In many multiple testing problems, point hypotheses are tested.Yet many researchers feel that point hypotheses (especially hypotheses of null effects) are unrealistic in most if not all situations, because a null hypothesis, for example that the mean of a control group is equal to the mean of a treatment group, is never (or almost never) exactly true. There are two ways of interpreting the test of a single hypothesis concerning the value of a parameter: as a test of a point null hypothesis at level, or as a corresponding test of a pair of hypotheses concerning the sign of the parameter, each such directional hypothesis at level (in a symmetric case) and at some level in a nonsymmetric case. In multiple testing, the relationship between the two approaches (point hypothesis vs. directional pair) is more complex, and depends on the characteristics of the multiple procedure. In stepwise tests, it is not even clear that the familywise error rate remains . (See Finner, 1999.) Even if it does, the relations between the directional and nondirectional levels is more complex than in testing single hypotheses. Some examples of relationships will be discussed, and possible compromise procedures, taking both the direction and nondirectional points of view into consideration, will be explored.

References:

1. Finner, H. (1999). Stepwise multiple test procedures and controlof directional errors. Annals of Statistics 27, 274-289.