Tukey rapid test

The rapid test Tukey , as Tukey's Quick Test , Tukey's pocket test , Tukey Duckworth test or Tail Count test called, is a statistical test , with the two independent samples for differences in the situation can be compared to their elements. The test does not require a normal distribution of the data and is therefore a non-parametric method . It is comparatively easy to carry out and is therefore particularly suitable for a quick estimate. The test is named after John W. Tukey , who described it in 1959.

Requirements and hypotheses

The two samples to be examined, the elements of which are at least ordinally scaled , must have been collected independently and randomly and each contain at least five elements ( ). In addition, the size of the two samples should not deviate too much from one another; Tukey gives a ratio of the larger to the smaller sample of less than 1.33. The frequency distributions of both samples should be comparable. The test's null hypothesis assumes that the two samples do not differ in terms of the position of their elements. The alternative hypothesis assumes a difference in position. ${\displaystyle n\geq 5}$ ${\displaystyle H_{0}}$ ${\displaystyle H_{A}}$

execution

To carry out the test, the elements of both samples are first sorted together; their association with one of the two samples is retained. It is then checked whether both samples are shifted against each other and thus one sample contains the highest and the other the lowest of all values. If this is not the case, i.e. the highest and lowest of all values ​​are in the same sample, the test cannot be used.

The number of elements by which the two samples are shifted is then determined at the ends of the sorted totality of the elements of both samples. On the one hand, the elements are counted that are smaller in the sample with the smallest common element than the smallest element of the other sample, and on the other hand, the elements in the sample with the largest common element are larger than the largest element of the other sample. Identical values ​​in both samples that lie at the beginning of the respective shifted end are counted as 0.5.

The sum of these two numbers is the test statistic , for which the following applies approximately: ${\displaystyle T}$

• when the null hypothesis is to accept ${\displaystyle T<7}$ ${\displaystyle H_{0}}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 7<T<10}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =5\,\%}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 10<T<13}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =1\,\%}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 13\leq T}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =0{,}1\,\%}$

For samples where the ratio of the sample sizes exceeds 1.33, Tukey suggested a correction value that is calculated from the sample sizes and subtracted from the test statistic . ${\displaystyle T}$

In a further development of the test published by Neave, the individual value of the combined elements of both samples is omitted if the test is otherwise carried out the same, the non-consideration of which maximizes the test statistics. For the test statistic modified in this way, the following applies approximately: ${\displaystyle T_{N}}$

• when the null hypothesis is to accept ${\displaystyle T_{N}<10}$ ${\displaystyle H_{0}}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 10<T_{N}<13}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =5\,\%}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 13<T_{N}<16}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =1\,\%}$
• in the case of the null hypothesis on the significance level is to be rejected ${\displaystyle 16\leq T_{N}}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha =0{,}1\,\%}$

Historical information

John Tukey published the test he developed in 1959 in response to a challenge that Walter Eric Duckworth had formulated three years earlier during a meeting of the Royal Statistical Society . Henry R. Neave described a modification of the test in 1966, which in this form is also known as the Tukey-Neave test . Two years later he also studied together with Clive WJ Granger by Monte Carlo simulations , the statistical power of the test in the original version and the modified version compared to other two-sample tests. William P. Dunlap and colleagues carried out further power analyzes with larger sample sizes in the 1990s. Corrected and expanded versions of the tables published by Tukey with critical values ​​for the exact determination of the significance level were later published. In 1971, Wilfred Westlake also described a test that can be viewed as a one-sided version of the Tukey rapid test.

Alternative procedures

The test of choice for comparing two independent samples is the two-sample t-test when the data are normally distributed . If the data are not normally distributed, the Wilcoxon-Mann-Whitney test and the median test can be used as non-parametric methods . The advantage of the Tukey rapid test compared to these tests is that it is easy to carry out, which can also be carried out without a computer or pocket calculator and, if the sample size is moderate, with mental arithmetic .

Individual evidence

1. ↑ John W. Tukey: A Quick, Compact, Two-Sample Test to Duckworth's Specifications. In: Technometrics. 1 (1) / 1959, pp. 31-48, doi : 10.1080 / 00401706.1959.10489847 JSTOR 1266308
2. ^ Steve Selvin: A Biostatistics Toolbox for Data Analysis. Cambridge University Press, Cambridge 2015, ISBN 1-10-711308-3 , p. 300
3. ^ Henry R. Neave: A Development of Tukey's Quick Test of Location. In: Journal of the American Statistical Association. 61 (316) / 1966, pp. 949-964, doi : 10.1080 / 01621459.1966.10482186 JSTOR 2283191
4. ^ Henry R. Neave, Clive WJ Granger: A Monte Carlo Study Comparing Various Two-Sample Tests for Differences in Mean. In: Technometrics. 10 (3) / 1968, pp. 509-522, doi : 10.1080 / 00401706.1968.10490598 JSTOR 1267105
5. ^ William P. Dunlap, Tammy Greer, Gregory O. Beatty: A Monte-Carlo Study of Type I Error Rates and Power for Tukey's Pocket Test. In: Journal of General Psychology. 123 (4) / 1996, pp. 333-339, doi : 10.1080 / 00221309.1996.9921285
6. ^ Daniel J. Gans: Corrected and Extended Tables for Tukey's Quick Test. In: Technometrics. 23 (2) / 1981, pp. 193-195, doi : 10.1080 / 00401706.1981.10486265 JSTOR 1268038
7. ^ WJ Westlake: A One-Sided Version of the Tukey-Duckworth Test. In: Technometrics. 13 (4) / 1971, pp. 901-903, doi : 10.1080 / 00401706.1971.10488864 JSTOR 1266969