F-Distribution and factorial ANOVA

Klinkenberg

University of Amsterdam

2024-09-30

F-distribution

Ronald Fisher

The F-distribution, also known as Snedecor’s F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is, in probability theory and statistics, a continuous probability distribution. The F-distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see F-test.

Wikipedia

Sir Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962), known as R.A. Fisher, was an English statistician, evolutionary biologist, mathematician, geneticist, and eugenicist. Fisher is known as one of the three principal founders of population genetics, creating a mathematical and statistical basis for biology and uniting natural selection with Mendelian genetics.

Wikipedia

Ronald Fisher

Analysing variance

Decomposing variance example of height for males and females.

Population distribution

layout(matrix(c(2:6,1,1,7:8,1,1,9:13), 4, 4))

n  = 56    # Sample size
df = n - 1 # Degrees of freedom

mu    = 120
sigma = 15

IQ = seq(mu-45, mu+45, 1)

par(mar=c(4,2,0,0))  
plot(IQ, dnorm(IQ, mean = mu, sd = sigma), type='l', col="red")

n.samples = 12

for(i in 1:n.samples) {
  
  par(mar=c(2,2,0,0))  
  hist(rnorm(n, mu, sigma), main="", cex.axis=.5, col="red")
  
}layout(matrix(c(2:6,1,1,7:8,1,1,9:13), 4, 4))

n  = 56    # Sample size
df = n - 1 # Degrees of freedom

mu    = 120
sigma = 15

IQ = seq(mu-45, mu+45, 1)

par(mar=c(4,2,0,0))  
plot(IQ, dnorm(IQ, mean = mu, sd = sigma), type='l', col="red")

n.samples = 12

for(i in 1:n.samples) {
  
  par(mar=c(2,2,0,0))  
  hist(rnorm(n, mu, sigma), main="", cex.axis=.5, col="red")
  
}layout(matrix(c(2:6,1,1,7:8,1,1,9:13), 4, 4))

n  = 56    # Sample size
df = n - 1 # Degrees of freedom

mu    = 120
sigma = 15

IQ = seq(mu-45, mu+45, 1)

par(mar=c(4,2,0,0))  
plot(IQ, dnorm(IQ, mean = mu, sd = sigma), type='l', col="red")

n.samples = 12

for(i in 1:n.samples) {
  
  par(mar=c(2,2,0,0))  
  hist(rnorm(n, mu, sigma), main="", cex.axis=.5, col="red")
  
}

Population distribution

F-statistic

F=MSmodelMSerror=between group var.within group var.=ExplainedUnexplained=SIGNALNOISE

The F-statistic represents a signal to noise ratio by defiding the model variance component by the error variance component.

A samples

Let’s take two sample from our normal population and calculate the F-value.

x.1 = rnorm(n, mu, sigma)
x.2 = rnorm(n, mu, sigma)

F=MSmodelMSerror=SIGNALNOISE=522.11296.77=1.76

More samples

let’s take more samples and calculate the F-value every time.

n.samples = 1000

f.values = vector()

for(i in 1:n.samples) {
  
  x.1 = rnorm(n, mu, sigma); x.1
  x.2 = rnorm(n, mu, sigma); x.2

  data <- data.frame(group = rep(c("s1", "s2"), each=n), score = c(x.1,x.2))
    
  f.values[i] = summary(aov(lm(score ~ group, data)))[[1]]$F[1]

}

k = 2
N = 2*n

df.model = k - 1
df.error = N - k

hist(f.values, freq = FALSE, main="F-values", breaks=100)
F = seq(0, 6, .01)
lines(F, df(F,df.model, df.error), col = "red")n.samples = 1000

f.values = vector()

for(i in 1:n.samples) {
  
  x.1 = rnorm(n, mu, sigma); x.1
  x.2 = rnorm(n, mu, sigma); x.2

  data <- data.frame(group = rep(c("s1", "s2"), each=n), score = c(x.1,x.2))
    
  f.values[i] = summary(aov(lm(score ~ group, data)))[[1]]$F[1]

}

k = 2
N = 2*n

df.model = k - 1
df.error = N - k

hist(f.values, freq = FALSE, main="F-values", breaks=100)
F = seq(0, 6, .01)
lines(F, df(F,df.model, df.error), col = "red")

More samples

F-distribution

So if the population is normally distributed (assumption of normality) the f-distribution represents the signal to noise ratio given a certain number of samples (dfmodel=k−1) and sample size (dferror=N−k).

