NEYMAN–PEARSON THEORY

Formulate your alternative hypotheses at the same time you set forth the hypothesis that is of chief concern to you.

When the objective of our investigations is to arrive at some sort of conclusion, then we need not only have a single primary hypothesis in mind but 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 mathematical principle, 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 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 protection; indeed, some individuals may actually come down with the disease as a result of taking the vaccine. Many factors over which we have no control, such as the weather, may result in none of the subjects, even those who received only placebo, contracting the disease during the study period. All sorts of outcomes are possible.

The tests are being conducted in accordance with regulatory agency guidelines. Our primary hypothesis H is that the new vaccine can cut the number of infected individuals in half. From the regulatory agency’s perspective, the alternative hypothesis A1 is that the new vaccine offers no protection or, A2, ...

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