The Joint Assessment of Normality of Several Independent Samples

Statistical methods are presented for the joint assessment of the supposed normality of a collection of independent (small) samples, which may derive from populations having differing means and variances. The procedures are based on the use of the W statistic (Shapiro and Wilk, 1965) as a measure of departure from normality. Two modes of combination of a collection of W statistics are considered, namely the standardized mean of the normal transforms of W and the sum of the x 2(2) transforms of the significance levels of w; and these are proposed for use in conjunction with the probability plotting of the collection of these transforms. Tables and formulae for practical implementation are provided. Some summary empirical sampling results are given on the comparative sensitivities of these procedures, along with detailed consideration of several specific examples which illustrate the additional informative value of probability plotting. The proposed techniques appear to have substantial data analysis value ...

[1]  E. S. Pearson,et al.  On questions raised by the combination of tests based on discontinuous distributions. , 1950, Biometrika.

[2]  I. J. Good,et al.  On the Weighted Combination of Significance Tests , 1955 .

[3]  M. Zelen,et al.  The Weighted Compounding of Two Independent Significance Tests , 1959 .

[4]  E. S. Pearson.  THE PROBABILITY INTEGRAL TRANSFORMATION FOR TESTING GOODNESS OF FIT AND COMBINING INDEPENDENT TESTS OF SIGNIFICANCE , 1938 .

[5]  R. Gnanadesikan,et al.  Probability plotting methods for the analysis of data. , 1968, Biometrika.

[6]  R. Gnanadesikan,et al.  Probability Plots for the Gamma Distribution , 1962 .

[7]  R. Fisher.  Statistical Methods for Research Workers , 1971 .

[8]  S. Shapiro,et al.  An Analysis of Variance Test for Normality (Complete Samples) , 1965 .

[9]  S. Shapiro,et al.  A Comparative Study of Various Tests for Normality , 1968 .

[10]  B WILKINSON,et al.  A statistical consideration in psychological research. , 1951, Psychological bulletin.

[11]  W. Allen Wallis,et al.  Compounding Probabilities from Independent Significance Tests , 1942 .

[12]  H. O. Lancaster.  The combination of probabilities arising from data in discrete distributions. , 1949, Biometrika.

[13]  M. B. Wilk,et al.  Approximations for the Null Distribution of the W Statistic , 1968 .

[14]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[15]  Allan Birnbaum,et al.  Combining Independent Tests of Significance , 1954 .