For a bivariate normal distribution, the distribution of correlation coefficients is given by
(1)
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(2)
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(3)
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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)
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(5)
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(6)
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(7)
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where . If the variates are uncorrelated, then
and
(8)
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(9)
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so
(10)
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(11)
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But from the Legendre duplication formula,
(12)
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so
(13)
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(14)
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(15)
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(16)
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The uncorrelated case can be derived more simply by letting be the true slope, so that
. Then
(17)
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is distributed as Student's t with degrees of freedom.
Let the population regression coefficient
be 0, then
, so
(18)
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and the distribution is
(19)
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Plugging in for and using
(20)
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(21)
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(22)
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gives
(23)
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(24)
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(25)
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(26)
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so
(27)
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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)
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(29)
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(30)
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Let . For even
, the exponent
is an integer so, by the binomial
theorem,
(31)
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and
(32)
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(33)
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For odd , the integral is
(34)
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(35)
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Let so
, then
(36)
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(37)
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But is odd, so
is even. Therefore
(38)
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Combining with the result from the cosine integral gives
(39)
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Use
(40)
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and define , then
(41)
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(In Bevington 1969, this is given incorrectly.) Combining the correct solutions
(42)
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If , a skew distribution is obtained, but the variable
defined by
(43)
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is approximately normal with
(44)
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(45)
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(Kenney and Keeping 1962, p. 266).
Let be the slope of a best-fit line, then the multiple correlation
coefficient is
(46)
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where is the sample variance.
On the surface of a sphere,
(47)
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where is a differential solid angle.
This definition guarantees that
. If
and
are expanded in real spherical
harmonics,
(48)
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(49)
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Then
(50)
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The confidence levels are then given by
(51)
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(52)
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(53)
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(54)
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(55)
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where
(56)
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(Eckhardt 1984).