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Dr. Sinn, PSYC 301 2 Way ANOVA
Factorial Design Two Way ANOVAs
• 2 Independent Variables• Examples
– IV#1 IV#2 DV
– Drug Level Age of Patient Anxiety Level
– Type of Therapy Length of Therapy Anxiety Level
– Type of Exercise Type of Diet Weight Change
– Toy Color Gender Satisf. with Toy
• Key Advantages– Compare relative influences on DV
– Examine interactions between IV
Dr. Sinn, PSYC 301 2 Way ANOVA
Example Two Way ANOVAs
• Toy Study– IV: Toy Color (Blue, Pink)– IV: Gender (Boy, Girl)– DV: Satisfaction with Toy
• Terms
– Factors: __ * ___ * ___
– Levels (a,b)
– Design: ___ x ___
– Main Effect, collapsing
– Interaction
Dr. Sinn, PSYC 301 2 Way ANOVA
Main Effects Two Way ANOVAs
• Main Effect for Toy Color?– Compare Column
Means
Toy Color
Blue (1)
Pink (2)
Sex
Boy (1)
7
6
5
2
3
4
Girl (2)
4
5
6
12
10
11
M=5.5 M=7.0
M=8.0
M=4.5• Main Effect for
Gender?– Compare Row
Means
Dr. Sinn, PSYC 301 2 Way ANOVA
Interactions- Cell Means Two Way ANOVAs
Toy Color
Blue (1) Pink (2)
Sex
Boy (1)
7
6
5
2
3
4
Girl (2)
4
5
6
12
10
11
M=6 M=3
M=5 M=11
Graph cell means to examine
possibility of interaction
Dr. Sinn, PSYC 301 2 Way ANOVA
Interactions-Graph Two Way ANOVAs
• General rule of life: – If two lines cross, it probably means something.
0
2
4
6
8
10
12
Blue Toy Pink Toy
Boy
Girl
Non-parallel lines suggests interaction.
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS: Analysis, Step #1 Two Way ANOVAs
• Go to Analyze, General Linear Model, Univariate
• Move DV to Dependent Variable
• Move 2 IVs to Fixed Faxtors
Step #1
Step #2
Step #3
Step #4
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS: Analysis, Step #2 Two Way ANOVAs
• Select Plots; Graph sample means with two IVs
• If one IV has more levels, put on Horizontal Axis
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS: Analysis, Step #3 Two Way ANOVAs
• Select Options
• Ask for Descriptive Statistics
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS: Analysis, Step #4 Two Way ANOVAs
• Select Post Hoc• Do Post Hoc (SNK) for IVs with 3+ levels• Not required in this example; both IVs have only 2 levels:
color (blue & pink), sex (boy & girl)
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS Output #1 Two Way ANOVAs
Descriptive Statistics
Dependent Variable: satisf
6.00 1.000 3
5.00 1.000 3
5.50 1.049 6
3.00 1.000 3
11.00 1.000 3
7.00 4.472 6
4.50 1.871 6
8.00 3.406 6
6.25 3.194 12
sexboy
girl
Total
boy
girl
Total
boy
girl
Total
colorblue
pink
Total
Mean Std. Deviation N
Between-Subjects Factors
blue 6
pink 6
boy 6
girl 6
1
2
color
1
2
sex
ValueLabel N
Dr. Sinn, PSYC 301 2 Way ANOVA
Tests of Between-Subjects Effects
Dependent Variable: satisf
104.250a 3 34.750 34.750 .000
468.750 1 468.750 468.750 .000
6.750 1 6.750 6.750 .032
36.750 1 36.750 36.750 .000
60.750 1 60.750 60.750 .000
8.000 8 1.000
581.000 12
112.250 11
SourceCorrected Model
Intercept
color
sex
color * sex
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .929 (Adjusted R Squared = .902)a.
SPSS Output #2: Two Way ANOVAs
BG
WG
Dr. Sinn, PSYC 301 2 Way ANOVA
SPSS Output #3: Two Way ANOVAs
blue pinkcolor
2
4
6
8
10
12
Es
tim
ate
d M
arg
ina
l M
ea
ns sex
boy
girl
Estimated Marginal Means of satisf
Note: post-hoc tests are needed only when you have 3+ levels of an IV (here we don’t).
Dr. Sinn, PSYC 301 2 Way ANOVA
Write-up Two Way ANOVAs
• The hypotheses were supported. [1] There was a
main effect for toy color. Pink toys (M=7.00)
elicited significantly more satisfaction than blue toys
(M=5.5), F(1,8) = 6.750, p≤ .05. [2]There was also
a main effect for sex. Girls were significantly more
satisfied (M=8.00) than boys (M=4.50),
F(1,8)=36.750, p≤ .05.
Dr. Sinn, PSYC 301 2 Way ANOVA
Write-up (cont.)Two Way ANOVAs
• [3] Additionally, there was a significant interaction
between color and sex, F(1,8) = 60.75, p≤.05.
