# Multiple-correlation coefficient

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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)