Chapter 11 One-Way Between-Subjects ANOVA
Usage: To check for a difference between three or more group means.
Requirements: Three or more variables that are interval/ratio.
Note: This is basically the same as the independent groups t-test, but with more than two means.
Steps to conducting an one-way between-subjects ANOVA:
- Write out null and alternative hypotheses.
- Determine critical F-value for rejection of null hypothesis.
A. Use degrees of freedom (df) and alpha level.
- Calculate F-value for your data.
A. You must first find the sums of squares and mean squares.
- Compare your calculated F-value to the critical F-value.
A. If your calculated value is greater than the critical value, you reject the null hypothesis.
B. If your calculated value is less than the critical value, you fail to reject the null hypothesis.
- Calculate effect size (eta squared \(\eta^2\)).
- Run post hoc tests if neccessary.
- Calculate confidence interval (CI).
11.1 Example Setup
11.1.1 Hypotheses
\(\Large H_0: \mu_a = \mu_b = \mu_c\)
\(\Large H_1: the\ three\ means\ are\ not\ equal\)
11.1.2 Critical F-Value (CV)
\(\Large N = number\ of\ participants\)
\(\Large n_{group} = number\ of\ participants\ in\ group\)
\(\Large k = number\ of\ groups\)
\(\Large df_{between} = k - 1\)
\(\Large df_{within} = N - k\)
\(\Large df_{total} = N - 1\)
\(\Large \alpha = .05\)
\(\Large CV = [check\ table]\)
11.1.3 Calculate F
\(\Large T_{group} = \sum X_{group}\)
\(\Large SS_{total} = \sum X^2 - \frac{(\sum X)^2}{N}\)
\(\Large SS_{within} = \sum X^2 - \frac{\sum T^2}{n}\)
\(\Large SS_{between} = \frac{\sum T^2}{n} - \frac{(\sum X)^2}{N}\)
\(\Large MS_{between} = \frac{SS_{between}}{df_{between}}\)
\(\Large MS_{within} = \frac{SS_{within}}{df_{within}}\)
\(\Large F = \frac{MS_{between}}{MS_{within}}\)
11.1.4 Compare Observed F to CV
If F is greater than the CV, reject the null hypothesis.
If F is less than the CV, fail to reject the null hypothesis.
11.1.5 Calculate Effect Size
\(\Large \eta^2 = \frac{SS_{between}}{SS_{total}}\)
11.1.6 Post Hoc Testing
Which means are different? Run Tukey’s HSD Test.
\(\Large q = [check\ table]\)
\(\Large CD = q\sqrt{\frac{MS_{within}}{n}}\)
Check to see if any of the differences between the means is greater than the CD.
11.1.7 Calculate Confidence Interval (CI)
\(\Large CI = (\bar X_a - \bar X_b)\ \pm\ CD\)
\(\Large CI = (\bar X_a - \bar X_c)\ \pm\ CD\)
\(\Large CI = (\bar X_b - \bar X_c)\ \pm\ CD\)
11.2 Tips
Draw a table like the one below to organize your calculations as you go.
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Between | 50 | 2 | 25 | 12.5 |
| Within | 30 | 15 | 2 | - |
| Total | 80 | 17 | - | - |
F is actually closely related to t: \(\Large F = t^2\)