PERCENTAGE BEND CORRELATION
Name:
PERCENTAGE BEND CORRELATION (LET)
Type:
Purpose:
Compute the percentage bend correlation for a variable.
Description:
Mosteller and Tukey (see Reference section below) define
two types of robustness:
- resistance means that changing a small part, even by a
large amount, of the data does not cause a large change
in the estimate
- 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 Pearson correlation coefficient is an optimal estimator for
Gaussian data. However, it is not resistant and it does not
have robustness of efficiency.
The percentage bend correlation estimator, discussed in
Shoemaker and Hettmansperger and also by Wilcox, is both
resistant and robust of efficiency. The rationale and
derivation for this estimate is given in these references.
The percentage bend correlation between two variables X
and Y is computed as follows:
- Set m =
(1-
n) + 0.5. Round m down to
the nearest integer.
- Let
for i = 1, ...,
n where Mx.
is the median of X.
- Sort the Wi in ascending order.
= W(m) (i. e., the mth order
statistic). W(m) is the estimate of the
(1- ) quantile of W.
- Sort the X values. Compute the number of values of
that are less than -1
and the number that are greater than +1 and store in
i1 and i2 respectively. Then compute
- Repeat the above calculations on the Y variable.
Store corresponding quantities in
,
,
and Vi.
- Define the function
- Compute
Ai =
i
(Ui)
Bi =
i
(Vi)
- Compute the percentage bend correlation
The value of
is selected
between 0 and 0.5. Higher values of
result in a
higher breakdown point at the expense of lower efficiency.
Syntax:
LET <par> = PERCENTAGE BEND CORRELATION
<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 percentage
bend correlation is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = PERCENTAGE BEND CORRELATION Y1 Y2
LET A = PERCENTAGE BEND CORRELATION Y1 Y2 SUBSET TAG > 2
Note:
To set the value of
,
enter the command
where <value> is greater than 0 and less than or equal to
0.5. The default value for
is 0.1.
Note:
Support for the percentage bend correlation has been added to the
following plots and commands:
PERCENTAGE BEND CORRELATION PLOT
CROSS TABULATE PERCENTAGE BEND CORRELATION PLOT
BOOTSTRAP PERCENTAGE BEND CORRELATION PLOT
JACKNIFE PERCENTAGE BEND CORRELATION PLOT
PERCENTAGE BEND CORRELATION INTERACTION STATISTIC PLOT
Default:
Synonyms:
Related Commands:
PERCENTAGE BEND MIDVARIANCE
|
= Compute the percentage bend midvariance of a variable.
|
BIWEIGHT CORRELATION
|
= Compute a biweight correlation estimate of a variable.
|
WINSORIZED CORRELATION
|
= Compute a Winsorized correlation estimate of a
variable.
|
CORRELATION
|
= Compute the correlation between two variables.
|
RANK CORRELATION
|
= Compute the rank correlation between two variables.
|
VARIANCE
|
= Compute the variance of a variable.
|
STATISTIC PLOT
|
= Generate a statistic versus group plot for a given
statistic.
|
CROSS TABULATE PLOT
|
= Generate a statistic versus group plot for a given
statistic and two group variables.
|
BOOTSTRAP PLOT
|
= Generate a bootstrap plot for a given statistic.
|
INFLUENCE CURVE
|
= Generate an influence curve for a given statistic.
|
INTERACTION STATISTIC PLOT
|
= Generate an interaction plot for a given statistic.
|
Reference:
"Robust Estimates of and Tests for the One- and Two-Sample
Scale Models", Shoemaker and Hettmansperger, Biometrika 69,
1982, pp. 47-54.
"Introduction to Robust Estimation and Hypothesis Testing",
Rand Wilcox, Academic Press, 1997.
"Data Analysis and Regression: A Second Course in Statistics",
Mosteller and Tukey, Addison-Wesley, 1977, pp. 203-209.
Applications:
Implementation Date:
Program 1:
SKIP 25
READ MATRIX IRIS.DAT Y1 Y2 Y3 Y4 X
LET M = CREATE MATRIX Y1 Y2 Y3 Y4
SET CORRELATION TYPE PERCENTAGE BEND
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 SCALE FACTOR 2
MULTIPLOT 2 1
BOOTSTRAP SAMPLES 500
BOOTSTRAP PERCENTAGE BEND CORRELATION PLOT Y1 Y2
X1LABEL DISPLACEMENT 12
X1LABEL B025 = ^B025, B975=^B975
HISTOGRAM YPLOT
END OF MULTIPLOT
MOVE 50 96
JUSTIFICATION CENTER
TEXT PERCENTAGE BEND CORRELATION BOOTSTRAP: IRIS DATA
Date created: 8/12/2002
Last updated: 4/4/2003
Please email comments on this WWW page to
alan.heckert@nist.gov.
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