Null Hypothesis - Terminology

Terminology

Much of the terminology used in connection with null hypotheses derives from the immediate relation to statistical hypothesis testing; part of this terminology is outlined here, but see this list of definitions for a more complete set.

Simple hypothesis

Any hypothesis which specifies the population distribution completely.

Composite hypothesis

Any hypothesis which does not specify the population distribution completely.

A point hypothesis is more complicated to describe. The term arises in contexts where the set of all possible population distributions is put in parametric form. A point hypothesis is one where exact values are specified for either all the parameters or for a subset of the parameters. Formally, the case where only a subset of parameters is defined is still a composite hypothesis; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic (the distribution under the null hypothesis) does not depend on the parameters whose values have not been specified under the point null hypothesis. Careful treatments of point hypotheses for subsets of parameters do consider them as composite hypotheses and study how the p-value for a fixed critical value of the test statistic varies with the parameters that are not specified by the null hypothesis.

A one-tailed hypothesis is a hypothesis in which the value of a parameter is specified as being either:

• above or equal to a certain value, or
• below or equal to a certain value.

An example of a one-tailed null hypothesis would be that, in a medical context, an existing treatment, A, is no worse than a new treatment, B. The corresponding alternative hypothesis would be that B is better than A. Here if the null hypothesis weren't rejected (i.e. there is no reason to reject the hypothesis that A is at least as good as B), the conclusion would be that treatment A should continue to be used. If the null hypothesis were rejected, i.e. there is evidence that B is better than A, the result would be that treatment B would be used in future. An appropriate hypothesis test would look for evidence that B is better than A, not for evidence that the outcomes of treatments A and B are different. Formulating the hypothesis as a "better than" comparison is said to give the hypothesis directionality.