Correlation Coefficient--Bivariate Normal Distribution

For a bivariate normal distribution, the distribution of correlation coefficients is given by

			(1)
			(2)
			(3)

where  is the population correlation coefficient,  is a hypergeometric function, and  is the gamma function (Kenney and Keeping 1951, pp. 217-221). The moments are

			(4)
			(5)
			(6)
			(7)

where . If the variates are uncorrelated, then  and

			(8)
			(9)

so

			(10)
			(11)

But from the Legendre duplication formula,

(12)

so

			(13)
			(14)
			(15)
			(16)

The uncorrelated case can be derived more simply by letting  be the true slope, so that . Then

(17)

is distributed as Student's t with  degrees of freedom. Let the population regression coefficient  be 0, then , so

(18)

and the distribution is

(19)

Plugging in for  and using

			(20)
			(21)
			(22)

gives

			(23)
			(24)
			(25)
			(26)

so

(27)

as before. See Bevington (1969, pp. 122-123) or Pugh and Winslow (1966, §12-8). If we are interested instead in the probability that a correlation coefficient would be obtained , where  is the observed coefficient, then

			(28)
			(29)
			(30)

Let . For even , the exponent  is an integer so, by the binomial theorem,

(31)

and

			(32)
			(33)

For odd , the integral is

			(34)
			(35)

Let  so , then

			(36)
			(37)

But  is odd, so  is even. Therefore

(38)

Combining with the result from the cosine integral gives

(39)

Use

(40)

and define , then

(41)

(In Bevington 1969, this is given incorrectly.) Combining the correct solutions

(42)

If , a skew distribution is obtained, but the variable  defined by

(43)

is approximately normal with

			(44)
			(45)

(Kenney and Keeping 1962, p. 266).

Let  be the slope of a best-fit line, then the multiple correlation coefficient is

(46)

where  is the sample variance.

On the surface of a sphere,

(47)

where  is a differential solid angle. This definition guarantees that . If  and  are expanded in real spherical harmonics,

			(48)
			(49)

Then

(50)

The confidence levels are then given by

			(51)
			(52)
			(53)
			(54)
			(55)

where

(56)

(Eckhardt 1984).

REFERENCES:

Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969.

Eckhardt, D. H. "Correlations Between Global Features of Terrestrial Fields." Math. Geology 16, 155-171, 1984.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966.