Two-Way ANOVA - webs.wofford.edu 11_Two... · Golf anyone? A sports psychologist is interest in comparing the effects of two instructional methods on golf performance. 100 novice

Two-Way ANOVA

Chapter 11

Golf anyone?

A sports psychologist is interest in comparing the

effects of two instructional methods on golf performance.

100 novice golfers (50 boys and 50 girls)

Method I Method II

65 55

IV: Method Type

DV: Golf proficiency

Gender Effects

100 novice golfers

50 Boys 50 Girls

60 60

IV: Sex

DV: Golf proficiency

Can we ignore sex in our interpretation of the Method Type effect?

Factorial Design

Method

I II

65 55

60

60

Girls

Boys

55 65

75 45

20

30

40

50

60

70

80

Girls Boys

I

II

The instructional method interacts with sex

“Interaction” Defined

An interaction is present when the effects of one IV

depend upon a second IV

“Interaction effect”: The effect of each IV across the levels of the other IV

When there is an Ia, the effect of one IV depends on the level of the other IV

20

30

40

50

60

70

80

Girls Boys

I

II

Factorial designs:

Designs with more than 1 IV

Levels of processing lab IV: conditions (3 levels: letter, rhyme, sentence)

IV: correct response (2 levels: yes, no)

DV: accuracy

Therapy and disorder IV: therapy (3 levels: psychoanalysis, behavioral, none)

IV: disorder (3 levels: depression, anxiety, schizophrenia)

Complete factorial design All levels of each IV are paired w/ all levels of other IV

Incomplete factorial design Not all levels of each IV are paired

Factorial notation # levels of IV1 x # levels of IV2

E.g.: 2 x 2 design vs. 3 x 2 design vs. 3 x 2 x 4 design etc.!!

Outcomes from Factorial ANOVA

B1 B2

A1 30 30

A2 40 40

Experiment has two factors, A and B

Each has 2 levels (so, 2 x 2 ANOVA)

A1 Mean=30

A2 Mean=40

B1 Mean

=35

B2 Mean

=35

10 point difference

No Difference

Main Effect of A

No Main Effect of B

Data show main effect of A

No main effect of B

No interaction

=70=70


B1 B2

A1 30 40

A2 40 50


2 x 2 ANOVA

A1 Mean=35

A2 Mean=45

B1 Mean

=35

B2 Mean

=45

10 point difference

10 point difference

Main Effect of A

Main Effect of B

Data show main effect of A

Data show main effect of B

No interaction

=80=80


B1 B2

A1 10 20

A2 20 10


2 x 2 ANOVA

A1 Mean=15

A2 Mean=15

B1 Mean

=15

B2 Mean

=15

0 point difference

No Main Effect of A

0 point Difference

No Main Effect of B

No main effect of A

No main effect of B

Data show an interaction

2040

Line graphs:

main effects and interaction

10

20

30

40

50

1 2

10

20

30

40

50

60

1 2

0

5

10

15

20

25

1 2

Yerkes-Dodson relationship:

arousal v. performance

0

1

2

3

4

5

6

7

8

9

10

Low Med High

Easy

Difficult

Studies of 9/11: Talarico & Rubin (2003)

Duke undergraduates tested day after 9/11

Tested again 1, 6, or 32 weeks later

Compared FB to everyday memory event 2 days prior to 9/11

Results: Both types of memory

declined over time

Talarico & Rubin (2003)

Phenomenological properties differ Belief: “I believe the memory occurred in this way”

Recollection: “I feel as though I’m reliving it”

Remember: “I remember the details”

Vividness: “I can see it as if it were happening now”

Two-way ANOVA

2 IVs or factors and 1 dependent variable

Examples:

Stroop list (3 levels) and gender (2 levels) DV: Time

Temperature (3 levels) and humidity (2 levels) DV: Thinking/working proficiency

Vitamin C (3 levels) and susceptibility (2 levels) DV: How many times go to doctor

WM complexity (2 levels) and WM domain (2 levels) DV: Accuracy

Note-taking method (3 levels) and gender (2 levels) DV: Change in GPA (Spring GPA – Fall GPA)

What are benefits of this design?

