Dear Colleagues,
I have a few questions related to Pearson r when rho = zero.
First question,
I recall E(R-square)=k/(n-1), where k=numbers of predictor, and n=sample
size. Is it a correct expression?
Could you advise me the source of this formula?
Second question,
Assuming the above formula is correct, E(R-square)= 1/(n-1), when there is
one predictor.
(a) The above result can be rewritten to E(r-square) = 1/(n-1) or E(r) =
Sqrt {1/(n-1)}.
(b) As demonstrated elsewhere, Pearson r is an unbiased estimator only when
rho =0 or rho=1. Therefore, E(r) = rho = 0.
The statements of (a) and (b) are not consistent! Any comment?
Third question,
According to R. A. Fisher (1915), the standard error of Pearson r
approximates sqrt {(1 - rho-square)/n} or sqrt (1/n) when n increases.
When testing if Pearson r is different from zero with small sample sizes,
the standard error chosen in the t-test is sqrt {(1 - rho-square)/(n-2)} or
sqrt {1/(n-2)}. Where is the source about the standard error used in the
t-test?
Fourth question,
Following question 3, given the null hypothesis rho=0, why do we use sample
Pearson r, instead of rho, to calculate the standard error, sqrt {(1 -
r-square)/(n-2)}? It is not my impression that this t-test is conducted
based on a posterior probability distribution.
Fifth question 5,
Why don't we just use Fisher's z z-test to conduct the above hypothesis?
Thanks for your inputs.
Best,
Peter Chen
============================
Peter Chen
Department of Psychology
Colorado State University
Fort Collins, CO 80523-8033
[EMAIL PROTECTED] (home)
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Office Voice: 970-491-2143
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http://www.colostate.edu/~chenp
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