DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE

DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE (LET)

Let Subcommand

Compute the difference between the percentage bend midvariances for two response variables.

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 - \( \beta n \) + 0.5]. This is the value of (1 - \( \beta n \) + 0.5) rounded down to the nearest integer.

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

3. Sort the W_i in ascending order.

4. \( \hat{\omega}_{\beta} \) = W_m (i. e., the m-th order statistic). W_m is the estimate of the (1 - \( \beta \) ) quantile of W.

5. \( Y_i = \frac{X_i - M}{\hat{\omega}_{\beta}} \)

6. \( A(i) = \log(x_{i}) \)

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

8. \( s_{pb} = \frac{n \hat{\omega}_{\beta} \sum_{i=1}^{n}{\left( \Psi(Y_{i}) \right) ^{2}}} {(\sum_{i=1}^{n}{a_i})^2} \) \end{document}

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.

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.

LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2
LET A = DIFFERENCE OF PERCENTAGE BEND MIDVARIANCE Y1 Y2 SUBSET X > 1

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.

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

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 median absolute deviation.
DIFFERENCE OF AAD	= Compute the difference of the average absolute deviation.
DIFFERENCE OF SD	= Compute the difference of standard deviations.
STATISTICS PLOT	= Generate a statistic versus subset plot.
BOOTSTRAP PLOT	= Generate a bootstrap plot.
TABULATE	= Perform a tabulation for a specified statistic.

Shoemaker and Hettmansperger (1982), "Robust Estimates of and Tests for the One- and Two-Sample Scale Models," Biometrika, Vol. 69, pp. 47-54.

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

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

Data Analysis

2003/03

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