Factorial Design Two Way ANOVAs

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


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


Main Effects Two Way ANOVAs

• Main Effect for Toy Color?– Compare Column

Means

Toy Color

Blue (1)

Pink (2)

Sex

Boy (1)

765

234

Girl (2)

456

121011

M=5.5 M=7.0

M=8.0

M=4.5• Main Effect for

Gender?– Compare Row

Means


Interactions- Cell Means Two Way ANOVAs

Toy Color

Blue (1) Pink (2)

Sex

Boy (1)

765

234

Girl (2)

456

121011

M=6 M=3

M=5 M=11

Graph cell means to examine

possibility of interaction


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.


SPSS: Data Input #1 Two Way ANOVAs


SPSS: Data Input #2 Two Way ANOVAs


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



• Select Plots; Graph sample means with two IVs• If one IV has more levels, put on Horizontal Axis



• Select Options• Ask for Descriptive Statistics



• 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)


SPSS Output #1 Two Way ANOVAs

Descriptive Statistics

Dependent Variable: satisf

6.00 1.000 35.00 1.000 35.50 1.049 63.00 1.000 3

11.00 1.000 37.00 4.472 64.50 1.871 68.00 3.406 66.25 3.194 12

sexboygirlTotalboygirlTotalboygirlTotal

colorblue

pink

Total

Mean Std. Deviation N

Between-Subjects Factors

blue 6pink 6boy 6girl 6

12

color

12

sex

ValueLabel N


Tests of Between-Subjects Effects


104.250a 3 34.750 34.750 .000468.750 1 468.750 468.750 .000

6.750 1 6.750 6.750 .03236.750 1 36.750 36.750 .00060.750 1 60.750 60.750 .0008.000 8 1.000

581.000 12112.250 11

SourceCorrected ModelInterceptcolorsexcolor * sexErrorTotalCorrected 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


SPSS Output #3: Two Way ANOVAs

blue pinkcolor

2

4

6

8

10

12

Est

imat

ed M

argi

nal M

eans sex

boygirl

Estimated Marginal Means of satisf

Note: post-hoc tests are needed only when you have 3+ levels of an IV (here we don’t).


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.


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.


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


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


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


Tests of Between-Subjects Effects


104.250a 3 34.750 34.750 .000468.750 1 468.750 468.750 .000

6.750 1 6.750 6.750 .03236.750 1 36.750 36.750 .00060.750 1 60.750 60.750 .0008.000 8 1.000

581.000 12112.250 11

SourceCorrected ModelInterceptcolorsexcolor * sexErrorTotalCorrected 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


Age & Intelligence (2-way ANOVA)

TaskFluid Crystalized

Age

65 10510095100

10095110100

75 85909585

10595100105

85 85807580

10595100100


Important Means

• Main effect for Task?• Main effect for Age?• Graph it


Calculate degrees of freedom by hand:

• dfA

• dfB

• dfA*B

• dfError

• dfTotal


Complete Table with these SS

• SSTask = 759.375

• SSAge = 452.083

• SSTask*Age = 356.250

• SSError = 406.250


SPSS Data Entry


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


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.


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.


Interpreting 2-way Outcomes


Interpreting 2-way Outcomes (cont.)


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.


Is there a sig. difference in funniness?Yet another excuse for a 1-way Anova

#1#2

#3

Rate each video on the following scale:

Not funny 1 2 3 4 5 6 7 Funny

Factorial Design Two Way ANOVAs

Documents

Transcript of Factorial Design Two Way ANOVAs