The document discusses statistical analysis of factorial designs, including 2x2 factorial designs. It covers reviewing interactions through examining simple effects and main effects. It also discusses kinds of factorial designs, including between-groups, within-groups, and mixed designs. Additionally, it provides examples of interpreting interactions and main effects, constructing F-tests, and using LSD to describe interactions by comparing cell means in 2x2 designs.

1. Statistical Analysis of Factorial
Designs
 Review of Interactions
 Kinds of Factorial Designs
 Causal Interpretability of Factorial Designs
 The F-tests of a Factorial ANOVA
 Using LSD to describe the pattern of an interaction

2. Review #1-- interaction
Task Presentation
Paper Computer
Task Difficulty
Easy 90 70
Hard 40 60
1) Put in a <, > or =
to indicate the pattern
of the simple effect of
Task Presentation for
Easy Tasks
2) Put in a <, > or =
to indicate the pattern
of the simple effect of
Task Presentation for
Hard Tasks
3) Is there an interaction? Is the simple effect of Task
Presentation the same for Easy and for Hard Tasks ???
4) Describe the pattern of the interaction...
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3. Review #1-- 1st main effects
Task Presentation
Paper Computer
Task Difficulty
Easy 90 70
Hard 40 60
1) Compute the marginal
means for Task
Presentation.
3) Is there a main effect of
Task Presentation ???
4) Is the pattern of the main effect for Task Presentation descriptive or
potentially misleading (is the pattern of the main effect or Task Presentation
the same as the patterns of the simple main effects of Task Presentations for
both Easy and Hard Tasks) ?
2) Put in a <, > or = to
indicate the pattern of the
main effect of Task
Presentation
5) Describe the main effect
65 65
=
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4. Review #1-- 2nd main effects
Task Presentation
Paper Computer
Task Difficulty
Easy 90 70
Hard 40 60
1) Compute the marginal
means for Task Difficulty
3) Is there a main effect of
Task Difficulty ???
4) Is the pattern of the main effect of Task Difficulty descriptive or potentially
misleading (is the pattern of the main effect or Task Difficulty the same as the
patterns of the simple main effects of Task Difficulty for both Computer and
Paper Presentations) ?
2) Put in a <, > or = to
indicate the pattern of the
main effect of Task
Difficulty
5) Describe the main effect
80
50
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5. “Kinds” of 2-factor Designs
BETWEEN GROUPS FACTORIAL DESIGN:
• each IV uses a between groups comparison
• each participant completes only one condition of the design
WITHIN-GROUPS FACTORIAL DESIGN:
• each IV uses a within-groups comparison
• each participant completes all conditions of the design
MIXED FACTORIAL DESIGN:
• one IVs uses a between groups comparison and the other IV
uses a within-groups comparison.
• each participant completes both conditions of the within-
groups IV, but completes only one condition of the between
groups IV.
• it is important to specify which IV is “BG” and which is “WG”

6. Between groups factorial design  experimental or natural grps
designs used to study “differences”
Each participant is in only one condition, having a particular
combination of Initial Diagnosis and Type of Treatment.
Type of Treatment
Initial Diagnosis
Individual Group
Therapy Therapy
Clients diagnosed Clients diagnosed
Depression as depressed who as depressed who
are treated with are treated with
individual therapy group therapy
Clients diagnosed Clients diagnosed
Social Anxiety with social anxiety with social anxiety
who are treated with who are treated with
individual therapy group therapy

7. Mixed group factorial design  natural groups designs used to
study “different changes” or
“changing differences”
Species was a between groups IV (a turtle can only be a
member of one species). Each turtle participated in both
the mid-morning & dusk conditions of the Time of Day IV.
Species of Turtle
Time of Day
Snapping Turtle Painted Turtle
Each snapping turtle Each painted turtle
Mid-morning completed a trial completed a trial
during mid-morning during mid-morning
Each snapping turtle Each painted turtle
Evening completed a trial completed a trial
during the evening during the evening