Gender: Main effect

3030N =

Gender

WomenMen

Ch

an

ge

in

GP

A

1.2

1.0

.8

.6

.4

.2

0.0

-.2

Note-taking method: Main effect

202020N =

Note-Taking methods

ControlMethod 2Method 1

Ch

an

ge

in

GP

A

1.2

1.0

.8

.6

.4

.2

0.0

-.2

Note-taking x Gender: Interaction

101010 101010N =

Note-Taking methods


Ch

an

ge

in

GP

A

1.2

1.0

.8

.6

.4

.2

0.0

-.2

Gender

Men

Women

2-way ANOVA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M1 M2 Cntrl

Male

Female

3x2 ANOVA

Note Taking

M1 M2 Control

Gender Male .335 .640 .165 .380

Female .170 .305 .105 .193

.253 .472 .135

Male main

effect

Female Main

effect

M1 M2 Control

ME ME Main effect (ME)

Hypotheses

Main effect for gender

H0: µM = µF

H1: µM ≠ µF

Main effect for note-taking

H0: µM1 = µM2 = µC

H1: at least 1 mean different

Interaction of gender and note-taking

H0: Mean differences explained by ME

H1: Interaction between factors

SPSS outputDescriptive Statistics

Dependent Variable: Change in GPA

.3350 .22858 10

.6400 .17764 10

.1650 .14916 10

.3800 .26993 30

.1700 .18288 10

.3050 .19214 10

.1050 .14615 10

.1933 .18880 30

.2525 .21853 20

.4725 .24893 20

.1350 .14699 20

.2867 .24938 60

Note-Taking methods

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Gender

Men

Women

Total

Mean Std. Deviation N

Tests of Between-Subjects Effects


1.889a 5 .378 11.463 .000 .515

4.931 1 4.931 149.582 .000 .735

.523 1 .523 15.856 .000 .227

1.174 2 .587 17.809 .000 .397

.193 2 .096 2.921 .062 .098

1.780 54 .033

8.600 60

3.669 59

Source

Corrected Model

Intercept

GENDER

METHOD

GENDER * METHOD

Error

Total

Corrected Total

Type III Sum

of Squares df Mean Square F Significance Eta Squared

R Squared = .515 (Adjusted R Squared = .470)a.

2-way ANOVA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M1 M2 Cntrl

Male

Female

A two-way ANOVA was conducted to test the effects of note-taking methods (method 1, method 2, control) and gender (male, female) on the change in GPA.

A significant main effect of note-taking was found, F (2, 54) = 17.81, p < .001. The change in GPA was significantly different after using method 1 (M = .25, SD = .23), method 2 (M = .47, SD = .25), and control (M = .14, SD = .15) note-taking methods.

Men (M = .38, SD = .27) were found to perform significantly better overall compared to women (M = .19, SD = .19), F (1, 54) = 15.86, p < .001.

No interaction between note-taking and gender was found, F(2, 54) = 2.92, p = .06.

Note-taking x Gender (w/ interaction)

101010 101010N =

Note-Taking methods


Ch

an

ge

in

GP

A

1.2

1.0

.8

.6

.4

.2

0.0

-.2

Gender

Men

Women

2-way ANOVA: w/ interaction

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M1 M2 Cntrl

Male

Female

3x2 ANOVA: w/ interaction

Note Taking

M1 M2 Control

Gender Male .335 .305 .165 .268

Female .170 .640 .105 .305

.253 .472 .135

Male main

effect

Female Main

effect

M1 M2 Control

ME ME Main effect (ME)

