W8: One-Way Within-Subjects ANOVA

Question 1

What are alternative names of dependent group

Answer 1

1. “WIthin-Subject”
2. “Repeated-measures” (Only for time points)

Question 2

What are some ways we can use dependent groups

Answer 2

Same people:

1. Repeated occasion
2. Different condition

Different People:

1. Matched group (triad like mother-child)

Question 3

Do the interval levels have to be same in a dependent group design?

Answer 3

Yes.

• Intervals have to be the same for all participants
  
  • However, unequal spacing is just aceptable as long as each participant’s interval is the same.

Question 4

If a participants drops out of a dependent group study, how?

Answer 4

• Sample is invalid; or
• Adjustments have to be made

Question 5

What is the problem with small sample size in relation to identfiying normality?

Answer 5

Small samples makes it dificult to identify any systematic non-normality

Question 6

What are we interested in in repeated measures design

Answer 6

We are often interested in

• Understand the rate of change occuring
• Does the differences in rate of change reflect sampling variability or changes in experimental design?
  
  • And if changes is in experimental design, how best can we represent that rate of change occuring (Polynomial Contrasts)

Question 7

Conceptually, what does univariate and multivariate mean?

Answer 7

Univariate

• Treating each level separately
• Associated with CIs and (to a smaller degree) H₀

Multivariate

• Combining multiple levels
• Asscociated with H₀ (Omnibus)

*Notes

• Number of factors is irrelevant
• Still one IV in both approaches

Question 8

What are the assumptions and hence the difference in the assumptions of the univariate and multivariate approach. Which is better?

Answer 8

• Univariate:
  
  • Sphericity
  • Indepndence
  • Normality
• Multivariate
  
  • No Sphericity
  • Independence
  • Normaliy

If sphericity is met, univariate > multivariate

Question 9

Which variate does Hedges’ g and Bonnett’s delta assess in one-way within-subjects ANOVA

Answer 9

• Both Hedges’ g and Bonett’s delta estimate the effect size and associated confidence interval using a univariate method
• Multivariate method is only used in the omnibus test of the null hypothesis.

Question 10

What is the similarity between univariate and multivariate approach in a one-way within-subject anova

Answer 10

• Both assume independence of observation (which violated) and normality

Question 11

What is sphericity

Answer 11

Variance of all possible difference scores between pairs of three or more within-subject conditions/levels being homogenous at a population level

Question 12

How do we caclulate sphericity in sample data?

Answer 12

Its assessment in sample data requires calculating difference scores between all possible pairs of within-subject groups

Question 13

What is compound symmetry. And how does it relate to sphericity

Answer 13

Covariance martix with:

• (a) same variance in each diagonal
• (b) same variance in each off-diagonal
  
  • Hence, it is based on observed scores

Compound symmetry is a sufficient, not necessary, condition for sphericity

Question 14

Can if infer sphericity from compound symmetry?

Answer 14

if compound symmetry is not being met, we can infer sphercitiy is not being met.

Question 15

What is the calculation of sphericity. What are some properties: PP/SS; Range; And estimating it

Answer 15

Sphericity is calculated from the observed covariance matrix by ε (episolon)

• ε is a parameter when cacluated on population covariance matrix and
• ε is a statistic when estimated on sample covariance matrix
• ε ranges from 1/k to 1
• In sample data, ε is estimated using
  
  • Greenhouse-Geisser esimator (More conservative); or
  • Huynh-Feldt estimator (Less conservative)
  • Both are used to make adjustment to null hypothesis tests under the univariate approach.

Question 16

What is an important thing to take note in R in dependent group vs indepndent group

Answer 16

For the post-processing phase with ANOVA,

Dependent group:

• Have to add an ordered factor for the time points

Question 17

In multivariate investigation of dependent group, what do we look for?

Answer 17

Look for Pillai.

• Pillai result is the multivariate test result of the omnibus H₀ test that the within-subject means for all levels ar the same
• If p < .05, that means the data is not consistent with the means for all levels of the within-subject factors being equal.

Question 18

What is a more informative way of investigating mean differences over time?

Answer 18

Using a set of focued linear contrasts called orthogonal polynomial contrasts

• Decomposition of observed change over time into constituent component rates of change over time

Question 19

Using polynomial orthogonal contrasts, can the intervals between time be different?

Answer 19

Yes… but adjustments have to be made

• Week 1,2,3,4,5
  
  • No Adjustment
• Week 1,2,5,6,7
  
  • Adjustment

Question 20

How many orthogonal polynomial contrast are there in 5 groups

Answer 20

K-1.

• 5 Groups = 4 Polynomial Contrasts
  
  • Linear
  • Quadratic
  • Cubic
  • Quardic

Question 21

is the initial investigation in R uni or multivariate in one-way within-subjects ANOVA

Answer 21

Normally multivariate

• Pillai is a multivariate test.

However, we can use either, depending on focus of research

Question 22

Under what variate can interval estimators for dependent groups be calculated. What is the caveat?

Answer 22

• Multivariate approach does not include confidence intervals. It only assess the omnibus H₀
• Univariate approach includes confidence intervals
  
  • Hence, coverage rates of intervals may be affected by assumption of sphericity not being met

Question 23

If a within-subjects design is being analysed, must we use polynomial linear contrasts?

Answer 23

Not a must.

We May.

Question 24

What does linear, quadratic, quartic refer and not refer to:

Answer 24

• Refer to:
  
  • Types of polynomial linear contrasts
• Not refer to
  
  • Levels of within-subject factor

Question 25

If there are four levels in a within-subjects one-way ANOVA, how many orthogonal linear contrasts can be tested on the set of sample means

Answer 25

Maximum number of orthogonal linear contrasts: 3

k-1 fundemental differences

Question 26

Do the orthogonal polynomial contrasts rely on sphericity?

Answer 26

Not necessarily. They can be either:

• Multivaiate
  
  • Which will give us p-values
• Univariate
  
  • Which will give us confidence intervals and effect sizes

Question 27

In mean contrast differences (both observed and standardised) in a polynomial linear contrast, which one assumes sphericity?

Answer 27

Assumes Sphericity

• Observed contrast mean and its CI
• Standardized contrast mean and its CI
  
  • Labelled “Sphericity-assumed” (Not Hedges g)

Does not assume sphericity

• Standardised contrast mean and its CI
  
  • “Bonett’s Delta” (note, it’s still univariate)

Question 28

What happens if the linear contrasts are not orthogonal

Answer 28

If it is both unbalanced and unorthogonal

• Overlap in difference
• Transitivity does not apply

Question 29

Are orthogonal polynomials (linear, cubic, quadratic,…) linear contrasts?

Answer 29

Yes.

They are all still linear contrasts and should sum to 0

Question 30

What does the observed linear contrast mean difference reflect in Week 1,2,3,4,5

Answer 30

It reflects length of the trial

• Weeks 1 AND Weeks 5
• It doesn’t say what’s in between

Question 31

How do we transformed contrast mean difference to standardised mean difference?

Answer 31

Standardised Mean = (Raw Mean Difference / Standard Deviation)