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Dataplot Vol 1 Vol 2

COEFFICIENT OF VARIATION TEST

Name:
    COEFFICIENT OF VARIATION TEST
Type:
    Analysis Command
Purpose:
    Perform either a one sample coefficient of variation test or a two sample coefficient of variation test.
Description:
    The coefficient of variation is defined as the ratio of the standard deviation to the mean

      \( \mbox{cv} = \frac{\sigma} {\mu} \)

    where σ and μ denote the population standard deviation and population mean, respectively. The sample coefficient of variation is defined as

      \( \mbox{cv} = \frac{\bar{x}} {s} \)

    where s and \( \bar{x} \) denote the sample standard deviation and sample mean respectively.

    The coefficient of variation should typically only be used for ratio data. That is, the data should be continuous and have a meaningful zero. Although the coefficient of variation statistic can be computed for data that is not on a ratio scale, the interpretation of the coeffcient of variation may not be meaningful. Currently, this command is only supported for non-negative data. If the response variable contains one or more negative numbers, an error message will be returned.

    The one sample coefficient of variation tests whether the coefficient of variation is equal to a given value. Note that this can be for either a single sample or for the common coefficient for multile groups of data (it is assummed the groups have equal population coefficient of variation values).

      H0: γ = γ0
      Ha: γγ0

    The test statistic is

      \( \sum_{i}^{k}{\frac{(n_{i} - 1) u_{i}} {\theta_{0}}} \)

    where

      k = the number of groups
      \( u_{i} \) = \( \frac{c_{i}^{2}} {1 + c_{i}^{2} (n_{i} - 1)} \)
      ci = coefficient of variation for the i-th group
      ni = sample size for the i-th group
      θ0 = \( \frac{ \gamma_{0}^{2}} {1 + \gamma_{0}^{2}} \)

    where γ is the common coefficient of variation and γ0 is the hypothesized value.

    This statistic is compared to a chi-square with \( \sum_{i}^{k}{n_{i} - 1} \) degrees of freedom. The most common usage is the case for a single group (i.e., k = 1).

    The two sample coefficient of variation tests whether two distinct samples have equal, but unspecified, coefficients of variations. As with the single sample case, each of the two samples can consist of either a single group or multiple groups of data.

      H0: γ1 = γ2
      Ha: γ1γ2

    The test statistic is

      \( F = \frac {\mbox{NUM}} {\mbox{DENOM}} \)

    where

      \( \mbox{NUM} = \frac {\sum_{i}^{k}{(n_{1i} - 1) u_{1i}}} {\sum_{i}^{k}{n_{1i} - 1}} \)

      \( \mbox{DENOM} = \frac {\sum_{i}^{k}{(n_{2i} - 1) u_{2i}}} {\sum_{i}^{k}{n_{2i} - 1}} \)

    where

      k1 = the number of groups for sample one
      k2 = the number of groups for sample two
      \( u_{ri}\) = \( \frac{c_{ri}^{2}} {1 + c_{ri}^{2}*(n_{ri} - 1)/n_{ri}} \)
      \( c_{ri} \) = coefficient of variation for the i-th group and the r-th sample
      \( n_{ri} \) = the sample size for the i-the group and the r-th sample
      r = 1, 2 (i.e., the two samples)

    when k1 = k2 = 1, the test simplifies to

      \( F = \frac{c_{1}^{2}/(1 + c_{1}^{2}(n_{1} - 1)/n_{1})} {c_{2}^{2}/(1 + c_{2}^{2}(n_{2} - 1)/n_{2})} \)

    This statistic is compared to the F distribution with \( \sum_{i=1}^{k_{1}}{n_{1i} -1} \) and \( \sum_{i=1}^{k_{2}}{n_{2i} -1} \) degrees of freedom.

    The test implemented here was proposed by Forkman (see the References below). There are a number of alternative tests (see the paper by Krishnamooorthy and Lee in the References section). Simulations by Forkman and also by Krishnamoorthy and Lee indicate that the Forkman test has good nominal coverage and reasonable power.

