls 1978), and iguanids (Waldschmidt 1979). Generally, 2 methods of evaluating home areas (home range or territory size) have been used: nonstatistical (polygon) techniques and statistical models that calculate home areas with the assumption that the locational data follow some probabilistic distribution (Stickel 1954). The elliptical home-range models proposed by Jennrich and Turner (1969) and Koeppl et al. (1975) have gained general acceptance (Hawes 1977, Randolph 1977, O'Farrell 1978, Zach and Falls 1978, Gavin 1979, Inglis et al. 1979, Waldschmidt 1979, Danner and Smith 1980, Wasserman 1980). These models use similar algorithms that are dependent upon the assumption that the locational data follow a bivariate normal distribution. The G, and G2 statistics (Sokal and Rohlf 1969:113) were used to evaluate conformance of locational data to the bivariate normal distribution (Zach and Falls 1978, Waldschmidt 1979). However, skewness (GI) and kurtosis (G2) represent properties of the marginal (univariate) distributions and the accordance of marginal distributions to univariate normality does not generally imply joint conformance to the bivariate normal distribution (Neter and Wasserman 1974:395). This paper presents a technique for evaluating the compliance of locational data to a bivariate normal distribution. In addition to a description of the method and its application, 2 example sets of locational data obtained from telemetrymediated direct observations of marked Columbian white-tailed deer are evaluated. Moore and Stubblebine (1981) described the theoretical basis for a chisquare test of multivariate normality with applications in economics. Here I present a technique, derived from the Moore and IPresent address: Department of Biological Sciences, Southeastern Louisiana University, Hammond, LA 70402. J. Wildl. Manage. 47(3):1983 613 This content downloaded from 157.55.39.111 on Sat, 17 Sep 2016 05:33:05 UTC All use subject to http://about.jstor.org/terms 614 A TEST FOR HOME RANGE MODELS * Smith

Normality tests and transformations

Powerful univariate and multivariate normality tests are surveyed. The advantages and disadvantages of each procedure are exposed, furthermore, a few transformation methods which can be used for classification purposes in case of non-normality, are discussed.

Tests for multivariate normality with pearson alternatives

We consider a multivariate Pearson family of distributions. Certain parametric restrictions lead to the multivariate normal distribution. Using this fact we propose a number of asymptotically efficient tests. Through Monte Carlo experiments these tests are compared with some of the existing test procedures. A table is provided from which finite-sample critical points can be obtained.

Two statistics for testing for multivariate normality

Malkovich & Afifi (1973) proposed measures of multivariate skewness and kurtosis. In the present paper, distributional properties of the two statistics are examined. A method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8, for dimensions 2, 3 and 4, and, in principle, for all higher dimensions. The percentage points are derived from appropriate transformations of the asymptotic distributions. A simulation study shows the approximations to be good for sample sizes of 25 and above.

Some Techniques for Assessing Multivarate Normality Based on the Shapiro- Wilk W

SUMMARY Shapiro and Wilk's (1965) W test is a powerful procedure for detecting departures from univariate normality. The present paper extends the application of W to testing multivariate normality, and also to Healy's (1968) test ba'sed on squared radii. Three examples illustrate the approach, and also the utility of careful scrutiny of lower- dimensional subsets of the data where otherwise unsuspected departures from normality may appear. A fair number of test procedures for multivariate normality have been proposed in the literature. Cox and Small (1978) reviewed most of these recently, and I shall not cover the same ground again. Procedures may be said to concentrate either on combinations of univariate tests of normality (Small, 1979; Malkovich and Afifi, 1973), or on the geometrical properties in Rm of two or more variates taken together (Healy, 1968; Cox and Small, 1978), in particular, probability plots of squared radii from the centroid of the data in the metric defined by the sample covariance matrix (Healy, 1968; Small, 1978). Cox and Small (1978) considered departures represented by curvature in the variate-variate plots. Certain general difficulties appear. Quite frequently, suggested test statistics have intractable null hypothesis distributions. Convincing power studies are rare, due to the large variety of possible types of departure from normality and the associated expense of Monte Carlo computer studies. Outliers, too, have unpredictable effects-for example, Campbell (1980) pointed out the sensitivity of estimated covariance matrices to outliers, and recommended robust estimation of these matrices. Plotting is a valuable aid, but geometry in m dimensions (m > 3) is hard to depict graphically; also, if m is large, the number of possible plots of subsets of variates multiplies rapidly, making for difficulties in interpretation (and heavy computation). The present paper does not pretend to solve all these problems. Fortunately, however, many of the above conditions present related symptoms; for example curvature in the xl V. x2 plot is quite likely to manifest itself as a skew departure from 1-normality in xl or x2 (or both), and a data transformation may correct matters. I shall show that Shapiro and Wilk's (1965) W test furnishes a set of univariate test statistics which show low correlation even when the parent variates are quite highly correlated, which makes the computation of a combined test fairly simple. I also apply the W test to plots of squared radii, and extend the treatment to subsets of variates. Three detailed illustrative examples are given.

On a measure of multivariate skewness and a test for multivariate normality

We consider an extension of Pearson measure of skewness to a multivariate case and apply the proposed measure to a test of multivariate normality.

A bayesian procedure for mastery decisions based on multivariate normal test data

A Bayesian framework for making mastery/nonmastery decisions based on multivariate test data is described in this study. Overall, mastery is granted (or denied) if the posterior expected loss associated with such action is smaller than the one incurred by the denial (or grant) of mastery. An explicit form for the cutting contour which separates mastery and nonmastery states in the test score space is given for multivariate normal test scores and for a constant loss ratio. For multiple cutting scores in the true ability space, the test score cutting contour will resemble the boundary defined by multiple test cutting scores when the test reliabilities are reasonably close to unity. For tests with low reliabilities, decisions may very well be based simply on a suitably chosen composite score.