8. Mixed group factorial design  experimental designs used to
increase data collection efficiency
or statistical power
Type of Evidence was a between groups IV -- people can’t
read the “same study” twice & give independent ratings. Each
participant rated the guilt of both Defendants -- a within-groups IV
--as they would in this type of case.
Type of Evidence
Defendant
DNA Eye Witness
Major Actor DNA evidence was An eye witness testified
presented against to seeing the major
the major actor commit the crime
Conspirator DNA evidence was An eye witness testified
presented against to seeing the conspirator
the conspirator commit the crime

9. Within-groups factorial design – experimental designs used to
increase data collection efficiency
or statistical power
Each participant completed four trials, one of each
combination of Retention Interval and Word Type.
Retention Interval
Word Type
Immediate Test Delayed Test
The test was given The test was given
Familiar immediately after the 5 minutes after the
study of a list of study of a list of
40 familiar words. 40 familiar words.
The test was given The test was given
Unfamiliar immediately after the 5 minutes after the
study of a list of study of a list of
40 unfamiliar words. 40 unfamiliar words.

10. Within-groups factorial design – natural designs to study
“changing changes”
Each participant was observed in both School & Home (WG)
settings both when they were 12 &16 (WG)
Age
Setting
12 years old 16 years old
School Participants were Participants were
observed in a school observed in a school
setting at age 12. setting at age 16.
Home Participants were Participants were
observed in a home observed in a home
setting at age 12. setting at age 16.

11. Practice Identifying Types of Factorial Designs - answers next page
The purpose of the study was to examine the possible influence of two
variables upon maze-learning by rats, length of the maze (either 10 feet or 30
feet) and the size of the reward (either 1sugar pellet or 5 sugar pellets).
Here are three “versions” of the study tell which is BG, WG & MG
a. Each rat completed one trial. Each was assigned to either
the longer or the shorter maze, and also assigned to receive
either 1 or 5 sugar pellets upon completing the maze.
b. Each rat completed two trials in either the longer or the
shorter maze. Following one trial in the assigned maze, each
received 1 pellet reward, after the other trial they received the
5 pellets.
c. Each rat completed four trials, two in the shorter maze and
two in the longer maze. Each received 1 pellet after one of the
short-maze trials and 5 pellets after the other, and also 1
pellet after one of the long-maze trials and 5 pellets after the
BG
MG
WG

12. Another Example -- 3 versions of the same study
The researcher wanted to investigate infant's startle responses
to loud sounds. The two variables of interest were the Position
of the Sound (in front of versus behind the infant) and the Type
of Sound (a hand-clap versus deep male voice saying "Hey").
Here are three “versions” of the study tell which is BG, WG & MG
a. Each infant completed trials all involving a hand-clap or
all involving the voice saying "Hey". During some of the trials, the
appropriate type of sound was made in front of the infant. During
other trials, the appropriate type of sound was made behind the infant.
b. Each infant had some trials during which the sound was made in
front of then and some during which the sound was made behind them.
Some of the sounds were the hand-clap and the others were the voice
saying "Hey".
c. Each infant always heard either the hand-clap or the “Hey”, and
whatever sound they heard was always played either in front of them
or behind them.
BG
MG
WG

13. Remember about the causal interpretation of effects
of a factorial design
Start by assessing the causal interpretability of each main effect
Remember, in order to causally interpret an interaction, you must be able to
casually interpret BOTH main effects.
For each of the following: Tell the IVs and tell what effects could be causally
interpreted (assuming proper RA, IV manip. and confound control were used):
1. Male and female participants who were African
American, Mexican American, or European American
were asked to complete a questionnaire about
satisfaction with their Senators.
2. Children played with either a toy gun, a toy car or a
puzzle, some while their parents were in the room and
some not. The DV was the amount of aggressive
behavior they exhibited.
3. Participants played with either a simple puzzle or a
complex puzzle in pairs made up of two boys, two girls
or one boy & one girl.
zilcho-causo
Puzzle type
only.
All three !

