Discussion
The Fisher transformation is an approximate variance-stabilizing transformation for r when X and Y follow a bivariate normal distribution. This means that the variance of z is approximately constant for all values of the population correlation coefficient ρ. Without the Fisher transformation, the variance of r grows smaller as |ρ| gets closer to 1. Since the Fisher transformation is approximately the identity function when |r| < 1/2, it is sometimes useful to remember that the variance of r is well approximated by 1/N as long as |ρ| is not too large and N is not too small. This is related to the fact that the asymptotic variance of r is 1 for bivariate normal data.
The behavior of this transform has been extensively studied since Fisher introduced it in 1915. Fisher himself found the exact distribution of z for data from a bivariate normal distribution in 1921; Gayen, 1951 determined the exact distribution of z for data from a bivariate Type A Edgeworth distribution. Hotelling in 1953 calculated the Taylor series expressions for the moments of z and several related statistics and Hawkins in 1989 discovered the asymptotic distribution of z for virtually any data.
Read more about this topic: Fisher Transformation
Famous quotes containing the word discussion:
“We cannot set aside an hour for discussion with our children and hope that it will be a time of deep encounter. The special moments of intimacy are more likely to happen while baking a cake together, or playing hide and seek, or just sitting in the waiting room of the orthodontist.”
—Neil Kurshan (20th century)
“If we had had more time for discussion we should probably have made a great many more mistakes.”
—Leon Trotsky (1879–1940)
“My companion and I, having a minute’s discussion on some point of ancient history, were amused by the attitude which the Indian, who could not tell what we were talking about, assumed. He constituted himself umpire, and, judging by our air and gesture, he very seriously remarked from time to time, “you beat,” or “he beat.””
—Henry David Thoreau (1817–1862)