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: 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

# 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)

p <- recordPlot()
p

# add grid
segments(1:21-.5, 3, 1:21-.5, 7.5, col='grey')

p <- recordPlot()

\({MS}_{within}\)

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

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

p
# Personal means
segments(1:n-offset, mean_pp, 1:n+offset, mean_pp, col='orange', lwd=3)
# The lines show the within deviation from the personal mean.
segments(1:n-offset, mean_pp, 1:n-offset, some_alc, col='orange', lwd=3)
segments(1:n, mean_pp,        1:n,        none_alc,   col='orange', lwd=3)
segments(1:n+offset, mean_pp, 1:n+offset, much_alc,   col='orange', lwd=3)

# Add personal means symbols
for(i in 1:n) { text(i-(offset/2), mean_pp[i], expression(bar(X)[i]), pos=2, cex=.5)}

p <- recordPlot()

\({MS}_{between}\)

\[{MS}_{between} = \frac{{SS}_{{between}}}{{DF}_{{between}}}\]

\[{DF}_{between}-{DF}_{{within}} \\ {SS}_{between} = {SS}_{total} - {SS}_{within}\]

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

p
# Add the no alcohol mean
lines(c((1),(n)),rep(mean(none_alc),2),col='green',lwd=2)
text(n+offset,mean(none_alc),expression(bar(X)[none]),pos=4)

# With the bit alcohol mean
lines(c((1-offset),(n-offset)),rep(mean(some_alc),2),col='red',lwd=2)
text(n+offset,mean(some_alc),expression(bar(X)[some]),pos=4)

# With the much alcohol mean
lines(c((1+offset),(n+offset)),rep(mean(much_alc),2),col='blue',lwd=2)
text(n+offset,mean(much_alc),expression(bar(X)[much]),pos=4)

# The lines show the model deviation from the total mean.
segments(1:n,        mean(total), 1:n,        mean(none_alc),   col='green')
segments(1:n-offset, mean(total), 1:n-offset, mean(some_alc), col='red')
segments(1:n+offset, mean(total), 1:n+offset, mean(much_alc),   col='blue')

p <- recordPlot()

\({MS}_{error}\)

\[\frac{{SS}_{error}}{{DF}_{error}} \\ {DF}_{error} = (n-1)(k-1)\]

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")

F.values = seq(0, 100, .01)

plot(F.values, df(F.values, DF_model, DF_error), type = "l", ylab="density", main = "H0 and HA F-distribution")

critical.value = qf(.95, DF_model, DF_error)

critical.range = seq(critical.value, 100, .01)

polygon(c(critical.range,rev(critical.range)), 
        c(critical.range*0, rev(df(critical.range, DF_model, DF_error, ncp = 5))), col = "darkorange")

lines(F.values, df(F.values, DF_model, DF_error, ncp = 5))

polygon(c(critical.range,rev(critical.range)), 
        c(critical.range*0, rev(df(critical.range, DF_model, DF_error))), col = rgb(0,1,0,1), density = 40, angle = -45)

p.range = seq(F, 100, .01)

polygon(c(p.range,rev(p.range)), 
        c(p.range*0, rev(df(p.range, DF_model, DF_error))), col = rgb(0,0,1,.6), density = 20)

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