17. Factorial ANOVA

Independent factorial

Author

Affiliation

Johnny van Doorn

University of Amsterdam

Published

13 October 2026

In this lecture we aim to:

ANOVA with 2 categorical predictors
Discuss interaction
Demonstration in JASP

Reading: Chapter 13 (§13.1–13.14)

Updated slides

Lecture 12
- \(t\)-test = ANOVA = regression
Lecture 13
- Why don’t the parts add up?
- Exam note: checking assumptions

DSUJ in ANS

Not required, just extra practice
Exercises from the book (Smart Alex), as SSR exam questions
Do them as many times as you want
Made with help from an LLM, keeping as much human text as possible
- Questions 77% human-written, model answers 84%
Examples of added text
- Andy’s “I” becomes “Andy Field”
- Instructions (“Use alpha = 0.05.”, “how to” links)
- Wrong options for multiple-choice questions
- Bridging sentences in model answers

Independent factorial ANOVA

Two or more independent variables with two or more categories. One dependent variable.

Independent factorial ANOVA

The independent factorial ANOVA analyses the variance of multiple independent variables (Factors) with two or more categories.

Effects and interactions:

1 dependent/outcome variable
2 or more independent/predictor variables
- 2 or more cat./levels

Applet

Link

Assumptions

Continuous variable
Random sample
Normally distributed
- Q-Q plot
Equal variance within groups
- Ratio of observed sd’s
- Welch correction only exists for one-way ANOVA

Formulas

Variance	Sum of squares	df	Mean squares	F-ratio
Model	\(\text{SS}_{\text{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_{model}-1\)	\(\frac{\text{SS}_{\text{model}}}{\text{df}_{\text{model}}}\)	\(\frac{\text{MS}_{\text{model}}}{\text{MS}_{\text{residual}}}\)
\(\hspace{2ex}A\)	\(\text{SS}_{\text{A}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_A-1\)	\(\frac{\text{SS}_{\text{A}}}{\text{df}_{\text{A}}}\)	\(\frac{\text{MS}_{\text{A}}}{\text{MS}_{\text{residual}}}\)
\(\hspace{2ex}B\)	\(\text{SS}_{\text{B}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_B-1\)	\(\frac{\text{SS}_{\text{B}}}{\text{df}_{\text{B}}}\)	\(\frac{\text{MS}_{\text{B}}}{\text{MS}_{\text{residual}}}\)
\(\hspace{2ex}A \times B\)	\(\text{SS}_{A \times B} = \text{SS}_{\text{model}} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}}\)	\(df_A \times df_B\)	\(\frac{\text{SS}_{A \times B}}{\text{df}_{A \times B}}\)	\(\frac{\text{MS}_{A \times B}}{\text{MS}_{\text{residual}}}\)
Residual	\(\text{SS}_{\text{residual}} = \sum{s_k^2(n_k-1)}\)	\(N-k_{model}\)	\(\frac{\text{SS}_{\text{residual}}}{\text{df}_{\text{residual}}}\)
Total	\(\text{SS}_{\text{total}} = \text{SS}_{\text{model}} + \text{SS}_{\text{residual}}\)	\(N-1\)	\(\frac{\text{SS}_{\text{total}}}{\text{df}_{\text{total}}}\)

Variance in factorial designs

Example

In this example we will look at the amount of accidents in a car driving simulator while subjects were given varying doses of speed and alcohol.

Dependent variable
- Accidents
Independent variables
- Speed dose
  - None / Some / Much
- Alcohol dose
  - None / Some / Much
\(\alpha = 0.005\)

Data

Data (.csv)

SS full model

Variance	Sum of squares	df	Mean squares	F-ratio
Model	\(\text{SS}_{\text{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_{model}-1\)	\(\frac{\text{SS}_{\text{model}}}{\text{df}_{\text{model}}}\)	\(\frac{\text{MS}_{\text{model}}}{\text{MS}_{\text{residual}}}\)

Predicts group means for each cell of the design:

     speed  alcohol accidents  n
1 none (S) none (A)    2.1060 20
2 some (S) none (A)    2.9445 20
3 much (S) none (A)    3.8880 20
4 none (S) some (A)    3.4435 20
5 some (S) some (A)    4.7625 20
6 much (S) some (A)    5.5790 20
7 none (S) much (A)    5.2970 20
8 some (S) much (A)    6.5125 20
9 much (S) much (A)    7.5720 20

SS full model visual

SS residual

Variance	Sum of squares	df	Mean squares	F-ratio
Residual	\(\text{SS}_{\text{residual}} = \sum{s_k^2(n_k-1)}\)	\(N-k\)	\(\frac{\text{SS}_{\text{residual}}}{\text{df}_{\text{residual}}}\)

