Adding further factors to the design

In experiments with 8 experimental units, it is possible to independently estimate the main effects of up to 7 factors. The design is again based on a complete factorial design for 3 factors.

    Term    
Run   A     B     C     AB     AC     BC     ABC     Response  
1 -1 -1 -1 +1 +1 +1 -1 y—-
2 -1 -1 +1 +1 -1 -1 +1 y—+
3 -1 +1 -1 -1 +1 -1 +1 y-+-
4 -1 +1 +1 -1 -1 +1 -1 y-++
5 +1 -1 -1 -1 -1 +1 +1 y+—
6 +1 -1 +1 -1 +1 -1 -1 y+-+
7 +1 +1 -1 +1 -1 -1 -1 y++-
8 +1 +1 +1 +1 +1 +1 +1 y+++

All terms in the table above are orthogonal (the columns of ±1 are uncorrelated). We therefore allocate the levels of additional factors (up to 4 of them) according to the ±1 values in interaction columns. This ensures that the extra factors are orthogonal to A, B, C and each other, but each extra factor is confounded with some interaction between A, B and C.

Confounding

Consider adding two factors to the complete design, defined by

D  =   AB

E  =   AC

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

This design is called a 25-2 fractional factorial design.

All five main effects are orthogonal. The confounding can again be determined algebraically by manipulating the letters for the effects and using the fact that each squared term is 1. For example,

A   =   BD   =   CE   =   ABCDE

This means that A is confounded with the BD interaction, the CE interaction and the 5-factor ABCDE interaction.

estimate of main effect of A   =   (true main effect of A)   +   (BD interaction effect)   +   (CE interaction effect)   +   (ABCDE interaction effect)

In a similar way,

Each main effect is confounded with three interactions.

In general

The same principles guide the design of experiments for 2k experimental units where k > 3. They should again be based on the complete design for k factors, with additional factors having their levels defined by high-order interactions between the original k factors.

Design with five factors

The diagram below allows selection of the two terms that are confounded with the main effects for D and E.

Click the green and red headings at the top of the table to select the terms that are confounded with the main effect for D and E. Observe that all main effects are confounded with 2-factor interactions whatever interactions between A, B and C are confounded with D and E.

Note how the different treatments used for experimental units in the design are described by the 'words' under the table.