At 12:56 PM -0500 10/20/00, dennis roberts wrote:
>randomly independent events have the p value being the multiplication of
>each event's p value ... so ... p for getting a head in a good coin ....
>is .5 ... 2 in a row = .25 ... etc.
This is wrong. In general you cannot multiply the p-values from
independent events to obtain the p-value of the combined event.
Example: You toss 220 heads on 400 trials of a fair coin. The two-tailed
p-value for this event is almost exactly 0.05 [J.O. Berger and M.
Delampady, Statistical Science 2, 317-352 (1987)]. Suppose you then
independently toss 180 heads on an additional 400 trials. Again, the
two-tailed p-value is 0.05. However, the combined experiment is 400
heads on 800 trials, for which the two-tailed p-value is 1.0, not 0.05^2.
Similar examples can be given for one-tailed p-values.
If you must use p-values and must combine them from independent
experiments, you need to use the methods of meta-analysis. Not that I
recommend using either p-values or meta-analysis (I don't).
Contrary to popular belief, observed p-values are not probabilities.
They cannot be probabilities because they do not obey the rules of the
probability calculus, as the example shows. They are, well, p-values.
That said, I wonder if you haven't confused p-values and probabilities.
It is true that if you toss N heads in a row with a fair coin, the
probability of that event is 0.5^N. It is also true that this
probability happens to be numerically equal to the one-tailed p-value
for tossing N heads in a row. So in this particular case it happens that
the one-tailed p-value for the combined event is numerically equal to
the product of the individual p-values. However, this has nothing to do
with combining p-values. It is a consequence of the fortuitous numerical
equality between the p-value and the probability in this special case,
and the fact that independent probabilities do multiply to get the joint
probability. Put another way, there is really no "tail" in this special
case. The entire contribution to the p-value comes from the probability
of obtaining the actually observed data, not from outcomes out in the
tail that might have been observed but were not.
Bill
--
Bill Jefferys/Department of Astronomy/University of Texas/Austin, TX 78712
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