18. RM ANOVA

Repeated & Mixed

Johnny van Doorn

University of Amsterdam

14 October 2026

In this lecture we aim to:

  • Refresh some concepts from previous lectures
  • Introduce RM ANOVA
  • Introduce Mixed ANOVA
  • Show these in JASP

Reading: Chapter 14 (§14.1–14.18)

ANOVA
One-way repeated

One-way repeated measures ANOVA

The one-way repeated measures ANOVA analyses the variance of the model while reducing the error by the within person variance.

  • 1 dependent/outcome variable
  • 1 independent/predictor variable
    • 2 or more levels
  • All with same subjects

Applet

Link

Assumptions

  • Continuous dependent variable
  • Normally distributed
    • Q-Q plots (or Shapiro-Wilk)
  • Equality of variance of the within-group differences
    • Mauchly’s test of Sphericity
    • Same caution as Levene’s test (see also Misconception Mutt 14.1)
    • Plus caution as with any p-value (black-and-white decision making)
    • Greenhouse-Geisser correction is more robust
    • See Field 14.5, table 14.2, Jane Superbrain boxes 14.2 and 14.3
    • Always met when having only 2 levels

Formulas

Variance Sum of Squares df Mean Squares F-ratio
Between \({SS}_{{between}} = {SS}_{{total}} - {SS}_{{within}}\) \({df}_{{total}}-{df}_{{within}}\) \(\frac{{SS}_{{between}}}{{df}_{{between}}}\)  
Within \({SS}_{{within}} = \sum{s_i^2(n_i-1)}\) \((n_i-1)n\) \(\frac{{SS}_{{within}}}{{df}_{{within}}}\)  
• Model \({SS}_{{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k-1\) \(\frac{{SS}_{{model}}}{{df}_{{model}}}\) \(\frac{{MS}_{{model}}}{{MS}_{{residual}}}\)
• Residual \({SS}_{{residual}} = {SS}_{{within}} - {SS}_{{model}}\) \((n-1)(k-1)\) \(\frac{{SS}_{{residual}}}{{df}_{{residual}}}\)  
Total \({SS}_{{total}} = s_{grand}^2(N-1)\) \(N-1\) \(\frac{{SS}_{{total}}}{{df}_{{total}}}\)  

Where \(n_i\) is the number of observations per person and \(k\) is the number of conditions. These two are equal for a one-way repeated ANOVA. Furthermore \(n\) is the number of subjects per condition and \(N\) is the total number of data points \(n \times k\).

Example

Measure driving ability in a driving simulator. Test in three consecutive conditions where participants come back to attend the next condition.

  • Alcohol none
  • Alcohol some
  • Alcohol much
  • \(\alpha = 0.005\)

The data

Data (.csv)

Wide vs. long data formats

General flow RM ANOVA

  • We look at total variance
  • We divide it:
    • Within subject variance
    • Between subject variance

How much of the within subject variance can we explain by looking at alcohol condition?

Total variance (SS total) - visual

Within subject variance (\(SS_{within}\))

\({SS}_{{within}} = \sum{s_i^2(n_i-1)}\)

[1] 11.54771

Within subject variance (\(SS_{within}\)) - visual

Within subject variance (\(SS_{within}\)) - data

Alcohol model SS (\(SS_{model}\))

\({SS}_{model} = \sum{n_k(\bar{X}_k-\bar{X})^2}\)

[1] 9.543261

Alcohol model SS visual

Full model SS visual

Full model residual SS visual (\(SS_{residual}\))

Variance in the alcohol data

We use full model residual to compute F for alcohol

F ratio

\(F = \frac{{MS}_{{model}}}{{MS}_{{residual}}}\)

# Calculate F statistic
fStat <- MS_model / MS_error
fStat
[1] 19.04416

P-value

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

JASP

ANOVA factorial repeated

Factorial repeated measures ANOVA

The factorial repeated measures ANOVA analyses the variance of the model while reducing the error by the within person variance.

  • 1 dependent/outcome variable
  • 2 or more independent/predictor variable
    • 2 or more levels
  • All with same subjects

Assumptions

Same as one-way repeated measures ANOVA

Example

In this example we will again look at the amount of accidents in a car driving simulator while subjects were given varying doses of speed and alcohol. But this time we let participants partake in all conditions. Every week subjects returned for a different experimental condition.

  • Dependent variable
    • Accidents
  • Independent variables
    • Speed
      • None
      • Some
      • Much
    • Alcohol
      • None
      • Some
      • Much

Every participant is measured in every cell of the design

person 1_1 1_2 1_3 2_1 2_2 2_3 3_1 3_2 3_3
1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
2 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
3 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
20 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Data

Data (.csv)

Balanced by design

Alcohol / speed none some much
none 20 20 20
some 20 20 20
much 20 20 20
  • RM ANOVA only uses participants with all observations
  • So every cell holds the same people, always balanced
    • Between-subjects groups (mixed design) can still differ in size
  • Balanced, so alcohol and speed are uncorrelated

Mixed design ANOVA

Mixed design

The mixed ANOVA analyses the variance of the model while reducing the error by the within person variance.

  • 1 dependent/outcome variable
  • 1 or more independent/predictor variable with different subjects
    • 2 or more levels
  • 1 or more independent/predictor variable with same subjects
    • 2 or more levels

Assumptions

Same as repeated measures ANOVA and same as factorial ANOVA.

Example

  • Dependent variable
    • Accidents
  • Independent variables
    • Speed (same subjects)
      • None
      • Small
      • Large
    • Alcohol (same subjects)
      • None
      • Small
      • Large
    • Daytime
      • Morning
      • Evening

Data

Data (.csv)

Further reading

Oliver Twisted from Chapter 14

JASP

Closing

Recap

  • In a repeated measures design we can account for baseline differences, to reduce the overall model error
  • With within-subjects predictors, we need to check:
    • Sphericity (equal variances of difference scores)
  • With between-subjects predictors, we need to check:
    • Equal variances of groups

Next Time

  • Non-parametric tests: rank-based equivalents of old friends (lecture 20)

Labcoat Leni 14.1

Perham & Sykora (2012): Are splattered cadavers distracting?

Is recall affected by the sound played while learning letter sequences?

  • Two within-subject factors: sound (quiet, liked or disliked music) × position in the list (1–8)
  • quiet_pos1 to disliked_pos8: probability of correctly recalling a letter

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

Contact

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Scientific & Statistical Reasoning