Week 4 - ANOVA III

Question 1

• Explain what higher-order designs are (x1)

Answer 1

• More than 2 independent variables (factors)

Question 2

• Explain what research questions higher order designs can address (x2)

Answer 2

How do the factor influence the DV,

Independently and/or interactively

Question 3

• Explain why higher-order designs are useful 
(x1)

Answer 3

• Allow for designs with higher external validity

Question 4

• Indicate the general (x1) and specific (x1)notations for a factorial design with Factor A (with 2 groups/levels), Factor B (with 4 groups/levels), and Factor C (with 3 groups/levels) 


Answer 4

3-way factorial ANOVA

2 (a1, a2) x 4 (b1, b2, b3 b,4) x 3 (c1, c2, c3)

Question 5

• Identify the different omnibus effects that may be found in 3-way factorial ANOVA, and explain what research questions are addressed with each 
(x2, x3, x3)

Answer 5

Main effects x3
*Are the marginal means for that factor different (averaged over levels of other factors)?
Two-way interaction x3
*Does effect of one factor change depending on level of another (average over levels of a third)?
*Are the simple effect the same?
Three-way interaction x1
*Does a 2-way interaction hold at each level of third variable?
*Starts comparing unique cell means

Question 6

• Explain how three-way interactions are graphed 
(x3)

Answer 6

Two graphs - one each for least important variable/second moderator
*Plot 2-way interactions at each level of 3rd factor
Focal on the x-axis
First moderator as line variable

Question 7

• Explain how variance is partitioned in a 3-way ANOVA (x4, x2, x4, x2)


Answer 7

Main effects variance:
   *Due to alpha
   *Due to beta
   *Due to gamma
3-way interaction:
   *Due to alpha-beta-gamma
2-way interaction:
   *Due to alpha-beta
   *Due to beta-gamma
   *Due to alpha-gamma
Error/residual:
   *Due to e

Question 8

• Explain the structural model in 3-way factorial ANOVA 


Answer 8

Xijkl = μ. + αj + βk  + γl + αβjk  + βγkl + αγjl + αβγjkl + eijklo	Xijkl - score for person i, in j/k/l conditions
μ. - grand mean
αj - effect of being in A
βk - effect of being in B
γl - effect of being in C
αβjk - interaction of AxB
βγkl - interaction of BxC
αγjl - interaction of AxC
αβγjkl - interaction of AxBxC
eijkl - error associated with being in that level of each factor

Question 9

• Identify the formulae for total (x1) and factor (x4) degrees of freedom in 3-way factorial ANOVA 


Answer 9

dftotal = N - 1 
df factor always (# levels - 1):
   *dfj = j - 1
   *dfk = k - 1
   *dfl = l - 1

Question 10

• Identify the formulae for interaction degrees of freedom in 3-way factorial ANOVA 
(x5)

Answer 10

df interaction always a product of df for factors involved:

• dfjk = (j - 1)(k - 1)
• dfjl = (j - 1)(l - 1)
• dfkl = (k - 1)(l - 1)
• dfjkl = (j - 1)(k - 1)(l - 1)

Question 11

• Identify the formulae for error degrees of freedom in 3-way factorial ANOVA 
(x2)

Answer 11

df error always (N - #cells) or ([n - 1][# of cells])

dferror = N - jkl

Question 12

• In the omnibus summary table for 3-way factorial ANOVA, explain what the SS and MS values represent (x1)

Answer 12

The variance, and indexed (weighted by df) variance in each factor, 2-way interaction, and the 3-way interaction

Question 13

• In the omnibus summary table for 3-way factorial ANOVA, explain how F is calculated for each effect 
(x1)

Answer 13

MS for each effect/interaction is divided by the pooled error term (MSerror)

Question 14

• In a 3-way factorial ANOVA, explain what each of the following tell you: 

o A significant F test for the main effect of Factor A (x2)


Answer 14

That there is differences among the marginal means (from the grand mean) of Factor A,
Collapsed across levels of other 2 factors

Question 15

• In a 3-way factorial ANOVA, explain what each of the following tell you: 

o A significant F test for the A x B 2-way interaction
 (x2)

Answer 15

Effect of at least one factor is different at different levels of the other factor
• Ignoring (averaging across) the third factor

Question 16

• In a 3-way factorial ANOVA, explain what each of the following tell you: 

o A significant F test for the A x B x C 3-way interaction

Answer 16

That a 2-way interaction is different at different levels of the third factor
o Simple interaction of AxB for C1 is different from simple interaction for C2, etc..

