Week 10 - RM ANOVA

Question 1

• Explain the 2 sets of limitations associated with between-participants designs

Answer 1

Ps can only serve in one cell, as otherwise violate independence of scores in factorial ANOVA
Any diffs between Ps is viewed as error

Question 2

• Explain how within-participants designs can overcome the 2 limitations of between-Ps designs 
(x2)

Answer 2

Calculate and remove (from treatment and error) any variance due to dependence/individual difference (e.g., maybe more variance in the group, than between them)
o Thus, we can reduce our error term and increase POWER

Question 3

• Describe the sources of systematic variance in within-participants designs 
(x3)

Answer 3

No within-cell variance, as only 1 data point in each, so:
Between-Ps variance -individual difference (partitioned out of error first)
Within-Ps variance - between-treatment (effect) and error

Question 4

• Describe the sources of error/unexplained/residual variance in within-participants designs 
(x2)

Answer 4

It’s the interaction of participant and treatment

ie, inconsistencies of treatment effect/factor between Ps at each level of treatment

Question 5

• Compare the sources of error variance in within-participants ANOVA and between-participants ANOV A 


Answer 5

In between:
o Total Variance = Between group (treatment effect) + within group (error)
In within:
o Total Variance = Between Ps/group (individual differences) + within Ps/group (treatment and error)

Question 6

• Explain how between-participants variance and within-participants variance are used in within-participants ANOVA 


Answer 6

Between Ps is individual diffs - so is removed from both treatment and error???
Within-Ps diffs

Question 7

• State the formula for the calculated F ratio in 1-way within-participants ANOVA

Answer 7

Treatment divided by

Treatment x Ps interaction

Question 8

• Explain how the formula for the calculated F ratio in 1-way within-participants ANOVA is similar (x1) and different (x1) to the formula for calculated F in 
1-way between-participants ANOVA


Answer 8

Still equates to MStreat/MSerror

Just had individual diffs removed

Question 9

• Identify the structural model for within-participants ANOVA 
(x1)
And define components (x5)

Answer 9

Xij = μ + πi + τj + eij
For i cases and j treatments:
Xij, any DV score is a combination of:
o μ - the grand mean,
o πi - variation due to the i-th person (μi - μ) (think p for pi and Ps)
o τj - variation due to the j-th treatment (μj - μ)
o eij - error - variation associated with the i-th cases in the j-th treatment – error = πτij (interaction, plus chance)

Question 10

For RM ANOVA, define total variability (x1)

Answer 10

Deviation of each observation from grand mean

Question 11

For RM ANOVA, define variability due to factor (x1)

Answer 11

Deviation of factor group means from grand mean (μj - μ)

Question 12

For RM ANOVA, define variability due to Ps (x2)

Answer 12

Deviation of each participant’s mean from the grand mean (μi - μ)

Question 13

For RM ANOVA, define error (x3)

Answer 13

Changes (inconsistencies) in effect of factor across participants
Estimated variance due to individual difference averaged over treatment levels
(TR x P interaction)

Question 14

• List the formulae for various degrees of freedom in 1-way within-participants ANOVA 


Answer 14

n = number of Ps
N = number of observations
j = number of conditions

dftotal = nj-1 = N-1 
dfP = n-1 
dftr = j–1 
dferror = (n-1)(j-1) 
   o	Error df is different from between-participants anova – because is now interaction of participant factor x treatment factor

Question 15

• When looking at the summary table for the omnibus test in 1-way within-participants ANOVA, 
explain which parts of the output you need to report (x2), and which parts you can ignore (x1) 


Answer 15

Report df and F for treatment and error
Ignore output for between-subjects factor (estimated variance due to individual difference averaged over treatment levels)

Question 16

• Explain the similarities in following up a significant main effect in 1-way between-participants ANOVA and 1-way within-participants ANOVA 
(x1)

Answer 16

If more than 2 levels, both need follow up comparisons

Question 17

• Explain the differences in following up a significant main effect in 1-way between-participants ANOVA and 1-way within-participants ANOVA 


Answer 17

Between-Ps ANOVA uses pooled error term from omnibus tests (due to assumed homogeneity of variance)
Within-Ps partitions out and ignores main Ps effect - computes an error term estimating inconsistency as participants change over WS levels
• We expect inconsistency in TR effect x Ps, so in simple comparisons use only data for conditions involved in comparison & calculate separate error terms each time

Question 18

• Explain how systematic/treatment variance and error/residual variance is partitioned in
 2-way within-participants ANOVA. (x10)

Answer 18

Between Ps variance
Within-Ps variance, made up of:
Between-treatment variance
   *Main effect A
   *Main effect B
   *Interaction A x B
Residuals:
   *A x P interaction
   *B x P interaction
   *A x B x P interaction

Question 19

• Explain the similarities/diffs in how systematic/treatment variance and error/residual variance is partitioned in
2-way within-participants ANOVA and 2-way between

Answer 19

In between, partitioned into treatment factors, interaction, and error
In within, still have factor and interaction, but each has own corresponding error term

Question 20

• Explain how error terms are calculated in omnibus 2-way within-participants ANOVA tests 


Answer 20

Corresponds to an interaction between the effect due to participants, and the treatment effect
• Main effect of A - error term is MSAxP (SSA/dfA x P)
• Main effect of B - error term is MSBxP (SSB/dfB x P)
• AxB interaction - error term is MSABxP (SSAB/dfAB x P)

Question 21

• Explain how error terms are calculated in 2-way within-participants ANOVA follow-up tests 


Answer 21

Simple effects error term is MSA at B1xP
o The interaction between the A treatment and participants, at B1

