Transcript Factorial ANOVA

Inferential Statistics IV:
Factorial ANOVA
Michael J. Kalsher
Department of
Cognitive Science
MGMT 6970
PSYCHOMETRICS
© 2014, Michael Kalsher
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Outline
• Review of Important Definitions and
Concepts
• Introduction to factorial ANOVA
– Two-way Independent-groups ANOVA
• Homework Problems
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Review of Definitions
• Factorial ANOVA
– An ANOVA with more than one discrete IV and one continuous
DV. Each discrete IV is called a factor.
• Main Effect
– The omnibus test of a single factor ignoring any other factors is called a
main effect.
• Interaction
– The omnibus test of a moderator effect is called an interaction (effect).
• Moderator Effect
– A relationship involving three variables in which the relationship
between two of the variables differs depending upon the third variable.
A moderator effect exists when the relationship between two variables
changes for different values of a third variable.
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What is ANOVA?: A Quick Review
Analysis of variance (ANOVA) is a method of testing the null
hypothesis that three or more means are roughly equal.
Like the t-test, ANOVA produces a test statistic termed
the F-ratio.
Systematic Variance (SSM)
Unsystematic Variance (SSR)
The F-ratio tells us only that the experimental manipulation
has had an effect—not where the effect has occurred.
-- Planned comparisons
-- Post-hoc tests
-- Purpose of follow-up tests? Control Type I error rate at 5%.
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Separating Out The Variance
SST
SST = Sums of Squares Total
SSm = Sums of Squares Model
(Systematic Variance)
SSR = Sums of Squares Error
(Unsystematic Variance or error)
SSM
SSR
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Two-way
Independent-groups
ANOVA
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Factorial ANOVA: What is it?
A procedure that designates a single, continuous dependent
variable and uses two, or more, independent variables
(discrete) to gain an understanding of how the independent
variables influence the dependent variable individually and
interactively.
This operation requires the use of the General Linear
Models Univariate command.
In this module, we’ll consider both main effects and
interactions
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Assumptions
The two-way independent groups ANOVA test requires
the following statistical assumptions:
1. Random and independent sampling.
2. Data are from normally distributed populations.
Note: This test is robust against violation of this assumption if n > 30 for all groups.
3. Variances in these populations are roughly equal
(Homogeneity of variance).
Note: This test is robust against violation of this assumption if all group sizes are equal.
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Main Effect:
An outcome that represents a
consistent difference between levels of a factor
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Interaction:
If the effect of one factor depends
on the different levels of a second factor, then there
is an interaction between the factors
Interaction Effect
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Main Effects and Interactions: An Example
A researcher evaluates a new treatment
for aggression. She recruits 100
children (50 boys and 50 girls) from a
local elementary school.
Half of the boys are randomly assigned
to the experimental group (i.e., a
rational-emotive-therapy) and the other
half to a control group. We do the same
with the girls.
• All children in the treatment group receive RET therapy to lower aggression.
• All children in the control group participate in a group-based game that has
nothing to do with aggression.
After one month, all of the children are evaluated using a 1 to 5 scale
of aggressive behavior in the classroom.
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Main Effects:
When there are differences among the levels of a single factor, ignoring the
other factor, we say there is a main effect of that factor.
• Ignoring treatment condition, are there gender differences in aggressive
behavior? This is the main effect of gender.
• Ignoring gender, is there evidence that the therapy condition decreases
aggressive behavior compared to the control condition? This is the main effect
of treatment.
Interaction Effects
Do boys and girls respond differently to the same therapy? This is the
interaction between gender and treatment.
In our study of children’s aggression, the presence of an interaction suggests
the therapy affects boys and girls differently. “It depends” is another way of
describing an interaction: the effect of factor 1 depends on the level of factor 2.
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A main effect of gender, no effect of
treatment; no interaction
6
5
M = 5.0 M = 5.0
aggression
4
therapy
3
M = 3.0 M = 3.0
control
2
1
0
boys
girls
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A main effect of treatment, no main effect of
gender, no treatment-gender interaction.
