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Repeated half fractional design vs. whole design in 2 blocks

gchesterton

Suppose I have a budget for 32 runs for 5 categorical 2-level factors (full factorial of 2^5). Suppose I want to do a screening design, where I am interested in main effects and 2-factor interactions. So a screening design is ok. JMP lets me specify the choice of design. Let's talk about option (A) vs. (B):

A) Suppose I specify that I do the 32 runs in two blocks of 16, for a resolution 5 design where I can estimate all the 2-factor interactions.

Suppose instead I tell JMP that I want only a 16 run design, ignoring for the moment the blocking factor. In that case, I have a half fractional design of 16 runs. But since I have a budget of 32 runs, suppose I just repeat that 16-run design a second time, on my second block (of participants), randomizing accordingly, and treat that second 16-run instance as a second block.

What, then, is the fundamental difference between (A) and (B)? In both cases I can estimate the main effects and all the 2-factor interactions, and the design resolution seems to be the same. I suspect that (A) is the correct way to go, but I can't quite figure out what I'm losing if I were to do (B).

Thanks!

Accepted answers

Dan_Obermiller

As long as you limit your model to two-way interactions or lower, these designs are equivalent. If you are uncomfortable with the assumption that 3-way interactions and higher are negligible, then go with the full factorial approach.

To see how I reached this conclusion, I created the full factorial design in two blocks. I then created the 16 run fractional factorial design. Chose Augment Design, checking the box to put the new runs in a new block (this adds the blocking variable which should be there and allows appropriate comparisons) and Replicate.

Finally, use JMP's Compare Designs. The results are in the attached journal.

Compare Designs Journal.jrn

All comments

Dan_Obermiller

Finally, use JMP's Compare Designs. The results are in the attached journal.

Compare Designs Journal.jrn

Mark_Bailey

I agree with @Dan_Obermiller that the two designs are equivalent with respect to their performance when estimating the given model, but they are not the same design. They will exhibit different confounding among higher order terms and block but those differences should be unimportant.

Full factorial design for 5 two-level categorical factors in two blocks:

Screen Shot 2020-11-05 at 6.07.39 AM.png

Replicated half fraction design for t two-level categorical factors as two blocks:

Screen Shot 2020-11-05 at 6.07.56 AM.png

gchesterton

Thanks @DanP and @Mark_Bailey -- it was the fact that the designs look different that was leading me to wonder what I was missing with the repeated 16-run design. It sounds like I'm not missing anything for my purposes, if I care only about main effects and two-factor interactions. I assume they do this with equal power?

gchesterton

@Dan_Obermiller that was meant for you. Apologies for confusing @DanP

statman

Just to add a possible different perspective...

If you create the blocks using JMP it will, by default, consider the block a random effect. This will allow the block effect to be assigned and hence removed from the error estimate thereby reducing the MS error and increasing the likelihood of detecting factor effects (increasing the MSfactor/MSerror or F-Ratio).

If, however, you can identify and assign the factors that make up the noise in the blocks (you do this intentionally), you can then treat the block as a fixed effect in the model. This now allows for the estimation of not only the block, but the block by factor interactions. In a RCBD, you would get full resolution of the block effect. This is an extremely important opportunity. Why? The block by factor interactions are estimates of noise by factor interactions. A significant noise by factor interaction sounds like this...The effect of your design factor depends on noise. That would be a serious problem since you are, at some point, trying to set the design factors to their optimum levels. What optimum levels you would choose would depend on noise. This is the robust design problem. Treating the block as a fixed effect, you can now quantify and estimate the robustness of your design to noise, (essentially robust design is the absence of noise by factor interactions...your design performs consistently over changing noise.)

In your case, you would have resolution V of the design factors (2^5-1) able to separate 1st and 2nd order effects AND you would get all Block and Block by factor effects. 31 DF's

Y = (A+B+C+D+E+AB+AC+AD+AE+BC+BD+BE+CD+CE+DE)(1+Block)+Block