The F-distibution therefore is different for different sample sizes and number of groups.

(d1x)d1d2d2(d1x+d2)d1+d2xB(d12,d22)

Warning

Formula not exam material

F-distribution

F-distribution

Animated F-distrigutions

Independent factorial ANOVA

Two or more independent variables with two or more categories. One dependent variable.

Independent factorial ANOVA

The independent factorial ANOVA analyses the variance of multiple independent variables (Factors) with two or more categories.

Effects and interactions/moderation:

  • 1 dependent/outcome variable
  • 2 or more independent/predictor variables
    • 2 or more cat./levels

Flow chart

Flow chart statistical test selection

Assumptions

  • Continuous variable
  • Random sample
  • Normaly distributed
    • Shapiro-Wilk test
  • Equal variance within groups
    • Levene’s test

Example

In this example we will look at the amount of accidents in a car driving simulator while subjects where given varying doses of speed and alcohol.

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

person alcohol speed accidents
1 1 1 0
2 1 2 2
3 1 3 4
4 2 1 6
5 2 2 8
6 2 3 10
7 3 1 12
8 3 2 14
9 3 3 16

Data

 

Effects

  • Total
    • F=MSmodelMSerror
  • Main effects
    • F=MSgoupAMSerror
    • F=MSgoupBMSerror
  • Interaction/moderation
    • F=MSA×BMSerror

MS=Mean Squares

MS=SSdf

SS=Sums of Squares

df=degrees of freedom

SS model

SSmodel=494.22048

SS error

SSerror=66.34642

SS A Speed

SSspeed=128.1639233

SS B Alcohol

SSalcohol=364.14583

SS AB Alcohol x Speed

Variance Sum of squares df Mean squares F-ratio
AB SSA×B=SSmodel−SSA−SSB dfA×dfB SSABdfAB MSABMSerror

SSspeed×alcohol=1.9107267

Mean Squares

Mean squares for:

  • Speed
  • Alcohol
  • Speed × Alcohol

FSpeed=MSSpeedMSerrorFAlcohol=MSAlcoholMSerrorFAlcohol×Speed=MSAlcohol×SpeedMSerror

Interaction

FAlcohol×Speed=MSAlcohol×SpeedMSerror=0.480.39=1.23

P-value

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 R2 also called eta squared η2.

Effect sizes of post-hoc comparisons

  • Cohen’s r gives the effect size for a specific comparison
    • rContrast=t2t2+df

End

Contact

  • Klinkenberg
  • ln.AvU@grebneknilK.S
  • ShKlinkenberg

CC BY-NC-SA 4.0

CC BY-NC-SA 4.0

Statistical Reasoning 2024-2025

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F-Distribution and factorial ANOVA Klinkenberg University of Amsterdam 2024-09-30

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  • F-Distribution and factorial ANOVA
  • F-distribution
  • Ronald Fisher
  • Analysing variance
  • Population distribution
  • Population distribution
  • F-statistic
  • A samples
  • More samples
  • More samples
  • F-distribution
  • F-distribution
  • F-distribution
  • Independent factorial ANOVA
  • Independent factorial ANOVA
  • Flow chart
  • Assumptions
  • Example
  • person alcohol speed...
  • Data
  • Effects
  • SS model
  • SS error
  • SS A Speed
  • SS B Alcohol
  • SS AB Alcohol x Speed
  • Mean Squares
  • Interaction
  • P-value
  • Slide 30
  • Post-Hoc
  • Effect size
  • End
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