Six factors

When six or more factors are used in an experiment with 16 runs, it is impossible to avoid confounding 2-factor interactions with other 2-factor interactions. However designs for up to eight factors can be constructed in which the main effects are not confounded with any 2-factor interactions. These are therefore resolution IV designs.

To design a 26-2 fractional factorial experiment for 6 factors, we again start with a complete 24 factorial design for factors A, B, C and D. The levels of factors E and F should be given by 3-factor interactions between A, B, C and D. For example,

E   =   ABC

F   =   BCD

Note that:

If E or F is confounded with the 4-factor interaction ABCD, it is impossible to avoid confounding at least one main effect with a 2-factor interaction. Both additional main effects should be confounded with 3-factor interactions.

Seven factors

Again the three additional factors, E, F and G should be confounded with 3-factor interactions between A, B, C and D. Any 3-factor interactions can be chosen. For example,

E   =   ABC

F   =   BCD

G  =   ACD

This is again a resolution IV design since no main effects are confounded with 2-factor interactions (though several 2-factor interactions are again confounded with each other).

Eight factors

In order to get a resolution IV design (no main effects confounded with 2-factor interactions), it is necessary to define the levels four additional factors from the four 3-factor interactions between A, B, C and D,

E   =   ABC

F   =   BCD

G  =   ACD

H  =   ABD

The diagram below can be used to investigate the effects that are confounded in fractional factorial designs for seven factors in 16 runs.

The red, green and blue heading rows are used to specify the terms from the complete factorial in A, B, C and D that are used to define the levels of factors E, F and G. Click in these heading rows to set the levels of E, F and G.

For any definition of E, F and G, the columns of interactions listed in the heading are all confounded with each other. Observe that: