# 2: Tests for Nominal Variables

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• 2.1: Exact Test of Goodness-of-Fit
The main goal of a statistical test is to answer the question, "What is the probability of getting a result like my observed data, if the null hypothesis were true?" If it is very unlikely to get the observed data under the null hypothesis, you reject the null hypothesis.  Exact tests, such as the exact test of goodness-of-fit, are different. There is no test statistic; instead, you directly calculate the probability of obtaining the observed data under the null hypothesis.
• 2.2: Power Analysis
Many statistical tests have been developed to estimate the sample size needed to detect a particular effect, or to estimate the size of the effect that can be detected with a particular sample size. In order to do a power analysis, you need to specify an effect size. This is the size of the difference between your null hypothesis and the alternative hypothesis that you hope to detect. For applied and clinical biological research, there may be a very definite effect size that you want to detect.
• 2.3: Chi-Square Test of Goodness-of-Fit
Use the chi-square test of goodness-of-fit when you have one nominal variable with two or more values. You compare the observed counts of observations in each category with the expected counts, which you calculate using some kind of theoretical expectation. If the expected number of observations in any category is too small, the chi-square test may give inaccurate results, and you should use an exact test instead.
• 2.4: G–Test of Goodness-of-Fit
Use the G–test of goodness-of-fit when you have one nominal variable with two or more values (such as male and female, or red, pink and white flowers). You compare the observed counts of numbers of observations in each category with the expected counts, which you calculate using some kind of theoretical expectation.
• 2.5: Chi-square Test of Independence
Use the chi-square test of independence when you have two nominal variables, each with two or more possible values. You want to know whether the proportions for one variable are different among values of the other variable.
• 2.6: G–Test of Independence
To use the G–test of independence when you have two nominal variables and you want to see whether the proportions of one variable are different for different values of the other variable. Use it when the sample size is large.
• 2.7: Fisher's Exact Test
Use Fisher's exact test when you have two nominal variables. You want to know whether the proportions for one variable are different among values of the other variable.
• 2.8: Small Numbers in Chi-Square and G–Tests
Chi-square and G–tests of goodness-of-fit or independence give inaccurate results when the expected numbers are small. When the sample sizes are too small, you should use exact tests instead of the chi-square test or G–test.
• 2.9: Repeated G–Tests of Goodness-of-Fit
You use the repeated G–test of goodness-of-fit when you have two nominal variables, one with two or more biologically interesting values (such as red vs. pink vs. white flowers), the other representing different replicates of the same experiment (different days, different locations, different pairs of parents). You compare the observed data with an extrinsic theoretical expectation (such as an expected 1: 2: 1 ratio in a genetic cross).
• 2.10: Cochran-Mantel-Haenszel Test
Use the Cochran–Mantel–Haenszel test (which is sometimes called the Mantel–Haenszel test) for repeated tests of independence. The most common situation is that you have multiple 2×2 tables of independence; you're analyzing the kind of experiment that you'd analyze with a test of independence, and you've done the experiment multiple times or at multiple locations.

This page titled 2: Tests for Nominal Variables is shared under a not declared license and was authored, remixed, and/or curated by John H. McDonald via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.