Week 11 - Mixed ANOVA

Question 1

• Explain what types of factors are included in mixed/split-plot ANOVA (x2)

Answer 1

o WP - within participant

o BP - between participant

Question 2

• What types of research questions can be addressed with mixed ANOVA?

Answer 2

Those where manipulation of both IVs impossible/unethical, e.g. brain injury

Question 3

• Explain how mixed ANOVA is different from mixed model within-participants ANOVA 
(x1, x2)

Answer 3

• Mixed anova has a BP factor and a WP factor.
• Mixed model within-participants ANOVA is the normal way of doing WP ANOVA (where you evaluate sphericity and report an adjusted F, such as GG) – in contrast to MANOVA

Question 4

• Explain the main strengths of mixed ANOVA (x2) 


Answer 4

Can include factors in RM design that are unethical to manipulate
And exclude potential carry-over effects with BP factor_

Question 5

• Explain the difference between fixed and random factors in mixed ANOVA 


Answer 5

Both within-Ps and between Ps IVs are fixed factors (levels chosen by us)
While the Ps factor is still random (not pre-selected for particular combination of factors)

Question 6

• Explain what is meant by: One factor is nested under another factor (x1)

Answer 6

The (random) Ps factor is nested under/within levels of the between-Ps factor (independent group factor)

Question 7

• Explain what is meant by: One factor is crossed with another factor (x1)

Answer 7

The (random) Ps factor is crossed with levels of the within-Ps factor (RM factor)

Question 8

• In a 2-way mixed factorial ANOVA, indicate the sources of systematic and error variance for each effect (3 levels. x1, x3, x4)


Answer 8

SStotal divided into:

SSbetweenPs (independent groups factor)

• SSgroup (treatment)
• SSpswithingroups (error)

SSwithinPs (RM factor)

• SSblock (factor effect)
• SSbxg (interaction of block and group)
• SSbxpswithing (block by Ps within groups - error)

Question 9

Explain the diffs in error terms in between-groups, RM and Mixed ANOVA (x3)

Answer 9

o In between Ps ANOVA, we had one error term – deviations from cell means
o RM factor has three separate error terms, for each omnibus effect (TR x Ps interaction)
o While in mixed ANOVA, we have two error terms – one for between groups factor, and other for main effect of RM factor and the interaction

Question 10

What omnibus tests do you do in a 2-way Mixed ANOVA? (x3)

Answer 10

o Main effect of group
o Main effect of block
o Group x block interaction

Question 11

What 2 error terms are need for omnibus tests in 2-way mixed ANOVA? (x3, x3)

Answer 11

One for between-Ps factor (Ps within groups)
*Deviations of P’s mean from group mean - variability within the group
One for within-Ps factor and the 2-way interaction (interaction between block and participants within groups)
*Inconsistencies in effect of the within-subjects/RM/block factor across Ps, adjusted for between-group diffs in Ps

Question 12

• Explain how the F ratio is calculated in a 2-way mixed factorial design for: 

o The effect of the between-participants factor


Answer 12

Ratio of variability among group means divided by variability within groups
(the familiar ANOVA: MStreat/MSerror)

Question 13

• Explain how the F ratio is calculated in a 2-way mixed factorial design for: 

o The effect of the within-participants factor


Answer 13

Ratio of variability among means for within-Ps factor levels divided by variability in within-Ps factor effect

Question 14

• Explain how the F ratio is calculated in a 2-way mixed factorial design for: 

o The interaction of the between-participants factor and the within-participants factor

Answer 14

Ratio of variability among cell means for WPF levels within each group (adjusted for MEs) divided by variability in WP effect

Question 15

• Indicate the formulae for degrees of freedom for:
o Total effects
o All between-participants effects
o All within-participants effects

Answer 15

g = number of groups
b = number of levels of block/RM factor
n = number of Ps

df-total = gbn - 1

df-betweenPs = gn - 1

• df-group = g - 1
• df-Pswithing = df-betweenPs - df-group

df-withinPs = df-total - df-betweenPs
*df-block = b - 1
*df-bg (interaction) = (b - 1)(g - 1)
df-bxPswithing (error) = df-withinPs - df-block - df-bg

