BIWEIGHT MIDCOVARIANCE

BIWEIGHT MIDCOVARIANCE (LET)

Let Subcommand

Compute the biweight midcovariance between two variables.

Mosteller and Tukey (see Reference section below) define two types of robustness:
1. resistance means that changing a small part, even by a large amount, of the data does not cause a large change in the estimate

2. robustness of efficiency means that the statistic has high efficiency in a variety of situations rather than in any one situation. Efficiency means that the estimate is close to optimal estimate given that we know what distribution that the data comes from. A useful measure of efficiency is:
   Efficiency = (lowest variance feasible)/ (actual variance)

Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency.

The standard covariance estimate is the optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The rank covariance statistic is one example of a robust estimate of correlation.

The biweight midcovariance estimator is both resistant and robust of efficiency. Mosteller and Tukey recommend using the MAD or interquartile range for exploratory work where moderate efficiency in a variety of situations is adequate. The biweight midcovariance estimator can be considered for situations where high performance is needed.

The biweight midcovariance estimate is defined as:

\( s_{bxy} = \frac{n\sum_{i=1}^{n}{a_i(X_i - M_x)(1 - u_{i}^{2})^2 b_i(Y_i - M_y)(1 - v_{i}^{2})^2}} {(\sum_{i=1}^{n}{a_i(1 - u_{i}^{2})(1 - 5u_{i}^{2})}) (\sum_{i=1}^{n}{b_i(1 - v_{i}^{2})(1 - 5v_{i}^{2})})} \)

where

M_x = median X
M_y = median Y
u_i = (X_i - M_x)/[9*MAD]
v_i = (Y_i - M_y)/[9*MAD]
a_i = 1 if -1 <= u_i <= 1
b_i = 1 if -1 <= v_i <= 1
MAD = median absolute deviation

LET <par> = BIWEIGHT MIDCOVARIANCE <y1> <y2>
                            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
              <y2> is the second response variable;
              <par> is a parameter where the computed biweight midcovariance is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = BIWEIGHT MIDCOVARIANCE Y1
LET A = BIWEIGHT MIDCOVARIANCE Y1 SUBSET TAG > 2

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

The VARIANCE-COVARIANCE MATRIX command generates pairwise covariance estimates of the columns in a matrix. By default, this command generates the standard covariance estimate. The command
SET COVARIANCE TYPE <type>

can be used to specify an alternate covariance measure to compute in the COVARIANCE MATRIX command. The following types are supported:

DEFAULT	-	use the standard estimate
BIWEIGHT	-	use the biweight midcovariance estimate
WINSOR	-	use the Winsorized covariance estimate
RANK	-	use the rank covariance estimate

None

None

BIWEIGHT MIDCORRELATION	= Compute a biweight midcocorelation estimate between two variable.
BIWEIGHT MIDVARIANCE	= Compute a biweight midvariance estimate of a variable.
BIWEIGHT SCALE	= Compute a biweight scale estimate of a variable.
BIWEIGHT LOCATION	= Compute a biweight location estimate of a variable.
BIWEIGHT CONFIDENCE LIMITS	= Compute a biweight based confidence interval.
WINSORIZED COVARIANCE	= Compute the Winsorized covariance between two variables.
COVARIANCE	= Compute the covariance between two variables.
RANK COVARIANCE	= Compute the rank covariance between two variables.

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.

Robust Data Analysis

2002/7

SKIP 25
READ IRIS.DAT Y1 Y2 Y3 Y4 X
LET M = CREATE MATRIX Y1 Y2 Y3 Y4
SET COVARIANCE TYPE BIWEIGHT
LET B = COVARIANCE MATRIX Y1 Y2 Y3 Y4
    

 
SKIP 25
READ IRIS.DAT Y1 Y2 Y3 Y4 X
.
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT 2 1
BOOTSTRAP SAMPLES 500
BOOTSTRAP BIWEIGHT MIDCOVARIANCE PLOT Y1 Y2
X1LABEL B025 = ^B025, B975=^B975
HISTOGRAM YPLOT
END OF MULTIPLOT
MOVE 50 96
JUSTIFICATION CENTER
TEXT BIWEIGHT MIDCOVARIANCE BOOTSTRAP: IRIS DATA