6
5
M = 5.0
M = 5.0
aggression
4
therapy
3
M = 3.0
M = 3.0
control
2
1
0
boys
girls
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A main effect of treatment, a main effect of
gender, no treatment-gender interaction
6
5
M = 5.0
aggression
4
M = 4.0
therapy
3
control
M = 3.0
2
M = 2.0
1
0
boys
girls
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An interaction between treatment and gender;
main effect of treatment; no main effect of gender
6
5
M = 5.0
aggression
4
M = 4.0
therapy
3
M = 3.0
control
2
M = 2.0
1
0
boys
girls
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An interaction between treatment and gender;
no main effect of treatment or gender
6
5
M = 5.0
M = 5.0
4
therapy
3
2
control
M = 2.0
M = 2.0
1
0
boys
girls
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Factorial ANOVA:
An Example
Imagine the following scenario:
– A researcher compares the appetite suppressing effects of fenfluramine,
amphetamine, and saltwater injections in 15-day old rats. The results
show that fenfluramine produces greater appetite suppression than
saltwater, but fenfluramine and amphetamine do not differ.
– The researcher further hypothesizes that the relationship between type of
drug and appetite suppression varies depending on whether the rats are
5 or 15-days-old.
This suggests the existence of a moderator effect: a relationship
involving three variables in which the relationship between two of the
variables differs depending upon the third variable.
If the relationship between drug and percent weight gain varies
depending on the age of the rats, then age moderates the relationship
between type of drug and weight gain.
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Defining the Null & Alternative Hypotheses
The Age Main Effect: The omnibus test evaluates whether the percent
weight gain is different for 5-day-old rats than for 15-day-old rats
H0: µ5 = µ15
H1: µ5 = µ15
The Drug Main Effect: The omnibus test evaluates whether there are
differences in percentage weight gain depending on whether the rat is injected
with fenfluramine, amphetamine, or saltwater in a population of 5 and15-day-old rats
H0: µF = µA = µS
H1: not all µs are equal
The Age-Drug Interaction: If the proposed moderator effect exists, the
relationship between type of drug and percent weight gain should be
different depending on age, and the relationship between age and percent
weight gain should be different depending on the type of drug.
H0: (µ5F - µ15F) = (µ5A - µ15A)) = (µ5S - µ15S))
H1: differences between µs at different ages vary depending on type of drug
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Graphical Depiction of
Main Effect of Age
Mean percent weight
gain is different for 5day-old rats than for
15-day-old rats
14
13
Mean = 13
5 Day-Olds
12
15 Day-Olds
11
Mean = 11
10
Amphetamine
Fenfluramine
Saltwater
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Graphical Depiction of Main
Effect of Type of Drug
14
Mean percent weight gain is different
depending on the type of drug
13
Mean = 13
12
5 Day-Olds
15 Day-Olds
11
10
Mean = 11
Mean = 11
Amphetamine
Fenfluramine
Saltwater
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Graphical Depiction of an Interaction between
Age and Type of Drug, but no Main Effects
16
15
14
5 Day-Olds
13
15 Day-Olds
12
11
10
Amphetamine
Fenfluramine
Saltwater
Type of Drug
Age
Fenfluramine
Amphetamine
Saltwater
5 day-old
15
13
11
13
15 day-old
11
13
15
13
13
13
13
13
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Graphical Depiction of an Interaction
and Main Effect for Type of Drug
Saltwater
16
15
Fen
Amph
Mean = 12
=
=
Mean = 14
Mean = 13
•
14
5 Day-Olds
13
12
15 Day-Olds
•
15 Day-Olds
11
13, 13, 13
Mean = 13
10
Amphetamine
Fenfluramine
5 Day-Olds
=
11, 13, 15
Mean = 13
Saltwater
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Two-way Independent-groups
ANOVA: Steps in the Analysis
-- Three F tests required: one for each factor (Factor A; Factor B), and
a third for the interaction (A x B).
-- The numerator differs for the three F tests, but the denominator for all
three is the MSWG.
-- Calculate the sum of squares for factor A (SSA), factor B (SSB), the
interaction (SSAxB), and the error term (SSWG).
-- Convert each factor’s SS to the average sums of square or “mean
squares” (MS) by dividing by the appropriate degrees of freedom).
