Week 4 - ANOVA III Flashcards
• Explain what higher-order designs are (x1)
• More than 2 independent variables (factors)
• Explain what research questions higher order designs can address (x2)
How do the factor influence the DV,
Independently and/or interactively
• Explain why higher-order designs are useful (x1)
• Allow for designs with higher external validity
• 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)
3-way factorial ANOVA
2 (a1, a2) x 4 (b1, b2, b3 b,4) x 3 (c1, c2, c3)
• 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)
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
• Explain how three-way interactions are graphed (x3)
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
• Explain how variance is partitioned in a 3-way ANOVA (x4, x2, x4, x2)
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
• Explain the structural model in 3-way factorial ANOVA
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
• Identify the formulae for total (x1) and factor (x4) degrees of freedom in 3-way factorial ANOVA
dftotal = N - 1 df factor always (# levels - 1): *dfj = j - 1 *dfk = k - 1 *dfl = l - 1
• Identify the formulae for interaction degrees of freedom in 3-way factorial ANOVA (x5)
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)
• Identify the formulae for error degrees of freedom in 3-way factorial ANOVA (x2)
df error always (N - #cells) or ([n - 1][# of cells])
dferror = N - jkl
• In the omnibus summary table for 3-way factorial ANOVA, explain what the SS and MS values represent (x1)
The variance, and indexed (weighted by df) variance in each factor, 2-way interaction, and the 3-way interaction
• In the omnibus summary table for 3-way factorial ANOVA, explain how F is calculated for each effect (x1)
MS for each effect/interaction is divided by the pooled error term (MSerror)
• 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)
That there is differences among the marginal means (from the grand mean) of Factor A,
Collapsed across levels of other 2 factors
• 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)
Effect of at least one factor is different at different levels of the other factor
• Ignoring (averaging across) the third factor
• 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
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..
• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o A main effect (x4)
When it is significant
And has >2 levels,
Do main effect comparisons/t-tests/contrasts
(same as 2-way ANOVA)
• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o An omnibus two-way interaction (x4)
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
• In a 3-way factorial ANOVA, identify the circumstances in which you would follow up:
o An omnibus three-way interaction (x1)
If it is significant
• In a 3-way factorial ANOVA, identify the steps in following up an omnibus main effect (x2)
Main effect comparisons
*protected/t-tests or linear contrasts, exactly as in 2-way ANOVA
• In a 3-way factorial ANOVA, identify the steps in following up an omnibus 2-way interaction
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,
• 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)
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
• In a 3-way factorial ANOVA, explain what simple interaction effects are, including:
o What they test (x3)
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
• In a 3-way factorial ANOVA, explain what simple interaction effects are, including:
o Why we need them
(x4)
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
• In a 3-way factorial ANOVA, explain what simple interaction effects are, including:
o How they are tested
(x1)
F test
• 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)
Omnibus: ignores levels of the third factor
Simple: test the 2-way interaction at each level of the third factor
• 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)
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
• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o What question they answer
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)
• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o How they differ from simple effects
Simple effects average across levels of 3rd factor
Simple simple effects tests diffs in cell means AT each level of 3rd factor
• Explain what simple simple effects are in a 3-way factorial ANOVA, including:
o What error term is used (x2)
Pooled MSerror from the omnibus ANOVA table
rather than from separate 1-way experiments
• 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)
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
• 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)
That the cell means differ
• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o What question they answer (x2)
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?
• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o How they differ from simple comparisons
(x2)
They need to be computed at each level of a third factor
Which simple comparisons don’t have
• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o What error term is used
MSerror, in linear contrasts (t-tests) perhaps with Bonferroni adjustment
• Explain what simple simple comparisons are in a 3-way factorial ANOVA, including:
o How to determine how many simple simple comparisons to test
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
• 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)
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