## Correlation Coefficient--Gaussian Bivariate Distribution

For a Gaussian Bivariate Distribution, the distribution of correlation Coefficients is given by

 (1)

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

 (2) (3) (4)

where . If the variates are uncorrelated, then and
 (5)

so

 (6)

But from the Legendre Duplication Formula,
 (7)

so
 (8)

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

 (9)

is distributed as Student's t-Distribution with Degrees of Freedom. Let the population regression Coefficient be 0, then , so
 (10)

and the distribution is
 (11)

Plugging in for and using
 (12)

gives
 (13)

so
 (14)

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
 (15)

Let . For Even , the exponent is an Integer so, by the Binomial Theorem,
 (16)

and
 (17)

For Odd , the integral is
 (18)

Let so , then
 (19)

But is Odd, so is Even. Therefore
 (20)

Combining with the result from the Cosine Integral gives

 (21)

Use
 (22)

and define , then

 (23)

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

 (24)

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

 (25)

is approximately normal with
 (26) (27)

(Kenney and Keeping 1962, p. 266).

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

 (28)

where is the sample Variance.

On the surface of a Sphere,

 (29)

where is a differential Solid Angle. This definition guarantees that . If and are expanded in Real Spherical Harmonics,

 (30) (31)

Then
 (32)

The confidence levels are then given by

where
 (33)

(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. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

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

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