PARTIAL KENDALLS TAU CORRELATION

PARTIAL KENDALLS TAU (LET)

Let Subcommand

Compute the partial Kendall's tau correlation coefficient between two variables given the effect of a third variable.

Kendall's tau coefficient is a measure of concordance between two paired variables. Given the pairs (X_i,Y_i) and (X_j,Y_j), then
\( \frac{Y_j - Y_i}{X_j - X_i} \) > 0 - pair is concordant

\( \frac{Y_j - Y_i}{X_j - X_i} \) < 0 - pair is discordant

\( \frac{Y_j - Y_i}{X_j - X_i} \) = 0 - pair is considered a tie

X_i = X_j - pair is not compared

Kendall's tau is computed as

\( \tau = \frac{N_c - N_d}{N_c + N_d} \)

with N_c and N_d denoting the number of concordant pairs and the number of discordant pairs, respectively, in the sample. Ties add 0.5 to both the concordant and discordant counts. There are \( \left( \begin{array}{c} n \\ 2 \end{array} \right) \) possible pairs in the bivariate sample.

A perfect linear relationship between the ranks yields a Kendall's tau 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 Kendall's tau correlation is the Kendall's tau 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 Kendall's tau correlation can be computed based on standard Kendall's tau correlations between the three variables as follows:

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

with \( \tau_{xy} \) denoting the Kendall's tau correlation between x and y.

As with the standard Kendall's tau 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 KENDALLS TAU 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 Kendall's tau correlation is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = PARTIAL KENDALLS TAU 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 Kendall's tau correlation absolute value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the absolute value of the partial Kendall's tau 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 KENDALLS TAU CORRELATION Y1 Y2 Z
LET A = PARTIAL KENDALLS TAU CORRELATION Y1 Y2 Z SUBSET TAG > 2
LET A = PARTIAL KENDALLS TAU 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

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

The following output is generated.

 PARAMETERS AND CONSTANTS--

    C1      --         0.4610
    C2      --         0.4610