FLIGNER POLICELLO TEST

FLIGNER POLICELLO TEST

Analysis Command

Perform a two sample Fligner-Policello test for equal medians.

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.

<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.

<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.

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

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

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).

None

FLIGNER POLICELLO is a synonym for FLIGNER POLICELLO TEST

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

Two Sample Analysis

2023/08:

 
. 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