Two-Level Factorials (cont.)

Transcription

1 TwoLevel Factorials (cont.) Blocking and Confounding Fractional Factorials The concept of design Resolution 1

2 EE290H F05 2 What if we need to clean the reactor every four runs? Run P T F PT PF TF PTF block I II II I II I I II This block will bias the PTF interaction. The generator is 4=123. Blocking

3 A simple Example in Blocking Do 4 experiments, at 2 levels of b, and estimate the effects of x 1 and x 2 x 2 b x 1 3

4 Blocking What if we need to clean the reactor every two runs? Run P T F PT PF TF PTF b 1 b 2 block II IV III I III I II IV This will bias the PTF, TF interactions but also the P effect!! We used two generators: 4=123 (b 1 ) 5=23 (b 2 ) 45 = 123x23 = = 1II =1 4

5 Blocking For two runs/block (4 blocks) better to use this blocking: Run P T F PT PF TF PTF b 1 b 2 block IV I II III III II I IV This blocking will bias the PT, PF and TF We used two generators: b 1 =PT b 2 =PF => b 1 b 2 = PTPF = ITF = TF The main effects are still recognizable! 5

6 EE290H F05 6 Run A B C D block Examples on Blocking 2 4 design (16 runs) in 2 blocks of 8 runs: Need one generator: I = ABCDb (i.e. block=5)

7 EE290H F05 7 Run A B C D b1 b2 Examples on Blocking 2 4 design (16 runs) in 4 blocks of 4 runs: Need two generators: b 1 =124, b 2 =134

8 Can we measure the effect of the added variable? x 2 x 3 x 1 8

9 Large Experiments have many insignificant Effects... Effects of 2 5 experiment Residuals after eliminating insignificant effects 9

10 Large experiments can be cut to half without significant conflicts... 10

11 Fractional Factorials Do I need 8 runs to estimate 4 significant parameters? 2 31 fractional factorial (half fraction) The generating relation is F=PT P=TF T=PF F=PT (I=PTF) P = P TF T = T PF F = F PT I = average PTF/2 Run P T F PT Since TF, PF, PT and PTF are insignificant, this design works. (This is a resolution III design.) 11

12 2 31 Fractional Factorial The 2 31 fractional factorial is a complete factorial for any 2 of the 3 variables! (7) (8) (8) (3) (4) (3) (5) (6) (5) (3) (8) (1) (2) (2) (5) (8) (5) (2) (3) (2) 12

13 Combining Half Fractions Run P T F PT PF TF PTF P = P TF T = T PF F = F PT I = average PTF/2 Run P T F PT PF TF PTF P = P TF T = T PF F = F PT I = average PTF/2 13

14 Designs of "Resolution R" No pfactor confounded with anything less than Rp factors. 31 Example: I = PTF is a resolution III design. 2III Example: onehalf fractional of the LPCVD experiment I = PTF => 4 runs! P = (P TF) Peff = 0.32 T = (T PF) Teff = 0.97 F = (F PT) Feff = 0.15 avg = (Avg PTF/2 ) Avg =

15 EE290H F05 15 Run A B C D E 2 51 Fractional Factorial

16 Obtaining the Highest Possible Resolution Write a full factorial for the first k1 variables Associate the kth variable with / the interaction of the k 1 variables. A fractional factorial of of resolution R contains complete factorials in in every set of of R1 variables! 16

17 Example: Modeling Plasma Etch Anisotropy Parameter Range Units RF Power Watts Pressure mtorr Electrode Spacing cm CCl 4 Flow sccm He Flow sccm O 2 Flow 1020 sccm The original experiment included 2 61 = 32 runs (plus 3 center point replications). A 2 62 = 16 quarter fraction was used for anisotropy, because the measurements were costly and noisy. 17

18 Example: Designing the 2 62 quarter fraction Run = = IV Rf Pr El F1 F2 F3 18

19 Run Rf Blocked Fractional Factorial Experiments Fractional factorials can be blocked. Just choose interactions that are unimportant. 5 = = Pr El F1 F2 F3 B1B2B IV 19

20 Typical Designs are available... 20

21 Resolving Ambiguities in Fractional Factorials Upon the completion of a fractional factorial, selected confoundings can be clarified with selected additional runs. 5 = = 234 Suppose that 23 is significant. Since l 23 = ( ) we need a minimum of 2 additional runs. Because of potential blocking between the main experiment and the supplement, we actually need 3 additional runs. One choice is: Rf Pr El F1 F2 F Results R 1 R 2 R 3 Finally, solve the system: M 0.5l l l 64 = R 1 M 0.5l l l 64 = R 2 M 0.5l l l 64 = R 3 l 23 l 15 l 64 = l 23conf 21

22 Conclusion Factorial experiments can accommodate blocking, if one controls the conflicts in estimating effects. Fractional 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. (chapter 13 BHH / chapter 12 Montgomery / chapter 7 draft textbook) 22