MEDD8815 Chap.8 One-Way and Two-Way Analysis of Variance
One-Way and Two-Way Analysis of Variance
Comparing several means at once
The final session generalises the two-sample comparison to several groups. Analysis of variance judges whether sample means differ by more than the variation inside the groups would lead you to expect.
Running a t test for every pair would inflate the chance of a false alarm, so ANOVA replaces them with one overall F test, followed by an adjusted analysis of which means differ.
The one-way model and its table
The chapter sets out the model, the null hypothesis that every group shares one population mean, and the conditions: independent random samples, Normal populations and a common standard deviation, checked with the rule that the largest sample standard deviation should be no more than twice the smallest.
It builds the ANOVA table from group summaries on an AskSia-authored three-group data set, showing how the total sum of squares divides into a between-group part and a within-group part, and how their mean squares form the F statistic.
Contrasts and multiple comparisons
A significant F test says the means are not all equal but not which ones differ.
Questions fixed in advance are answered with contrasts, combinations of means whose coefficients sum to zero, tested with t on the error degrees of freedom and adjusted with Bonferroni when there are several. When no pattern was specified, pairwise methods such as Tukey and Scheffe control the familywise error rate.
The chapter works both and shows the SPSS Contrasts and Post Hoc routes.
Two factors and interaction
A two-way design crosses two factors, which is more efficient than separate studies and is the only way to see an interaction.
The chapter reads a profile plot in which the lines are not parallel, fills and reads a two-way ANOVA table, and follows the slides' cautions: plot the cell means, read the interaction row first, and interpret cell means when interaction is present. It also shows that the three tests of a balanced two-by-two design reappear as three contrasts in a one-way analysis of its four cells.
What this chapter covers
- 01
The idea of analysis of variance
- 02
The one-way ANOVA model and its conditions
- 03
The F statistic and the ANOVA table
- 04
Why many t tests inflate false alarms
- 05
Planned contrasts and their t tests
- 06
Bonferroni, Tukey and Scheffe comparisons
- 07
Two-way designs, main effects and interaction
- 08
Reading the two-way ANOVA table and profile plot
Worked example · free
Complete an ANOVA table from partial output
- 1Find the three degrees of freedom from I and N.
- 2Compute both mean squares and F.
- 2Compare with the critical value and state what follows next.
Key terms
- Analysis of Variance
- A method for comparing several population means by comparing the variation among sample means with the variation within the samples.
- Mean Square Error
- The within-group variance estimate in ANOVA, the error sum of squares divided by N minus I; its square root is the pooled standard deviation.
- F Statistic
- The ratio of the between-group mean square to the within-group mean square, compared with an F distribution to test equality of means.
- Contrast
- A combination of population means whose coefficients add to zero, used to answer a question about the means that was fixed before the data were seen.
- Familywise Error Rate
- The probability of at least one false rejection across a whole set of tests, which grows as more unadjusted tests are run.
- Bonferroni Procedure
- An adjustment for k tests that multiplies each P-value by k, capped at 1, so the familywise error rate stays at or below alpha.
- Interaction
- A situation in a two-way design in which the effect of one factor depends on the level of the other, shown by non-parallel lines in a profile plot.
One-Way and Two-Way Analysis of Variance FAQ
Why not just run a t test for each pair of groups?
Each test carries its own chance of a false alarm, so running many of them makes at least one spurious significant result increasingly likely. ANOVA tests all the means with one F test, and follow-up comparisons use methods that adjust for how many comparisons are made.
When do I use a contrast rather than Tukey or Scheffe?
Use a contrast for a question you fixed before the data were collected, such as whether any feedback beats none. Use Tukey to compare every pair after a significant F, and Scheffe when you want protection across any contrast, including ones suggested by the data.
What should I do when the interaction is significant?
Describe the cell means rather than the main effects. The main effect of a factor averages over the levels of the other factor, and when the effect changes from one level to another that average can hide or distort what actually happens in each combination.
Assessment move
Build one ANOVA table completely by hand from group means, standard deviations and sizes, then check it in SPSS One-Way ANOVA. Next, draw a profile plot for a two-way example and say in one sentence whether the lines suggest interaction before reading the table. Both skills are what the final homework and quiz material rest on.
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