The UNIVARIATE Procedure

Shapiro-Wilk Statistic

If the sample size is less than or equal to 2000 and you specify the NORMAL option, PROC UNIVARIATE computes the Shapiro-Wilk statistic, (also denoted as to emphasize its dependence on the sample size ). The statistic is the ratio of the best estimator of the variance (based on the square of a linear combination of the order statistics) to the usual corrected sum of squares estimator of the variance (Shapiro and Wilk; 1965). When is greater than three, the coefficients to compute the linear combination of the order statistics are approximated by the method of Royston (1992). The statistic is always greater than zero and less than or equal to one .

Small values of lead to the rejection of the null hypothesis of normality. The distribution of is highly skewed. Seemingly large values of (such as 0.90) may be considered small and lead you to reject the null hypothesis. The method for computing the -value (the probability of obtaining a statistic less than or equal to the observed value) depends on . For , the probability distribution of is known and is used to determine the -value. For , a normalizing transformation is computed:

The values of , , and are functions of obtained from simulation results. Large values of indicate departure from normality, and because the statistic has an approximately standard normal distribution, this distribution is used to determine the -values for .

EDF Goodness-of-Fit Tests

When you fit a parametric distribution, PROC UNIVARIATE provides a series of goodness-of-fit tests based on the empirical distribution function (EDF). The EDF tests offer advantages over traditional chi-square goodness-of-fit test, including improved power and invariance with respect to the histogram midpoints. For a thorough discussion, refer to D’Agostino and Stephens (1986).

The empirical distribution function is defined for a set of independent observations with a common distribution function . Denote the observations ordered from smallest to largest as . The empirical distribution function, , is defined as

Note that is a step function that takes a step of height at each observation. This function estimates the distribution function . At any value , is the proportion of observations less than or equal to , while is the probability of an observation less than or equal to . EDF statistics measure the discrepancy between and .

The computational formulas for the EDF statistics make use of the probability integral transformation . If is the distribution function of , the random variable is uniformly distributed between 0 and 1.

Given observations , the values are computed by applying the transformation, as discussed in the next three sections.

PROC UNIVARIATE provides three EDF tests:

• Kolmogorov-Smirnov

• Anderson-Darling

• Cramér-von Mises

The following sections provide formal definitions of these EDF statistics.

Kolmogorov D Statistic

The Kolmogorov-Smirnov statistic () is defined as

The Kolmogorov-Smirnov statistic belongs to the supremum class of EDF statistics. This class of statistics is based on the largest vertical difference between and .

The Kolmogorov-Smirnov statistic is computed as the maximum of and , where is the largest vertical distance between the EDF and the distribution function when the EDF is greater than the distribution function, and is the largest vertical distance when the EDF is less than the distribution function.

PROC UNIVARIATE uses a modified Kolmogorov statistic to test the data against a normal distribution with mean and variance equal to the sample mean and variance.

Anderson-Darling Statistic

The Anderson-Darling statistic and the Cramér-von Mises statistic belong to the quadratic class of EDF statistics. This class of statistics is based on the squared difference . Quadratic statistics have the following general form:

The function weights the squared difference .

The Anderson-Darling statistic () is defined as

Here the weight function is .

The Anderson-Darling statistic is computed as

Cramér-von Mises Statistic

The Cramér-von Mises statistic () is defined as

Here the weight function is .

The Cramér-von Mises statistic is computed as

proc univariate data=test;
   histogram / gumbel(edfnsamples=5000);
run;