Full Factorial Designs Explained

A full factorial design runs every combination of the levels of every factor. With two levels per factor, called low and high, and k factors, that is 2^k runs.

Add centre points (replicated runs with every factor at its midpoint) and the total goes up by however many you choose.

Coding the levels

In the table above each factor is shown in coded units: −1 for the low level and +1 for the high. Coding puts every factor on the same scale, so a coefficient for temperature in degrees and a coefficient for time in minutes can be compared directly. The midpoint is 0.

Estimating effects

The main effect of a factor is the average response at its high level minus the average response at its low level. In a full factorial every factor is high in exactly half the runs and low in the other half, so each main effect uses all the data.

An interaction effect is estimated the same way, using the product of the coded columns. For factors A and B the AB column is +1 when A and B have the same sign and −1 when they differ. A large AB effect means the effect of A depends on where B is set.

For a 2-factor design the model is

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2 + \varepsilon$

where the coefficients are half the corresponding effects, because a change from −1 to +1 spans two coded units.

What full factorials give you

• All main effects, uncorrelated with each other (the design is orthogonal).
• All interactions, up to the highest order, each estimated independently.
• Good precision, because every run contributes to every estimate.

Replication and centre points

A 2-factor, 2-level factorial has 4 runs and a model with 4 terms, so there is nothing left to estimate error from. Replication fixes that. Centre points are a cheap way: running 3 to 5 replicates at the midpoint gives a pure error estimate and, by comparing their average to the factorial average, a check for curvature that a two-level model cannot represent.

When a full factorial is not the right tool

• Many factors. At 6 factors you need 64 runs, and most of them estimate high-order interactions that are rarely important. A fractional factorial uses a carefully chosen fraction.
• You expect curvature. Two levels can only fit straight lines. To model a peak or a valley use a response-surface design: see central composite vs Box-Behnken.

Build a randomized full factorial in seconds with the free design generator, or see the practical guide to DoE for the bigger picture.