For a Gaussian Bivariate Distribution, the distribution of correlation Coefficients is given by
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
where . If the variates are uncorrelated, then and
so
But from the Legendre Duplication Formula,
|
(7) |
so
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
gives
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
Let
. For Even , the exponent is an Integer so, by the Binomial
Theorem,
|
(16) |
and
For Odd , the integral is
Let so
, then
But is Odd, so
is Even. Therefore
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
(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,
Then
|
(32) |
The confidence levels are then given by
where
|
(33) |
(Eckhardt 1984).
See also Fisher's z'-Transformation, Spearman Rank Correlation Coefficient,
Spherical Harmonic
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.
© 1996-9 Eric W. Weisstein
1999-05-25