PARTIAL RANK CORRELATION

PARTIAL RANK CORRELATION (LET)

Let Subcommand

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

The Spearman rank correlation coefficient is computed as
\( r = 1 - 6 \sum_{i=1}^{N}{\frac{D_{i}}{N(N^2 -1)}} \)

where D_i is the difference between the ranks assigned to the corresponding pairs and N is the sample size. Ties are assigned average ranks.

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

Partial rank 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 rank correlation can be computed based on standard rank 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 rank 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.

LET <par> = PARTIAL RANK 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 rank correlation is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = PARTIAL RANK 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 rank correlation absolute value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the absolute value of the partial rank 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 A = PARTIAL RANK CORRELATION Y1 Y2 Z
LET A = PARTIAL RANK CORRELATION Y1 Y2 Z SUBSET TAG > 2
LET A = PARTIAL RANK CORRELATION ABSOLUTE VALUE Y1 Y2 Z

The three variables must have the same number of elements.

The command PARTIAL CORRELATION MATRIX can be used to compute the matrix of partial correlations. This command has options to compute several robust forms of the partial correlation including the Spearman rank correlation discussed here.

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

None

None

RANK CORRELATION	= Compute the rank correlation.
PARTIAL CORRELATION	= Compute the partial correlation.
PARTIAL CORRELATION MATRIX	= Compute the matrix of partial correlations.
PARTIAL KENDALL TAU CORRELATION	= Compute the Kendall tau partial correlation.
CORRELATION MATRIX	= Compute the matrix of correlations.
CORRELATION	= Compute the 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 rank correlation m1 m2 m3
let c2 = partial rank correlation absolute value m1 m2 m3
print c1 c2
    

The following output is generated:

 
 PARAMETERS AND CONSTANTS--

    C1      --         0.6603
    C2      --         0.6603