Syntax 1:
    ONE SAMPLE COEFFICIENT OF VARIATION TEST <y> <x> <gamma0>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x> is the optional group-id variable;
                <gamma0> is a parameter that specifies the hypothesized value;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a two-tailed test.

    If there are no groups in the data, the group-id variable can be omitted.

    The <gamma0> can either be given on this command or specified before entering this command by entering

      LET GAMMA0 = <value>

    If the <x> variable is given, it should have the same number of rows as the <y> variable.

Syntax 2:
    ONE SAMPLE COEFFICIENT OF VARIATION <LOWER/UPPER> TAILED
                            TEST <y> <x> <gamma0>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x> is the optional group-id variable;
                <gamma0> is a parameter that specifies the hypothesized value;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a one-tailed test. If LOWER is entered, then the alternate hypothesis is

      Ha: gamma < gamma0

    If UPPER is entered, then the alternative hypothesis is

      Ha: gamma > gamma0

    If there are no groups in the data, the group-id variable can be omitted.

    The <gamma0> can either be given on this command or specified before entering this command by entering

      LET GAMMA0 = <value>

    If the <x> variable is given, it should have the same number of rows as the <y> variable.

Syntax 3:
    TWO SAMPLE COEFFICIENT OF VARIATION TEST <y1> <x1> <y2> <x2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> is the first response variable;
                <x1> is the optional first group-id variable;
                <y2> is the second response variable;
                <x2> is the optional second group-id variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a two-tailed test.

    If there are no groups in the data, the group-id variables can be omitted. However, if a group-id variable is specified for one response variable, it should also be specified for the second response variable. If one of the response variables has groups but the other response variable does not, then a group-id variable can be created that has all values equal to 1.

    The <y1> and <x1> variables should have the same number of rows. Likewise the <y2> and <x2> variables should have the same number of rows. However, <y1> and need not have the same number of rows.

Syntax 4:
    TWO SAMPLE COEFFICIENT OF VARIATION <LOWER/UPPER> TAILED
                            TEST <y1> <x1> <y2> <x2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> is the first response variable;
                <x1> is the optional first group-id variable;
                <y2> is the second response variable;
                <x2> is the optional second group-id variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a one-tailed test. If LOWER is entered, then the alternate hypothesis is

      Ha: gamma1 < gamma2

    If UPPER is entered, then the alternative hypothesis is

      Ha: gamma1 > gamma2

    If there are no groups in the data, the group-id variable can be omitted.

    If there are no groups in the data, the group-id variables can be omitted. However, if a group-id variable is specifiend for one response variable, it should also be specified for the second response variable. If one of the response variables has groups but the other response variable does not, then a group-id variable can be created that has all values equal to 1.

    The <y1> and <x1> variables should have the same number of rows. Likewise the <y2> and <x2> variables should have the same number of rows. However, <y1> and <y2> need not have the same number of rows.

Examples:
    LET GAMMA0 = 0.1
    ONE SAMPLE COEFFICIENT OF VARIATION TEST Y GAMMA0
    ONE SAMPLE COEFFICIENT OF VARIATION TEST Y X GAMMA0
    ONE SAMPLE COEFFICIENT OF VARIATION UPPER TAILED TEST ...
                Y X GAMMA0
    ONE SAMPLE COEFFICIENT OF VARIATION TEST Y X GAMMA0 ...
                SUBSET X > 2

    TWO SAMPLE COEFFICIENT OF VARIATION TEST Y1 Y2
    TWO SAMPLE COEFFICIENT OF VARIATION TEST Y1 X1 Y2 X2
    TWO SAMPLE COEFFICIENT OF VARIATION LOWER TAILED TEST Y1 Y2

Note:
    A table of confidence limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9.
Note:
    In addition to the COEFFICIENT OF VARIATION CONFIDENCE LIMIT command, the following commands can also be used:

