9: Categorical Data
- Page ID
- 45189
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No doubt you have already been introduced to chi-square \(\left(\chi^{2}\right)\) tests (click here correct pronunciation), particularly if you’ve had a genetics class, but perhaps you were not told why you were using the \(\chi^{2}\) test, as opposed to some other test, for example t-test, ANOVA, or linear regression.
Chi-square analyses are used in situations of discrete (i.e., categorical or qualitative) data types. When you can count the number of “yes” or “no” outcomes from an experiment, then you are talking about a \(\chi^{2}\) problem. In contrast, continuous (i.e., quantitative) data types for outcome variables would require you to use the \(t\)-test (for two groups) or the ANOVA-like procedures (for two or more groups). Chi-square tests can be applied when you have two or more treatment groups.
Two kinds of chi-square analyses
(1) We ask about the “fit” of our data against predictions from theory. This is the typical chi-square that student’s have been exposed to in biology lab. If outcomes of an experiment can be measured against predictions from some theory, then this is a goodness of fit (gof) \(\chi^{2}\). Goodness of fit is introduced in Section 9.1.
(2) We ask whether the outcomes of an experiment are associated with a treatment. These are called contingency table problems, and they will be the subject of the next lecture. The important distinction here is that there exists no outside source of information (“theory”) available to make predictions about what we would expect. Contingency tables are introduced in Section 9.2.
- 9.1: Chi-square test and goodness of fit
- Chi-square as a measure of the “fit” of data against theoretical predictions or external expectations of frequency of experimental outcomes.
- 9.2: Chi-square contingency tables
- Using categorical data as intrinsic models to generate tests of hypotheses via \(\chi^{2}\), when no theory or extrinsic model is available as a guide.
- 9.3: Yates continuity correction
- Corrections for calculated Pearson’s test statistic \(\chi^{2}\) close to the critical value, when you have only one degree of freedom. Includes discussion of the Yates and Serra corrections.
- 9.4: Heterogeneity chi-square tests
- Pooling multiple data sets to test against the same theoretical distribution, to create a more powerful test of the null hypothesis.
- 9.5: Fisher exact test
- Fisher’s exact test as an alternative to the chi-square test, for contingency tables with low expected values or DF = 1. Reasons it is often suited for analysis of biomedical research experiments.
- 9.6: McNemar's test
- Matched pair case-control experiments and lack of independence between sampling units. How McNemar's test and unconditional paired tests can be used as an alternative to the chi-square test in these scenarios.