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Dataplot Vol 1 Vol 2

COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS

Name:
    COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates a confidence interval for the coefficient of quartile 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.

    Another alternative is the coefficient of quartile dispersion which is defined as

      \( \mbox {cqv} = \frac{Q3 - Q1}{Q3 + Q1} \)

    with Q1 denoting the lower quartile (the 25-th percentile) and Q3 denoting the upper quartile (the 75-th percentile).

    These coefficients 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 quartile dispersion confidence limit is from the Bonett paper (see References below). See the Bonett paper for the derivation and formula for this interval.

    Bonett recommends the coefficient of variation for normal (or nearly normal) data, the coefficient of dispersion for moderately non-normal data, and the coefficient of quartile dispersion given here for more extreme non-normal data.

Syntax 1:
    <LOWER/UPPER> COEFFICIENT OF QUARTILE 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 QUARTILE 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 QUARTILE DISPERSION CONFIDENCE ...
                  LIMITS Y1 Y2 Y3

    is equivalent to

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

    You can also use the TO syntax as in

      COEFFICIENT OF QUARTILE 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 QUARTILE 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 QUARTILE DISPERSION CONFIDENCE LIMITS Y1
    COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS Y1 ...
                SUBSET TAG > 2
    MULTIPLE COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS
                Y1 TO Y5
    REPLICATED COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS
                Y X
Note:
    This statistic is based on the lower and upper quartiles. Note that there are various methods for defining percentiles of the data. Using different definitions of percentiles can result in slightly different values of the statistic and the confidence intervals.

    Although Bonett does not specify which method he uses to compute the Q1 and Q3 values, it appears as if he is taking the mean of the two closest points. To use this method to compute Q1 and Q3, enter the command

      SET QUANTILE METHOD AVERAGE

    Dataplot provides three additional methods (called R6, R7, and R8). Enter HELP QUANTILE for details on how these methods compute the quantiles. To use one of these methods, enter one of the following commands

      SET QUANTILE METHOD R6
      SET QUANTILE METHOD R7
      SET QUANTILE METHOD R8

    The default method in Dataplot is R6.

    As the sample size gets larger, this should be less of an issue.

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

      LET ALPHA = 0.05

      LET A = LOWER COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMIT Y
      LET A = UPPER COEFFICIENT OF QUARTILE 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:
    COEFFICIENT OF QUARTILE VARIATION CONFIDENCE LIMITS
    QUARTILE COEFFICIENT OF DISPEERSION CONFIDENCE LIMITS
    QUARTILE COEFFICIENT OF VARIATION CONFIDENCE LIMITS
Related Commands: References:
    Bonett (2006), "Confidence Interval for a Coefficient of Quartile Variation", Computational Statistics and Data Analysis, Vol. 50, pp. 2953-2957.

    Bonett and Seier (2006), "Confidence Interval for a Coefficient of Dispersion", Biometrical Journal, Vol. 48, No. 1, pp. 144-148.

Applications:
    Confirmatory Data Analysis
Implementation Date:
    2017/12
Program 1:
     
    . Following data from Bonett paper
    .
    LET Y = DATA 0.2 0.5 1.1 1.4 1.8 2.3 2.5 2.7 3.5 4.4 4.6 5.4 5.4 ...
            5.7 5.8 5.9 6.0 6.6 7.1 7.9
    .
    SET WRITE DECIMALS 5
    SET QUANTILE METHOD AVERAGE
    COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS Y
    .
    LET ALPHA = 0.05
    LET CQV  = COEFFICIENT OF QUARTILE DISPERSION Y
    LET LCDL = LOWER COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMIT Y
    LET UCDL = UPPER COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMIT Y
    .
    PRINT CQV LCDL UCDL
        
    The following output is generated
                Two-Sided Confidence Limit for the Coefficient of
                               Quartile Dispersion
     
     
    Response Variable: Y
     
    Summary Statistics:
    Number of Observations:                  20
    Sample Lower Quartile:                   2.05000
    Sample Upper Quartile:                   5.85000
    Quantile Method: Average
    Sample Coefficient of Quartile Disp:     0.48101
     
     
     
    ---------------------------------------------------------------
      Confidence       Coefficient of          Lower          Upper
       Value (%)  Quartile Dispersion          Limit          Limit
    ---------------------------------------------------------------
            80.0              0.48101        0.34624        0.61870
            90.0              0.48101        0.28164        0.76061
            95.0              0.48101        0.25607        0.83655
            99.0              0.48101        0.20089        1.06636
            99.9              0.48101        0.15464        1.38523
    
