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
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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
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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
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where is the variance of
and
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is the variance of with respect to the regression.
The sample analogue of the multiple-correlation coefficient is
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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=12840