12: One-way Analysis of Variance
- Page ID
- 45210
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We left off with two-group experiments in Chapter 10 where we introduced two-sample tests of the null hypothesis of no difference between the middles for each group (if means, t-tests; if medians, Wilcoxon test).
As review, please revisit what we mean by independent variables (statistical jargon for “different treatments, like a placebo vs. aspirin therapy”) and dependent variables (statistical jargon for “response or outcome of the experiment was recorded as number of living or dead subjects”).
Variables are independent in the sense that the values are not related to the experiment’s outcome — we select the levels of the variables. For example, we select to study green vs. red leaves (the variable is “leaf color”, and there are only two levels or states of the variable: green & red). In contrast, we denote the values of the response variable as dependent because the particular values that the variable will take depend on the experiment.
It’s rare that you, as a researcher, would only be interested in comparing two samples or two groups of data for which a treatment has been applied in an experiment or investigation. More often, inferences are drawn on multiple samples (more than two groups) and an experiment involves multiple groups (one or more controls plus one or more experimental treatments).
Previously, we have discussed data sets with only one or two samples or populations (e.g. one- and two-sample t-tests, Mann-Whitney tests). Now we want to extend the discussion of statistics to situations where we may have more than two samples or populations. We introduce the ANalysis Of VAriance (ANOVA).
Importantly, we will see that one- and two-sample tests are just simple cases of ANOVA. Thus, use of ANOVA should be your preference, even if you have just two groups.
- 12.1: The need for ANOVA
- The increasing rate of error when a series of t-tests is used to compare data from 3 or more groups, and why this creates a need for ANOVA. Brief discussion of other post-hoc tests that account for the multiple comparison problem.
- 12.2: One-way ANOVA
- Underlying principles of how the Analysis of Variance test is constructed, and how to perform the calculations needed for the test.
- 12.3: Fixed effects, random effects, and ICC
- Distinctions between fixed-effect and random-effect experimental models, and the different ANOVA models appropriate for each. The concept of intraclass correlation coefficient in settings where individuals are measured more than once.
- 12.4: ANOVA from "sufficient statistics"
- Shortcuts for calculating ANOVA from "sufficient statistics", when only descriptive statistics (means, standard deviations, and sample sizes) are available and raw data values are not.
- 12.5: Effect size for ANOVA
- Estimations of effect size using parameters calculated from the ANOVA.
- 12.6: ANOVA post-hoc tests
- One-way ANOVA in combination with post-hoc tests as a way of avoiding the multiple comparison problem and increase in Type I error. Discussion of types of post-hoc tests: Tukey’s Test, Dunnett’s Test, and the Bonferroni \(t\) test.
- 12.7: Many tests, one model
- Brief overview of the general Linear Model and how various tests (t-tests, ANOVA, linear and multiple regression) are special cases of this model. Implementing the Linear Model using R.