Page 90 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

2.3 Summarising the Data 69

STATISTICA and SPSS afford the possibility of computing partial correlations
as indicated in Commands 2.9. For the previous example, the partial correlation of
PRTG and ARTG, given PRT and ART, is 0.79. We see, therefore, that PRT and
ART can “explain” about 20% of the high correlation (0.99) of those two variables.
Another measure of association for continuous variables is the multiple
correlation coefficient, which measures the degree of association of one variable Y
in relation to a set of variables, X 1, X 2, …, X n, that linearly “predict” Y. Details on
multiple correlation will be postponed to Chapter 7.

Commands 2.9. SPSS, STATISTICA, MATLAB and R commands used to obtain
measures of association for continuous variables.

SPSS Analyze; Correlate; Bivariate | Partial
Statistics; Basic Statistics/Tables;
STATISTICA Correlation matrices (Quick |Advanced;
Partial Correlations)
MATLAB corrcoef(x) ; cov(x)

R cor(x,y) ; cov(x,y)

Partial correlations are computed in MATLAB and R as part of the regression
functions (see Chapter 7).

2.3.5 Measures of Association for Ordinal Variables

2.3.5.1 The Spearman Rank Correlation

When dealing with ordinal data the correlation coefficient, previously described,
can be computed in a simplified way. Consider the ordinal variables X and Y with
ranks between 1 and N. It seems natural to measure the lack of agreement between
X and Y by means of the difference of the ranks d i = x i − y i for each data pair (x i, y i).
Using these differences we can express 2.18 as:
∑ n x 2 + ∑ n y 2 − ∑ n d 2
= i 1
r = = i i 1 i = i 1 i . 2.20
2 ∑ n = x i 2 ∑ n i 1 y i 2
= i 1
Assuming the values of x i and y i are ranked from 1 through N and that there are
no tied ranks in any variable, we have:

n
∑ i= 1 i ∑ n i= 1 y = (N − N / ) 12 .
2
3
2
x =
i
Applying this result to 2.20, the following Spearman’s rank correlation (also
known as rank correlation coefficient) is derived: