The NPAR1WAY Procedure

Fligner-Policello Test

If you specify the FP option, PROC NPAR1WAY computes the Fligner-Policello location test for two-sample data (Fligner and Policello 1981). The null hypothesis for the test is upper H 0 colon theta Subscript x Baseline equals theta Subscript y Baseline, where theta Subscript x and theta Subscript y are the population medians of the two classes. The Fligner-Policello 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. For more information, see Hollander and Wolfe (1999) and Juneau (2007).

The Fligner-Policello test is based on placement scores (Orban and Wolfe 1979). The placement of an observation upper X Subscript i from class X, upper P left-parenthesis upper X Subscript i Baseline right-parenthesis, is defined as the number of observations in class Y that are less than upper X Subscript i. If there are ties, the placement of upper X Subscript i is adjusted by adding half the number of observations in class Y that are equal to upper X Subscript i. The placement of an observation upper Y Subscript j from class Y, upper P left-parenthesis upper Y Subscript j Baseline right-parenthesis, is defined in the same way. The placements can be expressed as

StartLayout 1st Row 1st Column upper P left-parenthesis upper X Subscript i Baseline right-parenthesis 2nd Column equals 3rd Column sigma-summation Underscript j equals 1 Overscript n Subscript y Endscripts left-parenthesis normal upper I left-parenthesis upper Y Subscript j Baseline less-than upper X Subscript i Baseline right-parenthesis plus 0.5 normal upper I left-parenthesis upper Y Subscript j Baseline equals upper X Subscript i Baseline right-parenthesis right-parenthesis 2nd Row 1st Column upper P left-parenthesis upper Y Subscript j Baseline right-parenthesis 2nd Column equals 3rd Column sigma-summation Underscript i equals 1 Overscript n Subscript x Endscripts left-parenthesis normal upper I left-parenthesis upper X Subscript i Baseline less-than upper Y Subscript j Baseline right-parenthesis plus 0.5 normal upper I left-parenthesis upper X Subscript i Baseline equals upper Y Subscript j Baseline right-parenthesis right-parenthesis EndLayout

where normal upper I left-parenthesis dot right-parenthesis is an indicator function and n Subscript x and n Subscript y denote the number of observations in class X and class Y, respectively.

The average placements for class X and class Y are computed as

StartLayout 1st Row 1st Column upper P overbar Subscript x 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript i equals 1 Overscript n Subscript x Baseline Endscripts upper P left-parenthesis upper X Subscript i Baseline right-parenthesis right-parenthesis slash n Subscript x 2nd Row 1st Column upper P overbar Subscript y 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript j equals 1 Overscript n Subscript y Baseline Endscripts upper P left-parenthesis upper Y Subscript j Baseline right-parenthesis right-parenthesis slash n Subscript y EndLayout

The Fligner-Policello test statistic is computed as

z equals left-parenthesis sigma-summation Underscript j equals 1 Overscript n Subscript y Baseline Endscripts upper P left-parenthesis upper Y Subscript j Baseline right-parenthesis minus sigma-summation Underscript i equals 1 Overscript n Subscript x Baseline Endscripts upper P left-parenthesis upper X Subscript i Baseline right-parenthesis right-parenthesis slash left-parenthesis 2 StartRoot upper V Subscript x Baseline plus upper V Subscript y Baseline plus upper P overbar Subscript x Baseline upper P overbar Subscript y Baseline EndRoot right-parenthesis

where

StartLayout 1st Row 1st Column upper V Subscript x 2nd Column equals 3rd Column sigma-summation Underscript i equals 1 Overscript n Subscript x Endscripts left-parenthesis upper P left-parenthesis upper X Subscript i Baseline right-parenthesis minus upper P overbar Subscript x Baseline right-parenthesis squared 2nd Row 1st Column upper V Subscript y 2nd Column equals 3rd Column sigma-summation Underscript j equals 1 Overscript n Subscript y Endscripts left-parenthesis upper P left-parenthesis upper Y Subscript j Baseline right-parenthesis minus upper P overbar Subscript y Baseline right-parenthesis squared EndLayout

and the standard deviation of the placements is StartRoot upper V Subscript x Baseline slash left-parenthesis n Subscript x Baseline minus 1 right-parenthesis EndRoot for class X and StartRoot upper V Subscript y Baseline slash left-parenthesis n Subscript y Baseline minus 1 right-parenthesis EndRoot for class Y.

Under the null hypothesis, the Fligner-Policello statistic has an asymptotic standard normal distribution. PROC NPAR1WAY provides one- and two-sided asymptotic p-values for the Fligner-Policello test. For the one-sided test, PROC NPAR1WAY displays the right-sided p-value when the test statistic z is greater than its null hypothesis expected value of 0. PROC NPAR1WAY displays the left-sided p-value when the test statistic z is less than or equal to 0. The one-sided p-value upper P 1 left-parenthesis z right-parenthesis can be expressed as

upper P 1 left-parenthesis z right-parenthesis equals StartLayout Enlarged left-brace 1st Row normal upper P normal r normal o normal b left-parenthesis upper Z greater-than z right-parenthesis normal i normal f z greater-than 0 2nd Row normal upper P normal r normal o normal b left-parenthesis upper Z less-than z right-parenthesis normal i normal f z less-than-or-equal-to 0 EndLayout

where Z has a standard normal distribution. The two-sided p-value upper P 2 left-parenthesis z right-parenthesis is computed as Prob(|Z| > |z|).

When you specify the FP option, PROC NPAR1WAY displays a "Fligner-Policello Placements" table and a "Fligner-Policello Test" table. The "Fligner-Policello 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 "Fligner-Policello Test" table contains the test statistic z and the corresponding one- and two-sided p-values. 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 Fligner-Policello placements.

PROC NPAR1WAY computes the Fligner-Policello 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 larger of the two classes as the reference class X. If both classes have the same number of observations, PROC NPAR1WAY uses the class that appears second in the "Fligner-Policello Placements" table as the reference class. (By default, the table displays class levels in the order in which they appear in the input data set. If you specify the ORDER=FORMATTED option, the table displays class levels in order of their formatted value.)

You can specify the reference class by using the FP(REFCLASS=) option. REFCLASS=1 specifies the first class that is listed in the "Fligner-Policello Placements" table, and REFCLASS=2 specifies the second class. REFCLASS='class-value' identifies the reference class by the formatted value of the CLASS variable.

Last updated: December 09, 2022