ANOVA

Repeated & Mixed

Klinkenberg

University of Amsterdam

18 oct 2022

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

  • Uni- or Multivariate
  • Continuous dependent variable
  • Normaly distributed
    • Shapiro-Wilk
  • Equality of variance within groups
    • Mauchly’s test of Sphericity

Uni- or Multi- descision tree

Decision tree

  1. Field: 15.5.2, Output 15.2
  2. Field: Output 15.4
  3. Field: Jane Superbrain 15.2, Output 15.2 GG and HF.
  4. Field: Jane Superbrain 15.2, Sample size \(n\) is larger than \(a\) (number of levels) + 10
  5. Field: 15.5.4, Output 15.2
  6. Field: 15.5.4, Output 15.4
  7. Field: 15.5.4, Output 15.4

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

MS total

# Assign to individual variables
none_alc = data$none_alc
some_alc = data$some_alc
much_alc = data$much_alc
total    = c(none_alc,some_alc,much_alc)

\({MS}_{total} = \frac{{SS}_{{total}}}{{DF}_{{total}}} = s_{grand}^2\)

MS_total = var(total); MS_total
[1] 0.9410458

SS total

\({DF_{total}} = N-1\)

\({SS}_{{total}} = s_{grand}^2(N-1)\)

N = length(total)
DF_total = N - 1
SS_total = MS_total * DF_total; SS_total
[1] 55.5217
sum((total - mean(total))^2)
[1] 55.5217

SS total visual

MS within

\({MS}_{within} = \frac{{SS}_{{within}}}{{DF}_{{within}}} \\ {DF}_{within} = (n_i-1)n\)

n.i = 3  # Number of mesurements per individual (none, some, much)
n   = 20 # Number of mesurements per group

DF_within = (n.i - 1) * n
DF_within
[1] 40

SS within

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

var_pp = apply(cbind(none_alc, some_alc, much_alc),1,var)
ss_pp  = var_pp * (n.i - 1)

SS_within = sum(ss_pp); SS_within
[1] 48.45032
mean_pp = apply(cbind(none_alc, some_alc, much_alc),1,mean)

sum(c((none_alc - mean_pp)^2, 
      (some_alc - mean_pp)^2,
      (much_alc - mean_pp)^2))
[1] 48.45032

SS within data

SS within visual

MS between

\({MS}_{between} = \frac{{SS}_{{between}}}{{DF}_{{between}}}\)

\({DF}_{between}-{DF}_{{within}} \\ {SS}_{between} = {SS}_{total} - {SS}_{within}\)

SS_between = SS_total - SS_within
SS_between
[1] 7.071382
DF_between = DF_total - DF_within
DF_between
[1] 19

MS model

\({MS}_{model} = \frac{{SS}_{{model}}}{{DF}_{{model}}} \\ {DF}_{model} = k-1\)

k = 3
DF_model = k - 1
DF_model
[1] 2

SS model

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

# SS model
n_k1 = length(none_alc)
n_k2 = length(some_alc)
n_k3 = length(much_alc)

# Calculate sums of squares for the model
SS_k1 = n_k1 * (mean(none_alc) - mean(total))^2
SS_k2 = n_k2 * (mean(some_alc) - mean(total))^2
SS_k3 = n_k3 * (mean(much_alc) - mean(total))^2

SS_model = sum(SS_k1, SS_k2, SS_k3)
SS_model
[1] 38.63266

SS model visual

MS error

\(\frac{{SS}_{error}}{{DF}_{error}}\)

\({DF}_{error} = (n-1)(k-1)\)

DF_error = DF_within - DF_model
DF_error
[1] 38

SS error

\({SS}_{error} = {SS}_{within} - {SS}_{model}\)

SS_error = SS_within - SS_model
SS_error
[1] 9.817655

F ratio

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

# Calculate mean squares
MS_model = SS_model / DF_model
MS_error = SS_error / DF_error

# Calculate F statistic
F = MS_model / MS_error
F
[1] 74.76537

Visualize

library('visualize')
visualize.f(F, DF_model, DF_error, section="upper")

Contrast

Planned comparisons

  • Exploring differences of theoretical interest
  • Higher precision
  • Higher power

Post-Hoc

Unplanned comparisons

  • Exploring all possible differences
  • Adjust T value for inflated type 1 error

Effect size

General effect size measures

  • Amount of explained variance \(R^2\) also called eta squared \(\eta^2\).
  • Omega squared \(\omega^2\)

Effect sizes of contrasts or post-hoc comparisons

  • Cohen’s \(r\) gives the effect size for a specific comparison
    • \(r_{Contrast} = \sqrt{\frac{t^2}{t^2+{df}}}\)
    • \(r_{Contrast} = \sqrt{\frac{F(1,{df}_R)}{F(1,{df}_R)+{df}_R}}\)

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
  • 2 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
    • Gender
      • Males
      • Females

person gender 1_1 1_2 1_3 2_1 2_2 2_3 3_1 3_2 3_3
1 males 1
2 males 2
3 males 3
4 males 4
5 males 5
6 males 6
7 males 7
8 males 8
9 males 9
10 females 1
12 females 2
13 females 3
14 females 4
15 females 5
16 females 6
17 females 7
18 females 8
20 females 9

Data

End

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