Lecture 4 - Factorial ANOVA

Question 1

What is the difference between all previous types of ANOVA and Factorial ANOVA?

Answer 1

In all previous forms of ANOVA, there was only one IV. If an experiment though has at least 2 IV, then it is a factorial design (compared to a linear one)

Question 2

What are the different type of Factorial Designs?

Answer 2

• Indepdendent Factorial Design: Many IV’s have been measured using different people for each IV (between-group measurement)
• Repeated-Measures Factorial Design: Many IV’s have been measured using the same people for each IV (within-group measurement)
• Mixed Design: Some IV’s are measured with the same people, some are measured with different people

Question 3

Some notes on different type of ANOVA tests?

Answer 3

• If a test say one-way ANOVA, then it means that there is one IV
• If a test says two-way ANOVA, then there’s 2 IV’s, and so on…

Question 4

Experiment example

Answer 4

A researcher wanted to test how alcohol (placebo, low dose, high dose) and facetype (attractive or unattractive) affect the perceived attractiveness of other people
- IV’s: facetype and alcohol levels
- DV’s: attractiveness

Question 5

Linear model

How can we apply this info in the linear model?

Answer 5

• First, for coding reasons, we assign 1 = Attractive, 0 = Unattractive. And, high alcohol dose = 1, placebo = 0.
  See Equation for 1 for the linear model
  See Equation 2 for all the equations giving us the b-values

Question 6

Interaction Plots

Question 7

Interaction Plots

What are interaction plots and why do we use them?

Answer 7

An interaction plot is a representation of how different levels of one independent variable affect the relationship between another independent variable and a dependent variable.
(See slide 2 for examples)

Question 8

Interaction Plots

How can we interpret Interaction Plots?

Answer 8

• If there are non-parallel lines, then there’s an interaction between any 2 means
• If there are parallel lines, there is NO interaction between any 2 means
  !!! IN GENERAL: the more non-parallel two lines, the stronger the interaction between the 2 means. Crossing lines though don’t always indicate an interaction !!!

Question 9

Interaction Plots

What are some other graphs we can use to represent the same effects?

Answer 9

See slide 3
NOTE: In slide 2, the right image no significant effect, since attractiveness increases as alcohol increases, and this is true for both types of faces.
The equivalent of the right graph in slide 2 is the graph in Slide 4.
In the equivalent box graph, although the overall levels of attractiveness increase, the difference alcohol levels in attractive and unattractive faces remains stable across all three alcohol conditions. Therefore, there is no significant interaction (see notes on slide 4 as well)

Question 10

Simple Effects Analysis

Question 11

Simple Effects Analysis

What is Simple Effects Analysis?

Answer 11

Looks at the effect of an individual varaible at individual levels of the other IV
e.g. IV = type of face (attractive or unattractive)
- effect on high dose of alcohol
- effect on low dose of alcohol
- effect on placebo group
In the example just mentioned, what we do is:
- we take the average rating of unattractive faces and compare to that of attractive faces after a placebo drink
- Then, we take the average rating of unattractive faces and compare to that of attractive faces after the low dose drink
- Then, we take the average rating of unattractive faces and compare to that of attractive faces after the high dose drink

Question 12

Simple Effects Analysis

What F-statistics do we get from this analysis and what do they indicate?

Answer 12

e.g. Assume we just have this analysis for attractive-unattractive faces and placebo-high dose alcohol groups
!!! We get 2 F-statistics !!!
- One F-Statistic: If differences in ratings for attract.-unattract. faces are significant for those in the placebo group
- Other F-Statistic: If differences in ratings for attract.-unattract. faces are significant for those in the high-dose group
(Of course, this is also applicable the other way around: high-dose group-placebo group for attractive faves, high-dose group-placebo group for unattractive faces. Slide 5 shows both different possibilities)

Question 13

F-Statistic

Question 14

F-Statistic

How is the Total Sum of Squares (SST) broken down in Factorial ANOVA?

Answer 14

See slide 6

Question 15

F-Statistic

How is SST calculated?

Answer 15

See Equation 3

Question 16

F-Statistic

How is SSM calculated?

Answer 16

See Equation 4
- k refers to the amount of groups
- g: represents a specific group
- ng: number of scores in each group (group sample size)
- df = k - 1

Question 17

F-Statistic

How is SSA calculated?

Answer 17

Assume Variable A = type of face. Ignore the dose of alcohol people drank and place all ratings of unattractive faces to one group and all ratings of attractive faces to another group (See image 1).
Then we apply Equation 5. The two groups which are being summed are the groups of unattractive faces and attractive faces (just the groups in image 1)

Question 18

F-Statistic

How is SSB calculated?

Answer 18

Same process as for SSA, only Variable B is alcohol so since you have three conditions, you’ll also have three groups (group1: all ratings of placebo drinks, group2: all ratings of low dose drinks, group3: all ratings of high-dose drinks)

Question 19

F-Statistic

How is SSAxB calculated?

Answer 19

SSAxB = SSM - SSA - SSB

Question 20

F-Statistic

How is SSR calculated?

Answer 20

See Equation 6
-sg represents variance within a group
- Also represents individual differences in performance or variance that can’t be explained

Question 21

F-Statistic

How is the F-Statistic calculated?

Answer 21

each effect in a factorial design has it’s own F-Statistic: e.g. in a 2-way design we compute F for the 2 main effects and interaction:
- FA: MSA/MSR = (SSA/dfA)/(SSR/dfR)
- FB: MSB/MSR = (SSB/dfB)/(SSR/dfR)
- FAxB: MSAxB/MSR = [(SSM - SSA - SSB)/dfB]/(SSR/dfR)

Question 22

Assumptions

Question 23

Assumptions

What are the Assumptions for Factorial ANOVA?

Answer 23

Same as for standard ANOVA

Question 24

Assumptions

What do we do if one of the assumptions is violated?

Answer 24

• If homogeneity of variance is violated then there are corrections based on the Welch Test
• Else, bootstrap the post hoc tests so that these will be robust
• Also, ask for CI’s and p=values for parameter estimates that are robust

Question 25

How do we report results?

Answer 25

• There was a significant effect of the amount of alcohol consumed on ratings of attractiveness of faces F(2,42) = 6.04, P = 0.005, partial ω^2 = 0.12 (0.05;0.43. Bonferroni post hoc tests revealed that the attractiveness ratings were significantly higher after a high dose than
  after a placebo drink (p = 0.004). The attractiveness ratings were not significantly different after a low dose compared to a placebo (p = 0.231), or a high dose compared to a low dose (p = 0.312).
• Attractive faces were rated significantly higher than unattractive faces, F(1, 42) = 15.58, p <0.001, partial ω^2 = 0.17 (0.01;0.36)
• There was a significant interaction between the amount of alcohol consumed and the type of face of the person rated on attractiveness, F(2, 42) = 8.51, p = 0.001, partial ω^2 = 0.238 (0.04;0.43). This effect indicates that ratings of unattractive and attractive faces were affected differently by alcohol. Simple effects analysis revealed that ratings of attractive faces were significantly
  higher than of unattractive faces in the placebo group, F(1, 42) = 24.15, p < 0.001, and in the low-dose group, F(1, 42) = 7.72, p = 0.008, but not in the the high-dose group, F(1, 42) = 0.73, p
  = 0.398.