24 oct 2018

Inhoud

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

  1. Field: 14.2.3 p 546, Output 14.2 p 560
  2. Field: Output 14.4 p 562
  3. Field: Jane Superbrain 14.2 p 548, Output 14.2 p 560. GG and HF.
  4. Field: Jane Superbrain 14.2 p 548. Sample size \(n\) is larger than \(a\) (number of levels) + 10
  5. Field: 14.2.5 p 548, Output 14.2 p 560
  6. Field: 14.2.5 p 548, Output 14.4 p 562
  7. Field: 14.2.5 p 548, Output 14.4 p 562

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

# Set offset
offset = .2

# Calculate n
n = length(none_alc)

# Create plot
plot(none_alc,
     xlab = 'participants',
     ylab = 'Brokken',
     xlim = c(.5,22),
     ylim = c(3,7.5),
     col='green')
points((1:n)-offset, some_alc, col='red')
points((1:n)+offset, much_alc,   col='blue')

# Add the total mean
lines(c((1-offset),(n+offset)),rep(mean(total),2),col='black',lwd=2)

segments(1:n,        mean(total), 1:n,        none_alc)
segments(1:n-offset, mean(total), 1:n-offset, some_alc)
segments(1:n+offset, mean(total), 1:n+offset, much_alc)

text(n+offset,mean(total),expression(bar(X)[grand]),pos=4)