Introduction to 2-way ANOVAIntroduction to 2-way ANOVA

Statistics

Spring 2005

TerminologyTerminology

2-Way ANOVA means 2 independent variables 1 dependent variable

3X4 ANOVA means 2 independent variables 1 dependent variable one IV has 3 levels one IV has 4 levels

HYPOTHESES TESTEDHYPOTHESES TESTEDin 2-WAY ANOVAin 2-WAY ANOVA

No differences for IV #1 (A - 3 levels) H0: MA1 = MA2 = MA3

No differences for IV #2 (B - 4 levels) H0: MB1 = MB2 = MB3 = MB4

No interaction At least one MAiBj MAmBn

These are called “Main Effects”

EXAMPLEEXAMPLE One might suspect that level of

education and gender both have significant impacts on salary. Using the data found inCensus90 condensed.savdetermine if this statement is true.

Dependent Variable

INCOME

(ratio level data)

Independent Variables

GENDER (2 levels)

EDUCAT (6 levels)

= .05

No differences for GENDER (2 levels)H0: MMale = MFemale

No differences for EDUCATION (6 levels)H0: MB1 = MB2 = MB3 = MB4 = MB5 = MB6

No interactionAt least one MAiBj MAmBn

HYPOTHESES TESTEDfor a 2X6 ANOVA

To run the test of these hypotheses in SPSS…..

Analyze General Linear Model Univariate

NOTE: Use this method of analysis when both IV’s are not repeated measures.

Estimated Marginal Means of Wage or salary income, 1989

EDUCAT

Graduate Degree

Bachelors Degree

Some college

HS diploma

HS - no diploma

<= 9th grade

Est

ima

ted

Ma

rgin

al M

ea

ns

70000

60000

50000

40000

30000

20000

10000

0

Sex

Male

Female

Tests of Between-Subjects Effects

Dependent Variable: Wage or salary income, 1989

5.946E+10a 11 5405064664 12.887 .000

1.781E+11 1 1.781E+11 424.577 .000

1.214E+10 1 1.214E+10 28.951 .000

2.884E+10 5 5768187295 13.752 .000

6215428787 5 1243085757 2.964 .012

1.976E+11 471 419428810.4

4.921E+11 483

2.570E+11 482

SourceCorrected Model

Intercept

SEX

EDUCAT

SEX * EDUCAT

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .231 (Adjusted R Squared = .213)a.

Tests of Between-Subjects Effects

Dependent Variable: Wage or salary income, 1989

5.946E+10a 11 5405064664 12.887 .000

1.781E+11 1 1.781E+11 424.577 .000

1.214E+10 1 1.214E+10 28.951 .000

2.884E+10 5 5768187295 13.752 .000

6215428787 5 1243085757 2.964 .012

1.976E+11 471 419428810.4

4.921E+11 483

2.570E+11 482

SourceCorrected Model

Intercept

SEX

EDUCAT

SEX * EDUCAT

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .231 (Adjusted R Squared = .213)a.

Source SS df MS F pGender 12,142,860,574 1 12,142,860,574 28.95 0.000Education 28,840,936,475 5 5,768,187,295 13.75 0.000Gender X Education 6,215,428,787 5 1,243,085,757 2.96 0.012Residual 197,550,969,717 471 419,428,810

Table 1Results of the 2-way ANOVA (Gender X Education) for Income.

No differences for GENDER(2 levels)

H0: MMale = MFemale

No differences for EDUCATION (6 levels)

H0: MB1 = MB2 = MB3 = MB4 = MB5 = MB6

No interactionAt least one MAiBj MAmBn

HYPOTHESES TESTEDfor a 2X6 ANOVA

Reject H0

(F(1,471)=29.95: p=.000)

Reject H0

(F(5,471)=13.75: p=.000)

Reject H0

(F(5,471)=2.96: p=.012)

Types of 2-Way ANOVA designsTypes of 2-Way ANOVA designs

Both IV’s are between subjects(i.e. not-repeated measures)

Both IV’s are within subjects(i.e. repeated measures)

One IV is between subjects, the other IV is within subjects

Both IV’s are between subjects(i.e. not-repeated measures)

Analyze General Linear Model Univariate

Both IV’s are within subjects(i.e. repeated measures)

