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TwoWay ANOVA

In a twoway ANOVA there is one dependent variable and multiple
independent variables. The twoway ANOVA can not only determine
the main effect of contributions of each IV but also identifies
if there is a significant interaction effect between them.
Linear Model

y_{ijk} = μ + α_{i} + β_{j} + γ_{ij} + e_{ijk}
Where,
μ = grand mean value
α_{i} = main effect for ith row.
β_{j} = main effect for jth column.
γ_{ij} = interaction effect for ith row and jth column
SPSS: Analyze>General Linear Model>Univariate
Test statistic: F test.
H_{0}: no interaction effects
H_{0}: γ_{ij} = 0, all i and j
H_{0}: no row effects
H_{0}: α_{i} = 0, all i
H_{0}: no column effects
H_{0}: β_{j} = 0, all j
The 2way ANOVA can be used to analyze factorial designs.
Factorial Design

A factorial experiment is an experiment whose design consists of
two or more factors, each with discrete possible levels and whose
experimental units take on all possible combinations of these
levels across all such factors
Consider a 2 x 4 factorial design:
1 2 3 4 < Factor A
X ? ? ? ?
Y ? ? ? ?
Error
^

Factor B
The mathematics involved in analyzing the 2way ANOVA is along
the same lines as the 1way ANOVA. However, the computations
involved are tedious and best handled by a statistical package
like SPSS.
df SS MS F
   
Factor A c  1 SSX MSSX = SSX/(c  1) MSSX/MSSE
Factor B r  1 SSY MSSY = SSY/(r  1) MSSY/MSSE
Fac A * Fac B (c  1)(r  1) SS(XY) MSS(XY) = SS(XY) MSS(XY)/MSSE

(c  1)(r  1)
Error n  cr SSE MSSE = SSE/(n  cr)
Total n  1
c = number of columns.
r = number of rows.
Eaxample:
Consider an experiment with 4 different brands of golf ball
(A, B, C, D) and 2 different clubs (driver and 5 iron). The
average distance hit is computed for all possible combinations.
Brand
A B C D
Club X 228.43 233.73 243.10 229.75
Club Y 171.30 182.68 167.18 160.50
The SPSS printout looks like:
df SS MS F p
    
Model 7 33659.80875 4808.54411 140.35 <0.0001
df SS MS F p
    
Club 1 32093.11125 32093.11125 936.75 <0.0001
Brand 3 800.73625 266.91208 7.79 0.0008
Club*Brand 3 765.96125 255.32042 7.45 0.0011
Total 7 33659.80875 *** ***
*** Total ≠ column totals.
The model tells us that at least 2 of the means are different.
Brand and Club interact so there is no point in testing the
main effects. Instead it is necessary to compare the treatment
means to learn the nature of the interaction.