Date: Dec 13, 2012 10:43 AM
Author: Luis A. Afonso
Subject: What changes a NHST into a No-NHST?

What changes a NHST into a No-NHST?

Some class of Statistics users claim that Null Hypotheses Statistical Tests are useless because a reject result does´nt inform how much data differs from a null one. They are right: The insignificance of Significance tests as it was said . . .
A parameter´s test statistics is in general expressed as
________W = (Obs - Common)/s______NHST
Obs = observed value compared with Common, the Null Hypotheses, H0, value.
Let be W0 and W1 the critical values (at a preset alpha) bounding the no-rejection interval. Then the set of values less than W0 or larger than W1 do contain the Rejection values of W.
A simple and sufficiently known ?mending?:
Subtracting a positive D to the numerator we, if so, intend to warrant that at least the Obs is larger than the H0 value by D. Then it must be NoNHST (my acronym):
_____ s * W0 < (Obs - H0) - D < s * W1____(1)
Then if D > 0 is inside the interval:
[(Obs - H0) - (s*W1), (Obs - H0) - (s*W0)]
we cannot warrant (at alpha significant level) at least D differs from H0 + D : to, in fact to get a gain D over H0 must 0 be at left to this interval.
Note that (1) defines a no-rejection D centered interval, with amplitude given by s * (W1 - W0),
_______[U - s * W1, U - s * W0]
Conclusion: Only if the observed difference noted U (U=Obs-H0) is larger than s*W1 our purpose is fulfilled.
Symmetric critical values
Let be s=1, W0= -1.96, W1= 1.96. We have
____ -1.960 < U - D < 1.960; Therefore:
__ U - 1.960 < D < U + 1.960.
We have not a positive gain unless U > s * W1, i.e., the left bound with 0 at its left. The no-rejection C.I. is centred at Obs-H0 and its semi-amplitude is W1 * s with W0= - W1.

Luis A. Afonso