Analyze General Linear Model Repeated Measures

Analyze General Linear Model Repeated Measures

Estimated Marginal Means of MEASURE_1

TRIAL

21

Est

ima

ted

Ma

rgin

al M

ea

ns

41.8

41.6

41.4

41.2

41.0

40.8

DAY

1

2

Tests of Within-Subjects Effects

Measure: MEASURE_1

.443 1 .443 .214 .647

.443 1.000 .443 .214 .647

.443 1.000 .443 .214 .647

.443 1.000 .443 .214 .647

64.123 31 2.068

64.123 31.000 2.068

64.123 31.000 2.068

64.123 31.000 2.068

2.872 1 2.872 1.971 .170

2.872 1.000 2.872 1.971 .170

2.872 1.000 2.872 1.971 .170

2.872 1.000 2.872 1.971 .170

45.178 31 1.457

45.178 31.000 1.457

45.178 31.000 1.457

45.178 31.000 1.457

5.686 1 5.686 4.719 .038

5.686 1.000 5.686 4.719 .038

5.686 1.000 5.686 4.719 .038

5.686 1.000 5.686 4.719 .038

37.351 31 1.205

37.351 31.000 1.205

37.351 31.000 1.205

37.351 31.000 1.205

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

SourceDAY

Error(DAY)

TRIAL

Error(TRIAL)

DAY * TRIAL

Error(DAY*TRIAL)

Type III Sumof Squares df Mean Square F Sig.

Tests of Between-Subjects Effects

Measure: MEASURE_1

Transformed Variable: Average

218728.237 1 218728.237 1867.847 .000

3630.156 31 117.102

SourceIntercept

Error

Type III Sumof Squares df Mean Square F Sig.

Source SS df MS F pSubjects 3630.16 31 117.10DAY 0.44 1 0.44 0.21 0.647Error(DAY) 64.12 31 2.07TRIAL 2.87 1 2.87 1.97 0.17Error(TRIAL) 45.18 31 1.46DAY * TRIAL 5.69 1 5.69 4.72 0.038Residual 37.35 31 1.21

Table 2Summary of the 2X2 repeated measures ANOVA (Day X Trial) for the test data.

One IV is between subjects, other IV is within subjects

Analyze General Linear Model Repeated Measures

Estimated Marginal Means of MEASURE_1

TRIAL

21

Est

ima

ted

Ma

rgin

al M

ea

ns

42.5

42.0

41.5

41.0

40.5

40.0

SEX

Female

Male

Tests of Within-Subjects Effects

Measure: MEASURE_1

8.144 1 8.144 4.773 .037

8.144 1.000 8.144 4.773 .037

8.144 1.000 8.144 4.773 .037

8.144 1.000 8.144 4.773 .037

.159 1 .159 .093 .762

.159 1.000 .159 .093 .762

.159 1.000 .159 .093 .762

.159 1.000 .159 .093 .762

51.193 30 1.706

51.193 30.000 1.706

51.193 30.000 1.706

51.193 30.000 1.706

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

SourceTRIAL

TRIAL * SEX

Error(TRIAL)

Type III Sumof Squares df Mean Square F Sig.

Tests of Between-Subjects Effects

Measure: MEASURE_1

Transformed Variable: Average

108429.072 1 108429.072 2027.326 .000

23.134 1 23.134 .433 .516

1604.514 30 53.484

SourceIntercept

SEX

Error

Type III Sumof Squares df Mean Square F Sig.

Source SS df MS F pSEX 23.13 1 23.13 0.43 0.516Error 1604.51 30 53.48TRIAL 8.14 1 8.14 4.77 0.037TRIAL * SEX 0.16 1 0.16 0.09 0.762Residual 51.19 30 1.71

Table 3.Analysis of the 2X2 ANOVA (Gender x Trial) with repeated measures on the second factor for the test data.

Source SS df MS F pSEX 23.13 1 23.13 0.43 0.516Error 1604.51 30 53.48TRIAL 8.14 1 8.14 4.77 0.037TRIAL * SEX 0.16 1 0.16 0.09 0.762Residual 51.19 30 1.71

Table 3.Analysis of the 2X2 ANOVA (Gender x Trial) with repeated measures on the second factor for the test data.

Source SS df MS F pGender 12,142,860,574 1 12,142,860,574 28.95 0.000Education 28,840,936,475 5 5,768,187,295 13.75 0.000Gender X Education 6,215,428,787 5 1,243,085,757 2.96 0.012Residual 197,550,969,717 471 419,428,810

Table 1Results of the 2-way ANOVA (Gender X Education) for Income.

Source SS df MS F pSubjects 3630.16 31 117.10DAY 0.44 1 0.44 0.21 0.647Error(DAY) 64.12 31 2.07TRIAL 2.87 1 2.87 1.97 0.17Error(TRIAL) 45.18 31 1.46DAY * TRIAL 5.69 1 5.69 4.72 0.038Residual 37.35 31 1.21

Table 2Summary of the 2X2 repeated measures ANOVA (Day X Trial) for the test data.

HYPOTHESES TESTEDHYPOTHESES TESTEDin 2-WAY ANOVAin 2-WAY ANOVA

No differences for IV #1 (A - 3 levels) H0: MA1 = MA2 = MA3

No differences for IV #2 (B - 4 levels) H0: MB1 = MB2 = MB3 = MB4

No interaction At least one MAiBj MAmBn

These are called “Main Effects”