      LET A = ONE SAMPLE COEFFICIENT OF VARIATION TEST ...
                  Y X
      LET A = ONE SAMPLE COEFFICIENT OF VARIATION TEST ...
                  CDF Y X
      LET A = ONE SAMPLE COEFFICIENT OF VARIATION TEST ...
                  PVALUE Y X
      LET A = ONE SAMPLE COEFFICIENT OF VARIATION LOWER ...
                  PVALUE Y X
      LET A = ONE SAMPLE COEFFICIENT OF VARIATION UPPER ...
                  PVALUE Y X

      LET A = TWO SAMPLE COEFFICIENT OF VARIATION TEST ...
                  Y1 Y2
      LET A = TWO SAMPLE COEFFICIENT OF VARIATION TEST ...
                  CDF Y1 Y2
      LET A = TWO SAMPLE COEFFICIENT OF VARIATION TEST ...
                  PVALUE Y1 Y2
      LET A = TWO SAMPLE COEFFICIENT OF VARIATION LOWER ...
                  PVALUE Y1 Y2
      LET A = TWO SAMPLE COEFFICIENT OF VARIATION UPPER ...
                  PVALUE Y1 Y2

    The LOWER PVALUE and UPPER PVALUE refer to the p-values based on lower tailed and upper tailed tests, respectively.

    For the one sample test, these statistics can be computed from summary data as well

      LET A = SUMMARY ONE SAMPLE COEFFICIENT OF VARIATION ...
                  TEST YMEAN YSD YN
      LET A = SUMMARY ONE SAMPLE COEFFICIENT OF VARIATION ...
                  CDF YMEAN YSD YN
      LET A = SUMMARY ONE SAMPLE COEFFICIENT OF VARIATION ...
                  PVALUE

    where YMEAN, YSD, and YN are arrays that contain the sample means, sample standard deviations, and sample sizes, respectively.

    In addition to the above LET commands, built-in statistics are supported for 20+ different commands (enter HELP STATISTICS for details).

Note:
    As mentioned above, there are a number of tests that have been proposed. For the two sample test with no groups for the samples, Dataplot also supports the Miller test. This test statistic is given by

      \( \frac {c_{1} - c_{2}} {\sqrt{\frac{c^{2}}{2(n_{1} - 1)} + \frac{c^{4}}{n_{1} - 1} + \frac{c^{2}}{2(n_{2} - 1)} + \frac{c^{4}}{n_{2} - 1}}} \)

    where

      n1 = = the sample size for sample one
      c1 = the sample coefficient of variation for sample one
      n2 = = the sample size for sample two
      c2 = the sample coefficient of variation for sample two
      c = \( \frac{(n_{1} - 1)c_{1} + (n_{2} - 1)c_{2}} {n_{1} + n_{2} - 2} \)

    To use the Miller test, enter the command (before the TWO SAMPLE COEFFICENT OF VARIATION TEST command)

      SET TWO SAMPLE COEFFICIENT OF VARIATION TEST MILLER

    To reset the default Forkman test, enter

      SET TWO SAMPLE COEFFICIENT OF VARIATION TEST FORKMAN
Default:
    None
Synonyms:
    None
Related Commands: References:
    Forkman (2009), "Estimator and Tests for Common Coefficients of Variation in Normal Distributions", Communications in Statistics - Theory and Methods, Vol. 38, pp. 233-251.

    Miller (1991), "Asymptotic Test Statistics for Coefficient of Variation", Communications in Statistics - Theory and Methods, Vol. 20, pp. 3351-3363.

    McKay (1932), "Distributions of the Coefficient of Variation and the Extended 't' Distribution", Journal of the Royal Statistical Society, Vol. 95, pp. 695-698.

    Krishnamoorthy and Lee (2014), "Improved Tests for the Equality of Normal Coefficients of Variation", Computational Statistics, Vol. 29, pp. 215-232.

