 Dataplot Vol 2 Vol 1

# BIWEIGHT MIDCORRELATION

Name:
BIWEIGHT MIDCORRELATION (LET)
Type:
Let Subcommand
Purpose:
Compute the biweight midcorrelation between two variables.
Description:
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 Pearson correlation coefficient is the optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The Spearman rank correlation is one example of a robust estimate of correlation.

The biweight midcorrelation estimator is another alternative correlation estimate. It is both resistant and robust of efficiency.

The biweight midcorrelation can be defined in terms of the biweight midvariance and the biweight midcovariance:

$$r_{b} = \frac{s_{bxy}} {\sqrt{s_{bxx} s_{byy}}}$$

where $$s_{bxy}$$ is the biweight midcovariance between X and Y and $$s_{bxx}$$ and $$s_{byy}$$ are the biweight midvariances of X and Y, respectively.

Syntax:
LET <par> = BIWEIGHT MIDCORRELATION <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 midcorrelation is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = BIWEIGHT MIDCORRELATION Y1 Y2
LET A = BIWEIGHT MIDCORRELATION Y1 Y2 SUBSET TAG > 2
Note:
Dataplot statistics can be used in a number of commands. For details, enter

Note:
The CORRELATION MATRIX command generates pairwise correlation estimates of the columns in a matrix. By default, this command generates the standard correlation estimate. The command

SET CORRELATION TYPE <type>

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

 DEFAULT - use the standard estimate BIWEIGHT - use the biweight midcorrelation estimate WINSOR - use the Winsorized correlation estimate RANK - use the rank correlation estimate
Default:
None
Synonyms:
None
Related Commands:
 BIWEIGHT MIDCOVARIANCE = Compute a biweight midcovariance estimate between two variables. 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 CORRELATION = Compute the Winsorized correlation of two variables. CORRELATION = Compute the correlation between two variables. RANK CORRELATION = Compute the rank correlation between two variables.
Reference:
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.

Applications:
Robust Data Analysis
Implementation Date:
2002/7
Program 1:

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

Program 2:

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 MIDCORRELATION PLOT Y1 Y2
X1LABEL B025 = ^B025, B975=^B975
HISTOGRAM YPLOT
END OF MULTIPLOT
MOVE 50 96
JUSTIFICATION CENTER
TEXT BIWEIGHT MIDCORRELATION BOOTSTRAP: IRIS DATA NIST is an agency of the U.S. Commerce Department.

Date created: 07/22/2002
Last updated: 11/02/2015