Null Hypothesis - Directionality

Directionality

Quite often statements of point null hypotheses appear not to have a "directionality", namely, that values larger or smaller than a hypothesized value are conceptually identical. However, null hypotheses can and do have "direction"—in many instances statistical theory allows the formulation of the test procedure to be simplified, thus the test is equivalent to testing for an exact identity. For instance, when formulating a one-tailed alternative hypothesis, application of Drug A will lead to increased growth in patients, then the true null hypothesis is the opposite of the alternative hypothesis, i.e. application of Drug A will not lead to increased growth in patients (a composite null hypothesis). The effective null hypothesis will be application of Drug A will have no effect on growth in patients (a point null hypothesis).

In order to understand why the effective null hypothesis is valid, it is instructive to consider the above hypotheses. The alternative predicts that exposed patients experience increased growth compared to the control group. That is,

H₁: μ_drug > μ_control (where μ = the patients' mean growth)

The true null hypothesis is:

H_T: μ_drug ≤ μ_control

The effective null hypothesis is:

H₀: μ_drug = μ_control

The reduction occurs because, in order to gauge support for the alternative, classical hypothesis testing requires calculating how often the results would be as or more extreme than the observations. This requires measuring the probability of rejecting the null hypothesis for each possibility it includes, and second to ensure that these probabilities are all less than or equal to the test's quoted significance level. For reasonable test procedures the largest such probability occurs on the region boundary H_T, specifically for the cases included in H₀ only. Thus the test procedure can be defined (that is the critical values can be defined) for testing the null hypothesis H_T exactly as if the null hypothesis of interest was the reduced version H₀.

Fisher said, "the null hypothesis must be exact, that is free of vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution", implying a more restrictive domain for H₀. According to this view, the null hypothesis must be numerically exact—it must state that a particular quantity or difference is equal to a particular number. In classical science, it is most typically the statement that there is no effect of a particular treatment; in observations, it is typically that there is no difference between the value of a particular measured variable and that of a prediction. The majority of null hypotheses in practice do not meet this "exactness" criterion. For example, consider the usual test that two means are equal where the true values of the variances are unknown—exact values of the variances are not specified.

Most statisticians believe that it is valid to state direction as a part of null hypothesis, or as part of a null hypothesis/alternative hypothesis pair. However, the results are not a full description of all the results of an experiment, merely a single result tailored to one particular purpose. For example, consider an H₀ that claims the population mean for a new treatmemnt is an improvement on a well-established treatment with population mean = 10 (known from long experience), with the one-tailed alternative being that the new treatment's mean > 10. If the sample evidence obtained through x-bar equals −200 and the corresponding t-test statistic equals −50, the conclusion from the test would be that there is no evidence that the new treatmnent is better than the existing one: it would not report that it is markedly worse, but that is not what this particular test is looking for. To overcome any possible ambiguity in reporting the result of the test of a null hypothesis, it is best to indicate whether the test was two-sided and, if one-sided, to include the direction of the effect being tested.

The statistical theory required to deal with the simple cases of directionality dealt with here, and more complicated ones, makes use of the concept of an unbiased test.