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Dataplot Vol 2 Auxiliary Chapter

DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE

Name:
    DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE (LET)
Type:
    Let Subcommand
Purpose:
    Compute the difference between the percentage bend midvariances for two response variables.
Description:
    The percentage bend midvariance estimator, discussed in Shoemaker and Hettmansperger and also by Wilcox, is both resistant and robust of efficiency.

    The percentage bend midvariance of a a variable X is computed as follows:

    1. Set m = [(1 - betan + 0.5]. This is the value of (1 - betan + 0.5 rounded down to the nearest integer.

    2. Let Wi = |Xi - M| for i = 1, ..., n where M is the median of X.

    3. Sort the Wi in ascending order.

    4. what(beta) = Wm (i. e., the mth order statistic). Wm is the estimate of the (1 - beta) quantile of W.

    5. Y(i) = (X(i) - M)/what(beta)

    6. a(i) = 1 if |Y(i)| < 1; = 0 if |Y(i)| >= 1

    7. phi(x) = max[-1, min(1,x)]

    8. s(pb) = n*what(beta)**2*SUM[{phi(Y(i))}**2]/(SUM[a(i)])**2

    The value of beta is selected between 0 and 0.5. Higher values of beta is selected result in a higher breakdown point at the expense of lower efficiency.

    For the differeence of percentage bend midvariances, the percentage bend midvariance is computed for each of two samples then their difference is taken.

Syntax:
    LET <par> = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE <y1> <y2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> is the first response variable;
    <y2> is the first response variable;
    <par> is a parameter where the computed difference of the percentage bend midvariances is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2
    LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 SUBSET X > 1
Note:
    To set the value of beta, enter the command

      LET BETA = <value>

    where is greater than 0 and less than or equal to 0.5. The default value for beta is 0.1.

Note:
    The difference of percentage bend midvariances statistic is also supported for the following commands:

      DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE PLOT Y1 Y2 X
      CROSS TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE PLOT Y1 Y2 X
      BOOTSTRAP DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE PLOT Y1 Y2
      TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 X
      CROSS TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 X
      LET Z = CROSS TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 X1 X2

    Enter

      HELP STATISTIC PLOT
      HELP CROSS TABULATE PLOT
      HELP BOOTSTRAP PLOT
      HELP TABULATE
      HELP CROSS TABULATE
      HELP CROSS TABULATE (LET)

    for more information on these plots and commands.

Default:
    None
Synonyms:
    None
Related Commands:
    PERCENTAGE BEND MIDVARIANCE = Compute the percentage bend midvariance.
    MAD = Compute the mad.
    AAD = Compute the aad.
    IQ RANGE = Compute the interquartile range.
    STANDARD DEVIATION = Compute the standard deviation.
    DIFFERENCE OF MAD = Compute the difference of the mad.
    DIFFERENCE OF AAD = Compute the difference of the aad.
    DIFFERENCE OF SD = Compute the difference of standard deviations.
    STATISTICS PLOT = Generate a statistic versus subset plot.
    CROSS TABULATE PLOT = Generate a statistic versus two subsets plot.
    BOOTSTRAP PLOT = Generate a bootstrap plot.
    TABULATE = Perform a tabulation for a specified statistic.
    CROSS TABULATE = Perform a cross tabulation for a specified statistic.
References:
    "Robust Estimates of and Tests for the One- and Two-Sample Scale Models", Shoemaker and Hettmansperger, Biometrika 69, 1982, pp. 47-54.

    "Introduction to Robust Estimation and Hypothesis Testing", Rand Wilcox, Academic Press, 1997.

    "Data Analysis and Regression: A Second Course in Statistics", Mosteller and Tukey, Addison-Wesley, 1977, pp. 203-209.

Applications:
    Data Analysis
Implementation Date:
    3/2003
Program: 
    SKIP 25 
READ IRIS.DAT Y1 TO Y4 X 
. 
LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 
TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 X 
. 
XTIC OFFSET 0.2 0.2 
X1LABEL GROUP ID 
Y1LABEL DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE 
CHAR X 
LINE BLANK 
DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE PLOT Y1 Y2 X 
CHAR X ALL 
LINE BLANK ALL 
BOOTSTRAP DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE PLOT Y1 Y2 X
    Dataplot generated the following output. 
           ***************************************************************
       **  LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2  **
       ***************************************************************
  
  
 THE COMPUTED VALUE OF THE CONSTANT A             =  0.51659620E+00
  
  
       ******************************************************************
       **  TABULATE DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 X  **
       ******************************************************************
  
  
                 *    Y1       AND Y2
     X           *    DIFFERENCE OF PERCENTAGE BEND MIDVARIANC
 **********************************************
     1.00000     *    0.153299E-01
     2.00000     *    0.201068
     3.00000     *    0.370212
  
       GROUP-ID AND STATISTIC WRITTEN TO FILE DPST1F.DAT
    plot generated by sample program

    plot generated by sample program

Date created: 3/27/2003
Last updated: 4/4/2003
Please email comments on this WWW page to alan.heckert@nist.gov.