The layout of a factorial design is typically organized in a table format. Each row of the table represents an experimental run, while each column represents a factor or the response variable. The levels of factors are indicated by symbols such as + and – for high and low levels, respectively. The response variable values corresponding to each experimental condition are recorded in the form of a sign table.
Consider a simple example layout for a two-factor factorial design with factors $A$ and $B$.
| Run | Factor A | Factor B | Response |
|---|---|---|---|
| 1 | – | – | $Y_1$ |
| 2 | + | – | $Y_2$ |
| 3 | – | + | $Y_3$ |
| 4 | + | + | $Y_4$ |
Layout of the Factorial Design: Two Factor in $n$ Replicates
Consider there are two factors and each factor has two levels in $n$ replicates. The layout of the factorial design will be as described below for $n$ replicates.
$y_{111}$ is the response from the first factor at the low level, the second factor at the low level, and the first replicate of the trial. Similarly, $y_{112}$ represents the second replicate of the same trial, and up to $n$th observation is $n$th trial at the same level of $A$ and $B$.
Geometrical Structure of Two-Factor Factorial Design
The geometrical structure of two factors (Factor $A$ and $B$), each factor has two levels, low ($-$) and high (+). Response 1 is at the low level of $A$ and a low level of $B$, similarly, response 2 is produced at a high level of $A$ and a low level of $B$. The third response is at a low level of $A$ and a high level of $B$, similarly, the 4th response is at a high level of $A$ and a high level of $B$.
Real Life Example
The concentration of reactant vs the amount of the catalyst produces some response, the experiment has three replicates.
Geometrical Structure of the Example
Factor Effects
\begin{align} A &=\frac{(a+ab)-((I) +b)}{2} = \frac{100+90-80-60}{2} = 25\\
B &= \frac{(b+ab) – ((I) +a) }{2} = \frac{60+90-80-100}{2} = -15\\
AB&=\frac{((I)+ab)-(a+b)}{2} = \frac{80+90-100-60}{2}=5
\end{align}
Minus 15 ($-15$) is the effect of $B$, which shows the change in factor level from low to high bringing on the average $-15$ decrease in the response.
Reference
Montgomery, D. C. (2017). Design and Analysis of Experiments. 9th ed, John Wiley & Sons.