Applications:
    Confirmatory Data Analysis
Implementation Date:
    2017/06
Program 1:
     
    . Step 1:   Read the data
    .
    skip 25
    read gear.dat y x
    skip 0
    set write decimals 6
    .
    . Step 2:   Define plot control
    .
    title case asis
    title offset 2
    label case asis
    .
    y1label Coefficient of Variation
    x1label Group
    title Coefficient of Variation for GEAR.DAT
    let ngroup = unique x
    xlimits 1 ngroup
    major x1tic mark number ngroup
    minor x1tic mark number 0
    tic mark offset units data
    x1tic mark offset 0.5 0.5
    y1tic mark label decimals 3
    .
    character X
    line blank
    .
    .
    . Step 3:   Plot the coefficient of variation over the batches
    .
    set statistic plot reference line average
    coefficient of variation plot y x
    .
    . Step 4:   Demonstrate the LET commands for the test statistics
    .           using raw data
    .
    let gamma0 = 0.005
    let statval = one sample coef of variation test         y x
    let statcdf = one sample coef of variation test cdf     y x
    let pvalue  = one sample coef of variation test pvalue  y x
    let pvall   = one sample coef of variation lower pvalue y x
    let pvalu   = one sample coef of variation upper pvalue y x
    print statval statcdf pvalue pvall pvalu
    .
    . Step 4:   Demonstrate the LET commands for the test statistics
    .           using summary data
    .
    set let cross tabulate collapse
    let ymean = cross tabulate mean y x
    let ysd   = cross tabulate sd   y x
    let yn    = cross tabulate size   x
    .
    let statval2 = summary one sample coef of variation test    ymean ysd yn
    let statcdf2 = summary one sample coef of variation cdf     ymean ysd yn
    let pvalue2  = summary one sample coef of variation pvalue  ymean ysd yn
    print statval2 statcdf2 pvalue2
    .
    . Step 5:   Hypothesis test for common coefficient of variation
    .
    let gamma0 = 0.005
    one sample coefficient of variation test y x
    one sample coefficient of variation upper tail test y x
    one sample coefficient of variation lower tail test y x
        
    plot generated by sample program

    The following output is generated.

     PARAMETERS AND CONSTANTS--
    
        STATVAL --     127.554980
        STATCDF --       0.994327
        PVALUE  --       0.011346
        PVALL   --       0.994327
        PVALU   --       0.005673
     
    
     PARAMETERS AND CONSTANTS--
    
        STATVAL2--     127.554980
        STATCDF2--       0.994327
        PVALUE2 --       0.011346
    
    
                Forkman One Sample Coefficient of Variation Test
     
    Response Variable: Y
    Group-ID Variable: X
     
    H0: Coefficient of Variation Equal       0.005000
    Ha: Coefficient of Variation Not Equal   0.005000
     
    Summary Statistics:
    Total Number of Observations:            100
    Number of Groups:                        10
    Number of Groups Included in Test:       10
    Sample Common Coefficient of Variation:  0.005953
     
    Test:
    Gamma0:                                  0.005000
    Test Statistic Value:                    127.554980
    Degrees of Freedom:                      90
    CDF Value:                               0.994327
    P-Value (2-tailed test):                 0.011346
    P-Value (lower-tailed test):             0.994327
    P-Value (upper-tailed test):             0.005673
     
     
                Two-Tailed Test
     
    H0: Gamma = Gamma0; Ha: Gamma <> Gamma0
    ---------------------------------------------------------------------------
                                            Lower          Upper           Null
       Significance           Test       Critical       Critical     Hypothesis
              Level      Statistic          Value          Value     Conclusion
    ---------------------------------------------------------------------------
              50.0%     127.554980      80.624665      98.649932         REJECT
              80.0%     127.554980      73.291090     107.565009         REJECT
              90.0%     127.554980      69.126030     113.145270         REJECT
              95.0%     127.554980      65.646618     118.135893         REJECT
              99.0%     127.554980      59.196304     128.298944         ACCEPT
              99.9%     127.554980      52.275778     140.782281         ACCEPT
     
     
                Forkman One Sample Coefficient of Variation Test
     
    Response Variable: Y
    Group-ID Variable: X
     
    H0: Coefficient of Variation Equal       0.005000
    Ha: Coefficient of Variation >           0.005000
     
    Summary Statistics:
    Total Number of Observations:            100
    Number of Groups:                        10
    Number of Groups Included in Test:       10
    Sample Common Coefficient of Variation:  0.005953
     
