1. Factorial experiments
Recall that in the Latin squares and Graeco-Latin squares videos I mentioned that one key assumption for both of those designs is no interaction between any combination of the blocking factors and the treatment. That's true, and was also foreshadowing, because there is an experiment type that considers interaction of factor variables called a factorial experiment.
2. Factorial designs
A factorial design is one in which two or more factor variables are combined and crossed in an experiment. Each combination of all the levels of all the factors will be considered separately as a factor affecting the outcome, meaning that we'll measure the outcome at every possible combination of all the factors. Let's dig into an example to see how this works.
3. Factorial example
Our hypothetical plant growth experiment will require testing all the combinations of levels of high or low water and high or low light. In the diagram, you can see that this includes low water/low light, low water/high light, high water/low light, and high water/high light. Every combination's effect on the outcome will be tested, and you can use Tukey's HSD test to specifically test differences between plant growth for each possible combination.
4. 2^k factorial experiments
For this course, we'll only be analyzing 2^k factorial experiments, a type where you have some number, k, of two-level factor variables. Usually, a 2^k experiment will have factors with a "high" and a "low" level, as we saw in the plant experiment we just discussed. Often, for reasons of cost, the high and low level of factor variables are chosen to be tested because the more levels a factor has, the more combinations begin to appear and the more money the experiment costs.
The simplest 2^k factor case is 2 factor variables with two levels each. All possible combinations will be tested, but any number, k, of variables can be selected. Often, running a 2^k experiment with many 2-level factor variables is a strategy employed to find the factors that are important in your experiment quickly. For example, you run six factor variables and examine results to find what is most affecting the outcome. Keep in mind though, that's 64 different combinations to model and explore!
5. Let's practice!
Let's dive into the exercises to see how to design, analyze, and evaluate factorial experiments.