Boys and girls appear equally satisfied with blue Boys and girls appear equally satisfied with blue
toys. Switching to pink toys, however, raised toys. Switching to pink toys, however, raised
satisfaction for girls but decreased satisfaction for satisfaction for girls but decreased satisfaction for
boys.boys. Sex accounted for only a small amount of
variance in satisfaction (η2 = .0601), but color (η2
= .3274) and the interaction (η2 = .5412) accounted
for a large amount of variance.
Dr. Sinn, PSYC 301 2 Way ANOVA
Two-Way ANOVA Cont.
• Announcements
• Review Study Guide for Final
• Homework: Influence Study
• Homework: Teamwork & Feedback Study, write-ups
• Explain Purpose of 2nd ANOVA Lab
• Studying for Final– Computational Review for Final
– Review Name That Stat Exercises
– Practice SPSS on computer
– Review Old Computations
Dr. Sinn, PSYC 301 2 Way ANOVA
Source of Variation Table for 2-way ANOVA
• Three possible influences on DV -- factors– A: IV #1
– B: IV #2
– C: Interaction
• Sum of Squares (SS) always given
• Calculating Degrees of Freedom by hand
– dfA = a-1
– dfB = b-1
– dfA*B = (a-1)*(b-1)
– dfwg = a * b * (n-1)
– dfTotal = N – 1
Dr. Sinn, PSYC 301 2 Way ANOVA
Table Reading Keys
1. Three F’s use same formula• MSBG / MSWG = MSSpecific Factor / MSError
• For example: MSA / MSError
2. Factor significant if p ≤ .05
3. MS = SS/df for each factor and error
Dr. Sinn, PSYC 301 2 Way ANOVA
Tests of Between-Subjects Effects
Dependent Variable: satisf
104.250a 3 34.750 34.750 .000
468.750 1 468.750 468.750 .000
6.750 1 6.750 6.750 .032
36.750 1 36.750 36.750 .000
60.750 1 60.750 60.750 .000
8.000 8 1.000
581.000 12
112.250 11
SourceCorrected Model
Intercept
color
sex
color * sex
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .929 (Adjusted R Squared = .902)a.
Source of Variation Table from Toy Study
BG
WG
Dr. Sinn, PSYC 301 2 Way ANOVA
Age & Intelligence (2-way ANOVA)
Task
Fluid Crystalized
Age
65 105
100
95
100
100
95
110
100
75 85
90
95
85
105
95
100
105
85 85
80
75
80
105
95
100
100
Dr. Sinn, PSYC 301 2 Way ANOVA
Important Means
• Main effect for Task?• Main effect for Age?• Graph it
Dr. Sinn, PSYC 301 2 Way ANOVA
Calculate degrees of freedom by hand:
• dfA
• dfB
• dfA*B
• dfError
• dfTotal
Dr. Sinn, PSYC 301 2 Way ANOVA
Complete Table with these SS
• SSTask = 759.375
• SSAge = 452.083
• SSTask*Age = 356.250
• SSError = 406.250
Dr. Sinn, PSYC 301 2 Way ANOVA
Check Output
• What means pertain to…– Effect of Task
– Effect of Age
– Effect of interaction
• Is there a ….– Main effect for Task
– Main effect for Age
– Interaction
• Is Post Hoc Required?• Explain graph• Do complete Write-up
Dr. Sinn, PSYC 301 2 Way ANOVA
2-way ANOVA: Age & Intelligence
• First, I’d like to thank my statistics teacher for devising such a creative, exciting, and enriching exercise. My life will never be the same.
Dr. Sinn, PSYC 301 2 Way ANOVA
2-way ANOVA: Age & Intelligence
• The hypotheses were supported.
• Participants scored significantly lower on tasks using fluid (M=89.58) rather than crystallized intelligence (M=100.83), F(1,18) = 33.46, p<=.05.
• In addition, participants aged 85 years scored lower (M=90.00) than those aged 75 years (M=95.00), who in turn scored lower than those aged 65 years (M=100.63), F(2,18)=10.015, p<=.05.
• Additionally, age interacted with type of task, F(2,18)=7.812, p<=.05. Although scores on crystallized tasks remain relatively constant, scores on fluid tasks decline with age.
I’m still smarter than you are, missy.
Dr. Sinn, PSYC 301 2 Way ANOVA
Bogus Winthrop Data – 2-way ANOVA
• Some of the hypotheses were supported. • There was a main effect for residence. On-campus
students earned higher GPAs (M=3.2545) than off-campus students (M=1.9667), F(1,14)=21.625,p<=.05.
• However, there was no main effect for program. GPAs for students in the control (M=2.5857), mentoring (M=2.5286), and study hall condition (M=2.9500) did not differ significantly, F(2,14)=.069, n.s.
• There was no interaction, F(2,14)=.205, n.s.• Residence accounted for a moderate amount of variance
in GPA, eta2 = .5132.• Overall, it appears residence, but not type of program,
affects GPA.