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One- and two-tailed tests
In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test is used if deviations of the estimated parameter in either direction from some benchmark value are considered theoretically possible; in contrast, a one-tailed test is used if only deviations in one direction are considered possible. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the normal distribution or "bell curve", pictured above right.
== Applications ==
One-tailed tests are used for asymmetric distributions that have a single tail, such as the chi-squared distribution, which are common in measuring goodness-of-fit, or for one side of a distribution that has two tails, such as the normal distribution, which is common in estimating location; this corresponds to specifying a direction. Two-tailed tests are only applicable when there are two tails, such as in the normal distribution, and correspond to considering either direction significant.〔Kock, N. (2015). (One-tailed or two-tailed P values in PLS-SEM? ) International Journal of e-Collaboration, 11(2), 1-7.〕〔Mundry, R., & Fischer, J. (1998). (Use of statistical programs for nonparametric tests of small samples often leads to incorrect P values: Examples from Animal Behaviour. ) Animal behaviour, 56(1), 256-259.〕〔Pillemer, D. B. (1991). (One-versus two-tailed hypothesis tests in contemporary educational research. ) Educational Researcher, 20(9), 13-17.〕
In the approach of Ronald Fisher, the null hypothesis H0 will be rejected when the ''p''-value of the test statistic is sufficiently extreme (vis-a-vis the test statistic's sampling distribution) and thus judged unlikely to be the result of chance. In a one-tailed test, "extreme" is decided beforehand as either meaning "sufficiently small" ''or'' meaning "sufficiently large" – values in the other direction are considered not significant. In a two-tailed test, "extreme" means "either sufficiently small or sufficiently large", and values in either direction are considered significant.〔John E. Freund, (1984) ''Modern Elementary Statistics'', sixth edition. Prentice hall. ISBN 0-13-593525-3 (Section "Inferences about Means", chapter "Significance Tests", page 289.)〕 For a given test statistic there is a single two-tailed test, and two one-tailed tests, one each for either direction. Given data of a given significance level in a two-tailed test for a test statistic, in the corresponding one-tailed tests for the same test statistic it will be considered either twice as significant (half the ''p''-value), if the data is in the direction specified by the test, or not significant at all (''p''-value above 0.5), if the data is in the direction opposite that specified by the test.
For example, if flipping a coin, testing whether it is biased ''towards'' heads is a one-tailed test, and getting data of "all heads" would be seen as highly significant, while getting data of "all tails" would be not significant at all (''p'' = 1). By contrast, testing whether it is biased in ''either'' direction is a two-tailed test, and either "all heads" or "all tails" would both be seen as highly significant data. In medical testing, while one is generally interested in whether a treatment results in outcomes that are ''better'' than chance, thus suggesting a one-tailed test; a ''worse'' outcome is also interesting for the scientific field, therefore one should use a two-tailed test that corresponds instead to testing whether the treatment results in outcomes that are ''different'' from chance, either better or worse.〔J M Bland, D G Bland (BMJ, 1994) ''Statistics Notes: One and two sided tests of significance''〕 In the archetypal lady tasting tea experiment, Fisher tested whether the lady in question was ''better'' than chance at distinguishing two types of tea preparation, not whether her ability was ''different'' from chance, and thus he used a one-tailed test.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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