SIEGEL TUKEY TEST

SIEGEL TUKEY TEST

Analysis Command

Perform a two sample Siegel-Tukey test that the standard deviations of two samples are equal.

The Siegel-Tukey test is computed as follows:
1. Combine the data from the two response variables and sort from smallest to largst.

2. Assign rank 1 to the smallest observation, rank 2 to the largest observation, rank 3 to next largest observation, rank 4 to the second smallest observation. Continue this pattern until all values are ranked.

   If there are an odd number of total observations, then omit the observation that is the median of the combined observations.

3. Perform a Mann-Whitney (rank sum) test on the ranked values.

<LOWER TAILED/UPPER TAILED> SIEGEL TUKEY 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 (i.e., \( \sigma_1 \) < \( \sigma_2 \)). If UPPER TAILED is specified, an upper tailed test is performed (i.e., \( \sigma_1 \) > \( \sigma_2 \)). If neither LOWER TAILED or UPPER TAILED is specified, a two-tailed test is performed (i.e., \( \sigma_1 \ne \sigma_2 \)).

<LOWER TAILED/UPPER TAILED> SIEGEL TUKEY 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 Siegel-Tukey tests for the listed variables. This syntax supports the TO syntax.

If LOWER TAILED is specified, a lower tailed test is performed (i.e., \( \sigma_1 \) < \( \sigma_2 \)). If UPPER TAILED is specified, an upper tailed test is performed (i.e., \( \sigma_1 \) > \( \sigma_2 \)). If neither LOWER TAILED or UPPER TAILED is specified, a two-tailed test is performed (i.e., \( \sigma_1 \ne \sigma_2 \)).

SIEGEL TUKEY TEST Y1 Y2
SIEGEL TUKEY TEST Y1 Y2 Y3
SIEGEL TUKEY TEST Y1 TO Y6
SIEGEL TUKEY TEST Y1 Y2 SUBSET Y2 > 0
LOWER TAILED SIEGEL TUKEY TEST Y1 Y2
UPPER TAILED SIEGEL TUKEY TEST Y1 Y2

The following parameters are saved after the Siegel Tukey 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 SIEGEL TUKEY TEST command, the following commands can also be used:
LET STATVAL = SIEGEL TUKEY TEST Y1 Y2
LET STATCDF = SIEGEL TUKEY TEST CDF Y1 Y2
LET PVALUE = SIEGEL TUKEY TEST PVALUE Y1 Y2
LET PVALUE = SIEGEL TUKEY LOWER TAIL TEST PVALUE Y1 Y2
LET PVALUE = SIEGEL TUKEY 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

None

BARTLETTS TEST	=	Perform a k-sample Bartlett test for homogeneous variances.
LEVENE TEST	=	Perform a k-sample Levene test for homogeneous variances.
SQUARED RANKS TEST	=	Perform a k-sample squared ranks test for homogeneous variances.
KLOTZ TEST	=	Perform a k-sample Klotz test for homogeneous variances.
RATIO OF SD CONFIDENCE LIMITS	=	Compute the confidence limtis for the ratio of two standard deviations.
SD CONFIDENCE LIMITS	=	Compute the confidence limits for the standard deviation.
CHI-SQUARE TEST	=	Performs a one sample chi-square test that the standard deviation is equal to a given value.
T TEST	=	Performs a two-sample t test for equal means.
CONFIDENCE LIMITS	=	Compute the confidence limits for the mean of a sample.
STANDARD DEVIATION	=	Computes the standard deviation of a variable.

Higgins (2004), "Introduction to Modern Nonparametric Statistics," Duxbury Press, pp. 52-53.

Confirmatory Data Analysis

2023/06:

 
. Step 1:   Define the data (example from Higgin's book)
.
let y1 = data 16.55 15.36 15.94 16.43 16.01
let y2 = data 16.05 15.98 16.10 15.88 15.91
.
. Step 2:   Compute Siegel Tukey test statistic
.
let statva  = siegel tukey test        y1 y2
let statcdf = siegel tukey test cdf    y1 y2
let pvalue  = siegel tukey test pvalue y1 y2
.
siegel tukey test y1 y2
    

The following output is generated

 PARAMETERS AND CONSTANTS--

    STATVA  --       24.00000
    STATCDF --        0.26543
    PVALUE  --        0.53087
  
             Two Sample Two-Sided Siegel Tukey Test
  
 First Response Variable: Y1
 Second Response Variable: Y2
  
 H0: Sigma1 = Sigma2
 Ha: Sigma1 not equal Sigma2
  
 Summary Statistics:
 Number of Observations for Sample 1:                  5
 Mean for Sample 1:                             16.05800
 Median for Sample 1:                           16.01000
 Standard Deviation for Sample 1:                0.47007
 Number of Observations for Sample 2:                  5
 Mean for Sample 2:                             15.98400
 Median for Sample 2:                           15.98000
 Standard Deviation for Sample 2:                0.09236
  
 Test (Small Sample, Exact):
 Test Statistic Value:                          24.00000
 CDF Value:                                      0.26543
 P-Value (2-tailed test):                        0.53087
 P-Value (lower-tailed test):                    0.26543
 P-Value (upper-tailed test):                    1.00000
  
  
             Two-Tailed Test: Exact - Small Sample
  
 H0: F(x) = G(x); Ha: F(x) <> G(x)  for some x
 ---------------------------------------------------------------------------
                                         Lower          Upper           Null
    Significance           Test       Critical       Critical     Hypothesis
           Level      Statistic      Value (<)      Value (>)     Conclusion
 ---------------------------------------------------------------------------
           80.0%       24.00000       21.00000       34.00000         ACCEPT
           90.0%       24.00000       20.00000       35.00000         ACCEPT
           95.0%       24.00000       18.00000       37.00000         ACCEPT
           99.0%       24.00000       16.00000       39.00000         ACCEPT