Date: Jun 19, 2013 2:49 AM
Author: Luis A. Afonso
Subject: Limits of the *Welch Correction"
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.

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Corollary:

Under sample-pair, each size n, the so-called Welch correction, niu, is so that: 0 <= niu <= 2*(n-1)

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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