# How to find the levels (+ or -) using the words in a Fractional Factorial?

I'm stuck on this question and would appreciate any help!

Q: Given the partial defining relation I = -ABD = -ACE = BCF for a $2^{6-3}$ fractional factorial design, obtain the complete defining relation and the 8 treatment combinations that are used in this design.

My approach: Now I have gotten the aliases and I know that from the 3 words that we are given, we have the following words: $I = -ABD = -ACE = BCF = BCDE = -ACDF = -ABEF$

Now here is where I run into trouble. When I am constructing the actual table (with 8 runs since $2^{3}=8$, I am not sure how I am supposed to figure out the levels of each factor since we are only testing 8 treatments compared to when we run the experiment where we test 64 treatments.

Any help would be very much appreciated! My textbook (Design and Analysis by Montgomery) did not explain very well how these levels are found.

Construct a design in enough of the factors to (say A, B, C,...) to create the correct number of runs. Then find an alias for each of the remaining factors in terms of the ones you have, and use the corresponding product to create the levels of that factor.

A different example: def rel: I = ABCD, so that D = ABC

A B C  D=ABC
- - -   -
+ - -   +
- + -   +
+ + -   -
- - +   +
+ - +   -
- + +   -
+ + +   +

• Thanks a lot! I understand how to make the table but I'm still confused on how you got the levels for each run? Can you please be a bit more specific? Thanks! Dec 9, 2014 at 4:13
• I believe I have to use the generators to "generate" the levels but no luck! Dec 9, 2014 at 4:21