ch14_GW_factorial_design_posting
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Transcript ch14_GW_factorial_design_posting
Two-Factor ANOVA
Outline
Basic logic of a two-factor ANOVA
Recognizing and interpreting main &
interaction effects
F-ratios
How to compute & interpret a two-way
ANOVA
Assumptions
Extension of Factorial ANOVA
Factorial Designs
Move beyond the one-way ANOVA to designs
that have 2+ IVs
The variables can have unique effects or can
combine with other variables to have a
combined effect
Why Should We Use a Factorial
Design?
We can examine the influence that each
factor by itself has on a behaviour, as well
as the influence that combining these
factors has on the behaviour
Can be efficient and cost-effective
Interpretation of Factorial
Designs
Two Kinds of Information:
1.
Main effect of an IV
– Effect that one IV has independently
of the effect of the other IV
– Design with 2 IVs, there are 2 main
effects (one for each IV):
Main Effect of Factor A (1st IV): Overall difference among the levels of A
collapsing across the levels of B.
Main Effect of Factor B (2nd IV): Overall difference among the levels of B
collapsing across the levels of A.
Interpretation of Factorial
Designs
Two Kinds of Information:
2.
Interaction
– Represent how independent variables work
together to influence behavior
– The relationship between one factor and the
DV change with, or depends on, the level of
the other factor that is present
– The influence of changing one factor is NOT
the same for each level of the other factor
Two-Way ANOVA
F= variance between groups
variance within groups
In a 2-way ANOVA, there are 3 F-ratios:
1. Main effect for Factor A
2. Main effect for Factor B
3. Interaction A x B
Guidelines for the Analysis of a
Factorial Design
First determine whether the interaction between
the independent variables is statistically
significant.
– If the interaction is statistically significant,
identify the source of the interaction by
examining the simple main effects
– Main effects should be interpreted cautiously
whenever an interaction is present in an
experiment
Then examine whether the main effects of each
independent variable are statistically significant.
Analysis of Main Effects
When a statistically significant main effect
has only 2 levels, the nature of the
relationship is determined in the same
manner as for the independent samples ttest
When a main effect has 3 or more levels,
the nature of the relationship is
determined using a Tukey HSD test
Effect Size
Three different values of ŋ2 are computed
ŋ2 for Factor A =
SSA_______
SStotal – SSB - SSAxB
ŋ2 for Factor B =
SSB_______
SStotal – SSA - SSAxB
ŋ2 for Factor AxB =
SSAxB______
SStotal – SSA - SSB
Effect Size – alternate formulas
2
2
2
A
SS A
SS A SS within
B
SS B
SS B SS within
AxB
SS AxB
SS AxB SS within
Assumptions
The observations within each sample must
be independent
DV is measured on an interval or ratio
scale
The populations from which the samples
are selected have must have equal
variances
The populations for which the samples are
selected must be normally distributed
Calculating 2 Factor Between
Subjects Design ANOVA by hand
Influence of a specific hormone on eating behaviour
IV (A): Gender
– Males
– Females
IV (B): Drug Dose
– No drug
– Small dose
– Large dose
DV: Eating consumption over a 48-hour period
The Data ….
Factor B – Amount of drug
No drug
Factor A - Gender
Male
Female
Small dose
Large dose
1
7
3
6
7
1
1
11
1
1
4
6
1
6
4
0
0
0
3
0
2
7
0
0
5
5
0
5
0
3
Homogeneity of variance
=
s2 largest =
s2 smallest
Satisfied
or violated???
Step 1: State the Hypotheses
Main Effect for Factor A
Main Effect for Factor B
Step 1: State the Hypotheses
Interaction between dosage & gender
Step 2: Compute df
Double Check:
dftotal= dfbetween + dfwithin
Step 3: Determine F-critical
Use the F distribution table
F Critical (df effect, df within)
Using = .05
Step 4: Calculate SS
SSTOTAL = 2 – G2
N
Step 4:
Calculate SS
SSBETWEEN Tx = T2 – G2
n
N
Step 4: Calculate SS
SSWITHIN TX = SS
inside each treatment
Double Check SS
SS Total SS Within Tx
SS BetweenTx
SS for Factor A
SS A = Trow2 – G2
nrow
N
SS for Factor B
SS B = TColumn2 – G2
nColumn
N
SS for Interaction
SS AxB SS Between SS A SS B
Step 5: Calculate MS for
Factor A
MSA = SSA
dfA
Step 5: Calculate MS for
Factor B
MSB = SSB
dfB
Step 5: Calculate MS for
Interaction
MSAxB = SSAxB
dfAxB
Step 5: Calculate MS Within
Treatments
MSwithin = SSwithin
dfwithin
Step 6: Calculate F ratios –
Factor A
MS A
F
MS within
Step 6: Calculate F ratios –
Factor B
MS B
F
MS within
Step 6: Calculate F ratios –
Interaction
MS AxB
F
MS within
Step 7: Summary Table
Source
Between Tx
Factor A
Factor B
Interaction
Within Tx
Total
SS
df
MS
F
Extension of Factorial ANOVA
1 factor is between subject & 1 factor is
within subject
e.g.: pre-post-control design
– All subjects are given a pre-test and a
post-test
– Participants divided into two groups
– Experimental group vs. control group
2 x 3 mixed design
Group
Therapy
Time
Control
Between-Subjects
Before
After
3 mos. after
Within-Subjects