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Dataplot Vol 1 Vol 2

COEFFICIENT OF DISPERSION CONFIDENCE LIMITS

Name:
    COEFFICIENT OF DISPERSION CONFIDENCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates a confidence interval for the coefficient of dispersion.
Description:
    The sample coefficient of variation is defined as the ratio of the standard deviation to the mean

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

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

    The coefficient of variation is sensitive to non-normality. An alternative statistic is the coefficient of dispersion which is defined as

      \( \mbox{COD} = \frac{\tau}{\eta} \)

    with \( \tau \) and \( \eta \) denoting the mean absolute difference from the mean and the median, respectively.

    The coefficients of variation and dispersion should typically only be used for ratio data. That is, the data should be continuous and have a meaningful zero. Although these statistics can be computed for data that is not on a ratio scale, the interpretation of them 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 method for computing the coefficient of dispersion confidence limit is from the Bonett paper (see References below). Dataplot uses a Fortran implementation of the R code given in the paper. See the Bonett paper for the derivation and formula for this interval.

    According to simulation studies by Bonett, the confidence interval tends to perform as well or better as the BCa bootstrap interval and significantly better than the percentile bootstrap. Bonett also recommends the coefficient of dispersion statistic for moderately non-normal data. For more extreme non-normality, large sample sizes may be required for decent performance. For the more extreme non-normal data sets, the quartile coefficient of dispersion may be preferred.

Syntax 1:
    <LOWER/UPPER> COEFFICIENT OF DISPERSION CONFIDENCE LIMITS <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax supports matrix arguments for the response variable.

Syntax 2:
    MULTIPLE <LOWER/UPPER> COEFFICIENT OF DISPERSION
                            CONFIDENCE LIMITS <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> .... <yk> is a list of 1 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax will generate a confidence interval for each of the response variables. The word MULTIPLE is optional. That is,

      MULTIPLE COEFFICIENT OF DISPERSION CONFIDENCE LIMITS ...
                              Y1 Y2 Y3

    is equivalent to

      COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y1 Y2 Y3

    You can also use the TO syntax as in

      COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y1 TO Y10

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax supports matrix arguments for the response variables.

Syntax 3:
    REPLICATED <LOWER/UPPER> COEFFICIENT OF DISPERSION
                            CONFIDENCE LIMITS <y> <x1> ... <xk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x1> .... <xk> is a list of 1 to 6 group-id variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a cross-tabulation of the <x1> ... <xk> and generates a confidence interval for each unique combination of the cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, six confidence intervals will be generated.

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax does not support matrix arguments.

Examples:
    COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y1
    COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y1 SUBSET TAG > 2
    MULTIPLE COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y1 TO Y5
    REPLICATED COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y X
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 DISPERSION CONFIDENCE LIMIT command, the following commands can also be used:

      LET ALPHA = 0.05

      LET A = LOWER COEFFICIENT OF DISPERSION CONFIDENCE LIMIT Y
      LET A = UPPPER COEFFICIENT OF DISPERSION CONFIDENCE LIMIT Y
      LET A = LOWER ONESIDED COEFFICIENT OF DISPERSION ...
                        CONFIDENCE LIMIT Y
      LET A = UPPER ONESIDED COEFFICIENT OF DISPERSION ...
                        CONFIDENCE LIMIT Y

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

Default:
    None
Synonyms:
    CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMITS
Related Commands: References:
    Bonett and Seier (2006), "Confidence Interval for a Coefficient of Dispersion", Biometrical Journal, Vol. 48, No. 1, PP. 144-148.

    Bonett (2006), "Confidence Interval for a Coefficient of Quartile Variation", Computational Statistics and Data Analysis, Vol. 50, pp. 2953-2957.

Applications:
    Confirmatory Data Analysis
Implementation Date:
    2017/11
Program 1:
     
    SKIP 25
    READ WEIBBURY.DAT Y
    .
    SET WRITE DECIMALS 5
    COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y
    .
    LET LCD  = COEFFICIENT OF DISPERSION Y
    LET LCDL = LOWER COEFFICIENT OF DISPERSION CONFIDENCE LIMIT Y
    LET UCDL = UPPER COEFFICIENT OF DISPERSION CONFIDENCE LIMIT Y
    .
    PRINT CD LCDL UCDL
     
        
    The following output was generated
                Two-Sided Confidence Limit for the Coefficient of Dispersion
     
     
    Response Variable: Y
     
    Summary Statistics:
    Number of Observations:                  20
    Sample Median:                           53.75000
    Sample Average Absolute Deviation:       6.26000
    Sample Coefficient of Dispersion:        0.11647
     
     
     
    ---------------------------------------------------------
      Confidence    Coefficient          Lower          Upper
       Value (%)  of Dispersion          Limit          Limit
    ---------------------------------------------------------
            50.0        0.11647        0.11024        0.14011
            80.0        0.11647        0.10048        0.16025
            90.0        0.11647        0.09354        0.17218
            95.0        0.11647        0.08982        0.17931
            99.0        0.11647        0.08297        0.19831
            99.9        0.11647        0.07036        0.22321
     
