Two-Level Factorials (cont.)

Transcription

1 EE0H 0 wolevel actorials (cont.) Blocking and Confounding ractional actorials he concept of design Resolution EE0H 0 Blocking What if we need to clean the reactor every four runs? block his block will bias the interaction. he generator is =.

2 EE0H 0 A simple Example in Blocking Do experiments, at levels of b, and estimate the effects of x and x x b x EE0H 0 Blocking What if we need to clean the reactor every two runs? his will bias the, interactions but also the effect!! We used two generators: = (b ) = (b ) = x = = = b b block V V

3 EE0H 0 or two runs/block ( blocks) better to use this blocking: b his blocking will bias the, and We used two generators: = (b ) = (b ) = x = = he main effects are still recognizable! b blck V V Blocking EE0H 0 0 A B C D block Examples on Blocking design ( runs) in blocks of runs: Need one generator: = (i.e. block=)

4 EE0H 0 Examples on Blocking design ( runs) in blocks of runs: Need two generators: B =, B = 0 A B C D B B EE0H 0 Can we measure the effect of the added variable? x x x

5 EE0H 0 Large Experiments have many insignificant Effects... Effects of experiment Residuals after eliminating insignificant effects EE0H 0 Large experiments can be cut to half without significant conflicts... 0

6 EE0H 0 ractional actorials Do need runs to estimate significant parameters? fractional factorial (half fraction) he generating relation is = = = = (=) = = = = avg x0. Since,, and are insignificant, this design works. (his is a resolution design.) EE0H 0 ractional actorial he fractional factorial is a complete factorial for any of the variables! () () () () () () () () () () () () () () () () () () () ()

7 EE0H 0 Combining Half ractions = = = = avg x0. = = = = avg x0. EE0H 0 Designs of "Resolution R" No pfactor confounded with anything less than Rp factors. Example: = is a resolution design. Example: onehalf fractional of the LCVD experiment = => runs! = ( ) eff = 0. = ( ) eff = 0. = ( ) eff = 0. = (Avg 0. ) Avg =. avg

8 EE0H 0 ractional actorial 0 A B C D E EE0H 0 Obtaining the Highest ossible Resolution Write a full factorial for the first k variables Associate the kth variable with / the interaction of the k variables. A fractional factorial of of resolution R contains complete factorials in in every set of of R variables!

9 EE0H 0 Example: Modeling lasma Etch Anisotropy arameter Range Units R ower 0000 Watts ressure 0000 morr Electrode Spacing.. cm CCl low 000 sccm He low 000 sccm O low 00 sccm he original experiment included = runs (plus center point replications). A = quarter fraction was used for anisotropy, because the measurements were costly and noisy. EE0H 0 Example: Designing the quarter fraction 0 = = V Rf r El

10 EE0H 0 ractional factorials can be blocked. Just choose interactions that are unimportant. = = 0 Blocked ractional actorial Experiments Rf r El BBB V EE0H 0 ypical Designs are available... 0

11 EE0H 0 Resolving Ambiguities in ractional actorials Upon the completion of a fractional factorial, selected confoundings can be clarified with selected additional runs. = = Suppose that is significant. Since l = ( ) we need a minimum of additional runs. Because of potential blocking between the main experiment and the supplement, we actually need additional runs. One choice is: Rf r El Results R R R inally, solve the system: M 0.l 0.l 0.l = R M 0.l 0.l 0.l = R M 0.l 0.l 0.l = R l l l = l conf EE0H 0 Conclusion actorial experiments can accommodate blocking, if one controls the conflicts in estimating effects. ractional factorial experiments take advantage of the insignificance of higher order terms, to accommodate many variables with few runs. Experiments can be done in stages, initially screening, and later analyzing important effects in detail. (see chapter in BHH or chapter in Montgomery)