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:
 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.
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
.
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
.
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



NIST is an agency of the U.S. Commerce Department.

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