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COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS

Name:
    COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates a confidence interval for the common coefficient of variation for two or more groups for normally distributed data.
Description:
    The coefficient of variation is defined as the ratio of the standard deviation to the mean

      \( \mbox{cv} = \frac{\sigma} {\mu} \)

    where \( \sigma \) and \( \mu \) denote the population standard deviation and population mean, respectively. The sample coefficient of variation is defined as

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

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

    The coefficient of variation should typically only be used for ratio data. That is, the data should be continuous and have a meaningful zero. Although the coefficient of variation statistic can be computed for data that is not on a ratio scale, the interpretation of the coeffcient of variation 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.

    For a single sample of normally distributed data, the command

    can be used to determine a confidence interval (this command supports six different methods for obtaining the confidence interval).

    The command documented here is for the case where there are two or more groups of data and we want to find a confidence interval for the common coefficient of variation for all groups. This command assumes that the data for each group is approximately normally distributed and that each group has the same coefficient of variation. For that reason, it is desirable to plot the coefficient of variation for each group (see the program example below) to check if this is a reasonable assumption.

    The biased common coefficient of variation is

      \( \gamma = \frac{\sum_{i}^{k}(N_{i}-1)C_{i}^{2}{}} {\sum_{i=1}^{k}{N_{i} - 1}} \)

    where

      k = the number of groups
      Ni = the sample size of the i-th group
      Ci = the sample coefficient of variation for the i-th group

    A bias correccted estimate of the common coefficient of variation is

      \( \gamma_{\mbox{BC}} = \frac{\gamma}{1 - 1/(4 \sum_{i=1}^{k}{(N_{i}-1}))} \)

    The confidence limits are computed as

      Lower Limit = \( \sqrt{ \frac{\sum_{i}^{k}{(N_{i} - 1) u_{i}}} {\chi_{1 - \alpha/2}^{2} - \sum_{i}^{k}{(N_{i} - 1) u_{i}}} } \)

      Upper Limit = \( \sqrt{ \frac{\sum_{i}^{k}{(N_{i} - 1) u_{i}}} {\chi_{\alpha/2}^{2} - \sum_{i}^{k}{(N_{i} - 1) u_{i}}} } \)

    where

      ui = \( \frac{c_{i}^{2}} {1 + c_{i}^{2}(N_{i} - 1)/N_{i}} \)
      DF = \( \sum_{i=1}^{k}{N_{i} - 1} \)
      CHSPPF = the percent point function of the chi-square distribution

    This confidence interval was proposed by Forkman (see References below).

Syntax 1:
    <LOWER/UPPER> COMMON COEFFICIENT OF VARIATION CONFIDENCE
                            LIMITS <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x> is the group-id 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 does not support matrix arguments for the response and group-id variables.

Syntax 2:
    MULTIPLE <LOWER/UPPER> COMMON COEFFICIENT OF VARIATION
                            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 can be used if the groups are in separate variables.

    You can use the TO syntax as in

      COEFFICIENT OF VARIATION 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.

Examples:
    COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 X1
    COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS Y1 X1 ...
                            SUBSET X1 > 2
    MULTIPLE COMMON COEFFICIENT OF VARIATION CONFIDENCE LIMITS ...
                            Y1 TO Y5
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 VARIATION CONFIDENCE LIMIT command, the following commands can also be used:

      LET ALPHA = 0.05

      LET A = COMMON COEFFICIENT OF VARIATION Y X
      LET A = COMMON BIAS CORRECTED COEFFICIENT OF VARIATION Y X
      LET A = LOWER COMMON COEFFICIENT OF VARIATION CONFIDENCE ...
                              LIMIT Y X
      LET A = UPPPER COMMON COEFFICIENT OF VARIATION ...
                              CONFIDENCE LIMIT Y X

    The LET ALPHA command specifies the significance level.

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

Note:
    Any groups that have only a single value or that have negative values will be omitted when computing the confidence interval. After removing any such groups, at least two groups must be available before the confidence limit will be generated.
Default:
    None
Synonyms:
    CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMITS
Related Commands: References:
    Forkman (2009), "Estimator and Tests for Common Coefficients of Variation in Normal Distributions", Communications in Statistics - Theory and Methods, Vol. 38, pp. 233-251.

    McKay (1932), "Distributions of the Coefficient of Variation and the Extended 't' Distribution", Journal of the Royal Statistical Society, Vol. 95, pp. 695-698.

Applications:
    Confirmatory Data Analysis
Implementation Date:
    2017/06
Program:
    . Step 1:   Create the data
    .
    skip 25
    read gear.dat y x
    skip 0
    set write decimals 6
    .
    . Step 2:   Define plot control settings and plot the data
    .
    title case asis
    title offset 2
    label case asis
    .
    y1label Coefficient of Variation
    x1label Group
    title Coefficient of Variation for GEAR.DAT
    let ngroup = unique x
    xlimits 1 ngroup
    major x1tic mark number ngroup
    minor x1tic mark number 0
    tic mark offset units data
    x1tic mark offset 0.5 0.5
    y1tic mark label decimals 3
    .
    character X
    line blank
    .
    set statistic plot reference line average
    .
    coefficient of variation plot y x
    .
    . Step 3:   Define plot control settings and plot the data
    .
    common coefficient of variation confidence limits y x
        
    plot generated by sample program
                Two-Sided Confidence Limits for the Common Coefficient
                      of Variation for Normally Distributed Data
     
    Method: Forkman
    Response Variable: Y
    Group-ID Variable: X
     
    Number of Distinct Groups:               10
    Number of Groups Included in Analysis:   10
    Common Coefficient of Variation:         0.005953
    Bias Corrected Estimate:                 0.005969
     
     
    ---------------------------------------------------------
      Confidence    Coefficient          Lower          Upper
       Value (%)   of Variation          Limit          Limit
    ---------------------------------------------------------
            50.0       0.005953       0.005686       0.006289
            80.0       0.005953       0.005445       0.006596
            90.0       0.005953       0.005309       0.006792
            95.0       0.005953       0.005196       0.006970
            99.0       0.005953       0.004985       0.007340
            99.9       0.005953       0.004759       0.007810
        

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Date created: 06/23/2017
Last updated: 06/23/2017

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