Question 17

• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o A main effect (x4)

Answer 17

When it is significant
And has >2 levels,
Do main effect comparisons/t-tests/contrasts
(same as 2-way ANOVA)

Question 18

• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o An omnibus two-way interaction (x4)

Answer 18

If significant,
Then test simple effects (with the F test),
(same as 2-way ANOVA)
• Simple effect tests of focal IV, even though you’re collapsing across/ignoring third factor

Question 19

• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o An omnibus three-way interaction (x1)

Answer 19

If it is significant

Question 20

• In a 3-way factorial ANOVA, identify the steps in following up an omnibus main effect (x2)

Answer 20

Main effect comparisons

*protected/t-tests or linear contrasts, exactly as in 2-way ANOVA

Question 21

• In a 3-way factorial ANOVA, identify the steps in following up an omnibus 2-way interaction 


Answer 21

Test simple effects (effect of focal level, at levels of B, across levels of C
*F test
If significant simple effect for a factor with > 2 levels, follow up with simple comparisons (effect of A at B1 [averaged across C])
*t-tests or linear contrasts,

Question 22

• In a 3-way factorial ANOVA, identify the steps in following up an omnibus 3-way interaction 
(x3, plus define each, and with tests for each)

Answer 22

Simple interaction effects (interactions within levels of a moderator)
o F tests
If simple interaction effects are significant, follow up with simple simple effects (effects of A, at different levels of B, at different levels C)
o F tests
If simple simple effects are significant with > 2 levels, follow up with simple simple comparisons (effects of A at particular levels of B, and of C)
o t-tests and linear contrasts

Question 23

• In a 3-way factorial ANOVA, explain what simple interaction effects are, including: 

o What they test (x3)

Answer 23

A two way interaction within a level of a moderator
• An AxB interaction for C1
• In contrast to a simple interaction at the other level of the moderator

Question 24

• In a 3-way factorial ANOVA, explain what simple interaction effects are, including: 

o Why we need them
 (x4)

Answer 24

Because a 3-way interaction doesn’t tell us AxB is different across levels of C,
If BxC is different across A,
Or if AxC is different across
They break down the 3-way interaction into a series of 2-way interactions at each level of the third factor

Question 25

• In a 3-way factorial ANOVA, explain what simple interaction effects are, including: 

o How they are tested
 (x1)

Answer 25

F test

Question 26

• In a 3-way factorial ANOVA, explain what simple interaction effects are, including: 

o How they are different from omnibus 2-way interactions in a 3-way factorial design (x2)

Answer 26

Omnibus: ignores levels of the third factor
Simple: test the 2-way interaction at each level of the third factor

Question 27

• In a 3-way factorial ANOVA, explain what simple interaction effects are, including: 

o How they are different from omnibus 2-way interactions in a 2-way factorial design (x3)

Answer 27

In 2-way designs, we’d run tests of the 3 factor separately,
So 2 different error terms
Might get same pattern of means, but different p-values

Question 28

• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o What question they answer

Answer 28

Whether there’s diffs in the effect of factor A at each level of factor B, at each level of factor C (i.e., within each combo of B & C)

Question 29

• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o How they differ from simple effects


Answer 29

Simple effects average across levels of 3rd factor

Simple simple effects tests diffs in cell means AT each level of 3rd factor

Question 30

• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o What error term is used (x2)

Answer 30

Pooled MSerror from the omnibus ANOVA table

rather than from separate 1-way experiments

Question 31

• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o how to determine how many simple simple effects to test
 (x3)

Answer 31

Focal IV still on the x-axis
So, theoretically, do as many as there are combinations of levels of other 2 factors
eg. the simple effects of A, at each level of B, separately for each level of C

Question 32

• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o What a significant F test for a simple simple effect means (e.g. significant simple simple effect of Factor A at first level of Factor B at first level of Factor C)

Answer 32

That the cell means differ

Question 33

• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o What question they answer (x2)

Answer 33

Like simple comparisons in 2-way:

Is there a difference in cell means of Factor A, at levels of factor B, at particular level of C?

Question 34

• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o How they differ from simple comparisons
 (x2)

Answer 34

They need to be computed at each level of a third factor

Which simple comparisons don’t have

Question 35

• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o What error term is used


Answer 35

MSerror, in linear contrasts (t-tests) perhaps with Bonferroni adjustment

Question 36

• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o How to determine how many simple simple comparisons to test


Answer 36

Exhaustive set explodes family-wise error (e.g., 7 omnibus tests, plus vast arrays of possible simple interactions, simple simple effects and follow up comparisons)
So do what hypotheses demand

Question 37

• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o What a significant t test for a simple simple comparison means (e.g., significant simple simple comparison of Factor A at first level of Factor B at first level of Factor C)

Answer 37

a