    Test:
    Gamma0:                                  0.005000
    Test Statistic Value:                    127.554980
    Degrees of Freedom:                      90
    CDF Value:                               0.994327
    P-Value (2-tailed test):                 0.011346
    P-Value (lower-tailed test):             0.994327
    P-Value (upper-tailed test):             0.005673
     
     
                Upper One-Tailed Test
     
    H0: Gamma = Gamma0; Ha: Gamma > Gamma0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (>)     Conclusion
    ------------------------------------------------------------
              50.0%     127.554980      89.334218         REJECT
              80.0%     127.554980     101.053723         REJECT
              90.0%     127.554980     107.565009         REJECT
              95.0%     127.554980     113.145270         REJECT
              99.0%     127.554980     124.116319         REJECT
              99.9%     127.554980     137.208354         ACCEPT
     
     
                Forkman One Sample Coefficient of Variation Test
     
    Response Variable: Y
    Group-ID Variable: X
     
    H0: Coefficient of Variation Equal       0.005000
    Ha: Coefficient of Variation <           0.005000
     
    Summary Statistics:
    Total Number of Observations:            100
    Number of Groups:                        10
    Number of Groups Included in Test:       10
    Sample Common Coefficient of Variation:  0.005953
     
    Test:
    Gamma0:                                  0.005000
    Test Statistic Value:                    127.554980
    Degrees of Freedom:                      90
    CDF Value:                               0.994327
    P-Value (2-tailed test):                 0.011346
    P-Value (lower-tailed test):             0.994327
    P-Value (upper-tailed test):             0.005673
     
     
                Lower One-Tailed Test
     
    H0: Gamma = Gamma0; Ha: Gamma < Gamma0
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%     127.554980      89.334218         ACCEPT
              80.0%     127.554980      78.558432         ACCEPT
              90.0%     127.554980      73.291090         ACCEPT
              95.0%     127.554980      69.126030         ACCEPT
              99.0%     127.554980      61.754079         ACCEPT
              99.9%     127.554980      54.155244         ACCEPT
        
Program 2:
     
    . Step 1:   Read the data
    .
    skip 25
    read auto83b.dat y1 y2
    skip 0
    set write decimals 6
    retain y2 subset y2 > 0
    .
    . Test for equal coefficient of variation
    .
    let statval = two sample coef of variation test         y1 y2
    let statcdf = two sample coef of variation test cdf     y1 y2
    let pvalue  = two sample coef of variation test pvalue  y1 y2
    let pvall   = two sample coef of variation lower pvalue y1 y2
    let pvalu   = two sample coef of variation upper pvalue y1 y2
    print statval statcdf pvalue pvall pvalu
    .
    . Test for equal coefficient of variation
    .
    two sample coefficient of variation test y1 y2
    two sample coefficient of variation lower tail test y1 y2
    two sample coefficient of variation upper tail test y1 y2
    set two sample coefficient of variation test miller
    two sample coefficient of variation test y1 y2
    two sample coefficient of variation lower tail test y1 y2
    two sample coefficient of variation upper tail test y1 y2
        
    The following output is generated.
     PARAMETERS AND CONSTANTS--
    
        STATVAL --       2.384724
        STATCDF --       0.999992
        PVALUE  --       0.000015
        PVALL   --       0.999992
        PVALU   --       0.000008
     
                Forkman Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 <> gamma2
     
    Sample One Summary Statistics:
    Total Number of Observations:            249
    Number of Groups Included:               1
    Sample Mean:                             20.144578
    Sample Standard Deviation:               6.414699
    Sample Coefficient of Variation:         0.318433
     
    Sample Two Summary Statistics:
    Total Number of Observations:            79
    Number of Included Groups:               1
    Sample Mean:                             30.481013
    Sample Standard Deviation:               6.107710
    Sample Coefficient of Variation:         0.200378
     
    Forkman Test Statistic Value:            2.384724
    Degrees of Freedom                       248
    Degrees of Freedom                       78
    CDF Value:                               0.999992
    P-Value (2-tailed test):                 0.000015
    P-Value (lower-tailed test):             0.999992
    P-Value (upper-tailed test):             0.000008
     
