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Example fractional factorial experiment
Montgomery gives the following example of a fractional factorial experiment. An engineer performed an experiment to increase the filtration rate (output) of a process to produce a chemical, and to reduce the amount of formaldehyde used in the process. The full factorial experiment is described in the Wikipedia page Factorial experiment. Four factors were considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D). The results in that example were that the main effects A, C, and D and the AC and AD interactions were significant. The results of that example may be used to simulate a fractional factorial experiment using a half-fraction of the original 24 = 16 run design. The table shows the 24-1 = 8 run half-fraction experiment design and the resulting filtration rate, extracted from the table for the full 16 run factorial experiment.



In this fractional design, each main effect is aliased with a 3-factor interaction (e.g., A = BCD), and every 2-factor interaction is aliased with another 2-factor interaction (e.g., AB = CD). The aliasing relationships are shown in the table. This is a resolution IV design, meaning that main effects are aliased with 3-way interactions, and 2-way interactions are aliased with 2-way interactions.



The analysis of variance estimates of the effects are shown in the table below. From inspection of the table, there appear to be large effects due to A, C, and D. The coefficient for the AB interaction is quite small. Unless the AB and CD interactions have approximately equal but opposite effects, these two interactions appear to be negligible. If A, C, and D have large effects, but B has little effect, then the AC and AD interactions are most likely significant. These conclusions are consistent with the results of the full-factorial 16-run experiment.



Because B and its interactions appear to be insignificant, B may be dropped from the model. Dropping B results in a full factorial 23 design for the factors A, C, and D. Performing the anova using factors A, C, and D, and the interaction terms A:C and A:D, gives the results shown in the table, which are very similar to the results for the full factorial experiment experiment, but have the advantage of requiring only a half-fraction 8 runs rather than 16.