The Fisher Transform of Pearson´s Correlation Coefficient

Objective
_To show if the Fisher transform over normal correlated samples of size n:
_____F = 0.5 * LOG [(1-r) / (1+r)]
has a normal distribution Z = N(m,s) where:
____m = 0.5 * LOG [(1-R) / (1+R)]
____s = 1 / sqr(n-3)
and R is the Population correlation coefficient, and r its estimative from the sample, i.e. , the K. Pearson´s product moment correlation

__r = ( [X*Y] - (1/n) [X][Y] ) / sqrt (varX * varY)
_____varX = [X*X] - (1/n) [X][X]
_____varY = [Y*Y] - (1/n) [Y][Y]
being [ ] the summation symbol (Gauss).


The Method to check normality

The real r.v. from -infinity to +infinity was divided in TEN intervals
(-inf. , -1.28155)__________interval no. 1
[-1.28155, -0.84162)_______________ 2
[-0.84162, -0.52440)_______________ 3
[-0.52440, -0.25335)_______________ 4
[-0.25335, -0.00000)_______________ 5
[+0.00000, +0.25335)_______________6
[+0.25335, +0.52440)_______________7
[+0.54440, +0.84162)_______________8
[+0.84162, +1.28155)_______________9
[+1.28155, +inf. )___________________10

each one associated to the (constant) probability 1/10 relative to the standard normal Distribution. (obtained by a software and checked by Abramowitz & Stegun),

*** Experiment nos. 1, 2 and 3

Input:
___ X(0,1), Y(0,1), size=20, R=+0.5

_I1____0.1080____0.1078____0.1077
_ 2____0.1061____0.1054____0.1054
_ 3____0.1040____0.1041____0.1044
_ 4____0.1035____0.1041____0.1030
_ 5____0.1017____0.1021____0.1020
_ 6____0.1002____0.1006____0.1003
_ 7____0.0991____0.0998____0.0996
_ 8____0.0969____0.0955____0.0965
_ 9____0.0925____0.0924____0.0929
_10___ 0.0880____0.0882____0.0882
***
CHI2 =1368______1377______1289

IT CAN BE STATED (with a very large confidence) that the product moment correlation of bivariate normal samples (of size 20, and R= +0.5) subjected to Fisher´s Transformation DOES NOT behaves as normal.
More:
The real distribution is neatly more concentrated around ZERO than N (0, 1) is.


____licas (Luis A. Afonso)
.

Relevant Pages

  • Re: Non-parametric correction for dependent variable
    ... >correlation is rather small, at 0.1. ... >look at ranks rather than original scores, ... >partial (Pearson) correlations from the matrix of r's. ... have a normal distribution. ...
    (sci.stat.math)
  • Re: Non-parametric correction for dependent variable
    ... Pearson's r does not "imply normality". ... shows the product moment correlation for any distributions. ... a Pearson r on the rank-transformed data -- can gain from ... correlations that could be computed within narrow age groups. ...
    (sci.stat.math)
  • Re: Hypothesis testing with proportions
    ... We made 45 gels from tissue samples of one patient which contain partially mutant ... the gels we calculated for each of the 45 gels the proportion of mutant DNA ... > requires the variable obey normal distribution. ... > proportion passes normality test, you can use t-test; ...
    (sci.stat.edu)
  • Re: advice for using right statistic method
    ... I thought that the performance should have a normal distribution for each ... I'm going to try to compute and plot some information about the residuals ... It is possible that you are mis-stating the assumption, ... resulting normality of residuals. ...
    (sci.stat.math)
  • Re: correlations of ordered residuals with their expected values.
    ... I suppose that if the correlation between the ordered residuals and ... it is evidence of normality. ... There will almost always be a pretty good correlation. ...
    (sci.stat.math)