21.4.3.1. Effect Analysis
An effect is said to be estimable if, and only if, there is a contrast in the data which has for its expectation the particular effect biased only by other effects which we choose to suppress. Experimental designs in which main effects are aliased with each other are of no interest. Hence, it is noted that the resolution of experimental designs is very important in the effect analysis process.
Considering \({{3}^{2}}\) full factorial design, this experiment has 1 degree-of-freedom for the main effects on factors A and B \(\left( {{A}_{L}},{{A}_{Q}},{{B}_{L}},{{B}_{Q}} \right)\) and 4 degree-of-freedom for the two-factor interaction AB \(\left( A{{B}_{L\times L}},A{{B}_{L\times Q}},A{{B}_{Q\times L}},A{{B}_{Q\times Q}} \right)\). Hence, this design clearly estimates the main effects and two-factor interaction effect.
Now, let’s consider the fractional factorial design of \(3_{III}^{3-1}\) with \(I=A{{B}^{2}}{{C}^{2}}\). The aliasing effects can be easily determined by multiplying \(I\) and \({{I}^{2}}\) by effects.
\(A\dot I=A^2B^2C^2=ABC\)
\(B\dot I=AB^3C^2=AC^2\)
\(C\dot I=AB^2C^3=AB^2\)
\(AB\dot I=A^2B^3C^2=AC\)
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\(A\dot I^2=A^3B^4C^4=BC\)
\(B\dot I^2=A^2B^5C^4=ABC^2\)
\(C\dot I^2=A^2B^4C^5=AB^2C\)
\(AB\dot I^2=A^3B^5C^4=BC^2\)
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These relations show that the \(3_{III}^{3-1}\) design estimates the aliased effects such as \(A+BC+ABC\), \(B+A{{C}^{2}}+AB{{C}^{2}}\), \(C+A{{B}^{2}}+A{{B}^{2}}C\) and \(AB+AC+B{{C}^{2}}\). From the definition of the Resolution III, \(3_{III}^{3-1}\) design should be used when the main effects are nearly independent from two-factor or higher effects in the system.