Statistical hypothesis testing · BEGINNER LESSON
One-sided test
A one-sided test is a statistical hypothesis test in which the alternative hypothesis points in only one direction, such as H0: θ ≤ θ0 versus H1: θ > θ0, or the reverse. It is used only for a scalar parameter when the question is about one direction of change, not both.
What it is
A one-sided test is a binary hypothesis test with a directional alternative. The null hypothesis says the parameter is at or beyond a boundary in one direction, and the alternative says it crosses that boundary in the chosen direction. In the sources, this means forms such as H0: θ ≤ θ0 versus H1: θ > θ0, or H0: θ ≥ θ0 versus H1: θ < θ0. The parameter θ is the unknown quantity being studied, and scalar means it is a single number, not a vector of many numbers. [sourceIndexes:[0,1,2]]
[1] Hypothesis Testing [2] An Investigator's Introduction to Statistical Considerations in ... [3] Hypothesis testing, type I and type II errors - PMC - NIH
How it works
A sample is collected, then a test statistic T(x) is computed from the data x. The test compares T(x) with a critical value or threshold, and values beyond the acceptance region lead to rejecting H0. A p-value is the smallest significance level α at which this rejection would happen for the observed data. Smaller p-values give stronger evidence against H0. [sourceIndexes:[0,1]]
[1] Hypothesis Testing [2] An Investigator's Introduction to Statistical Considerations in ...
Errors and operating characteristics
A one-sided test can still make two kinds of mistakes. A type I error is rejecting H0 when H0 is true; this is also called a false positive or false alarm, and its probability is α, the significance level. A type II error is failing to reject H0 when H1 is true; this is a miss or false negative. Power is 1 − β, the chance of detecting the alternative when it is true. In composite hypotheses, size is the worst-case type I error rate over all parameter values allowed by H0. [sourceIndexes:[0,1,2]]
[1] Hypothesis Testing [2] An Investigator's Introduction to Statistical Considerations in ... [3] Hypothesis testing, type I and type II errors - PMC - NIH
Why direction matters
The key feature of a one-sided test is that all of the allowable false-positive rate is placed on one tail of the distribution. That makes it appropriate only when a difference in one direction is the only meaningful question. If both directions matter, a two-sided test is the correct choice. The sources also note that one-sided and two-sided tests are defined for scalar parameter spaces and require at least one composite hypothesis. [sourceIndexes:[0,1,2]]
[1] Hypothesis Testing [2] An Investigator's Introduction to Statistical Considerations in ... [3] Hypothesis testing, type I and type II errors - PMC - NIH
Limitations
A one-sided test does not prove the null hypothesis when the p-value is large. A large p-value may mean the null is true, or it may mean the test has low power and could miss a real effect. One-sided tests are therefore narrower in scope than two-sided tests, and they can be inappropriate when an effect in either direction would matter. [sourceIndexes:[0,1,2]]
[1] Hypothesis Testing [2] An Investigator's Introduction to Statistical Considerations in ... [3] Hypothesis testing, type I and type II errors - PMC - NIH
SEE THE IDEA
One-sided test in three steps
Choose a null and alternative that point one way, such as greater than versus not greater than.
Summarize the sample with a test statistic T(x) and compare it with a threshold.
If the statistic falls in the rejection region, reject H0; otherwise do not reject H0.
A one-sided test moves from a directional question to a statistic, then to a decision based on one tail of the distribution.
The rejection region is placed on only one side because the hypothesis is directional.
TRY IT YOURSELF
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PUT IT TOGETHER
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.