Solved – When is a one-sided test used versus a two sided test in a Fisher’s Exact Test

contingency tablesfishers-exact-testhypothesis testing

Suppose I have a 2 by 2 table and I am trying to conduct a Fisher's Exact Test. I am wondering when the two-sided test is used versus a one-sided test. In programs like R, they have greater, less, and two-sided. In most cases, I would only use either greater or less depending on the direction. However, when is the two-sided version ever used? Thanks.

Best Answer

If you have a very strong assumption before exploring your data that the values of one group will be either lower or equal to the other group (alternative="less") or that the values of one group will be either higher or equal to the other group (alternative="greater") you can use a one-sided test to increase the power of this test:

A two-sided test:

If you are using a significance level of 0.05, [...] .025 is in each tail of the distribution of your test statistic [...] you are testing for the possibility of the relationship in both directions. [...] The mean is considered significantly different from x if the test statistic is in the top 2.5% or bottom 2.5% of its probability distribution, resulting in a p-value less than 0.05.

A one-tailed test:

When using a one-tailed test, you are testing for the possibility of the relationship in one direction and completely disregarding the possibility of a relationship in the other direction. [...] A one-tailed test will test either if the mean is significantly greater than x or if the mean is significantly less than x, but not both. [...] The one-tailed test provides more power to detect an effect in one direction by not testing the effect in the other direction. A discussion of when this is an appropriate option follows.

https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests/

Related Solutions

Solved – Fisher’s Exact Test in contingency tables larger than 2×2

The only problem with applying Fisher's exact test to tables larger than 2x2 is that the calculations become much more difficult to do. The 2x2 version is the only one which is even feasible by hand, and so I doubt that Fisher ever imagined the test in larger tables because the computations would have been beyond anything he would have envisaged.

Nevertheless, the test can be applied to any mxn table and some software including Stata and SPSS provide the facility. Even so, the calculation is often approximated using a Monte Carlo approach.

Yes, if the expected cell counts are small, it is better to use an exact test as the chi-squared test is no longer a good approximation in such cases.

Solved – One-sided Fisher’s exact test and its complement

The particular table helps a lot. The Fisher's exact test assigns probabilities to tables with these particular marginals using the hypergeometric distribution. In this case, we're thinking of drawing 9 balls from an urn (the cases) with 2852 white balls (exposed) and 2861 black balls (not exposed). The number of white balls drawn is the count for exposed cases. The distribution is:

0     1     2     3     4     5     6     7     8     9 
0.002 0.018 0.071 0.165 0.247 0.246 0.163 0.070 0.017 0.002

The one-sided test in your output is giving the probability of 2 or fewer:

0.002 + 0.018 + 0.071 = 0.0904

The one-sided test in the other direction would give the probability of 2 or more, which is 1 minus the probability of 0 or 1:

1 - (0.002 + 0.018) = 0.98

Note that the two-sided test is the probability of 0, 1, 2, 7, 8, or 9, which does come to 0.179.

So the p-values for the two one-tailed tests don't add to one, because they each include the particular observed value and the distribution is discrete.

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Solved – Fisher’s Exact Test in contingency tables larger than 2×2

Solved – One-sided Fisher’s exact test and its complement

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