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Dataplot Vol 2 Vol 1

STATISTIC ANDERSON DARLING

Name:
    <dist> ANDERSON DARLING
Type:
    LET Subcommand
Purpose:
    Compute the Anderson-Darling (A-D) goodness of fit statistic for a specified distribution for a response variable.
Description:
    The Anderson-Darling test is a goodness of fit statistic (see the documentation for the GOODNESS OF FIT command for details).

    Although this value is normally determined using the GOODNESS OF FIT command, for a limited number of distributions you can also generate this as a statistic LET subcommand. The advantage in this case is that you can use it with any of the commands that support built-in statistics (e.g., the STATISTIC PLOT or the TABULATION command). For example, if you have groups of data, you can use the TABULATE or STATISTIC PLOT commands to easily compare the goodness of fit across the groups.

    In order to compute the Anderson-Darling statistic, the distribution parameters are first computed using maximum likelihood. This command can also return the maximum likelihood estimates.

    This command is only supported for a subset of the distributions for which the Anderson-Darling statistic is supported with the GOODNESS OF FIT command. See the Note section below for a list of distributions supported by this command.

Syntax 1:
    LET <par> = <dist> ANDERSON DARLING <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <dist> is one of the supported distributions listed below;
                <par> is the parameter where the AD value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax is used for the case where there are no shape parameters.

Syntax 2:
    LET <par> = <dist> ANDERSON DARLING STATISTIC <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <dist> is one of the supported distributions listed below;
                <par> is the parameter where the AD value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax is used for those distributions that have a shape parameter. When the distribution has a shape parameter, then the word STATISTIC is required to distinguish this command from the ANDERSON DARLING PLOT command. Specifically it is used to distinguish the following two cases

      WEIBULL ANDERSON DARLING PLOT Y
      WEIBULL ANDERSON DARLING STATISTIC PLOT Y X

    The first command is the Weibull Anderson Darling plot (i.e., a plot of the Anderson-Darling goodness of fit across values of the shape parameter) while the second command plots the Anderson Darling statistic for Y for each group in X.

Syntax 3:
    LET <par> = <dist> ANDERSON DARLING LOCATION <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <dist> is one of the supported distributions listed below;
                <par> is the parameter where the location value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax returns the estimate of the location parameter rather than the AD value. Not all supported distributions have a location parameter.

Syntax 4:
    LET <par> = <dist> ANDERSON DARLING SCALE <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <dist> is one of the supported distributions listed below;
                <par> is the parameter where the scale value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax returns the estimate of the scale parameter rather than the AD value.

Syntax 5:
    LET <par> = <dist> ANDERSON DARLING SHAPE <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <dist> is one of the supported distributions listed below;
                <par> is the parameter where the shape value is saved;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax returns the estimate of the shape parameter rather than the AD value. Not all supported distributions have a shape parameter.

Examples:
    LET A = NORMAL ANDERSON DARLING Y
    LET A = NORMAL ANDERSON DARLING LOCATION Y
    LET A = NORMAL ANDERSON DARLING SCALE Y

    LET A = LOGISTIC ANDERSON DARLING Y

    LET A = WEIBULL ANDERSON DARLING STATISTIC Y
    LET A = WEIBULL ANDERSON DARLING SCALE Y
    LET A = WEIBULL ANDERSON DARLING SHAPE Y

Note:
    The following location/scale distributions are supported.

      LET A = DOUBLE EXPONENTIAL ANDERSON DARLING Y
      LET A = EXPONENTIAL ANDERSON DARLING Y
      LET A = GUMBEL ANDERSON DARLING Y
      LET A = LOGISTIC ANDERSON DARLING Y
      LET A = MAXWELL ANDERSON DARLING Y
      LET A = NORMAL ANDERSON DARLING Y
      LET A = RAYLEIGH ANDERSON DARLING Y
      LET A = UNIFORM ANDERSON DARLING Y

    The Maxwell and Rayleigh support the 2-parameter case rather than the 1-parameter case (i.e., the location parameter will be estimated).

    For the uniform distribution, the scale parameter will actually return the upper limit parameter.

    In addition, the following distributions with a single shape parameter are supported.

      LET A = BURR TYPE 10 ANDERSON DARLING STATISTIC Y
      LET A = FATIGUE LIFE ANDERSON DARLING STATISTIC Y
      LET A = FRECHET ANDERSON DARLING STATISTIC Y
      LET A = GAMMA ANDERSON DARLING STATISTIC Y
      LET A = GEOMETRIC EXTREME EXPONENTIAL ...
                      ANDERSON DARLING STATISTIC Y
      LET A = INVERTED GAMMA ANDERSON DARLING STATISTIC Y
      LET A = LOGISTIC EXPONENTIAL ANDERSON DARLING ...
                      STATISTIC Y
      LET A = LOGNORMAL ANDERSON DARLING STATISTIC Y
      LET A = WEIBULL ANDERSON DARLING STATISTIC Y

    Note that the above support the 2-parameter form of the distribution (i.e., the scale and shape parameters are estimated from the data and the location parameter is set to zero).

Note:
    The distribution parameters are estimated using maximum likelihood. For several distributions, you can choose an alternative estimation method using the command

      SET DISTRIBUTIONAL FIT TYPE <value>

    where <value> can be one of the following

      ML - use the default maximum likelihood (available for all supported distributions)
      MOMENT - use the moment estimates, available for uniform, Gumbel, 2-par Maxwell, 2-par gamma, 2-par inverted gamma, 2-par fatigue life
      MODIFIED MOMENT - use the modified moment estimates, available for 2-par Rayleigh
Note:
    Dataplot statistics can be used in 20+ commands. For details, enter

Default:
    None
Synonyms:
    AD is a synonym for ANDERSON DARLING
Related Commands: Reference:
    Stephens, M. A. (1974), "EDF Statistics for Goodness of Fit and Some Comparisons," Journal of the American Statistical Association, Vol. 69, pp. 730-737.

