DIFFERENCE OF QN

DIFFERENCE OF QN (LET)

Let Subcommand

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

The Q_n scale estimate is motivated by the Hodges-Lehmann estimate of location:
\( \hat{\mu} = \mbox{median} \frac{X_i + X_j} {2} \hspace{0.5in} 1 \le i \le j \le n \)

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

\( \mbox{median} |X_i - X_j| \hspace{0.5in} i \le j \)

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:

\( d \{ |X_i - X_j| \hspace{0.2in} i \le j \}_{(k)} \)

where d is a constant factor and k = \( \left( \begin{array}{c} h \\ 2 \end{array} \right) \) which is approximately \( \left( \begin{array}{c} n \\ 2 \end{array} \right) \). The value of h is [n/2]+1 (i.e., roughly half the number of obserations). In other words, we take k-th order statistic of the \( \left( \begin{array}{c} n \\ 2 \end{array} \right) \) 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.

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

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.
STATISTICS PLOT	= Generate a statistic versus subset plot.
BOOTSTRAP PLOT	= Generate a bootstrap plot.
TABULATE	= Perform a tabulation for a specified statistic.

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

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

Data Analysis

2003/04

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