Muhammad Imdad Ullah

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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The Factorial Experiment advantages and disadvantages over One-Factor-at-a-time Experiment.

Factorial Experiment Advantages

• Required Less Number of Observations
  Let $A$ and $B$ be two factors. The information on a factor can be obtained by varying that factor and keeping the other factor fixed.

Effect of changing factor $A = A_2 B_1 – A_1B_1$

Effect of changing factor $B = A_1B_2 – A _1 B_1$

Three treatment combinations are used for two effects for error estimation we need two replicates so six observations are needed.

In the case of factorial experiments, one more combination $A_2B_2$ is utilized and we get:

Two estimates of $A$ are: $=A_2B_1 – A_1B_1 \qquad \text{ and } \qquad =A_2B_2 – A_1B_2$

Two estimates of $B$ are: $=A_1B_2 – A_1B_1 \qquad \text{ and } \qquad =A_2B_2 – A_2B_1$
Thus, by using four observations we can get the estimates of the same precision under a factorial experiment.

• More Relative Efficiency
  In the case of two factors the relative efficiency of factorial design to one-factor-at-a-time experimental design is $\frac{6}{4}=1.5$
  This relative efficiency increases with the increase of the number of factors.
• Necessary When Interaction is Present
  When using a one-factor-at-a-time design and the response of $A_1B_2$ and $A_2B_1$ is better than $A_1B_1$, an experimenter logically concludes that the response of $A_2B_2$ would be even better than $A_1B_1$. Whereas, $A_2B_2$ is available in factorial design.
• Versatility
  Factorial designs are more versatile in computing effects. These designs provide a chance to estimate the effect of a factor at several levels of the other factor.

Factorial Experiment Advantages in simple words

The factorial Experiment Advantages without any statistical formula or symbol are:

• A factorial experiment is usually economical.
• All the experimental units are used in computing the main effects and interactions.
• The use of all treatment combinations makes the experiment more efficient and comprehensive.
• The interaction effects are easily estimated and tested through the usual analysis of variance.
• The experiment yields unbiased estimates of effects, which are of wider applicability.

Factorial Experiments Disadvantages

• A factorial experiment requires an excessive amount of experimentation when there are several factors at several levels. For example, for 8 factors, each factor at 2 levels, there will be 256 combinations. Similarly, for 7 factors each at 3 levels, there will be 2187 combinations.
• A large number of combinations when used decrease the efficiency of the experiment. The experiment may be reduced to a manageable size by confounding some effects considered of little practical consequence.
• The experiment setup and the resulting statistical analysis are more complex.

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R and Data Analysis

In factorial experiments, the effect of two or more factors, each at two or more discrete possible levels are simultaneously investigated for all possible combinations using some suitable basic experimental design.

The experiment allows us to estimate the effect of each factor and the interaction effect of factors on the response.

A factorial design was initially used in the 19th century at Rothamsted Experimental Station (one of the oldest research stations in the UK). Ronal Fisher and Frank Yates are the pioneers of factorial design.

Experiments are often planned to investigate the effects of (say), different rates of fertilizers, different dates of planting, different categories of education, different intensities of a stimulus, etc.

The independent variables such as fertilizers, planting, education, and stimulus, etc., are called factors. In contrast, the values such as rates, dates, categories, or intensities at which a factor is held fixed, are known as levels. In the figure below, the “Amount of Seeding” is a factor variable, while 60kg, 50kg, and 80kg are levels of the factor “Amount of Seeding”. There are three levels of this factor variable.

Types of Factorial Experiments

There are two types of factorial experiments

Full Factorial Experiment

The experimental units of such an experiment take on all possible combinations of all levels across all the factors. Therefore, a full factorial design is also called a fully crossed design.

Fractional Factorial Experiment

If the full factorial design includes too many combinations (runs) to be logically feasible, a fractional factorial may be used. The fractional factorial design may include half, one-third, etc. runs of a full factorial experiment.

In factorial experiments, we try to perform one rather than two, three, or more single-factor experiments. The single experiment involves a factorial set of treatments, that is, the treatments are all possible combinations of various levels of different factors.

The effect of a factor is defined to be the change in response produced by a change in the level of factors. This is called the main effect.

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