The smallest case of a $2^K$ factorial experiment is one in which 2 factors are of interest and each factor has two levels. This design is known as a $2^2$ factorial design. We are interested in Factor effects or Effects of Factors.
The level of the factors (say $a$' and$b$’) may be called the low and high or presences and absences.
In a factorial design, in each complete trial (or replicate of the experiment), all possible combinations of the levels of the factors are investigated. For example, if Factor-A has $a$'' levels and Factor-B has$b$” levels, then each replicate contains all “$ab$” treatment combinations. Two factors each at 2 levels are:
Factor Effects (or Effect of Factors)
A change in the quantity of response due to the change in the level of a factor is called the effect of that factor. Here we mean average effect.
Main Effects
A main effect of a factor is defined as a measure of the average change in effect produced by changing the level of the factor. It is measured independently from the effect of other factors. The main effect is the effect of the factor only. Main effects are sometimes regarded as an interaction of zero order. Frequently, the main effect refers to the primary factors of interest in the experiment.
Interaction Effects
Factors are said to interact when they are not independent. Interaction in a factorial experiment is a measure of the extent to which the effect of changing the levels of one or more factors depends on the levels of the other factors. Interactions between two factors are referred to as first-order interactions, those concerning three factors, as second-order interactions, and so on.
Example: Consider a two-factor factorial experiment. Consider an investigation into the effect of the concentration of reactant (Factor $A$) and the presence of catalysts on the reaction time of the chemical process (Factor $B$).
Solution of Example
Main Effects
\begin{align}
\text{Main effect of A} & = \text{Average response at high level of $A$} – \text{Average response at low level of $A$}\\
&=\frac{45+60}{2}-\frac{20+35}{2}=25
\end{align}
The results indicate that Increasing Factor-A from the low level to the high level causes an average response increase of 25 units.
\begin{align}
\text{Main effect of B}&=\text{Average response at high level of $B$} -\text{ Average response at low level of $B$}\\
&=\frac{35+60}{2}-\frac{20+45}{2}=15
\end{align}
Increasing Factor B from the low level to the high level causes an average response increase of 15 units.
Effect of AB Interaction
It is possible that the difference in response between the levels of a factor is not the same at all levels of the other factor(s), then there is an interaction between the factors. Consider
\begin{align}
\text{The effect of Factor $A$ (at low level of Factor $B$)} &= 50 – 20 = 30\\
\text{The effect of Factor $A$ (at high level of Factor $B$)}&= 15 – 40 = -25\\
\text{The effect of Factor $B$ (at low level of Factor $A$)} &= 40 – 20 = 20\\
\text{The effect of Factor $B$ (at high level of Factor $B$)} &= 15 – 50 = -35
\end{align}
Because the effect of Factor-$A$ depends on the level chosen for Factor-$B$, we see that there is interaction between $A$ and $B$. One can computer Effect of $AB$ interaction as described below:
Effect of AB Interaction
= Average difference between effect of $A$ at high level of $B$ and the effect of $A$ at low level of $B$.
The magnitude of the interaction effect is the average difference in these two A effects, or $AB=\frac{-25-30}{2}=\frac{-55}{2}$.
OR
= Average difference between effect of $B$ at high level of $A$ and the effect of $B$ at low level of $A$.
The magnitude of the interaction effect is the average difference in these two B effects, or $AB = \frac{-35-20}{2} = \frac{-55}{2}$.
The interaction is large in this experiment.