18. RM ANOVA

Repeated & Mixed

Johnny van Doorn

University of Amsterdam

2025-10-15

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

What have we learned so far?

  • Explaining variance…
  • Partitioning explained variance
  • For each predictor, what is its model sum of squares?
    • Divide by the error sum of squares of the full model (with all predictors) \(\rightarrow\) F-ratio

Having multiple predictors - interaction

Having multiple predictors - interaction

Assess with:

  • Descriptives plots
  • Conditional post hoc tests
  • Simple main effects

Having multiple predictors - dependence

Having multiple predictors - dependence

Avoid by:

  • Experimental manipulation
  • Randomization
  • Matching

Assess with:

  • ANOVA (if categorical + continuous)
  • Correlation (if continuous + continuous) \(\rightarrow\) block 3
  • Chi-squared (if categorical + categorical) \(\rightarrow\) block 3

Contrasts

Contrast 1: compare no alcohol to alcohol

Contrast 2: compare two alcohol conditions

Assumptions

  • Know which ones are relevant
  • Know how to assess them
  • How to apply corrections in JASP (for unequal variances, sphericity)

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

Assumptions

  • Continuous dependent variable
  • Normally distributed
    • Q-Q plots
    • Shapiro-Wilk
  • Equality of variance of the within-group differences
    • Mauchly’s test of Sphericity
    • 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 \({SS}_{{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k-1\) \(\frac{{SS}_{{model}}}{{DF}_{{model}}}\) \(\frac{{MS}_{{model}}}{{MS}_{{error}}}\)
• Error \({SS}_{{error}} = {SS}_{{within}} - {SS}_{{model}}\) \((n-1)(k-1)\) \(\frac{{SS}_{{error}}}{{DF}_{{error}}}\)  
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

The data

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 (\(SS_W\))

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

[1] 11.54771

Within subject variance (\(SS_W\)) - visual

Within subject variance (\(SS_W\)) - data

Alcohol model SS (\(SS_M\))

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

[1] 9.543261

Alcohol model SS visual

Full model SS visual

Full model error SS visual (\(SS_R\))

We use full model error to compute F for alcohol

F ratio

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

# 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 where given varying doses of speed and alcohol. But this time we lat participants partake in all conditions. Every week subjects returned for a different experimental condition.

  • Dependent variable
    • Accidents
  • Independent variables
    • Speed
      • None
      • Small
      • Large
    • Alcohol
      • None
      • Small
      • Large

person 1_1 1_2 1_3 2_1 2_2 2_3 3_1 3_2 3_3
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9

Data

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

Further reading

Oliver Twisted from Chapter 14

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

Contact

CC BY-NC-SA 4.0

Scientific & Statistical Reasoning