    
     PARAMETERS AND CONSTANTS--
    
        CQV     --        0.48101
        LCDL    --        0.25607
        UCDL    --        0.83655
        
Program 2:
     
    SKIP 25
    READ GEAR.DAT Y X
    .
    SET WRITE DECIMALS 5
    SET QUANTILE METHOD AVERAGE
    REPLICATED COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS Y X
        
    The following output is generated
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     1.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99350
     Sample Upper Quartile:                   1.00000
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00326
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00326        0.00105        0.00284
             90.0              0.00326        0.00091        0.00327
             95.0              0.00326        0.00082        0.00361
             99.0              0.00326        0.00057        0.00522
             99.9              0.00326        0.00035        0.00847
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     2.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99650
     Sample Upper Quartile:                   1.00100
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00225
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00225        0.00045        0.00291
             90.0              0.00225        0.00035        0.00379
             95.0              0.00225        0.00026        0.00500
             99.0              0.00225        0.00018        0.00711
             99.9              0.00225        0.00008        0.01651
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     3.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99250
     Sample Upper Quartile:                   0.99800
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00276
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00276        0.00074        0.00280
             90.0              0.00276        0.00061        0.00338
             95.0              0.00276        0.00061        0.00339
             99.0              0.00276        0.00044        0.00474
             99.9              0.00276        0.00024        0.00875
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     4.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99450
     Sample Upper Quartile:                   1.00100
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00326
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00326        0.00112        0.00266
             90.0              0.00326        0.00099        0.00301
             95.0              0.00326        0.00084        0.00356
             99.0              0.00326        0.00067        0.00444
             99.9              0.00326        0.00043        0.00696
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     5.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.98300
     Sample Upper Quartile:                   0.99700
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00707
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00707        0.00279        0.00597
             90.0              0.00707        0.00250        0.00665
             95.0              0.00707        0.00220        0.00755
             99.0              0.00707        0.00182        0.00915
             99.9              0.00707        0.00113        0.01470
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     6.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99150
     Sample Upper Quartile:                   1.00550
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00701
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00701        0.00234        0.00699
             90.0              0.00701        0.00200        0.00817
             95.0              0.00701        0.00190        0.00858
             99.0              0.00701        0.00136        0.01197
             99.9              0.00701        0.00072        0.02258
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     7.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99600
     Sample Upper Quartile:                   1.00300
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00350
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00350        0.00068        0.00514
             90.0              0.00350        0.00051        0.00686
             95.0              0.00350        0.00046        0.00752
             99.0              0.00350        0.00028        0.01263
             99.9              0.00350        0.00013        0.02730
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     8.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99700
     Sample Upper Quartile:                   1.00200
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00250
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00250        0.00077        0.00215
             90.0              0.00250        0.00067        0.00248
             95.0              0.00250        0.00066        0.00249
             99.0              0.00250        0.00042        0.00397
             99.9              0.00250        0.00031        0.00533
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     9.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99550
     Sample Upper Quartile:                   1.00050
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00251
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00251        0.00067        0.00246
             90.0              0.00251        0.00056        0.00295
             95.0              0.00251        0.00056        0.00297
             99.0              0.00251        0.00036        0.00456
             99.9              0.00251        0.00018        0.00899
      
      
                 Two-Sided Confidence Limit for the Coefficient of
                                Quartile Dispersion
      
      
     Response Variable: Y
     Factor Variable 1: X                     10.00000
      
     Summary Statistics:
     Number of Observations:                  10
     Sample Lower Quartile:                   0.99100
     Sample Upper Quartile:                   0.99700
     Quantile Method: Average
     Sample Coefficient of Quartile Disp:     0.00302
      
      
      
     ---------------------------------------------------------------
       Confidence       Coefficient of          Lower          Upper
        Value (%)  Quartile Dispersion          Limit          Limit
     ---------------------------------------------------------------
             80.0              0.00302        0.00073        0.00345
             90.0              0.00302        0.00058        0.00430
             95.0              0.00302        0.00047        0.00531
             99.0              0.00302        0.00034        0.00740
             99.9              0.00302        0.00018        0.01374
        
Date created: 12/12/2017
Last updated: 12/11/2023

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