     
     PARAMETERS AND CONSTANTS--
    
        COD     --        0.11647
        LCDL    --        0.08982
        UCDL    --        0.17931
        
Program 2:
     
    SKIP 25
    READ GEAR.DAT Y X
    .
    SET WRITE DECIMALS 5
    REPLICATED COEFFICIENT OF DISPERSION CONFIDENCE LIMITS Y X
        
    The following output was generated
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     1.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99850
     Sample Average Absolute Deviation:       0.00340
     Sample Coefficient of Dispersion:        0.00341
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00341        0.00318        0.00000
             80.0        0.00341        0.00273        0.00526
             90.0        0.00341        0.00249        0.00577
             95.0        0.00341        0.00229        0.00626
             99.0        0.00341        0.00195        0.00733
             99.9        0.00341        0.00195        0.00733
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     2.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99900
     Sample Average Absolute Deviation:       0.00370
     Sample Coefficient of Dispersion:        0.00370
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00370        0.00333        0.00000
             80.0        0.00370        0.00275        0.00615
             90.0        0.00370        0.00246        0.00689
             95.0        0.00370        0.00222        0.00761
             99.0        0.00370        0.00183        0.00929
             99.9        0.00370        0.00183        0.00929
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     3.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99600
     Sample Average Absolute Deviation:       0.00280
     Sample Coefficient of Dispersion:        0.00281
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00281        0.00251        0.00000
             80.0        0.00281        0.00205        0.00474
             90.0        0.00281        0.00183        0.00534
             95.0        0.00281        0.00165        0.00593
             99.0        0.00281        0.00135        0.00727
             99.9        0.00281        0.00135        0.00727
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     4.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99700
     Sample Average Absolute Deviation:       0.00320
     Sample Coefficient of Dispersion:        0.00321
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00321        0.00302        0.00000
             80.0        0.00321        0.00261        0.00486
             90.0        0.00321        0.00239        0.00530
             95.0        0.00321        0.00222        0.00573
             99.0        0.00321        0.00191        0.00664
             99.9        0.00321        0.00191        0.00664
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     5.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99450
     Sample Average Absolute Deviation:       0.00610
     Sample Coefficient of Dispersion:        0.00613
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00613        0.00568        0.00000
             80.0        0.00613        0.00482        0.00970
             90.0        0.00613        0.00438        0.01069
             95.0        0.00613        0.00402        0.01166
             99.0        0.00613        0.00340        0.01377
             99.9        0.00613        0.00340        0.01377
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     6.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99750
     Sample Average Absolute Deviation:       0.00740
     Sample Coefficient of Dispersion:        0.00742
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00742        0.00680        0.00000
             80.0        0.00742        0.00574        0.01181
             90.0        0.00742        0.00515        0.01309
             95.0        0.00742        0.00471        0.01439
             99.0        0.00742        0.00397        0.01709
             99.9        0.00742        0.00397        0.01709
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     7.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           1.00050
     Sample Average Absolute Deviation:       0.00550
     Sample Coefficient of Dispersion:        0.00550
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00550        0.00490        0.00000
             80.0        0.00550        0.00401        0.00931
             90.0        0.00550        0.00356        0.01048
             95.0        0.00550        0.00320        0.01161
             99.0        0.00550        0.00260        0.01427
             99.9        0.00550        0.00260        0.01427
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     8.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           1.00000
     Sample Average Absolute Deviation:       0.00280
     Sample Coefficient of Dispersion:        0.00280
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00280        0.00260        0.00000
             80.0        0.00280        0.00222        0.00436
             90.0        0.00280        0.00202        0.00479
             95.0        0.00280        0.00185        0.00521
             99.0        0.00280        0.00158        0.00612
             99.9        0.00280        0.00158        0.00612
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     9.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99800
     Sample Average Absolute Deviation:       0.00310
     Sample Coefficient of Dispersion:        0.00311
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00311        0.00286        0.00000
             80.0        0.00311        0.00240        0.00494
             90.0        0.00311        0.00217        0.00547
             95.0        0.00311        0.00199        0.00598
             99.0        0.00311        0.00167        0.00713
             99.9        0.00311        0.00167        0.00713
      
      
                 Two-Sided Confidence Limit for the Coefficient of Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     10.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Median:                           0.99600
     Sample Average Absolute Deviation:       0.00380
     Sample Coefficient of Dispersion:        0.00382
      
      
      
     ---------------------------------------------------------
       Confidence    Coefficient          Lower          Upper
        Value (%)  of Dispersion          Limit          Limit
     ---------------------------------------------------------
             50.0        0.00382        0.00340        0.00000
             80.0        0.00382        0.00279        0.00646
             90.0        0.00382        0.00248        0.00728
             95.0        0.00382        0.00224        0.00806
             99.0        0.00382        0.00182        0.00992
             99.9        0.00382        0.00182        0.00992
      
        

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Date created: 12/07/2017
Last updated: 12/07/2017

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