Lecture 13ANOVAs continued
We were playing with this example last week...
Placebo 1/2 Full
3 7 5
1 5 6
2 6 4
M M M
1 . 2 . = 3 . =
= 2 6 5
- The grand mean = M... = X / N = 39 / 9 = 4.33
Source of Variability SS df MS F
Between the Treatment 26.19 2 13.1 13.1
Groups
Within the 6 6 1
Treatment
Groups
Total 32.19 8
- F-crit (2,6; = .05) = 5.14
Some ANOVAphilosophy
- MS within groups is a measure of variability attributable to random error and fluctuations
- Let’s use to symbolize “random error”
- Since MS between groups was generated from the same people responsible for the random error,
some of MS between is also comprised of random error
- But some of MS between can also be explained by the effect of our treatment
- Let’s use to symbolize “the effect of treatment”
- F = ( + ) /
- What happens if there is no effect of treatment?
- In other words, what happens if = 0?
- F = ( + 0) /
• /
• 1
Theoretically, this is the lowest possible value for an F score
Some ANOVAphilosophy
- Consider the followingANOVAconducted on a single factor with only two levels, A& B:
- A= {1, 2, 3}, B = {4, 5, 6}
Source of Variability SS df MS F
Between the Treatment 13.5 1 13.5 13.5
Groups
Within the 4 4 1
Treatment
Groups
Total 17.5 5 Some ANOVAphilosophy
- What if we did a t-test on these data?
A B SS A = 2, dfA = 2
1 4 SS B df B
2 5 = 2, = 2
3 6 2
MA= 2 MB = 5 SP = 1
S( M 1 - M 2 ) = .8165
tobserved
= - 3.67
- Just for fun, square the t-observed
- t² = 3.67² = 13.5 = F
Some ANOVAphilosophy
- t² = F
- t (∞) = Z
- So all of the tests are related to one another:
• To fully understandANOVA, you have to understand t
• To fully understand t, you have to understand z
• To fully understand z, you have to understand variance, SS, and p-values
Afew more things: proportions of variance accounted for
- This is actually much less complicated than it was for the t-test where r² = t² / (t² + df)
- η² = explained variability / total variability
- What is our measure of explained variability?
- What is our measure of total variability?
Afew more things: proportions of variance accounted for
- η² = SS between / SS total
- calculate η² for our example:
Source of Variability SS df MS F
Between the Treatment 26.19 2 13.1 13.1
Groups
Within the 6 6 1
Treatment
Groups
Total 32.19 8
- 26.19 / 32.19 = .81 Afew more things: post hoc tests
- Asuccessful ANOVAlets you reject H 0 and conclude “at least two of the factor level means
differ from one another”
- But which ones differ?
- If your ANOVAonly has 2 factor levels, you don’t have to worry about this.
- Why?
Post hoc test: error
- Basically, you do a series of t-tests (or

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