Factorial designs for several factors

The experimental designs and analysis described in this section for three factors extend to experiments with four or more factors. A complete factorial design has the same number of replicates for each combination of levels for the different factors (treatments).

The number of possible treatment combinations in a factorial experiment increases very quickly as the number of controlled factors increases. The table below gives the number of treatments for different numbers of factors if each factor has two levels (top row) and if each factor has three levels (bottom row).

    Number of factors, k  
       2       3       4       5       6     
Levels per factor     2   4 8 16 32 64   2k
  3   9 27 81 243 729   3k

This is also the number of experimental units required if there is only a single replicate for each treatment. To keep the number of experimental units at a practical level, experiments involving several factors are therefore usually conducted with only two levels for each factor and a single replicate. If each of k factors in a factorial experiment has two levels, the experiment is said to have a 2k factorial design.

Experiments involving several two-level factors will be discussed more fully in a later section.