     
                Forkman Two Sample Test for Equal Coefficient of Variations
     
    H0: gamma1 = gamma2; Ha: gamma1 <> gamma2
    ---------------------------------------------------------------------------
                                            Lower          Upper           Null
       Significance           Test       Critical       Critical     Hypothesis
              Level      Statistic          Value          Value     Conclusion
    ---------------------------------------------------------------------------
              50.0%       2.384724       0.889585       1.140474         REJECT
              80.0%       2.384724       0.798111       1.280145         REJECT
              90.0%       2.384724       0.748580       1.373470         REJECT
              95.0%       2.384724       0.708464       1.461059         REJECT
              99.0%       2.384724       0.636939       1.652378         REJECT
              99.9%       2.384724       0.563977       1.913576         REJECT
     
     
                Forkman Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 < gamma2
     
    Sample One Summary Statistics:
    Total Number of Observations:            249
    Number of Groups Included:               1
    Sample Mean:                             20.144578
    Sample Standard Deviation:               6.414699
    Sample Coefficient of Variation:         0.318433
     
    Sample Two Summary Statistics:
    Total Number of Observations:            79
    Number of Included Groups:               1
    Sample Mean:                             30.481013
    Sample Standard Deviation:               6.107710
    Sample Coefficient of Variation:         0.200378
     
    Forkman Test Statistic Value:            2.384724
    Degrees of Freedom                       248
    Degrees of Freedom                       78
    CDF Value:                               0.999992
    P-Value (2-tailed test):                 0.000015
    P-Value (lower-tailed test):             0.999992
    P-Value (upper-tailed test):             0.000008
     
     
                Lower One-Tailed Test
     
    H0: gamma1 = gamma2; Ha: gamma1 < gamma2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       2.384724       1.005895         REJECT
              80.0%       2.384724       0.863240         REJECT
              90.0%       2.384724       0.798111         REJECT
              95.0%       2.384724       0.748580         REJECT
              99.0%       2.384724       0.664874         REJECT
              99.9%       2.384724       0.583430         REJECT
     
     
                Forkman Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 > gamma2
     
    Sample One Summary Statistics:
    Total Number of Observations:            249
    Number of Groups Included:               1
    Sample Mean:                             20.144578
    Sample Standard Deviation:               6.414699
    Sample Coefficient of Variation:         0.318433
     
    Sample Two Summary Statistics:
    Total Number of Observations:            79
    Number of Included Groups:               1
    Sample Mean:                             30.481013
    Sample Standard Deviation:               6.107710
    Sample Coefficient of Variation:         0.200378
     
    Forkman Test Statistic Value:            2.384724
    Degrees of Freedom                       248
    Degrees of Freedom                       78
    CDF Value:                               0.999992
    P-Value (2-tailed test):                 0.000015
    P-Value (lower-tailed test):             0.999992
    P-Value (upper-tailed test):             0.000008
     
     
                Forkman Two Sample Test for Equal Coefficient of Variations
     
    H0: gamma1 = gamma2; Ha: gamma1 <> gamma2
    ------------------------------------------------------------
                                            Lower           Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic          Value     Conclusion
    ------------------------------------------------------------
              50.0%       2.384724       1.005895         ACCEPT
              80.0%       2.384724       1.177041         ACCEPT
              90.0%       2.384724       1.280145         ACCEPT
              95.0%       2.384724       1.373470         ACCEPT
              99.0%       2.384724       1.571455         ACCEPT
              99.9%       2.384724       1.835646         ACCEPT
     
     
    THE FORTRAN COMMON CHARACTER VARIABLE TWO SAMP HAS JUST BEEN SET TO MILL
     
                Miller Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 <> gamma2
     
    Sample One Summary Statistics:
    Number of Observations:                    249
    Sample Mean:                               20.144578
    Sample Standard Deviation:                 6.414699
    Sample Coefficient of Variation:           0.318433
     
