Final Material Flashcards
Describe the two-factor mixed design ANOVA
- The two-factor mixed design has one between-subjects factor and one within-subjects factor
- Mixed designs are useful when it’s impossible or inadvisable to manipulate a factor within subjects
- Ex: the participant might be permanently changed by levels of the factor (e.g., removal of a brain region in rats) or the factor is an immutable characteristic of the participant (e.g., incurable medical condition)
What’s the notational system for the two-factor mixed design ANOVA?
A: between-subject factor
B: within-subject factor
sij: subject i in group j (e.g., s23 is subject 2 in level a3) Xijk: individual score of subject i in group j in level k of the within-subjects factor B
How many factors can you have in the between-subjects and within-subjects for a two-factor mixed design ANOVA
You can have more than one factor for both the between subjects and the within subjects
What are the different hypotheses for the two-factor mixed designs ANOVA?
- Main effect of the between-subjects factor A
H0A :μ1 =μ2 =…=μa
H1A : Not all μa’s are the same - Main effect of the within-subjects factor B
H0B :μ1 =μ2 =…=μb
H1B : Not all μb’s are the same - Interaction between factors A and B
H0AB: there is no interaction between AxB
H1AB: there is an interaction between AxB
What’s the F ratio formula for the between-subjects main effect of A in a two-factor mixed designs ANOVA?
F = MS(A) / MS(S/A)
What’s the F ratio formula for the within-subjects main effect of B in a two-factor mixed designs ANOVA?
F = MS(B) / MS(BxS/A)
What’s the F ratio formula for the within-subjects interaction effect of AxB in a two-factor mixed designs ANOVA?
F = MS(AB) / MS(BxS/A)
What are the assumptions for a mixed-design ANOVA?
- The assumptions of the mixed ANOVA contain assumptions from both between-subjects ANOVA and within-subjects ANOVA
Between-subjects assumptions: - Normal distribution of scores
- Homogeneity of variances (at each combination of levels of factors A and B) -> at the population level
- Independence of observations for between-subjects
- The within-subjects effect requires the assumption sphericity
- Tested only for the within-subject main effect
- Mauchly’s W test can be used to check the assumption of sphericity
- If sphericity is violated, the same ε is used for both within-subject effects
What are the effects of violations of the sphericity assumption?
- Changes the probability of making a Type I error rate
- The consequences are not appreciable for the test of the between-subject factor, but the probability of falsely rejecting the null hypothesis increases for the within-subject effects
- Using a more conservative F test (i.e., Greenhouse-Geisser or Huynh-Feldt corrections) solves this issue
What are the effects of violations of the homogeneity of variances assumption?
- Changes the probability of making a Type I error rate
- The consequences depend on equality of within-group sample sizes
- Type I error rate > αlpha when smaller groups have higher variability (increased rate of false positive findings)
- Type I error rate < αlpha when smaller groups have lower variability (lower power to detect effects)
What are the sources of variation in a two-factor mixed design ANOVA?
- Main effect of the between-subjects factor A
- Subject variation at levels of the between-subjects factor (S/A)
- Main effect of the within-subjects factor B
- Interaction between the between-subjects factor and the within-subjects factor AxB
- Interaction between the within-subjects factor and subjects nested in levels of the between-subjects factor (BxS/A)
What measure of effect size do we use for the main effect of the between-subject factor in a two-factor mixed design ANOVA?
Partial omega-squared (ωA2)
What measure of effect size do we use for the within-subject effects in a two-factor mixed design ANOVA?
Descriptive partial effect size measures (η2B & η2AxB)
R2 is the same as what?
η2
What does omnibus mean?
All encompassing, assessing that all means are equal
What’s the model?
Group membership
Why is the F ratio or F test considered an omnibus test?
- The F ratio or F test gives a global effect of the independent variable on the dependent variable (omnibus or overall test)
- It does not tell us which pairs of means are different
- We need to perform post hoc tests to make further inferences about which means are different
What are post hoc tests?
- Post hoc (a posteriori/unplanned) comparisons
- Decided upon after the experiment
- In the case of a one-way between-subjects ANOVA, used if 3 or more means were compared
- Examples of 2 post-hoc tests: Sheffé & Tukey’s Honestly Significant Difference (HSD) test
What’s Sheffé’s Test?
- Post hoc test
- Can be used if groups have different sample sizes
- Less sensitive to departures from the assumption of normality and equal variances in the population (violations of the assumptions)
- It’s the most conservative test (very unlikely to reject H0)
- This means it is a good choice if you wish to avoid Type I errors, but it has lower power to detect differences
- If the null hypothesis is true to the population then the conservative test is good but if the null hypothesis isn’t true to the population then it has less power
- Uses F ratio to test for a significant difference between any 2 means (e.g., H0 : μ1 = μ2)
- But, uses a larger critical value
How is the updated critical value for Sheffé’s Test obtained?
- Obtain the critical value of F with dfM =k−1 and dfR =N−k (in other words, obtain the critical value as usual)
- Then multiply this value by k − 1
- The SS for the specific comparisons need to be calculated based on which groups are being compared and the residual SS is simply the SSresidual from the main analysis
What are the steps in Sheffé’s Test?
- Calculate the SScomparison, MScomparison, and Fcomparison
- MScomparison = SScomparison
- Fcomparison = MScomparison/MSresidual
- Compare the observed Fcomparison to (k − 1)x(Fcritical(k−1),(N−k))
- If the observed Fcomparison is greater, conclude that the pair of means is significantly different from 0
- If not, the pair of means being compared is not significantly different
- The df comparison will always be 1 because you’re always comparing 2 groups at a time
- All SScomparison are calculated using means of each level of the independent variable
In Sheffé’s Test, what kind of coefficients does each group receive?
- The groups being compared receive coefficients of -1 and 1
- The group(s) excluded from this particular comparison receive a coefficient of 0
- Ex: if not comparing c3 -> give it 0 -> c1 = 1 and c3 = 0 then c2 = -1
Describe Tukey’s HSD test
- Typically used if groups have equal sample sizes and all comparisons represent simple differences between 2 means
- This test uses the studentized range statistic, Q
- The observed Q value is compared against a critical value of Q for α = .05 which is associated with k and N − k
- Call this critical value Qcrit
- Reject H0 : μg = μg′ , when the observed Q value is greater than or equal to Qcrit, the critical value
What does HSD mean?
Minimum absolute difference between 2 means required for a statistically significant difference