FA = MSA
MSWG
FB = MSB
MSWG
FAxB = MSAxB
MSWG
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Computing the Denominator: MSWG
SSWG = ( squared scores) – ( first group)2 + ( last group)2
n
n
Denominator degrees of freedom:
dfWG = N – k(q)
Note: k = number of groups in Factor A
q = number of groups in Factor B
MSWG =
SSWG
dfWG
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Numerator degrees of freedom:
MSA, MSB, MSAxB,
Main Effect of A MSA
Main Effect of B MSB
A x B Interacti o n MSAxB
degrees of freedom
dfA = k - 1
dfB = q - 1
dfAxB = (k – 1)(q – 1)
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Computing a Two-way
Independent-groups
ANOVA by hand
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Data Set
Subject
1
2
3
4
5
6
7
8
9
10
11
12
Type of
Drug
Age
Fen
Fen
Fen
Fen
Amph
Amph
Amph
Amph
Salt
Salt
Salt
Salt
5 Day
5 Day
15 Day
15 Day
5 Day
5 Day
15 Day
15 Day
5 Day
5 Day
15 Day
15 Day
Percent
Weight Gain
1
2
4
6
3
6
15
14
10
8
12
10
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Critical Values for F
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Computing the Two-way
Independent-groups
ANOVA using SPSS
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SPSS Data Editor
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1. Select WtGain as DV
2. Select Age and Drug as
the IVs (Fixed Factors).
3. Click on “Post-Hoc”
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Now … think carefully!
Which of the IVs requires a follow-up (post-hoc) test?
Which post-hoc procedure(s) will you choose? Why?
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Note that you can also get follow-up tests by clicking on
the “Options” button. This box also allows you to get
“Descriptive Statistics”, “Estimates of Effect Size”, and
“Homogeneity Tests”, if you want these …
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Descriptive
Statistics
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Main ANOVA Summary Table
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Post-hoc Tests: Age
Did we need to request
a post-hoc test for the
“Age” factor?
Why or why not?
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Post-hoc Tests: Type of Drug
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Reporting the Results
Mean Percent Weight Gain
The results showed a significant Age x Drug interaction, F(2,6) = 9.43, p < .05. As shown
in Figure 1, appetite suppression at both ages differs depending on type of drug injected.
Percent weight gain was different for fenfluramine than for amphetamine or saltwater.
Specifically, 15-day-old rats gained significantly more weight after receiving amphetamine
than 5-day-old rats, t(2) = 6.33, p<.05. The results showed a significant main effect of
Age, F(1,6) = 41.78, p < .05. The mean percent weight gain was significantly greater for
15-day-olds (M=10.2, SD=4.4)) than for 5-day-olds (M=5.0, SD=3.6). The results also
showed a significant main effect of Type of Drug, F(2,6) = 29.52, p < .05. Fenfluramine
was significantly more effective than either amphetamine or saltwater. Post-hoc
comparisons (Bonferroni) showed weight gain was significantly lower for rats injected with
fenfluramine (M=3.3, SD=2.2) than rats injected with either amphetamine (M=9.5, SD=5.9)
or saltwater (M=10.0, SD=1.6) which did not differ from each other (ps>.05)
15
10
5-Day
15-Day
5
0
Amphetamine
Fenfluramine
Saltwater
Figure 1. Mean percentage weight gain for 5-day-old and 15-day-old rats
as a function of injections of amphetamine, fenfluramine, or saltwater.
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Calculating Effect Size:
Two-way Independent-groups ANOVA - 2
2
effect
=
2Age =
SSM - (dfM x MSR)
SST + MSR
2Drug =
2AxB =
80.08 - (1 x 1.917)
113.167 - (2 x 1.917)
36.167 - (2 x 1.917)
240.917 + 1.917
240.917 + 1.917
240.917 + 1.917
78.163
242.834
= .32
109.33
242.834
= .45
32.33
242.834
= .13
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Practice Problem:
Goggles.sav
An researcher interested in the effects of alcohol on mate selection at
night clubs hypothesizes that after alcohol has been consumed,
subjective perceptions of physical attractiveness will become more
inaccurate. She is also interested in whether this effect differs for men
and women.
She selects 48 students for the study: 24 are males and 24 are
females. All participants are taken to a night club and are given either
non-alcoholic beer; 2 pints of “real” beer; or 4 pints of “real” beer.
Assume that the non-alcoholic beer tastes identical to the real beer.
At the end of the night, she photographs the person the participant
was with. She then has an independent panel of judges to assess the
attractiveness of the person in each photograph.
How will you analyze this problem?
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