COEFFICIENT OF DISPERSION CONFIDENCE LIMITS

COEFFICIENT OF DISPERSION CONFIDENCE LIMITS

Analysis Command

Generates a confidence interval for the coefficient of dispersion.

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.

<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.

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.

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.

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

A table of confidence limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9.

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).

None

CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMITS

COEFFICIENT OF DISPERSION	=	Compute the coefficient of variation.
COEFFICIENT OF VARIATION	=	Compute the coefficient of variation.
COEFFICIENT OF VARIATION CONFIDENCE LIMITS	=	Generate a confidence limit for the coefficient of variation.
CONFIDENCE LIMITS	=	Generate a confidence limit for the mean.
SD CONFIDENCE LIMITS	=	Generate a confidence limit for the standard deviation.
PREDICTION LIMITS	=	Generate prediction limits for the mean of one or more new observations.
TOLERANCE LIMITS	=	Generate a tolerance limit.

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.

Confirmatory Data Analysis

2017/11

 
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
    

 
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