14. Statistical Analysis of 2x2 Factorial Designs
Like a description of the results based upon inspection of the
means, formal statistical analyses of factorial designs has five
basic steps:
1. Tell IVs and DV 2. Present data in table or figure
3. Determine if the interaction is significant
• if it is, describe it in terms of one of the sets of simple effects.
4. Determine whether or not the first main effect is significant
• if it is, describe it
• determine if that main effect is descriptive or misleading
5. Determine whether or not the second main effect is significant
• if it is, describe it
• determine if that main effect is descriptive or misleading

15. Interpreting Factorial Effects
Important things to remember:
• main effects and the interaction are 3 separate effects each must
be separately interpreted -- three parts to the “story”
• most common error -- “interaction is different main effects”
• best thing -- be sure to carefully separate the three parts of
the story and tell each completely
• Be careful of “causal words” when interpreting main effects and
interactions (only use when really appropriate).
• caused, effected influenced, produced, changed ….
• Consider more than the “significance”
• consider effect sizes, confidence intervals, etc. when
describing the results

16. Statistical Analysis of a 2x2 Design
Task Presentation (a) SE of Presentation
Paper Computer for Easy Tasks
Task 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
SSPresentation SSDificulty SSInteraction
65 vs. 65 80 vs. 50 SEEasy vs. SEHard

17. Constructing F-tests for a 2x2 Factorial
FPresentation = ( SSPresentation / dfPresentation )
( SSError / dfError)
FDifficulty = ( SSDifficulty / dfDifficulty )
( SSError / dfError )
FInteraction = ( SSInteraction / dfInteraction )
( SSError / dfError)

18. 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

19. 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 compare
the 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

20. 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.

21. Applying lsdmmd to 2x2 BG ANOVA
Task Presentation
Paper Computer
Task Difficulty for the interaction
Easy 60 90 F(1,56) = 6.5, Mse= 300,
p = .023
Hard 60 70
Is there an Interaction? Based on what?
What info do we need to compute the LSDmmd?
k = 4 groups
n = (df + k) / k = (56 + 4) / 4 = 15
MSe = 6.5
df error = 56 (round down to 50)
Yes! F-test of Int

22. Simple effect of Task Presentation
SE of Task Presentation for Easy Tasks
SE of Task Presentation for Hard Tasks
Simple effects of Task Difficulty
SE of Task Difficulty for Paper Pres.
SE of Task Difficulty for Comp. Pres.
30 >
10 =
0
>
20
With an LSDmmd = 12.7

23. Applying lsdmmd to 2x2 BG ANOVA
Task Presentation
Paper Computer
Task Difficulty for Difficulty ME
Easy 60 90 75 F(1,56) = 4.5, p = .041
Hard 60 70 65 lsdmmd = 12.7
Is there a Task Difficulty main effect? Based on what?
Is main effect descriptive (unconditional) or potentially misleading (conditional)?
Simple effects of Task Difficulty
SE of Task Difficulty for Paper Pres.
SE of Task Difficulty for Comp. Pres.
Yes! F-test of ME
Descriptive only for Computer presentation; misleading for Paper
presentations.
0
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24. Applying lsdmmd to 2x2 BG ANOVA
Task Presentation
Paper Computer
Task Difficulty for Presentation ME
Easy 60 90 F(1,56) = 7.2, p = .011
Hard 60 70 lsdmmd = 12.7
60 80
Is there a Task Difficulty main effect? Based on what?
Is main effect descriptive (unconditional) or potentially misleading (conditional)?
Simple effects of Task Difficulty
SE of Task Presentation for Easy Tasks
SE of Task Presentation for Hard Tasks
Yes! F-test of ME
Descriptive only for Easy tasks; misleading for Difficult tasks.
30 <
10 =

25. 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”)