    Stephens, M. A. (1976), "Asymptotic Results for Goodness-of-Fit Statistics with Unknown Parameters," Annals of Statistics, Vol. 4, pp. 357-369.

    Stephens, M. A. (1977), "Goodness of Fit for the Extreme Value Distribution," Biometrika, Vol. 64, pp. 583-588.

    Stephens, M. A. (1977), "Goodness of Fit with Special Reference to Tests for Exponentiality," Technical Report No. 262, Department of Statistics, Stanford University, Stanford, CA.

    Stephens, M. A. (1979), "Tests of Fit for the Logistic Distribution Based on the Empirical Distribution Function," Biometrika, Vol. 66, pp. 591-595.

Applications:
    Distributional Modeling
Implementation Date:
    2015/2
Program 1:
     
    . Step 1:   Read the data
    .
    skip 25
    read nor.dat      y1
    read exp.dat      y2
    read weibbury.dat y3
    read lgn.dat      y4
    read gamma.dat    y5
    read frechet.dat  y6
    let y x = stack y1 y2 y3 y4 y5 y6
    skip 0
    .
    case asis
    label case asis
    title case asis
    title offset 2
    multiplot corner coordinates 2 2 98 98
    multiplot scale factor 2
    .
    . Step 2:   Plot normal a-d statistic
    .
    multiplot 2 2
    y1label Anderson-Darling Statistic
    x3label Datasets
    xlimits 1 6
    major xtic mark number 6
    minor xtic mark number 0
    x1tic mark offset 0.5 0.5
    x1tic mark label format alpha
    x1tic mark label content NOR.DAT sp()cr()sp()cr()EXP.DAT WEIBBURY.DAT ...
          sp()cr()sp()cr()LGN.DAT GAMMA.DAT sp()cr()sp()cr()FRECHET.DAT
    x1tic mark label size 1.2
    y1label displacement 12
    ylimits 0 1
    character X
    line blank
    .
    title Normal AD
    normal anderson darling plot y x
    .
    ylimits
    y1label Location
    title Normal AD Location
    normal anderson darling location plot y x
    .
    y1label Scale
    title Normal AD Scale
    normal anderson darling scale plot y x
    .
    
    plot generated by sample program
    .
    . Step 3:   Location/Scale distributions
    .
    multiplot 2 2
    label
    .
    ylimits 0 1
    title Normal AD
    normal anderson darling plot y x
    .
    ylimits 0 6
    title Exponential AD
    exponential anderson darling plot y x
    ylimits 0 1000
    title Double Exponential AD
    double exponential anderson darling plot y x
    ylimits 0  1.5
    title Gumbel (Maximum)
    gumbel anderson darling plot y x
    .
    end of multiplot
    .
    justification center
    move 50 3
    text Datasets
    direction vertical
    move 2 50
    text Anderson-Darling Statistic
    direction horizontal
    .
    
    plot generated by sample program
    multiplot 2 2
    x1tic mark label format numeric
    label
    .
    ylimits 0 20
    title Uniform AD
    uniform anderson darling plot y x
    .
    ylimits 0 1
    title Maxwell AD
    maxwell anderson darling plot y x
    .
    ylimits 0 10
    title Rayleigh AD
    rayleigh anderson darling plot y x
    .
    ylimits 0 1
    title Logistic
    logistic anderson darling plot y x
    .
    end of multiplot
    .
    justification center
    move 50 3
    text Datasets
    direction vertical
    move 2 50
    text Anderson-Darling Statistic
    direction horizontal
        
    plot generated by sample program
Program 2:
    . Step 1:   Read the data
    .
    skip 25
    read nor.dat      y1
    read exp.dat      y2
    read weibbury.dat y3
    read lgn.dat      y4
    read gamma.dat    y5
    read frechet.dat  y6
    let y x = stack y1 y2 y3 y4 y5 y6
    skip 0
    .
    case asis
    title case asis
    title offset 2
    .
    xlimits 1 6
    major xtic mark number 6
    minor xtic mark number 0
    x1tic mark offset 0.5 0.5
    x1tic mark label size 1.5
    character X
    line blank
    .
    . Step 3:   Weibull, Lognormal, Gamma, Fatigue Life
    .
    multiplot corner coordinates 2 2 98 98
    multiplot scale factor 2
    multiplot 2 2
    x1tic mark label format numeric
    label
    .
    ylimits 0 1
    title Weibull AD
    weibull anderson darling statistic plot y x
    . weibull ad statistic plot y x
    .
    ylimits 0 1
    title Lognormal AD
    lognormal anderson darling statistic plot y x
    ylimits
    .
    .           Note that gamma has problem with datasets 1 and 4
    .
    ylimits 0 1
    title Gamma AD
    gamma anderson darling statistic plot y x  subset x 2 3 5 6
    .
    ylimits 0 2
    title Fatigue Life AD
    fatigue life anderson darling statistic plot y x
    .
    end of multiplot
    .
    justification center
    move 50 3
    text Datasets
    direction vertical
    move 2 50
    text Anderson-Darling Statistic
    direction horizontal
        
    plot generated by sample program

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Date created: 02/09/2015
Last updated: 02/09/2015

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