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Science Forum Index » Statistics - Math Forum » Exact Level of Significance
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Message |
| Reef Fish |
 Posted: Sat Dec 30, 2006 11:26 am |
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Guest
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In the accepted usage of statistical concepts, the significance
level of a Hypothesis Test is
Pr( rejecting Ho ! Ho is true) = alpha = Pr( Type I error).
It is a level often taken as 0.05 or 0.01 BEFORE any data is observed.
The notion of a p-value is that it is NOT necessary to pre-determine
any alpha level to test a hypothesis, because the OBSERVED
significance level, or a p-value will enable one to decide on a
test no matter what the significant level is.
On the basis of the above, Keven E. Thorpe first argued that there
are TWO different approaches to p-value, the Fisherian way, and
the Neyman-Pearson way. I disagreed noting that
a p-value DOES NOT exist in the Fisherian way
because of the absence of the Alternative Hypothesis to make
"more extreme" well defined after the value of a test statistic has
been observed.
Kevin gave THIS as his example for the inequality used in the
inappropriate example:
KT> In the case of continuous data, the strict inequality does not
KT> affect the computation of a p-value, but for discrete data it
KT> does, which is why NP suggests randomization on the
KT> boundary.
That is the WRONG example to cite for the p-value which does
NOT exist in the Fisherian framework. For the Neyman-Pearson
definition, a p-value is well-defined thought it sometimes may
lead to unsatisfactory interpretation -- but that is the DEFECT of
the frequentist method of hypothesis testing.
The idea of randomization is NOT for p-values, but for those
pedants who insisists on using 0.05 (or some other equally
arbitrary rounded small values such as 0.01) for the significant
level of a test based on a discrete test statistic.
THAT is when the idea of randomization is used to make ALPHA
EXACTLY equal to some pre-scribed value, such as 0.05. It is
NOT used for p-values which does not require any tweaking of
fudging. Once the Test Statistic and the Alternative Hypothesis
are given, the p-value is uniquely defined as soon as the value of
the test statistic is observed.
Simple.
Fisherian ideas are as outdated as his idea of fiducial intervals.
It wasn't bad when an ape first turned erect and was groping
for some idea to pick the fruit off a tree.
Today we are well past the Age of the Statistical Apes.
-- Reef Fish Bob. |
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