Note-taking ANOVANote Taking

M1 M2 Control

Gender Male .335 .305 .165 .268

Female .170 .640 .105 .305

.253 .472 .135

Descriptive Statistics


.3350 .22858 10

.3050 .19214 10

.1650 .14916 10

.2683 .20064 30

.1700 .18288 10

.6400 .17764 10

.1050 .14615 10

.3050 .29254 30

.2525 .21853 20

.4725 .24893 20

.1350 .14699 20

.2867 .24938 60

Note-Taking methods

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Gender

Men

Women

Total


SPSS output (w/ interaction)Descriptive Statistics


.3350 .22858 10

.3050 .19214 10

.1650 .14916 10

.2683 .20064 30

.1700 .18288 10

.6400 .17764 10

.1050 .14615 10

.3050 .29254 30

.2525 .21853 20

.4725 .24893 20

.1350 .14699 20

.2867 .24938 60

Note-Taking methods

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Method 1

Method 2

Control

Total

Gender

Men

Women

Total




1.889a 5 .378 11.463 .000

4.931 1 4.931 149.582 .000

.020 1 .020 .612 .438

1.174 2 .587 17.809 .000

.695 2 .348 10.543 .000

1.780 54 .033

8.600 60

3.669 59

Source

Corrected Model

Intercept

GENDER

METHOD

GENDER * METHOD

Error

Total

Corrected Total

Type III Sum

of Squares df Mean Square F Significance


Write-up

A 3 (note-taking: method 1, method 2, and control) x 2 (gender: male, female) ANOVA was conducted.

A significant main effect of note-taking was found, F (2, 54) = 17.81, p < .001. The change in GPA was significantly different if method 1 (M = .25, SD = .23), method 2 (M = .47, SD = .25), or control method (M = .14, SD = .15) was used.

The main effect of gender was not found to be statistically significant, F (1, 54) = 0.61.

The interaction between the note-taking method and gender was found to be statistically significant, F (2, 54) = 10.54, p < .001.

The interaction suggests that performance for women was best when using note-taking method 2, while men were successful using method 1 or 2.

(Post-hoc tests would be necessary to make above statements!)

SPSS output (note-taking w/ interaction)



1.889a 5 .378 11.463 .000

4.931 1 4.931 149.582 .000

.020 1 .020 .612 .438

1.174 2 .587 17.809 .000

.695 2 .348 10.543 .000

1.780 54 .033

8.600 60

3.669 59

Source

Corrected Model

Intercept

GENDER

METHOD

GENDER * METHOD

Error

Total

Corrected Total

Type III Sum

of Squares df Mean Square F Significance


Source df SS MS F

Between

Within

Total

Factor A

Factor B

A x B

5

1

1.889

0.020 0.6120.020

2 1.174 17.809*

2 0.695 10.543*0.348

0.587

54 1.780 0.033

59 3.669

* p < .01

Partitioning the Variance in Factorial ANOVA

2-way ANOVA

Total Variability

Between-treatments

variability

Within-treatments

variability

Factor A

variability

Factor B

variability

Interaction

variability

atmentwithin tre

AxB)or Bor (A reatment

MS

MSF

t

Degrees of freedom

dftotal = N – 1

dfbetween = # cells -1

dfwithin = df each treatment

dfA = # rows – 1 (# levels for A)

dfB = # columns – 1 (#levels B)

dfAxB = dfbetween – dfA - dfB

dftotal = 30 – 1 = 29

dfbetween = 6 -1 = 5

dfwithin = 4+4+4+4+4+4 = 24

dfA = 2 – 1 = 1

dfB = 3 – 1 = 2

dfAxB = 5 – 1 – 2 = 2

Source Table

Source df SS MS F

Between

Within

Total

Factor A

Factor B

A x B

5

1

240

120 120 24*

2 60 30

2 60 30 6*

6*

24 120 5

29 360

* p < .01

Documents

Two-Way ANOVA - webs.wofford.edu 11_Two... · Golf anyone? A sports psychologist is interest in comparing the effects of two instructional methods on golf performance. 100 novice