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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