PARTIAL CORRELATION

PARTIAL CORRELATION (LET)

Let Subcommand

Compute the partial correlation coefficient between two variables given the effect of a third variable.

The standard correlation coefficient is a measure of the linear relationship between two variables. It is computed as:
\( r = \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{yy}}} \)

where

\( S_{xx} = \sum_{i=1}^{N}{(X_{i}-\bar{X})^2} \)
\( S_{yy} = \sum_{i=1}^{N}{(Y_{i}-\bar{Y})^2} \)
\( S_{xy} = \sum_{i=1}^{N}{(X_{i}-\bar{X}) (Y_{i} - \bar{Y})} \)

A perfect linear relationship yields a correlation coefficient of +1 (or -1 for a negative relationship) and no linear relationship yields a correlation coefficient of 0.

Partial correlation is the correlation between two variables after removing the effect of one or more additional variables. This command is specifcally for the the case of one additional variable. In this case, the partial correlation can be computed based on standard correlations between the three variables as follows:

\( r_{12.3} = \frac{r_{12} - r_{13}r_{23}}{\sqrt{(1 - r_{13}^2)(1 - r_{23}^2)}} \)

with r_xy denoting the correlation between x and y.

As with the standard correlation coefficient, a value of +1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear relationship, and a value of 0 indicates no linear relationship.

It may be of interest to determine if the partial correlation is significantly different than 0. The CDF value for this test is

CDF = FCDF(VAL,1,N-3)

where FCDF is the F cumulative distribution function with 1 and N - 3 degrees of freedom (N is the number of observations) and VAL = ABS((N-3)*R**2/(1 - R**2)) with R denoting the computed partial correlation. The pvalue is 1 - CDF.

LET <par> = PARTIAL CORRELATION <y1> <y2> <y3>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <y3> is the third response variable;
            <par> is a parameter where the computed partial correlation is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = PARTIAL CORRELATION ABSOLUTE VALUE <y1> <y2> <y3>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <y3> is the third response variable;
            <par> is a parameter where the computed partial correlation absolute value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the absolute value of the partial correlation coefficient. This is typically used in screening applications where there is an interest in identifying high magnitude correlations regardless of the direction of the correlation.

LET <par> = PARTIAL CORRELATION PVALUE <y1> <y2> <y3>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <y3> is the third response variable;
            <par> is a parameter where the computed partial correlation pvalue is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the pvalue (described above) of the partial correlation.

LET <par> = PARTIAL CORRELATION CDF <y1> <y2> <y3>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <y3> is the third response variable;
            <par> is a parameter where the computed partial correlation cdf is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the cdf (described above) of the partial correlation.

LET A = PARTIAL CORRELATION Y1 Y2 Z
LET A = PARTIAL CORRELATION Y1 Y2 Z SUBSET TAG > 2
LET A = PARTIAL CORRELATION ABSOLUTE VALUE Y1 Y2 Z
LET A = PARTIAL CORRELATION PVALUE Y1 Y2 Z
LET A = PARTIAL CORRELATION CDF Y1 Y2 Z

The three variables must have the same number of elements.

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

None

None

PARTIAL CORRELATION MATRIX	= Compute the matrix of partial correlations.
PARTIAL RANK CORRELATION	= Compute the partial rank correlation.
PARTIAL KENDALL TAU CORRELATION	= Compute the Kendall tau partial correlation.
CORRELATION MATRIX	= Compute the matrix of correlations.
CORRELATION	= Compute the correlation.
RANK CORRELATION	= Compute the rank correlation.
KENDALL TAU CORRELATION	= Compute the Kendall tau correlation.

Conover (1999), "Practical Nonparametric Statistics," Third Edition, Wiley, p. 327.

Peavy, Bremer, Varner, Hogben (1986), "OMNITAB 80: An Interpretive System for Statistical and Numerical Data Analysis," NBS Special Publication 701.

Linear Regression

2012/06

 
.  This data is from page 202 of
.
.  Peavy, Bremer, Varner, Hogben (1986), "OMNITAB 80:
.  An Interpretive System for Statistical and Numerical
.  Data Analysis," NBS Special Publication 701.
.
.  Original source of the data is from
.  Draper and Smith (1981), "Applied Regression Analysis",
.  Wiley, p. 373.
.
dimension 40 columns
.
read matrix m
42.2  11.2  31.9  167.1
48.6  10.6  13.2  174.4
42.6  10.6  28.7  160.8
39.0  10.4  26.1  162.0
34.7   9.3  30.1  140.8
44.5  10.8   8.5  174.6
39.1  10.7  24.3  163.7
40.1  10.0  18.6  174.5
45.9  12.0  20.4  185.7
end of data
.
set write decimals 4
let c1 = partial correlation m1 m2 m3
let c2 = partial correlation absolute value m1 m2 m3
let c3 = partial correlation cdf m1 m2 m3
let c4 = partial correlation pvalue m1 m2 m3
print c1 c2 c3 c4
    

The following output is generated.

 
 PARAMETERS AND CONSTANTS--

    C1      --         0.7442
    C2      --         0.7442
    C3      --         0.9767
    C4      --         0.0342