Experiments with many factors

The number of possible treatment combinations in a factorial experiment increases very quickly as the number of controlled factors increases, as illustrated in the table below.

   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

To keep the number of runs of the experiment to a practical level, experiments involving several factors are therefore usually conducted with only two levels for each factor.

If each of k factors in a factorial experiment has two levels, the experiment is called a 2k factorial experiment. If there are 4 or more factors, it is rare to use more than one replicate, so there are usually 2k runs in the design.

Coding of factor levels

It was explained earlier that a factor with two levels can be equivalently modelled as a numerical or categorical variable. In 2k factorial experiments, it is convenient to treat the factors as numerical variables with values -1 and +1 denoting the two levels for each factor

Sugar reduction study

A food manufacturer wanted to reduce the sugar content of a sauce without altering the taste and without any additive. The first part of a study into this problem was an experiment to evaluate the influence of Saltiness, Acidity, Thickness (viscosity) and Hotness (piquancy) on the perceived sweeness of the sauce.

Two levels for each of these four factors were chosen by the scientists to cover values that were considered to be feasible for the variables:

Factor  Low level   High level 
Saltiness     0.10 g% 0.20 g%
Acidity 0.00 g% 0.05 g%
Thickness 3.00 g% 4.00 g%
Hotness     0.05 g% 0.15 g%

The low level for each factor will be coded as -1 and the high level as +1. The table below shows the treatment levels for a 24 factorial experiment and average sweetness of each sauce as perceived by a panel of tasters.

 Saltiness   Acidity   Thickness   Hotness   Sweetness 
+1
+1
+1
+1
+1
+1
+1
+1
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
7.60
7.92
7.72
7.27
8.09
7.87
7.52
6.89
8.02
8.34
8.20
8.21
8.14
7.82
8.04
7.00