SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 1 Vol 2

KLOTZ TEST

Name:
    KLOTZ TEST
Type:
    Analysis Command
Purpose:
    Perform a Klotz test that k samples have equal variances.
Description:
    The F test is the standard parameteric test for testing the equality of variances for the two sample case.

    A Klotz test is a non-parametric alternative to the F test. It is based on the squares of normal scores. Normal scores are computed as

      \( A_{i} = \Phi^{-1} \left( \frac{R_i}{N+1} \right) \)

    where Ri is the rank of the i-th observation, N is the sample size, and \( \Phi^{-1} \) is the percent point function of the standard normal distribution.

    The advantage of many tests based on normal scores is that they perform well when the assumptions of the standard parametric test are satisfied while still providing protection when the assumptions are not satisfied.

    The Klotz test is defined as

      H0: The two populations have equal variances
      Ha: The two populations do not have equal variances
      Test Statistic: To compute the test statistic, for each sample first subtract the mean from each observation. Then compute the normal scores (Ai) for the combined sample. The test statistic is then

        T1 = NUM/DEN

      where

        \( \mbox{NUM} = \sum_{i=1}^{n_1}{A_{i}^2} - \left( \frac{n1}{n1+n2}\right) \sum_{i=1}^{n1+n2}{A_{i}^2} \)

        \( \mbox{DEN} = \sqrt{C \left( \sum_{i=1}^{n1+n2}{A_{i}^4} - \frac{1}{n1+n2} \left( \sum_{i=1}^{n1+n2}{A_{i}^2} \right) ^2 \right) } \)

        n1 = sample size 1
        n2 = sample size 2
        C = (n1*n2)/ ((n1+n2)* (n1+n2-1))

      Significance Level: \( \alpha \)
      Critical Region: For a two-tailed test:

        \( \mbox{T1} > \Phi^{-1}(1 - (\alpha/2)) \)

        \( \mbox{T1} < \Phi^{-1}(\alpha/2) \)

      For a lower-tailed test:

        \( \mbox{T1} < \Phi^{-1}(\alpha) \)

      For an upper-tailed test:

        \( \mbox{T1} > \Phi^{-1}(1 - \alpha) \)

      \( \Phi^{-1} \) is the percent point function of the standard normal distribution

      Conclusion: Reject the null hypothesis if the test statistic is in the critical region.

    The critical values based on the normal test are only approximate. Dataplot does not currently compute exact critical values.

Syntax 1:
    <LOWER TAILED/UPPER TAILED> KLOTZ TEST <y1> <y2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword for the two sample case;
                <y1> is the first response variable;
                <y2> is the second variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The LOWER TAILED and UPPER TAILED keywords are optional Only one can be specified and if neither is entered a two-tailed test will be performed.

Syntax 2:
    <LOWER TAILED/UPPER TAILED> KLOTZ TEST <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <LOWER TAILED/UPPER TAILED> is an optional keyword for the two sample case;
                <y1> ... <yk> is a list of 2 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The LOWER TAILED and UPPER TAILED keywords are optional Only one can be specified and if neither is entered a two-tailed test will be performed.

    This syntax will generate all the pairwise Klotz tests for the list of variables.

Examples:
    KLOTZ TEST Y1 Y2
    LOWER TAILED KLOTZ TEST Y1 Y2
    UPPER TAILED KLOTZ TEST Y1 Y2
    KLOTZ TEST Y1 TO Y10
Note:
    The Klotz test accepts matrix arguments. A matrix will act like a variable in that all the values in the matrix will be converted to a single variable. That is, it does not act on the rows or colums independently.

    The TO syntax is supported for this command.

Note:
    The squared ranks test is an alternative non-parametric test for comparing the variances of k populations.
Note:
    The following statistics are also supported:

      LET A = KLOTZ TEST Y1 Y2
      LET A = KLOTZ TEST CDF Y1 Y2
      LET A = KLOTZ TEST PVALUE Y1 Y2
      LET A = KLOTZ TEST LOWER TAILED PVALUE Y1 Y2
      LET A = KLOTZ TEST UPPER TAILED PVALUE Y1 Y2

    Enter HELP STATISTICS to see what commands can use these statistics.

Default:
    None
Synonyms:
    None
Related Commands: Reference:
    W. J. Conover (1999), "Practical Nonparameteric Statistics," Third Edition, Wiley, pp. 401-402.
Applications:
    Nonparametric Analysis
Implementation Date:
    2011/5
Program:
     
    . Step 1: Read Data (from p. 402 of Conover)
    .
    let y1 = data 10.8 11.1 10.4 10.1 11.3
    let y2 = data 10.8 10.5 11.0 10.9 10.8 10.7 10.8
    set write decimals 4
    .
    .  Step 2: Check the statistic
    .
    let stat  = klotz test y1 y2
    let cdf   = klotz test cdf y1 y2
    let pval  = klotz test pvalue y1 y2
    print stat pval cdf
    .
    .  Step 3: Perform Klotz test
    .
    klotz test y1 y2
        
    The following output is generated.
    PARAMETERS AND CONSTANTS--
    
        STAT    --         2.3447
        PVAL    --         0.0190
        CDF     --         0.9905
    
    
    
                Two Sample Two-Sided Klotz Test
     
    First Response Variable: Y1
    Second Response Variable: Y2
     
    H0: Var(Y1) = Var(Y2)
    Ha: Var(Y1) <> Var(Y2)
     
    Summary Statistics:
    Number of Observations for Sample 1:                  5
    Mean for Sample 1:                              10.7400
    Variance for Sample                              0.2430
    Number of Observations for Sample 2:                  7
    Mean for Sample 2:                              10.7857
    Variance for Sample                              0.0247
     
    Test (Normal Approximation):
    Test Statistic Value:                            2.3447
    CDF Value:                                       0.9904
    P-Value (2-tailed test):                         0.0190
    P-Value (lower-tailed test):                     0.9904
    P-Value (upper-tailed test):                     0.0095
     
     
                Two-Tailed Test: Normal Approximation
     
    H0: Var(Y1) = Var(Y2); Ha: Var(Y1) <> Var(Y2)
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic    Value (+/-)     Conclusion
    ------------------------------------------------------------
              80.0%         2.3447         1.2815         REJECT
              90.0%         2.3447         1.6448         REJECT
              95.0%         2.3447         1.9599         REJECT
              99.0%         2.3447         2.5758         ACCEPT
        

Privacy Policy/Security Notice
Disclaimer | FOIA

NIST is an agency of the U.S. Commerce Department.

Date created: 09/15/2011
Last updated: 10/19/2015

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