Information Technology Reference
In-Depth Information
To test that “80% of redheads are passionate,” we have two choices
depending on how “passion” is measured. If “passion” is an all-or-none
phenomenon, then we can forget about trying to formulate a null
hypothesis and instead test the binomial hypothesis that the probability
p
that a redhead is passionate is 80%. If “passion” can be measured on a
seven-point scale and we define “passionate” as “passion” greater than or
equal to 5, then our hypothesis becomes “the 20th percentile of redhead
passion exceeds 5.” As in the first example above, we could convert this to
a “null hypothesis” by subtracting five from each observation. But the
effort is unnecessary.
NEYMAN-PEARSON THEORY
Formulate your alternative hypotheses at the same time you set forth your
principal hypothesis.
When the objective of our investigations is to arrive at some sort of con-
clusion, then we need to have not only a hypothesis in mind, but also one
or more potential alternative hypotheses.
The cornerstone of modern hypothesis testing is the Neyman-Pearson
Lemma. To get a feeling for the working of this lemma, suppose we are
testing a new vaccine by administering it to half of our test subjects and
giving a supposedly harmless placebo to each of the remainder. We
proceed to follow these subjects over some fixed period and to note which
subjects, if any, contract the disease that the new vaccine is said to offer
protection against.
We know in advance that the vaccine is unlikely to offer complete pro-
tection; indeed, some individuals may actually come down with the disease
as a result of taking the vaccine. Depending on the weather and other
factors over which we have no control, our subjects, even those who
received only placebo, may not contract the disease during the study
period. All sorts of outcomes are possible.
The tests are being conducted in accordance with regulatory agency
guidelines. From the regulatory agency's perspective, the principal
hypothesis H is that the new vaccine offers no protection. Our alternative
hypothesis A is that the new vaccine can cut the number of infected indi-
viduals in half. Our task before the start of the experiment is to decide
which outcomes will rule in favor of the alternative hypothesis A and
which in favor of the null hypothesis H.
The problem is that because of the variation inherent in the disease
process, each and every one of the possible outcomes could occur regard-
less of which hypothesis is true. Of course, some outcomes are more likely
if H is true (for example, 50 cases of pneumonia in the placebo group and
