Analysis of Factorial Designs

• Statistical Analysis of 2x2 Designs

• Statistical Analysis of kxk Designs

Statistical Analysis of a 2x2 Design

Task Presentation (a) SE of Presentation

Paper Computer for Easy TasksTask Difficulty (b)

Easy 90 70 80 Hard 40 60 50

65 65 SE for Presentation for Hard Tasks

Presentation Difficulty Interaction Main Effect Main Effect Effect

FPresentation FDificulty FInteraction

65 vs. 65 80 vs. 50 SEEasy vs. SEHard

In a 2x2 Design, the Main effects F-tests are sufficient to tell us about the relationship of each IV to the DV…

• since each main effect involves the comparison of two marginal means -- the corresponding significance test

tells us what we need to know …

• whether or not those two marginal means are “significantly different”

• Don’t forget to examine the means to see if a significant difference is in the hypothesized direction !!!

Statistical Analyses Necessary to Describe Main Effects of a 2x2 Design

However, the F-test of the interaction only tells us whether or not there is a “statistically significant” interaction…

• it does not tell use the pattern of that interaction

• to determine the pattern of the interaction we have to compare the simple effects

• to describe each simple effect, we must be able to comparethe cell means

we need to know how much of a cell mean difference is “statistically significant”

Statistical Analyses Necessary to Describe the Interaction of a 2x2 Design

Using LSD to Compare cell means to describe the simple effects of a 2x2 Factorial design

• LSD can be used to determine how large of a cell mean difference is required to treat it as a “statistically

significant mean difference”

• Will need to know three values to use the computator

• dferror -- look on the printout or use N – 4

• MSerror – look on the printout

• n = N / 4 -- use the decimal value – do not round to the nearest whole number!

Remember – only use the lsdmmd to compare cell means. Marginal means are compared using the man effect F-tests.

What statistic is used for which factorial effects????

There will be 4 statistics

1. FAge

2. FGender

3. FInt

4. LSDmmd

Age

5

10

GenderMale Female

This design as 7 “effects”

1. Main effect of age

2. Main effect of gender

3. Interaction of age & gender

4. SE of age for males

5. SE of age for females

6. SE of gender for 5 yr olds

7. SE of gender for 10 yr olds

30 30 30 20 30 25 25 30

Effect Sizes for 2x2 BG Factorial designs

For Main Effects & Interaction (each w/ df=1)

r = [ F / (F + dferror)]

For Main Effects & Simple Effects

d = (M1 - M2 ) / Mserror

d² r = ----------

d² + 4 (This is an “approximation formula”)

“Larger” Factorial Designs

The simplest factorial design is a 2x2, which can be expanded in two ways:

1) Adding conditions to one, the other, or both IVs

In a kxk Design, the Main effects F-tests are sufficient to tell us about the relationship of each IV to the DV only for 2-condition main effects…

• since a 2-condition main effect involves the comparison of two marginal means -- the corresponding F-test tells us what we need to know – the two marginal means are different

• however, for a k-condition main effect, the F-test only tells us that there is a pattern of significant differences among the marginal means, but doesn’t tell us which means are significantly different

• for a k-condition main effect we need to use an LSDmmd to determine which pairs of marginal means are significantly different

Statistical Analyses Necessary to Describe Main Effects of a kxk Design

As with the 2x2 design, the interaction F-test for a kxk design only tells us whether or not there is a “statistically significant” interaction…

• it does not tell use the pattern of that interaction

• we need to use an LSDmmd to determine which pairs of cell means are significantly different

Be sure you are using the correct “n” when you compute LSDmmd

n = N / #conditions in that effect

Statistical Analyses Necessary to Describe the Interaction of a kxk Design

Effect Sizes for kxk BG Factorial designs

For Main Effects & Interaction (each w/ df=1)

r = [ F / (F + dferror)]

For specific comparisons among marginal or cell means

d = (M1 - M2 ) / Mserror

d² r = ----------

d² + 4 (This is an “approximation formula”)

What statistic is used for which factorial effects????

There will be 5 statistics

1. FGender

2. FAge

3. Age LSDmmd (n=N/3)

4. FInt

5. Int LSDmmd (n = N/6)

Age

5

10

15

GenderMale Female

15 “Effects” in this study1. Main effect of gender2. Main effect of age3. 5 vs. 10 marginals4. 5 vs. 15 marginals5. 10 vs. 15 marginals6. Interaction of age & gender7. 5 vs. 10 yr old males8. 5 vs. 15 yr old males9. 10 vs. 15 yr old males10. 5 vs. 10 yr old females11. 5 vs. 15 yr old females12. 10 vs. 15 yr old females13. male vs. female 5 yr olds14. male vs. female 10 yr olds15. male vs. female 15 yr olds

30 30 30

20 30 25

25 30 27.5

25 30 27.5

Back to 100 males and 100 females completed the task, either under instructions to work quickly, work accurately, to work as quickly as possible without making unnecessary errors or no instructions.

Instruction Quick Accurate Both NoneGender

Male

Female

For the interaction p = .03

• will we need an LSDmmd to compare cell means? why or why not?•

what will separate model “n” be?

For the main effect of instruction p = .02

• will we need an LSDmmd to compare marginal means? why or why not?

• what will separate model “n” be?

• will we need an LSDmmd to compare cell means? why or why not?

• what will separate model “n” be?

For the main effect of gender p = .02

• will we need an LSDmmd to compare marginal means? why or why not?

•

what will separate model “n” be?

•will we need an LSDmmd to compare cell means? why or why not?• what will separate model “n” be?

Yep! sig. Int & k = 8 !200 / 8 = 25

Yep! sig. ME & k = 4 !

200 / 4 = 50

Nope – k = 2 !

Yep! sig. Int !200 / 8 = 25

Yep! sig. Int !200 / 8 = 25