Post on 07-Feb-2016
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Monday, November 15
Analysis of Variance
Monday, November 15
The Analysis of Variance
Monday, November 15
The Analysis of VarianceANOVA
F = Between-group variance estimate
within-group variance estimate
SST = (X - XG)2
SSB = Ni (Xi - XG)2
SSW = (X1 - X1)2 +
(X2 - X2)2 + •••• (Xk - Xk)2
SST = SSB + SSW
_
_ _
__
_
Between-group variance estimate
within-group variance estimate
MSB = SSB / dfB
MSW = SSW / dfW
where
dfB = k-1 (k = number of groups)dfW = N - k
F =
Fisher’s Protected t-test
t = Xi - Xj
MSW ( 1/Ni + 1/Nj)
__
Where df = N - k
Est ω = dfB (F - 1)
dfB F + dfW
Est ω bears the same relationship to F that rpb bears to t.
The factorial design is used to study the relationship of two or more independent variables (called factors) to a dependent variable, where each factor has two or more levels. - p. 333
The factorial design is used to study the relationship of two or more independent variables (called factors) to a dependent variable, where each factor has two or more levels.
In this design, you can evaluate the main effects of each factor independently (essentially equivalent to doing one-way ANOVA’s for each of the factors independently), but you are also able to evaluate how the two (or more factors) interact.
0.5 1.0 1.5 2.0 2.5 3.0 3.5SES
30
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80RDG
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RACE
0.5 1.0 1.5 2.0 2.5 3.0 3.5SES
30
40
50
60
70
80MATH
1234
RACE
TOTAL VARIATION
Variation within groups (error)
Variation between groups
Variationfrom Factor 1
Variationfrom Factor 2
Variation from Factor 1 x 2 interaction
Partitioning variation in a 2x2 factorial design.
1.A Compute SST
B. Compute SSB
C. Subtract SSB from SST to obtain SSW (error) D. Compute SS1
E. Compute SS2
F. Compute SS1x2 by subtracting SS1 and SS2 from SSB
2. Convert SS to MS by dividing SS by the appropriate d.f.
3. Test MS1,MS2 and MS1x2 using F ratio.
More advanced ANOVA topics
• N-way ANOVA
• Repeated Measures designs
• Mixed models
• Contrasts