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

40

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80RDG

1234

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