Most screening experiments assess the effects of many factors and use far more than four runs of the experiment. However,

We will use simple experiments with only four runs to illustrate some principles for designing incomplete experiments

Complete experiment with two factors

A complete factorial experiment for two factors uses the same number of runs of the experiment with all possible combinations of levels for the two factors. For factors with two levels, this can be done in a single replicate of the experiment with four runs.

    Factor    
Run A B   Response  
1 -1 -1 y
2 -1 +1 y-+
3 +1 -1 y+-
4 +1 +1 y++

The main effects for both factors can be independently estimated and each is the difference between the response mean at the high and low levels of the factor,

Main effect of A   =   (y++ + y+- - y-+ - y) / 2

Main effect of B   =   (y++ + y-+ - y+- - y) / 2

Note that the main effects are based on the sum of the response values multiplied by the ±1 values for the factors.

Interaction

The total degrees of freedom for this experiment are (n - 1) = 3, so there is one degree of freedom left that allows us to estimate the interaction between the two factors.

AB interaction effect   =   (y++ + y - y+- - y-+) / 2

By rewriting the equation, this can be interpreted as:

AB interaction effect   =   ( (y++ - y-+) - (y+- - y) ) / 2

AB interaction effect   =   ( (y++ - y+-) - (y-+ - y) ) / 2


Whichever interpretation, the interaction is zero if each factor has the same effect for the high and low value of the other factor.

The interaction is found in a similar way to the main effects of the factors — it is based on the sum of the response values with ± signs. These signs can be found by multiplying the ±1 values for the two factors. Note that there are an equal number of each sign and that the interaction column below is uncorrelated with both factors.

    Factor   Interaction  
Run A B AB   Response  
1 -1 -1 +1 y
2 -1 +1 -1 y-+
3 +1 -1 -1 y+-
4 +1 +1 +1 y++