COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS

COEFFICIENT OF QUARTILE DISPERSION CONFIDENCE LIMITS

Analysis Command

Generates a confidence interval for the coefficient of quartile 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.

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.

<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.

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.

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.

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

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.

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

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).

None

COEFFICIENT OF QUARTILE VARIATION CONFIDENCE LIMITS
QUARTILE COEFFICIENT OF DISPEERSION CONFIDENCE LIMITS
QUARTILE COEFFICIENT OF VARIATION CONFIDENCE LIMITS

COEFFICIENT OF QUARTILE DISPERSION	=	Compute the coefficient of quartile dispersion.
COEFFICIENT OF DISPERSION	=	Compute the coefficient of variation.
COEFFICIENT OF VARIATION	=	Compute the coefficient of variation.
COEFFICIENT OF DISPERSION CONFIDENCE LIMITS	=	Generate a confidence limit for the coefficient of dispersion.
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 (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.

Confirmatory Data Analysis

2017/12

 
. 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
    

 
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