Statistics for Data Analyst

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.

R and Data Analysis

Test Preparation MCQs

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.

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