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”
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