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48 in the vaccine group), and others are more likely if the alternative
hypothesis is true (for example, 38 cases of pneumonia in the placebo
group and 20 in the vaccine group).
Following Neyman and Pearson, we order each of the possible out-
comes in accordance with the ratio of its probability or likelihood when
the alternative hypothesis is true to its probability when the principal
hypothesis is true. When this likelihood ratio is large, we shall say the
outcome rules in favor of the alternative hypothesis. Working downwards
from the outcomes with the highest values, we continue to add outcomes
to the
rejection
region of the test—so-called because these are the out-
comes for which we would reject the primary hypothesis—until the total
probability of the rejection region under the null hypothesis is equal to
some predesignated
significance level
.
To see that we have done the best we can do, suppose we replace one
of the outcomes we assigned to the rejection region with one we did not.
The probability that this new outcome would occur if the primary
hypothesis is true must be less than or equal to the probability that the
outcome it replaced would occur if the primary hypothesis is true. Other-
wise, we would exceed the significance level. Because of how we assigned
outcome to the rejection region, the likelihood ratio of the new outcome
is smaller than the likelihood ratio of the old outcome. Thus the probabil-
ity the new outcome would occur if the alternative hypothesis is true must
be less than or equal to the probability that the outcome it replaced would
occur if the alternative hypothesis is true. That is, by swapping outcomes
we have reduced the
power
of our test. By following the method of
Neyman and Pearson and maximizing the likelihood ratio, we obtain the
most powerful test at a given significance level.
To take advantage of Neyman and Pearson's finding, we need to have
an alternative hypothesis or alternatives firmly in mind when we set up a
test. Too often in published research, such alternative hypotheses remain
unspecified or, worse, are specified only
after
the data are in hand.
We
must specify our alternatives before we commence an analysis
, preferably at
the same time we design our study.
Are our alternatives one-sided or two-sided? Are they ordered or
unordered? The form of the alternative will determine the statistical
procedures we use and the significance levels we obtain.
Decide beforehand whether you wish to test against a one-sided or a two-
sided alternative.
One-Sided or Two-Sided
Suppose on examining the cancer registry in a hospital, we uncover the
following data that we put in the form of a 2 ¥ 2 contingency table.
