ANOVA (General)

All GLM models can be expressed in what general form?

Outcome = (model) + error

Every GLM (t-test, ANOVA, regression, correlation) is Outcome_i = model + error_i. Tests differ only in how the model is specified. The outcome must be numeric.

What is the null hypothesis in a one-way ANOVA with 3 groups?

µ1 = µ2 = µ3 (all group means equal the grand mean)

H0 says there is no difference between group means; the null model predicts every group equals the grand mean. H1 only says there is a difference somewhere, not where.

What does ANOVA compare to decide whether groups differ?

Between-group variance vs within-group variance (error)

F is a ratio of variance due to group differences to variance within groups. A large ratio is evidence against the null.

In Y_ij = µ + A_i + S(A)_ij, what does A_i represent?

The deviation of group i's mean from the grand mean

A-hat_i = Y_i. − µ-hat. For example, Grade 3's mean (11.5) minus the grand mean (10.64) gives A1 = 0.86. S(A) is the error: the person's deviation from their group mean.

How are degrees of freedom calculated in a one-way between-subjects ANOVA with N = 30 and 3 groups?

df_A = 2, df_error = 27

df_A = levels − 1 = 2. df_S(A) = participants − groups = 30 − 3 = 27. Mean square = SS / df.

Why do we divide sums of squares by degrees of freedom before forming F?

Because sums of squares depend on the number of levels and participants, so they can't be compared directly

Dividing SS by df gives mean squares, which are comparable. Squaring (not dividing) is what removes negative signs.

A one-way ANOVA gives a significant main effect with 3 groups. What does that tell you?

The groups differ somewhere, but not which ones; post-hoc tests are needed

The omnibus F only says at least one difference exists. Post-hoc pairwise tests (e.g. emmeans with Bonferroni) locate it. In the example only Grade 3 vs Grade 5 was significant.

Which is NOT one of the three GLM/ANOVA assumptions covered?

The dependent variable itself is normally distributed

The GLM assumes normal residuals, not a normal DV. i.i.d. is determined by design; homogeneity is checked with Levene's test and violin plots; normality with a QQ plot.

What can you do if ANOVA assumptions are violated in a one-way design?

Run ANOVA anyway, transform the DV / use a better model, or use Kruskal–Wallis

Options: carry on (with reduced power and more Type II error risk), transform the DV or use a more suitable model, or use a non-parametric test (Kruskal–Wallis for one-way ANOVA).

What is a main effect vs an interaction?

Main effect: one IV's effect averaged across the other IV's levels. Interaction: one IV's effect depends on the level of another

Main effects average across the levels of other factors. An interaction means the effect of A differs across levels of B (and vice versa), shown by non-parallel lines in an interaction plot.

In an interaction plot, which pattern indicates an interaction?

Non-parallel lines (e.g. crossing or converging)

Non-parallel lines = interaction. Parallel lines (flat or sloped) mean main effects only, with no interaction.

How many terms must be estimated for a 2 × 2 between-subjects ANOVA with 40 participants?

49

1 grand mean + 4 main-effect terms (2 per factor) + 4 interaction terms (1 per cell) + 40 error terms = 49.

In the Personality × Motivation example, AB11 (Introvert–Praise) = 49 − 38.5 − 2.1 − 4.95 = 3.45. What does 3.45 represent?

The part of the cell's deviation from the grand mean NOT explained by the two main effects

Main effects alone predict 45.55 for that cell, but its actual mean is 49. The leftover 3.45 is the interaction term.

What are the degrees of freedom for the main effects, interaction and error in a 2 × 2 between-subjects ANOVA with N = 40?

Main effects 1, interaction 1, error 36

df for each factor = levels − 1 = 1. df_AB = df_A × df_B = 1. Error = N − cells = 40 − 4 = 36. This matches F(1, 36) in the results.

What does sphericity assume, and when does it matter?

Equal variances of the differences between treatment levels; matters for within-subject factors with 3+ levels

Sphericity concerns the variances of pairwise condition differences (A−B, A−C, B−C). With only 2 levels there is just one difference, so it doesn't apply.

Mauchly's test gives p < .05 in a repeated-measures ANOVA. What should you do?

Report a Greenhouse–Geisser or Huynh–Feldt corrected F-test with corrected p-value

A violation makes the F-test too liberal (too many false positives). Correct the degrees of freedom: Greenhouse–Geisser is more conservative, Huynh–Feldt less so. In the music example Mauchly's p = .485, so no correction was needed.