<P> An example of Neyman--Pearson hypothesis testing can be made by a change to the radioactive suitcase example . If the "suitcase" is actually a shielded container for the transportation of radioactive material, then a test might be used to select among three hypotheses: no radioactive source present, one present, two (all) present . The test could be required for safety, with actions required in each case . The Neyman--Pearson lemma of hypothesis testing says that a good criterion for the selection of hypotheses is the ratio of their probabilities (a likelihood ratio). A simple method of solution is to select the hypothesis with the highest probability for the Geiger counts observed . The typical result matches intuition: few counts imply no source, many counts imply two sources and intermediate counts imply one source . Notice also that usually there are problems for proving a negative . Null hypotheses should be at least falsifiable . </P> <P> Neyman--Pearson theory can accommodate both prior probabilities and the costs of actions resulting from decisions . The former allows each test to consider the results of earlier tests (unlike Fisher's significance tests). The latter allows the consideration of economic issues (for example) as well as probabilities . A likelihood ratio remains a good criterion for selecting among hypotheses . </P> <P> The two forms of hypothesis testing are based on different problem formulations . The original test is analogous to a true / false question; the Neyman--Pearson test is more like multiple choice . In the view of Tukey the former produces a conclusion on the basis of only strong evidence while the latter produces a decision on the basis of available evidence . While the two tests seem quite different both mathematically and philosophically, later developments lead to the opposite claim . Consider many tiny radioactive sources . The hypotheses become 0, 1, 2, 3...grains of radioactive sand . There is little distinction between none or some radiation (Fisher) and 0 grains of radioactive sand versus all of the alternatives (Neyman--Pearson). The major Neyman--Pearson paper of 1933 also considered composite hypotheses (ones whose distribution includes an unknown parameter). An example proved the optimality of the (Student's) t - test, "there can be no better test for the hypothesis under consideration" (p 321). Neyman--Pearson theory was proving the optimality of Fisherian methods from its inception . </P> <P> Fisher's significance testing has proven a popular flexible statistical tool in application with little mathematical growth potential . Neyman--Pearson hypothesis testing is claimed as a pillar of mathematical statistics, creating a new paradigm for the field . It also stimulated new applications in statistical process control, detection theory, decision theory and game theory . Both formulations have been successful, but the successes have been of a different character . </P>

In the logic of statistical significance the elements include