A one-sided test has either an upper (greater) or a lower (lesser) alternative. This section describes one-sided tests with upper alternatives only. Corresponding results for one-sided tests with lower alternatives can be derived similarly.
For a one-sided test of with an upper alternative , an equivalent null hypothesis is with an upper alternative , where . A fixed-sample test rejects if the standardized test statistic , where is the sample estimate of and is the critical value.
The p-value of the test is given by , and the hypothesis is rejected if the p-value is less than . An upper confidence interval has the lower limit
The hypothesis is rejected if the confidence interval for the parameter does not contain zero—that is, if the lower limit is greater than 0.
With an alternative reference , , a Type II error probability is defined as
which is equivalent to
Thus, . Then, with ,
The drift parameter can be computed for specified and and the maximum information is given by
If the maximum information is available, then the required sample size can be derived. For example, in a one-sample test for the mean with a specific standard deviation , the sample size n required for the test is
On the other hand, if the alternative reference , standard deviation , and sample size n are all specified, then can be derived for a given and, similarly, can be derived for a given .
With an alternative reference , , the power is the probability of correctly rejecting the null hypothesis at :
A superiority trial that tests the response to a new drug is clinically superior to a comparative placebo control or active control therapy. If a positive value indicates a beneficial effect, a test for superiority has
where is the hypothesis of nonsuperiority and is the alternative hypothesis of superiority.
The superiority test rejects the hypothesis and declares superiority if the standardized statistic , where the critical value .
For example, if is the response difference between the treatment and placebo control groups, then a superiority trial can be
with a Type I error probability level and a power at .
A noninferiority trial does not compare the response to a new treatment with the response to a placebo. Instead, it demonstrates the effectiveness of a new treatment compared with that of a nonexisting placebo by showing that the response of a new treatment is not clinically inferior to the response of a standard therapy with an established effect. That is, this type of trial attempts to demonstrate that the new treatment effect is not worse than the standard therapy effect by an acceptable margin. These trials are often performed when there is an existing effective therapy for a serious disease, and therefore a placebo control group cannot be ethically included.
It can be difficult to specify an appropriate noninferiority margin. One practice is to choose with reference to the effect of the active control in historical placebo-controlled trials (Snapinn, 2000, p. 20). With this practice, there is some basis to imply that the new treatment is better than the placebo for a positive noninferiority trial.
If a positive value indicates a beneficial effect, a test for noninferiority has a null hypothesis and an alternative hypothesis , where is the specified noninferiority margin.
An equivalent test has
where the parameter , is the null hypothesis of inferiority, and is the alternative hypothesis of noninferiority,
The noninferiority test rejects the hypothesis and declares noninferiority if the standardized statistic , where the critical value .
For example, if is the response difference between the treatment and active control groups and is the noninferiority margin, then a noninferiority trial with a power at might be
where .