Completely Randomized Factorial (CRF) Design

A design with more than one factor is a factorial design if all factor-level combinations are used in the experiment.

Advantages

• Able to detect interactions between treatments
• More efficient (for all tests) than separate one-at-a-time experiments.

Hot Dog Example

Taste is investigated as we vary the type of meat (beef, chicken, or pork) and the level of sodium (Low, Medium, and High). There are two strategies:

• One-at-a-time method
• Factorial method

One-at-a-time strategy

Run two separate experiments, each with 9 subjects. Here is the ANOVA table:

Source      SS    df
Meat type   SST   2
Error       SSE   6
Total       TSS   8

Suppose the meat means are: B 8.2, C 7.8, P 7.2 and the sodium means are: L 7.2, M 8, H 8

Factorial strategy

Still using 18 subjects, but with 2 at each of 9 meat-sodium combinations. Here is the ANOVA table:

Source        SS     df
Meat type     SSA    2
Salt type     SSB    2
Meat*Salt     SSA*B  4
Error         SSE    9
Total         TSS    17

Note that the Error sum of squares has more degrees of freedom than with the one-at-a-time experiments.