    Sample Two Summary Statistics:
    Number of Observations:                    79
    Sample Mean:                               30.481013
    Sample Standard Deviation:                 6.107710
    Sample Coefficient of Variation:           0.200378
     
    Miller Test Statistic Value:               4.100052
    CDF Value:                                 0.999979
    P-Value (2-tailed test):                   0.000041
    P-Value (lower-tailed test):               0.999979
    P-Value (upper-tailed test):               0.000021
     
     
                Miller Two Sample Test for Equal Coefficient of Variations
     
    H0: gamma1 = gamma2; Ha: gamma1 <> gamma2
    ---------------------------------------------------------------------------
                                            Lower          Upper           Null
       Significance           Test       Critical       Critical     Hypothesis
              Level      Statistic          Value          Value     Conclusion
    ---------------------------------------------------------------------------
              50.0%       4.100052      -0.674490       0.674490         REJECT
              80.0%       4.100052      -1.281552       1.281552         REJECT
              90.0%       4.100052      -1.644854       1.644854         REJECT
              95.0%       4.100052      -1.959964       1.959964         REJECT
              99.0%       4.100052      -2.575829       2.575829         REJECT
              99.9%       4.100052      -3.290527       3.290527         REJECT
     
     
                Miller Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 < gamma2
     
    Sample One Summary Statistics:
    Number of Observations:                    249
    Sample Mean:                               20.144578
    Sample Standard Deviation:                 6.414699
    Sample Coefficient of Variation:           0.318433
     
    Sample Two Summary Statistics:
    Number of Observations:                    79
    Sample Mean:                               30.481013
    Sample Standard Deviation:                 6.107710
    Sample Coefficient of Variation:           0.200378
     
    Miller Test Statistic Value:               4.100052
    CDF Value:                                 0.999979
    P-Value (2-tailed test):                   0.000041
    P-Value (lower-tailed test):               0.999979
    P-Value (upper-tailed test):               0.000021
     
     
                Lower One-Tailed Test
     
    H0: gamma1 = gamma2; Ha: gamma1 < gamma2
    ------------------------------------------------------------
                                                            Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic      Value (<)     Conclusion
    ------------------------------------------------------------
              50.0%       4.100052       0.000000         REJECT
              80.0%       4.100052      -0.841621         REJECT
              90.0%       4.100052      -1.281552         REJECT
              95.0%       4.100052      -1.644854         REJECT
              99.0%       4.100052      -2.326348         REJECT
              99.9%       4.100052      -3.090232         REJECT
     
     
                Miller Two Sample Test for Equal Coefficient of Variations
     
    First Response Variable:  Y1
    Second Response Variable: Y2
     
    H0: Population Coefficients of Variation
        Are Equal (gamma1 = gamma2)
    Ha: gamma1 > gamma2
     
    Sample One Summary Statistics:
    Number of Observations:                    249
    Sample Mean:                               20.144578
    Sample Standard Deviation:                 6.414699
    Sample Coefficient of Variation:           0.318433
     
    Sample Two Summary Statistics:
    Number of Observations:                    79
    Sample Mean:                               30.481013
    Sample Standard Deviation:                 6.107710
    Sample Coefficient of Variation:           0.200378
     
    Miller Test Statistic Value:               4.100052
    CDF Value:                                 0.999979
    P-Value (2-tailed test):                   0.000041
    P-Value (lower-tailed test):               0.999979
    P-Value (upper-tailed test):               0.000021
     
     
                Miller Two Sample Test for Equal Coefficient of Variations
     
    H0: gamma1 = gamma2; Ha: gamma1 <> gamma2
    ------------------------------------------------------------
                                            Lower           Null
       Significance           Test       Critical     Hypothesis
              Level      Statistic          Value     Conclusion
    ------------------------------------------------------------
              50.0%       4.100052       0.000000         ACCEPT
              80.0%       4.100052       0.841621         ACCEPT
              90.0%       4.100052       1.281552         ACCEPT
              95.0%       4.100052       1.644854         ACCEPT
              99.0%       4.100052       2.326348         ACCEPT
              99.9%       4.100052       3.090232         ACCEPT
        

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