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UNC-Chapel Hill BIOS 662 - An analysis of variance test for normality

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An Analysis of Variance Test for Normality (Complete Samples)S. S. Shapiro; M. B. WilkBiometrika, Vol. 52, No. 3/4. (Dec., 1965), pp. 591-611.Stable URL:http://links.jstor.org/sici?sici=0006-3444%28196512%2952%3A3%2F4%3C591%3AAAOVTF%3E2.0.CO%3B2-BBiometrika is currently published by Biometrika Trust.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/bio.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgMon Oct 1 13:09:39 2007Biometrika (1965), 52, 3 and 2, p. 591 With 5 text-jgures Printed in &eat Britain An analysis of variance test for normality (complete samp1es)t BYS. S. SHAPIRO AND M. B. WILK General Electric Go. and Bell Telephone Laboratories, Inc. The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing trans- formations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in $2. Operational information and tables useful in employing the test are detailed in $3(which may be read independently of the rest of the paper). Some examples are given in $4. Section5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in $6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2.1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order models can be fitted and an 3'-type ratio used to evaluate the adequacy of the linear fit. t Part of this research was supported by the Office of Naval Research while both authors were at Rutgers University.This approach was investigated in preliminary work. While some promising results were obtained, the procedure is subject to the serious shortcoming that the selection of the higher-order model is, practically speaking, arbitrary. However, research is continuing along these lines. Another analysis of variance viewpoint which has been investigated by the present authors is to compare the squared slope of the probability plot regression line, which under the normality hypothesis is an estimate of the population variance multiplied by a constant, with the residual mean square about the regression line, which is another estimate of the variance. This procedure can be used with incomplete samples and has been described elsewhere (Shapiro & Wilk, 1965b). As an alternative to the above, for complete samples, the squared slope may be com- pared with the usual symmetric sample sum of squares about the mean which is independent of the ordering and easily computable. It is this last statistic that is discussed in the re- mainder of this paper. 2.2. Derivation of the W statistic Let m' = (ml,m,, ...,m,) denote the vector of expected values of standard normal order statistics, and let V = (vii) be the corresponding n x n covariance matrix. That is, if x, 6 x, 6 . . .x, denotes an ordered random sample of size n from a normal distribution with mean 0 and variance 1, then E(x)~= mi (i= 1,2,...,n), and cov (xi, xj) = vii (i,j = 1,2,...,n). Let y' = (y,, ...,y,) denote a vector of ordered random observations. The objective is to derive a test for the hypothesis that this is a sample from a normal distribution with unknown mean p and unknown variance a,. Clearly, if the {y,} are a normal sample then yi may be expressed as yi=p+rxi (i= 1,2,...,n). It follows from the generalized least-squares theorem


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