SS residual visual

SS A Speed

Variance	Sum of squares	df	Mean squares	F-ratio
\(\hspace{2ex}A\)	\(\text{SS}_{\text{A}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_A-1\)	\(\frac{\text{SS}_{\text{A}}}{\text{df}_{\text{A}}}\)	\(\frac{\text{MS}_{\text{A}}}{\text{MS}_{\text{residual}}}\)

none (S) some (S) much (S) 
3.615500 4.739833 5.679667

SS A Speed Visual

SS A Speed Residual Visual

SS B Alcohol

Variance	Sum of squares	df	Mean squares	F-ratio
\(\hspace{2ex}B\)	\(\text{SS}_{\text{B}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)	\(k_B-1\)	\(\frac{\text{SS}_{\text{B}}}{\text{df}_{\text{B}}}\)	\(\frac{\text{MS}_{\text{B}}}{\text{MS}_{\text{residual}}}\)

none (A) some (A) much (A) 
  2.9795   4.5950   6.4605

SS B Alcohol Visual

SS B Alcohol Residual Visual

SS AB Alcohol x Speed

Variance	Sum of squares	df	Mean squares	F-ratio
\(\hspace{2ex}A \times B\)	\(\text{SS}_{A \times B} = \text{SS}_{\text{model}} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}}\)	\(df_A \times df_B\)	\(\frac{\text{SS}_{A \times B}}{\text{df}_{A \times B}}\)	\(\frac{\text{MS}_{A \times B}}{\text{MS}_{\text{residual}}}\)

# Sums of squares for the interaction between speed and alcohol
ss.speed.alcohol <- ss.model - ss.speed - ss.alcohol
ss.speed.alcohol

[1] 1.910727

Using mean squares to compute \(F\)

For every \(F\)-statistic, we use the \(MS_{residual}\) of the full model:

\[\begin{aligned} F_{Speed} &= \frac{{MS}_{Speed}}{{MS}_{residual}} \\ F_{Alcohol} &= \frac{{MS}_{Alcohol}}{{MS}_{residual}} \\ F_{Alcohol \times Speed} &= \frac{{MS}_{Alcohol \times Speed}}{{MS}_{residual}} \\ \end{aligned}\]

Interaction

\[F_{Alcohol \times Speed}\]

N          <- length(accidents)
k.speed    <- 3
k.alcohol  <- 3
k.model    <- 9
df.speed   <- k.speed   - 1
df.alcohol <- k.alcohol - 1
df.speed.alcohol <- df.speed * df.alcohol

ms.speed.alcohol <- ss.speed.alcohol / df.speed.alcohol

df.error <- N - k.model
ms.error <- ss.error / df.error

\(P\)-value

Partition the explained variance (BONUS)

In reality, we do not know exactly how the shared variance is distributed across these three sources (IV1, IV2, shared)

Partition the explained variance (BONUS)

Fit a model with only IV1, then use that to compute the explained variance for IV2: \(SS_{\text{model 1}} = 70 \rightarrow SS_{IV2} = 130 - 70 = 60\)
Fit a model with only IV2, then use that to compute the explained variance for IV1: \(SS_{\text{model 2}} = 80 \rightarrow SS_{IV1} = 130 - 80 = 50\)
See also Oliver Twisted from Chapter 13

Contrast

Planned comparisons

Exploring differences of theoretical interest
Higher precision
Higher power

Post-Hoc

Unplanned comparisons

Exploring all possible differences
Adjust p-value for inflated type 1 error
- P(1 type-I error) = \(0.05\)
- P(not 1 type-I error) = \(0.95\)
- P(no type-I error in 5 tests) = \(0.95^5 = 0.774\)
Bonferroni more conservative, Tukey more permissive

Exam note: Know where the options are, exam question will inform you which correction to use

Effect size in ANOVA

General effect size measure: Partial omega squared \(\omega_p^2\)

How to interpret?

\(\omega^2\) and \(\omega_p^2\)	Magnitude of Effect
≥ 0.01	Small
≥ 0.06	Medium
≥ 0.14	Large

Rough guidelines, interpret in context (Kirk, 1996; DSUJ p. 461)

Effect size in follow-up tests

Effect sizes of contrasts or post-hoc comparisons: Cohen’s \(d\)

How to interpret?

Cohen’s d	Magnitude of Effect
≥ 0.2	Small
≥ 0.5	Medium
≥ 0.8	Large

Rough guidelines, interpret in context (Cohen, 1988, 1992; DSUJ p. 130)

JASP

Closing

Recap

With multiple predictor variables:

We are still explaining variance
We check the interaction
- Plots!
- F-value of interaction effect
- Conditional post-hoc tests
- Simple main effects

Next Time

RM ANOVA: within-subjects measurements (lecture 18)
Mixed ANOVA (lecture 18)

Recommended Exercises

Exercise 13.1
Exercise 13.3
Exercise 13.6
You do not need to compute effects sizes by hand!

Labcoat Leni 13.1

Davey et al. (2003): Don’t forget your toothbrush

Does the effect of the stop rule on checking depend on mood?

mood: negative, positive or neutral mood induction
stop_rule: “as many as you can” vs. “feel like continuing”
checks: number of things to check before going on holiday

Data (.jasp) · Results (.jasp) · Results (.html)

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