DIFFERENCE OF QN

DIFFERENCE OF QN (LET)

Let Subcommand

Compute the difference of the Q_n scale estimates for two response variables.

The Qn scale estimate is motivated by the Hodges-Lehmann estimate of location:

An analogous scale estimate can be obtained by replacing pairwise averages with pairwised distances:

This estimate has high efficiency for normal data (86%), but a breakdown point of only 29%. Rousseeuw and Croux proposed the following variation of this statistic:

where d is a constant factor and k = which is approximately /4 . The value of h is [n/2]+1 (i.e., roughly half the number of obserations). In other words, we take kth order statistic of the interpoint distances. The value of d is choosen to make Q_n a consistent estimator of scale. We use the value 2.2219 since this is the value that makes Q_n a consistent estimator for normal data.

Enter HELP QN SCALE for a more detailed discussion of the Q_n scale estimate.

For the difference of Q_n scale estimates, the Q_n scale estimate is computed for each of the two samples then their difference is taken.

LET <par> = DIFFERENCE OF QN <y>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <par> is a parameter where the computed difference of Q_n scale estimates is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = DIFFERENCE OF QN Y1
LET A = DIFFERENCE OF QN Y1 SUBSET TAG > 2

Dataplot uses code provided by Rousseeuw and Croux to compute the Q_n estimate. This algorithm uses an efficient computational method for computing Q_n.

The Rousseeuw and Croux article also proposes the S_n scale estimate. The article discusses the properties of both estimators in detail.

In addition, the Q_n statistic is supported for the following plots and commands
DIFFERENCE OF QN PLOT Y X
CROSS TABULATE DIFFERENCE OF QN PLOT Y X1 X2
BOOTSTRAP DIFFERENCE OF QN PLOT Y
JACKNIFE DIFFERENCE OF QN PLOT Y


TABULATE DIFFERENCE OF QN Y X
CROSS TABULATE SN Y X1 X2
LET Z = CROSS TABULATE DIFFERENCE OF QN Y X1 X2

None

None

QN SCALE	= Compute the Qn scale estimate of a variable.
SN SCALE	= Compute the Sn scale estimate of a variable.
DIFFERENCE OF MAD	= Compute the difference of the median absolute deviations between two variables.
DIFFERENCE OF IQ RANGE	= Compute the difference of iq ranges between two variables.
DIFFERENCE OF SD	= Compute the difference of standard deviations between two variables.
DIFFERENCE OF SN	= Compute the difference of the Sn scale estimates between two variables.
STATISTIC PLOT	= Generate a statistic versus subset plot.
CROSS TABULATE PLOT	= Generate a statistic versus subset plot (two subset variables).
BOOTSTRAP PLOT	= Generate a bootstrap plot for a statistic.

"Alternatives to the Median Absolute Deviation", Peter J. Rousseuw and Christophe Croux, Journal of the American Statistical Association, December, 1993, Vol. 88, No. 424, pp. 1273-1283.

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

Data Analysis

2003/4

SKIP 25
READ IRIS.DAT Y1 TO Y4 X
.
LET A = DIFFERENCE OF QN Y1 Y2
TABULATE DIFFERENCE OF QN Y1 Y2 X
.
MAJOR XTIC MARK NUMBER 3
MINOR XTIC MARK NUMBER 0
XTIC OFFSET 0.2 0.2
X1LABEL GROUP ID
Y1LABEL DIFFERENCE OF QN
CHARACTER X
LINE BLANK
DIFFERENCE OF QN PLOT Y1 Y2 X
CHARACTER X ALL
LINE BLANK ALL
BOOTSTRAP DIFFERENCE OF QN PLOT Y1 Y2 X
    

Dataplot generated the following output.

       **************************************
       **  LET A = DIFFERENCE OF QN Y1 Y2  **
       **************************************
  
  
 THE COMPUTED VALUE OF THE CONSTANT A             =  0.43340060E+00
  
  
       *****************************************
       **  TABULATE DIFFERENCE OF QN Y1 Y2 X  **
       *****************************************
  
  
                 *    Y1       AND Y2
     X           *    DIFFERENCE OF QN
 **********************************************
     1.00000     *   -0.506639E-06
     2.00000     *    0.206496
     3.00000     *    0.206497