# 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) |

**How to Cite This Entry:**

Multiple-correlation coefficient. A.V. Prokhorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Multiple-correlation_coefficient&oldid=12840