# Full Factorial Design

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Definition

Several Factors are investigated at several levels by running all combinations of Factors and levels in a full factorial designed experiment. The notation for a full factorial design with k Factors each at 2 levels is 2k. This number also gives the number of treatment combinations or runs in the experiment.

Examples

In order to increase the mileage of his car, a commuter decides to run a full factorial experiment. Three factors, A, B and C are under study, where each factor is set at two levels, Low (-) and High (+):
A: Air conditioner (Off/On)
B: Tire Pressure (Low/High)
C: Roof Rack (Off/On)

The full factorial design consists of all possible combinations of the six factor levels, giving the 23 = 8 treatments or runs as shown.

Data are collected over the next 8 days, while commuting to work over the same route with the same traffic conditions etc. to minimize their impact on the results. The resulting Anova output indicated that all three main effects are significant at the 10% significance level. A cube plot of the three factors shows that the best combination to achieve greatest mileage is Air Conditioner = Off (-), Tire Pressure = High (+) and Roof Rack = Off (-).

Application

If the design includes two or more replications of the design, then a lack-of-fit test can be done to check the validity of the model.

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