Two Factor Experiments

Two Factor Experiments, Factorial Designs, Main Effects, Interaction Effects, Factorial Experiments, Complete Block Design, Split Plot Design Strip Plot Design, Group Balanced Block in Split Plot Designs

Factorial Experiment Advantages and Disadvantages

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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.
Factorial Experiment Advantages


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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Factorial Experiment Made Easy

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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.

Factorial Experiment

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.

Design of Experiments

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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