Simple comparisons error term is MSACOMP at B1xP
o Interaction between the A treatment (only the data contributing to the comparison, ACOMP), and participants, at B1

Question 22

• Define fixed factors

Answer 22

You choose the levels of the IV

Eg, treatment is fixed

Question 23

• Define random factors 


Answer 23

Levels of IV chosen randomly

eg Ps - randomly assigned to conditions

Question 24

• List the 3 assumptions of the mixed-model approach 


Answer 24

Not dissimilar to between-participants assumptions:
Sample is randomly drawn from population
DV scores are normally distributed in the population
Compound symmetry

Question 25

What are the two assumptions that make up compound symmetry

Answer 25

o Homogeneity of variances in levels of repeated-measures factor
o Homogeneity of covariances
(equal correlations/covariances between pairs of levels)

Question 26

• For Mauchley’s test of sphericity: 


o Explain what question it tests
 (x3)

Answer 26

Compound symmetry assumption very restrictive.
Often violated, so instead asks:
Are the main diagonal (variances) and off-diagonal (covariances) roughly equal

Question 27

• For Mauchley’s test of sphericity: 

o Identify the statistic it uses
 (x1)
o explain what a significant result means (x1)

Answer 27

Chi-square

That sphericity assumption is violated

Question 28

• For Mauchley’s test of sphericity: 


o Indicate whether it is a robust test or not 
(x2)

Answer 28

No - often says everything is fine when sphericity violations present in data

Question 29

• Explain when violations of sphericity matter (x1)

Answer 29

In all within-participants designs/factors with 3+ levels

Question 30

• Explain when violations of sphericity do not matter (x2)

Answer 30

In between-participants designs, because treatments are unrelated (different participants in different treatments)
• The assumption of homogeneity of variance still matters though
When within-participant factors have 2 levels, because only one estimate of covariance can be computed

Question 31

• Explain why the sphericity assumption is important – what implications does it have for research? 


Answer 31

When violated, F-ratios are positively biased
• Critical values of F [based on df a – 1, (a – 1)(n – 1)] are too small
Therefore, probability of type-1 error increases
• So we need to adjust critical values of F

Question 32

• Explain what epsilon adjustments are (x1)

Answer 32

A value by which the degrees of freedom for the test of F-ratio is multiplied (0- 1)

Question 33

• Explain what epsilon adjustments do (x3)

Answer 33

Correct violations of sphericity

Equal to 1 when sphericity assumption is met (no adjustment), and

Question 34

• Explain why epsilon adjustments are important/useful 


Answer 34

Further epsilon is from 1, the more problem you have with sphericity violation for that tests, and more diff it makes to critical F

Question 35

• List and explain the 3 types of epsilon (x3, x4, x4)

And indicate which one is: rather liberal/lax, rather conservative, and just right 


Answer 35

Lower-bound epsilon
o Used for conditions of maximal heterogeneity, or worst-case violation of sphericity
o Often too conservative/increases Type 2

Greenhouse-Geisser epsilon
o Size of ε depends on degree of sphericity violation
o 1 ≥ ε ≥ 1/(k-1) : varies between 1 (sphericity intact) and lower-bound epsilon (worst-case violation)
o Generally recommended – not too stringent, not too lax

Huynh-Feldt epsilon
o An adjustment applied to the GG-epsilon
o Often results in epsilon exceeding 1, in which case reset to 1
o Used when “true value” of epsilon is believed to be ≥ .75 - too liberal

Question 36

• List the advantages of within-participants/repeated-measures designs 


Answer 36

More efficient
• n Ps in j treatments generate nj data points
• Simplifies procedure
More sensitive
• Estimate individual differences (SSparticipants) and remove from error term

Question 37

• List the disadvantages of within-participants/repeated-measures designs 


Answer 37

Restrictive statistical assumptions
Sequencing effects:
• Learning, practice – improved later regardless of manipulation
• Fatigue – deteriorating later regardless of manipulation
• Habituation – insensitivity to later manipulations
• Sensitisation – become more responsive to later manipulations
• Contrast – previous treatment sets standard to which react
• Adaptation – adjustment to previous manipulations changes reaction to later
• Direct carry-over – learn something in previous that alters later
• Etc!

Counterbalancing helps, but can still get treatment x order interactions

Question 38

• Explain the methodology that can be employed to reduce one of the disadvantages of within-participants designs (x4)

Answer 38

MANOVA - gets around restrictive assumptions (spec. sphericity/compound symmetry):
Weighting DV for each level of RM IV with coefficients
(as with scores for each IV in multiple regression),
To create a predicted DV score that maximises differences across the levels of the IV

Question 39

What are degrees of freedom in RM ANOVA?

Answer 39

Factor: number of levels -1
Factor error: (dfFactor) x (number of Ps -1)
Interaction: dfA x dfB
Interaction error: dfA x dfB x (number of Ps - 1)

Question 40

What is the main issue with using MANOVA? (x2)

Answer 40

• Instead of adapting model to observed DVs, selectively weight/discount them according to how well they fit existing model
• Atheoretical, over-capitalises on chance

Question 41

Under what conditions would it be ok to use MANOVA? (x2)

Answer 41

Where you’ve used a grab bag of levels (e.g., randomly selected 4 out of 100trials to analyse)
o When levels then have no real meaning, and some may be more error prone, e.g. emotion scales

Question 42

In RM ANOVA, what is the error term for ANY effect equal to (including main effects, interactions and follow-ups)

Answer 42

The interaction between that effect and the effect of participants (a random factor)