3^3 Factorial Experiment
Lecture - 33
A 3 ^3 factorial experiment in which three factors
(say, A, B, C), each at three levels are analyzed.
The three levels (usually) referred to as low, intermediate and high levels.
These levels are numerically expressed as 0, 1, and 2. In 3^3 factorial
experiment there are
total treatment combination have 26 degrees of
freedom. There are 3 main effects each have 2 degree of freedom, 3 two factor
interactions each have 4 degree of freedom and one three factor interaction
with 8 degrees of freedom.
That’s in
factorial
experiment:
A
treatment combination is denoted by X1, X2, X3, Xi i = 0, 1, 2 being level of the ith factor in a particular
combination and takes the values 0, 1, 2. In 3^3 treatment combinations are 000, 001, 002, …,
222. The
effects are represented by;The total degree of freedom (3^3 - 1) treatment combination is partitioned into (3^3-1) / (3-1) = 13
effects each consist of (3 – 1) independent
contrasts, each with 1 degree of freedom.
The 13 effects are:
Each main effects having 2 degrees of freedom can be further
partitioned AL, AQ, BL, BQ, CL. CQ into
each have 1 degree of freedom.
In 3^3 factorial experiment treatment
combination may be displayed as follows:
Generally;
Statistical
Model for
Factorial Experiment
The 3^3 factorial experiment is represented by the
following linear statistical model:
Yijkl =μ + αj + β j + (αβ) jk + γl + (αγ)jl + (βγ)kl + (αβγ)jkl+ ϵijkl
i= 1, 2, ..., r
j , k, l, 0, 1, 2.
Where:
αj : effect of factor A at ith level.
β j : effect of factor B at jth level.
γl : effect of factor C at lth level.
(αβ) jk : interaction effect of factor A at jth level and factor B at kth level called first order interaction.
in the same manner;
(αβγ)jkl : interaction effect of factor A at jth level, factor B at kth level and factor C at lth level called fsecond order interaction.
Analysis of 3 ^3 Factorial Experiment
Analysis
of
Factorial Experiment
Let Yijkl is
the yield of 3 factors each at three levels and each is replicated “r” times.
ANOVA Table:
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Test the highest order
(second order) interaction first.
If the highest order of interaction is non-significant,
then consider the lower order of interaction.
If no interaction is
significant, then test.
We cannot consider (αβγ)jkl in
the model because there is no degree of freedom for error. In these cases, it is
assumed that, there is no 3 factors interaction. Then
the degree of freedom is attributed to error degree of freedom and begins
testing with two way interactions. MSE contain random error and 3 factors
interaction.
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