2x2%20ANOVA%20BG%20

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2x2 ANOVA BG MG Models

• Kinds of Factorial Designs
• ANOVA for BG, MG WG designs
• Causal Interpretation of Factorial Results

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• When working with factorial designs we will
  consider two major types of IVs
• those that are investigated using a between
  groups comparison
• different participants in the conditions of the
  IV (BG)
• those that are investigated using a
  within-groups comparison
• the same participants are in all conditions of
  the IV (WG)
• Thus, there are three kinds of 2-way
  factorial designs
• All three kinds of these designs can involve any
  combination of manipulated IVs and/or selected
  IVs
• All three kinds of these factorial ANOVA models
  have 3 F-tests
• one for each of two Main Effects and one for the
  Interaction

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• BETWEEN GROUPS FACTORIAL DESIGN
• both of the IVs use between groups comparison
• each participant completes only one condition of
  the design
• WITHIN-GROUPS FACTORIAL DESIGN
• both of the IVs use within-groups comparison
• each participant completes all conditions of the
  design
• MIXED FACTORIAL DESIGN
• one of the IVs uses a between groups comparison
  and one of the IVs 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 uses a
  between groups comparison and which IV uses a
  within-groups comparison

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Between groups factorial design -- 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
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Mixed group factorial design Species was a
between groups IV (a turtle can only be a
member of one species). Each turtle
participated in both the mid-morming 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
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Within-groups factorial design -- 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.
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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 other.
BG
MG
WG
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Another Example -- 3 versions of the same
study The researcher wanted to investigate
infants 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 them 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.
MG
WG
BG
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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
All three !
Puzzle type only.
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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
FPresentation ME FDifficulty ME
FPresentation X Diffuclty Interaction
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Statistical Analyses Necessary to Describe Main
Effects of a 2x2 Design

• In a 2x2 Design, the Main effects 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
• Dont forget to examine the means to see if a
  significant difference is in the hypothesized
  direction !!!

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Statistical Analyses Necessary to Describe the
Interaction 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 us 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

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Using the Pairwise Computator LSDmmd to Compare
cell means to describe the simple effects of a
2x2 Factorial design
For a 2x2 BG Factorial Design
k 4 conditions
n N/4 20/4 5
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Using the Pairwise Computator LSDmmd to Compare
cell means to describe the simple effects of a
2x2 Factorial design
For a 2x2 MG Factorial Design
k 4 conditions
n N 20
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Applying mmdlsd to Simple Effects Describing the
Interaction Task
Presentation Fint 13.21, p .012
Paper Computer
LSDmmd 14 Task Difficulty Easy
90 70 Hard
40 60 for the following, tell
the size of the effect and apply the mmdlsd
Simple effect of Task Presentation SE Task
Presentation for Easy Tasks SE Task
Presentation for Hard Tasks Simple effects of
Task Difficulty SE of Task Difficulty for Paper
Pres. SE of Task Difficulty for Comp. Pres.
20 gt
- 20 lt
50
gt
10

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Applying mmdlsd to Simple Effects Checking if
Main effects are Descriptive/Misleading - 1
Task Presentation Fdif
1.21, p .72 Paper
Computer LSDmmd 14 Task Difficulty
Easy 90 70
Hard 40 60
65
Main effect of Task Presentation marginal mean
for Paper marginal mean for
Computer Simple effects of Task
Presentation SE of Task Presentation for
Hard SE of Task Presentation for Easy Main
effect of Task Presentation is .
65
20
gt
-20
lt
misleading
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Applying mmdlsd to Simple Effects Checking if
Main effects are Descriptive/Misleading - 2
Task Presentation Fdif
7.83, p .027
Paper Computer LSDmmd 14 Task
Difficulty Easy 90
70 Hard 40
60
80
Main effect of Task Difficulty marginal mean
for Easy marginal mean for
Hard Simple effects of Task Difficulty SE of
Task Difficulty for Paper Pres. SE of Task
Difficulty for Comp. Pres. Main effect of
Difficulty is .
50
50
gt
10

misleading