To correctly develop the alias structure of any design, follow the steps below. Video 10 demonstrates the following steps to develop the alias structure of a design systematically.
1. Calculate the total number of effects of the design (Video 9). Total number of effects in a factorial design is 2k -1.
2. Write all factorial effects (Video 9).
3. Multiply the main effects with all the defining relations, including their generalized interactions and place them side by side with an equal sign in-between (Table 10).
4. Multiply the remaining two-factor interactions with all the defining relations, including their generalized in….follow step 3 (Table 10).
Keep doing it until all the effects are in the table (e.g., Table 10). In the 25-2 quarter design, four columns multiplied by the eight rows of the table minus the Identity, I in the first cell of the Table 10 equal to all 31 effects.
Table 10. Alias structure of one-quarter design
Why the full alias structure?
Imagine in a musical band, the vocal and drummer are needed to coordinate/interaction to produce better-quality music. While the drummer can produce some appealing music alone, the vocal can only sing if the drummer is in there. Therefore, the data analysis results may show that the main effect of the drummer factor is significant while the main of the vocal factor is insignificant with respect to the audience rating. However, their interactions are observed to be significant too. In this scenario, if the two-factor interaction is aliased with the main effects, it will be impossible to independently know whether the interaction is significant or not. Without knowing all the alias structures of the design, the conclusions may be misleading. In this hypothetical music band, the “alias” can be thought of as if the drummer and the vocal are the same person. When the drummer (factor A) is aliased with the drummer and the vocal, they are the same person.