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Topic: Limits of the *Welch Correction"
Replies: 0

 Luis A. Afonso Posts: 4,758 From: LIsbon (Portugal) Registered: 2/16/05
Limits of the *Welch Correction"
Posted: Jun 19, 2013 2:49 AM

The so-called Welch correction for the two-normal samples difference niu= u/v, where
u = [ssdx/(nx*(nx-1))+ ssdy/(ny*(ny-1))]^2
which if the samples have the same size n:
u0= 1/(n^2*(n-1)^2)*(ssdx+ ssdy)^2
v0= (ssdx)^2 /[(n^2*(n-1)^3] + (ssdy)^2/[(n^2*(n-1)^3]
one have
(u0/v0)/(n-1) =
= 1+ 2*(ssdx*ssdy) /[(ssdx)^2 + (ssdy)^2] < 2.
__________
Corollary:
Under sample-pair, each size n, the so-called Welch correction, niu, is so that: 0 <= niu <= 2*(n-1)
________________

In practice the degrees of freedom are exclusively function of data. Note that:
__overestimating df we underestimate the amplitude of the C.I., which probably will be really wider: The danger is to reject unduly. We are more likely to found significant issues than it should be.
__on contrary underestimating df one can be sure that alpha is shorter than you got: a significant value, if so, got a reinforced truthfulness.

In fact the consideration above being correct does not include an important fact: the right tail fractiles of the Student t distribution (except for very small df) are sufficient close one another:

Table of Student t critical values
______df=10 ___ 2.228(.975)__ 2.764(.99)___
_______20______2.086_______ 2.528_____
_______30______2.042_______ 2.457_____
_______40______2.021_______ 2.423_____
_______60______2.000_______ 2.390_____
______120______1.980_______ 2.358__

Luis A. Afonso