Multiple-correlation coefficient
A measure of the linear dependence between one random variable and a certain collection of random variables. More precisely, if 
is a random vector with values in 
, then the multiple-correlation coefficient between 
and 
is defined as the usual correlation coefficient between 
and its best linear approximation 
relative to 
, i.e. as its regression relative to 
. The multiple-correlation coefficient has the property that if 
and if
 |
is the regression of 
relative to 
, then among all linear combinations of 
the variable 
has largest correlation with 
. In this sense the multiple-correlation coefficient is a special case of the canonical correlation coefficient (cf. Canonical correlation coefficients). For 
the multiple-correlation coefficient is the absolute value of the usual correlation coefficient 
between 
and 
. The multiple-correlation coefficient between 
and 
is denoted by 
and is expressed in terms of the entries of the correlation matrix 
, 
, by
 |
where 
is the determinant of 
and 
is the cofactor of 
; here 
. If 
, then, with probability 
, 
is equal to a linear combination of 
, that is, the joint distribution of 
is concentrated on a hyperplane in 
. On the other hand, 
if and only if 
, that is, if 
is not correlated with any of 
. To calculate the multiple-correlation coefficient one can use the formula
 |
where 
is the variance of 
and
 |
is the variance of 
with respect to the regression.
The sample analogue of the multiple-correlation coefficient 
is
 |
where 
and 
are estimators of 
and 
based on a sample of size 
. To test the hypothesis of no relationship, the sampling distribution of 
is used. Given that the sample is taken from a multivariate normal distribution, the variable 
has the beta-distribution with parameters 
if 
; if 
, then the distribution of 
is known, but is somewhat complicated.
References
| [1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |
| [2] | M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979) |
Comments
For the distribution of 
if 
see [a2], Chapt. 10.
References
| [a1] | T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1958) |
| [a2] | M.L. Eaton, "Multivariate statistics: A vector space approach" , Wiley (1983) |
| [a3] | R.J. Muirhead, "Aspects of multivariate statistical theory" , Wiley (1982) |
Multiple-correlation coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple-correlation_coefficient&oldid=47929