Dataplot Vol 2 Vol 1

# FLIGNER POLICELLO TEST

Name:
FLIGNER POLICELLO TEST
Type:
Analysis Command
Purpose:
Perform a two sample Fligner-Policello test for equal medians.
Description:
The Fligner-Policello test for equal medians is a nonparametric test based on placement scores. Given two respoonse variables, say Y1 and Y2, the placement score for $$Y1_i$$ is defined as the number of observations in $$Y2$$ that are less than $$Y1_i$$. Likewise, the placement score for $$Y2_i$$ is the number of observations in $$Y1$$ that are less than $$Y2_i$$. Values in $$Y2$$ that are equal to $$Y1_i$$ (or values in $$Y1$$ that are equal to $$Y2_i$$ add 0.5 rather than 1.

If the placement scores are in P1 and P2, the Fligner-Policello test statistic is

$$z = \frac {\sum_{j=1}^{n_{Y1}}{P(Y1_{j})} - \sum_{i=1}^{n_{Y2}}{P(Y2_{i})}} {2 \sqrt{V_{Y1} + V_{Y2} + \bar{P}_{1} \bar{P}_{Y2}}}$$

where $$\bar{P}_{Y1}$$ and $$\bar{P}_{Y2}$$ are the means of the placement scores

$$\bar{P}_{Y1} = \frac{\sum_{i=1}^{n_{Y1}}{P(Y1_{i})}} {n_{Y1}}$$

$$\bar{P}_{Y2} = \frac{\sum_{i=1}^{n_{Y2}}{P(Y2_{i})}} {n_{Y2}}$$

and where

$$V_{Y1} = \sum_{i=1}^{n_{Y1}}{(P(Y1_{i} - \bar{P}_{Y1})^2}$$

$$V_{Y2} = \sum_{i=1}^{n_{Y2}}{(P(Y2_{i} - \bar{P}_{Y2})^2}$$

The standard deviations of the placements are

$$SD_{P1} = \sqrt{\frac{V_{Y1}} {n_{Y1} - 1}}$$ $$SD_{P2} = \sqrt{\frac{V_{Y2}} {n_{Y2} - 1}}$$

The above test statistic is compared to a standard normal distribution.

Syntax 1:
<LOWER TAILED/UPPER TAILED> FLIGNER POLICELLO TEST <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If LOWER TAILED is specified, a lower tailed test is performed. If UPPER TAILED is specified, an upper tailed test is performed. If neither LOWER TAILED or UPPER TAILED is specified, a two-tailed test is performed.

Syntax 2:
<LOWER TAILED/UPPER TAILED> FLIGNER POLICELLO TEST <y1> ... <yk>
<SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of two or more response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax performs all the two-way Fligner-Policello tests for the listed variables. This syntax supports the TO syntax.

If LOWER TAILED is specified, a lower tailed test is performed. If UPPER TAILED is specified, an upper tailed test is performed. If neither LOWER TAILED or UPPER TAILED is specified, a two-tailed test is performed.

Examples:
FLIGNER POLICELLO TEST Y1 Y2
FLIGNER POLICELLO TEST Y1 Y2 Y3
FLIGNER POLICELLO TEST Y1 TO Y6
FLIGNER POLICELLO TEST Y1 Y2 SUBSET Y2 > 0
LOWER TAILED FLIGNER POLICELLO TEST Y1 Y2
UPPER TAILED FLIGNER POLICELLO TEST Y1 Y2
Note:
The following parameters are saved after the Fligner-Policello test is performed.

 STATVAL - value of the test statistic STATCDF - CDF of the test statistic PVALUE - p-value of the two tailed test statistic PVALUELT - p-value of the lower tailed test statistic PVALUEUT - p-value of the upper tailed test statistic CUTUPP90 - 90% upper critical value CUTUPP95 - 95% upper critical value CUTUP975 - 97.5% upper critical value CUTUPP99 - 99% upper critical value CUTUP995 - 99.5% upper critical value CUTUP999 - 99.9% upper critical value CUTLOW10 - 10% lower critical value CUTLOW05 - 5% lower critical value CUTLO025 - 2.5% lower critical value CUTLOW01 - 1% lower critical value CUTLO005 - 0.5% lower critical value CUTLO001 - 0.1% lower critical value
Note:
In addition to the FLIGNER POLICELLO TEST command, the following commands can also be used:

LET STATVAL = FLIGNER POLICELLO TEST Y1 Y2
LET STATCDF = FLIGNER POLICELLO TEST CDF Y1 Y2
LET PVALUE = FLIGNER POLICELLO TEST PVALUE Y1 Y2
LET PVALUE = FLIGNER POLICELLO LOWER TAIL TEST PVALUE Y1 Y2
LET PVALUE = FLIGNER POLICELLO UPPER TAIL TEST PVALUE Y1 Y2

In addition to the above LET commands, built-in statistics are supported for 30+ different commands (enter HELP STATISTICS for details).

Default:
None
Synonyms:
FLIGNER POLICELLO is a synonym for FLIGNER POLICELLO TEST
Related Commands:
 T TEST = Perform a 2-sample t-test for location RANK SUM TEST = Perform a 2-sample rank sum test for location MEDIAN TEST = Perform a k-sample medians test VAN DER WAERDEN TEST = Perform a k-sample Van Der Waerden test KRUSKAL WALLIS TEST = Perform a k-sample Kruskal-Wallis test
Applications:
Two Sample Analysis
Implementation Date:
2023/08:
Program:

. Step 1:   Read the data
.
skip 25
read shoemake.dat y1 y2
skip 0
.
. Step 2:   Generate the statistics
.
let statval = fligner policello test                        y1 y2
let statcdf = fligner policello test cdf                    y1 y2
let pvalue  = fligner policello test pvalue                 y1 y2
let pvallt  = fligner policello test lower tail pvalue      y1 y2
let pvalut  = fligner policello test upper tail pvalue      y1 y2
let statval = round(statval,2)
let statcdf = round(statcdf,2)
let pvalue  = round(pvalue,2)
let pvallt  = round(pvallt,2)
let pvalut  = round(pvalut,2)
.
print "Fligner-Policello:"
print "Test Statistic:                        ^statval"
print "Test Statistic CDF:                    ^statcdf"
print "Test Statistic P-Value:                ^pvalue"
print "Test Statistic Lower Tailed P-Value:   ^pvallt"
print "Test Statistic Upper Tailed P-Value:   ^pvalut"
.
. Step 3:   Perform the tests
.
fligner policello test                y1 y2
lower tailed fligner policello test   y1 y2
upper tailed fligner policello test   y1 y2

The following output is generated
Fligner-Policello:
Test Statistic:                        1.56
Test Statistic CDF:                    0.94
Test Statistic P-Value:                0.12
Test Statistic Lower Tailed P-Value:   0.94
Test Statistic Upper Tailed P-Value:   0.06

Two Sample Two-Sided Fligner Policello Test
Test for Equal Medians

First Response Variable:  Y1
Second Response Variable: Y2

H0: Median1 = Median2
Ha: Median1 not equal Median2

Summary Statistics:
Number of Observations for Sample 1:                 10
Mean for Sample 1:                              6.02100
Median for Sample 1:                            5.53000
Standard Deviation for Sample 1:                1.58184
Number of Observations for Sample 2:                 10
Mean for Sample 2:                              5.01900
Median for Sample 2:                            5.03500
Standard Deviation for Sample 2:                1.10440

Test Statistic Value:                           1.55802
CDF Value:                                      0.94039
P-Value (2-tailed test):                        0.11923
P-Value (lower-tailed test):                    0.94039
P-Value (upper-tailed test):                    0.05961

Two-Tailed Test: Normal Approximation

---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic      Value (<)      Value (>)     Conclusion
---------------------------------------------------------------------------
80.0%        1.55802       -1.28155        1.28155         REJECT
90.0%        1.55802       -1.64485        1.64485         ACCEPT
95.0%        1.55802       -1.95996        1.95996         ACCEPT
99.0%        1.55802       -2.57583        2.57583         ACCEPT

Two Sample Lower-Tailed Fligner Policello Test
Test for Equal Medians

First Response Variable:  Y1
Second Response Variable: Y2

H0: Median1 = Median2
Ha: Median1 < Median2

Summary Statistics:
Number of Observations for Sample 1:                 10
Mean for Sample 1:                              6.02100
Median for Sample 1:                            5.53000
Standard Deviation for Sample 1:                1.58184
Number of Observations for Sample 2:                 10
Mean for Sample 2:                              5.01900
Median for Sample 2:                            5.03500
Standard Deviation for Sample 2:                1.10440

Test Statistic Value:                           1.55802
CDF Value:                                      0.94039
P-Value (2-tailed test):                        0.11923
P-Value (lower-tailed test):                    0.94039
P-Value (upper-tailed test):                    0.05961

Lower-Tailed Test: Normal Approximation

------------------------------------------------------------
Lower           Null
Significance           Test       Critical     Hypothesis
Level      Statistic      Value (<)     Conclusion
------------------------------------------------------------
80.0%        1.55802       -0.84162         ACCEPT
90.0%        1.55802       -1.28155         ACCEPT
95.0%        1.55802       -1.64485         ACCEPT
99.0%        1.55802       -2.32635         ACCEPT

Two Sample Upper-Tailed Fligner Policello Test
Test for Equal Medians

First Response Variable:  Y1
Second Response Variable: Y2

H0: Median1 = Median2
Ha: Median1 > Median2

Summary Statistics:
Number of Observations for Sample 1:                 10
Mean for Sample 1:                              6.02100
Median for Sample 1:                            5.53000
Standard Deviation for Sample 1:                1.58184
Number of Observations for Sample 2:                 10
Mean for Sample 2:                              5.01900
Median for Sample 2:                            5.03500
Standard Deviation for Sample 2:                1.10440

Test Statistic Value:                           1.55802
CDF Value:                                      0.94039
P-Value (2-tailed test):                        0.11923
P-Value (lower-tailed test):                    0.94039
P-Value (upper-tailed test):                    0.05961

Upper-Tailed Test: Normal Approximation

------------------------------------------------------------
Upper           Null
Significance           Test       Critical     Hypothesis
Level      Statistic      Value (>)     Conclusion
------------------------------------------------------------
80.0%        1.55802        0.84162         REJECT
90.0%        1.55802        1.28155         REJECT
95.0%        1.55802        1.64485         ACCEPT
99.0%        1.55802        2.32635         ACCEPT


Date created: 08/04/2023
Last updated: 08/04/2023

Please email comments on this WWW page to alan.heckert@nist.gov.