Assigning levels of D to the runs

We now describe how a fourth factor can be added to a complete 23 factorial design. The best way to allocate the levels of the fourth factor, D, to the runs of the experiment is according to the ±1 values for the 3-factor interaction between A, B and C.

    Factor      
Run   A     B     C     D = ABC     Response  
1 -1 -1 -1 -1 y——
2 -1 -1 +1 +1 y—++
3 -1 +1 -1 +1 y-+-+
4 -1 +1 +1 -1 y-++-
5 +1 -1 -1 +1 y+—+
6 +1 -1 +1 -1 y+-+-
7 +1 +1 -1 -1 y++—
8 +1 +1 +1 +1 y++++

This design uses half of the runs needed for a complete factorial experiment with four factors, so it is a 24-1 fractional factorial design.

Confounding

From the definition of the levels of D, its main effect is clearly confounded with the 3-factor interaction between A, B and C.

D  =   ABC

When we estimate the main effect for D, what we measure is actually the sum of its true main effect and the effect of the ABC interaction.

estimate of main effect of D   =   (true main effect of D)   +   (ABC interaction effect)

An ABC interaction could therefore potentially obscure the main effect of D or incorrectly make it appear to be too large.

Several other effects are confounded in this design. They can be found using arithmetic on the columns of ±1, noting that squaring any column gives +1,

A2  =   B2  =   C2  =   D2  =   1

Therefore

A   =   AD2   =   (AD)D   =   (AD)(ABC)   =   A2BCD   =   BCD

and similarly,

B  =   ACD

C  =   ABD

AB  =   CD

AC  =   BD

AD  =   BC

ABCD  =   1

Each main effect is therefore only confounded with a 3-factor interaction and is therefore orthogonal to the 2-factor interactions. Interpretation of the estimated main effects in this design therefore do not rely on an assumption that the 2-factor interactions are negligible.

However the 2-factor interactions are confounded in pairs, so we cannot estimate their effects independently. For example, the estimated AB interaction effect could be caused by an interaction between C and D instead.

Since the main effects are not confounded with 2-factor interactions but the 2-factor interactions are confounded with each other, this design is called a resolution IV design.

Testing

It is necessary to assume that all 2-factor interactions are zero (or in practice, negligible) in order to get residual degrees of freedom for testing the significance of the main effects.

The diagram below allows selection of the term that is confounded with the main effect for D.

Click the red heading at the top of the table to select the term that is confounded with the main effect for D. Observe that none of the main effects are confounded with 2-factor interactions when D is confounded with ABC.