Question 16

• When following up the main effect of a between-participants factor in a mixed ANOVA:
o Explain what scores you are comparing


Answer 16

Means of group (main effect comparisons if 3+ levels)

Question 17

• When following up the main effect of a between-participants factor in a mixed ANOVA:
o Indicate the type of test you would use


Answer 17

Could use linear contrast,

But generally pair-wise comparisons (t-tests)

Question 18

• When following up the main effect of a between-participants factor in a mixed ANOVA:
o Indicate which error term you would use

Answer 18

Original error term from between-Ps main effect test

*MS-Pswithing

Question 19

• When following up the main effect of a within-participants factor in a mixed ANOVA:
o Explain what scores you are comparing


Answer 19

Block means

Question 20

• When following up the main effect of a within-participants factor in a mixed ANOVA:
o Indicate the type of test you would use


Answer 20

Could use linear contrast,

But generally run a 1-way within-Ps ANOVA on each pair of comparison blocks

Question 21

• When following up the main effect of a within-participants factor in a mixed ANOVA:
o Indicate which error term you would use

Answer 21

We have to get the error term for each comparison based upon only the data involved in that comparison:
SS-BcomparisonxPs

Question 22

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the within-participants factor:
o Explain what scores you are comparing

Answer 22

The cell means of block, at each level of group (are the simple effects of block)

Question 23

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the within-participants factor:
o Indicate the type of test you would use

Answer 23

Always run a 1-way within-Ps ANOVA on block, separately for each group

Question 24

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the within-participants factor:
o Indicate which error term you would use

Answer 24

Individual ANOVAs on block for each group creates a new error term that only considers the data involved in the simple effect we’re analysing
o Which is best because pooled error might overestimate error, even when df are adjusted

Question 25

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the between-participants factor:
o Explain what scores you are comparing


Answer 25

The cell means of group, at each level of block (are the simple effects of group)

Question 26

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the between-participants factor:
o List the two approaches that you could use


Answer 26

Run four 1-way between-participants anovas to compare groups at each of the four blocks (ie using separate errors) OR
Use same (new, special, pooled) error for all - averages error variance across all cells

Question 27

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the between-participants factor:
o Explain the strengths of the two approaches (x2, x2)

Answer 27

Separate 1-way ANOVAs is more precise - not homogenising the variance in groups
But using single pooled error tests all simple effects with same power, and applies logic familiar for between-groups ANOVA

Question 28

• When following up a significant 2-way interaction in a mixed ANOVA with the simple effects of the between-participants factor:
o Explain the weaknesses of the two approaches

Answer 28

Separate is damn time consuming!

Pooled error can conceal (often seen) heterogeneity of variances across the RM factor

Question 29

• When following up a significant simple effect of a between-participants factor with simple comparisons:
o Explain what scores you are comparing

Answer 29

If any of the simple effects of group at each level of block was significant, need to compare the block levels

Question 30

• When following up a significant simple effect of a between-participants factor with simple comparisons:
o Indicate the type of test you would use

Answer 30

Linear contrasts

Question 31

• When following up a significant simple effect of a between-participants factor with simple comparisons:
o Indicate which error term you would use

Answer 31

The same as the original 1-way ANOVA simple effect test (that we did for each block):
MS-Pwithing(@ block#)

Question 32

What are assumptions re DV and between-Ps factor in Mixed ANOVA? (x1, x1)

Answer 32

DV normally distributed

Homogeneity of variance (as with all between-Ps designs)

Question 33

What are assumptions re within-Ps factor in Mixed ANOVA? (x1, x1)

Answer 33

Homogeneity of variance: assume WPFxP interactions constant at all levels of between participant factor
o Interaction of Ps and within participants factor is consistent… Group factor doesn’t change the error term for the RM factor
Variance-covariance matrix same at all levels of WPF
Pooled (average) variance-covariance matrix exhibits compound symmetry (c.f. sphericity)
• Usual epsilon adjustments apply when within-Ps assumptions violated

Question 34

• In a two by three mixed ANOVA in which gender (male; female) serves as a between-participants variable and time of test (start of semester, mid-semester, end of semester) serves as a repeated measures variable, participant is crossed with ______ and nested within _______ .

Answer 34

Time of test

Gender