If you specify the FP option, PROC NPAR1WAY computes the FlignerPolicello location test for twosample data (Fligner and Policello, 1981). The null hypothesis for the test is , where and are the population medians of the two classes. The FlignerPolicello test assumes that the distribution in each class is symmetric around the class median, but it does not require that the two class distributions have the same form or that the class variances be equal. See Hollander and Wolfe (1999) and Juneau (2007) for more information.
The FlignerPolicello test is based on placement scores (Orban and Wolfe, 1979). The placement of an observation from class X, , is defined as the number of observations in class Y that are less than . If there are ties, the placement of is adjusted by adding half the number of observations in class Y that are equal to . The placement of an observation from class Y, , is defined in the same way. The placements can be expressed as






where is an indicator function and and denote the number of observations in class X and class Y, respectively.
The average placements for class X and class Y are computed as






The FlignerPolicello test statistic is computed as
where






and the standard deviation of the placements is for class X and for class Y.
Under the null hypothesis, the FlignerPolicello statistic has an asymptotic standard normal distribution. PROC NPAR1WAY provides one and twosided asymptotic pvalues for the FlignerPolicello test. For the onesided test, PROC NPAR1WAY displays the rightsided pvalue when the test statistic z is greater than its null hypothesis expected value of zero. PROC NPAR1WAY displays the leftsided pvalue when the test statistic z is less than or equal to zero. The onesided pvalue can be expressed as
where Z has a standard normal distribution. The twosided pvalue is computed as Prob(Z > z).
When you specify the FP option, PROC NPAR1WAY displays a “FlignerPolicello Placements” table and a “FlignerPolicello Test” table. The “FlignerPolicello Placements” table contains the following information for each of the two classes: number of observations, sum of the placements, average placement, and standard deviation of the placements. The “FlignerPolicello Test” table contains the test statistic z and the corresponding one and twosided pvalues. This table also displays the difference between the class placement sums, which is the numerator of the test statistic. When ODS Graphics is enabled and you specify the FP or PLOTS=FPBOXPLOT option, PROC NPAR1WAY provides a box plot of the FlignerPolicello placements.
PROC NPAR1WAY computes the FlignerPolicello difference (the numerator of the test statistic) as the placement sum for class Y minus the placement sum for class X (the reference class). By default, PROC NPAR1WAY uses the first class that is listed in the “FlignerPolicello Placements” table as class Y and the second class as the reference class X. The table displays class levels in the order in which they appear in the input data set. To use the first class (instead of the second class) as the reference class for the FlignerPolicello test, you can specify the FP(REFCLASS=1) option. Or you can specify the FP(REFCLASS=classvalue) option to identify the reference class by the formatted value of the CLASS variable.