How to Read the Aliases of Main Effects & Interactions Spreadsheet

Shown below are two spreadsheets that summarize how some of the main effects in a 2(7-4) experimental design (7 factors, 8 runs) are created from higher order interactions. Specifically, 8 runs can "accommodate" 3 completely crossed factors (23 = 8); the remaining 4 "extra" factors are created as aliases of the 3 two-way interactions, and the three-way interaction (the so-called "generating factors" or "design generators;" see also the Introductory Overview section).

The spreadsheet below (click the Generators of fractional design button on the Generators & aliases tab of the Design of an Experiment with Two-Level Factors dialog box) shows how the extra 4 factors are related to the respective two-way interactions and the three-way interaction.

The second spreadsheet (click the Aliasing of effects button on the Generators & aliases tab of the Design of an Experiment with Two-Level Factors dialog box) summarizes all aliases for all main effects and two-way interactions.

Shown in the column headers in the table above are the different identities of the four generating factors. These are found in the following manner. The generating factor of 4 is 12. Another way to express this relation is to write: 4 = 12, where both 4 and 12 stand for the columns of 1's and -1's associated with the 4 main effect and the 12 (factor 1 by factor 2) interaction. If one multiplies the columns 4 * 12 then the resulting column will contain all 1's (since -1 * -1 = +1); thus, I = 124 (where I stands for the column of all 1s).

The 2(7-4) design shown above can be completely summarized as I = 124 = 135 = 236 = 1237 ... and so on. These are the column headers in the spreadsheet above. Thus, if one wants to determine the aliases of factor 1 then all one needs to do is multiply out factor 1 with the respective column headers, e.g., 124; since 1 * 1 cancels each other out, this amounts to simply "crossing out" the respective effect from the column header. In that manner, one would arrive at 1 * 124 = 24; and one has thus identified the 24 interaction to be an alias of 1. By the same method, one may determine 1 * 135 = 35 and 1 * 236 = 1236, that is, the 35 interaction to be an alias of 1, as well as the 1236 interaction.

Therefore, the spreadsheet shown above summarizes all aliases of the main effects and two-way interactions. For additional information, refer to Box, Hunter, & Hunter (1978).