 Factorial Experiments

Factorial Experiments are experiments that investigate the effects of two or more factors or input parameters on the output response of a process.  Factorial experiment design, or simply factorial design, is a systematic method for formulating the steps needed to successfully implement a factorial experiment.  Estimating the effects of various factors on the output of a process with a minimal number of observations is crucial to being able to optimize the output of the process.

In a factorial design, the effects of varying the levels of the various factors affecting the process output is investigated. Each complete trial or replication of the experiment takes into account all the possible combinations of the varying levels of these factors.  Effective factorial design ensures that the least number of experiment runs are conducted to generate the maximum amount of information about how input variables affect the output of a process.

For instance, if the effects of two factors A and B on the output of a process are investigated, and A has 3 levels of intensity (e.g., weak, moderate, and strong presence) while B has 2 levels (weak and strong), then one would need to run 6 treatment combinations to complete the experiment, observing the process output for each of the combinations: weakA-weakB, weakA-strongB, moderateA-weakB, moderateA-strongB, strongA-weakB, strongA, strongB.

The amount of change produced in the process output for a change in the 'level' of a given factor is referred to as the 'main effect' of that factor. Table 1 shows an example of a simple factorial experiment involving two factors with two levels each. The two levels of each factor may be denoted as 'low' and 'high', which are usually symbolized by '-' and '+' in factorial designs, respectively.

Table 1. A Simple 2-Factorial Experiment

 A (-) A(+) B(-) 20 40 B(+) 30 52

The main effect of a factor is basically the 'average' change in the output response as that factor goes from '-' to '+'.  Mathematically, this is the average of two numbers: 1) the change in output when the factor goes from low to high level as the other factor stays low, and 2) the change in output when the factor goes from low to high level as the other factor stays high.

In the example in Table 1, the output of the process is just 20 (lowest output) when both A and B are at their '-' level, while the output is maximum at 52 when both A and B are at their '+' level. The main effect of A is the average of the change in output response when B stays '-' as A goes from '-' to '+', or (40-20) = 20, and the change in output response when B stays '+' as A goes from '-' to '+', or (52-30) = 22.  The main effect of A, therefore, is equal to 21.

Similarly, the main effect of B is the average change in output as it goes from '-' to '+' , i.e., the average of 10 and 12, or 11. Thus, the main effect of B in this process is 11. Here, one can see that the factor A exerts a greater influence on the output of process, having a main effect of 21 versus factor B's main effect of only 11.

It must be noted that aside from 'main effects', factors can likewise result in 'interaction effects.'  Interaction effects are changes in the process output caused by two or more factors that are interacting with each other.  Large interactive effects can make the main effects insignificant, such that it becomes more important for the engineer to pay attention to the interaction of the involved factors than to investigate them individually.

The running of factorial combinations and the mathematical interpretation of the output responses of the process to such combinations is the essence of factorial experiments.  It allows an engineer to understand which factors affect his or her process most so that improvements (or corrective actions) may be geared towards these.

See also:   2-Level Factorial Design; Factorial Design Tables; Example of a 2-Level Factorial Experiment