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Design and Analysis of Multi-Factored Experiments

Design and Analysis of Multi-Factored Experiments. Part I Experiments in smaller blocks. Design of Engineering Experiments Blocking & Confounding in the 2 k. Blocking is a technique for dealing with controllable nuisance variables Two cases are considered Replicated designs

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Design and Analysis of Multi-Factored Experiments

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  1. Design and Analysis of Multi-Factored Experiments Part I Experiments in smaller blocks DOE Course

  2. Design of Engineering ExperimentsBlocking & Confounding in the 2k • Blocking is a technique for dealing with controllable nuisance variables • Two cases are considered • Replicated designs • Unreplicated designs DOE Course

  3. Confounding In an unreplicated 2k there are 2k treatment combinations. Consider 3 factors at 2 levels each: 8 t.c.’s If each requires 2 hours to run, 16 hours will be required. Over such a long time period, there could be, say, a change in personnel; let’s say, we run 8 hours Monday and 8 hours Tuesday - Hence: 4 observations on each of two days. DOE Course

  4. (or 4 observations in each of 2 plants) (or 4 observations in each of 2 [potentially different] plots of land) (or 4 observations by 2 different technicians) Replace one (“large”) block by 2 smaller blocks DOE Course

  5. M T 1 a b ab c ac bc abc 1 2 3 Consider 1, a, b, ab, c, ac, bc, abc, M T M T 1 ab c abc a b ac bc 1 ab ac bc a b c abc Which is preferable? Why? Does it matter? DOE Course

  6. The block with the “1” observation (everything at low level) is called the “Principal Block” (it has equal stature with other blocks, but is useful to identify). Assume all Monday yields are higher than Tuesday yields by a (near) constant but unknown amount X. (X is in units of the dependent variable under study). What is the consequence(s) of having 2 smaller blocks? DOE Course

  7. M T 1 ab ac bc a b c abc Again consider Usual estimate: A= (1/4)[-1+a-b+ab-c+ac-bc+abc] NOW BECOMES DOE Course

  8. = (usual estimate) [x’s cancel out] Usual ABC = Usual estimate - x DOE Course

  9. We would find that we estimate A, B, AB, C, BC, ABC - X Switch M & T, and ABC - X becomes ABC + X Replacement of one block by 2 smaller blocks requires the “sacrifice” (confounding) of (at least) one effect. DOE Course

  10. M T 1 a b ab c ac bc abc M T M T 1 ab c abc a b ac bc 1 ab ac bc a b c abc Confounded Effects: Only C Only AB Only ABC DOE Course

  11. M T 1 a b ac ab c bc abc Confounded Effects: B, C, AB, AC (4 out of 7, instead of 1 out of 7) DOE Course

  12. Recall: X is “nearly constant”. If X varies significantly with t.c.’s, it interacts with A/B/C, etc., and should be included as an additional factor. DOE Course

  13. Basic idea can be viewed as follows: STUDY IMPORTANT FACTORS UNDER MORE HOMOGENEOUS CONDITIONS, With the influence of some of the heterogeneity in yields caused by unstudied factors confined to one effect, (generally the one we’re least interested in estimating- often one we’re willing to assume equals zero- usually the highest order interaction). We reduce Exp. Error by creating 2 smaller blocks, at expense of confounding one effect. DOE Course

  14. All estimates not “lost” can be judged against less variability (and hence, we get narrower confidence intervals, smaller  error for given  error, etc.) For large k in 2k, confounding is popular- Why? (1) it is difficult to create large homogeneous blocks (2) loss of one effect is not thought to be important (e.g. in 27, we give up 1 out of 127 effects- perhaps, ABCDEFG) DOE Course

  15. Partial Confounding 23 with 4 replications: Confound ABC Confound AB Confound AC Confound BC 1 ab ac bc a b c abc 1 ab c abc a b ac bc 1 b ac abc a ab c bc 1 a bc abc b ab c ac DOE Course

  16. Can estimate A, B, C from all 4 replications (32 “units of reliability”) AB from Repl. 1, 3, 4 AC from 1, 2, 4 BC from 1, 2, 3 ABC from 2, 3, 4 24 “units of reliability” DOE Course

  17. Example from Johnson and Leone, “Statistics and Experimental Design in Engineering and Physical Sciences”, 1976, Wiley: Dependent Variable: Weight loss of ceramic ware A: Firing Time B: Firing Temperature C: Formula of ingredients DOE Course

  18. Confound ABC 1 Confound AB 2 Only 2 weighing mechanisms are available, each able to handle (only) 4 t.c.’s. The 23 is replicated twice: Machine 1 Machine 2 Machine 1 Machine 2 1 ab ac bc a b c abc 1 ab c abc a b ac bc A, B, C, AC, BC, “clean” in both replications. AB from repl. ; ABC from repl. 2 1 DOE Course

  19. Multiple Confounding Further blocking: (more than 2 blocks) 24 = 16 t.c.’s Example: 1 2 3 4 1 cd abd abc a acd bd bc b bcd ad ac c d abcd ab R S T U DOE Course

  20. Imagine that these blocks differ by constants in terms of the variable being measured; all yields in the first block are too high (or too low) by R. Similarly, the other 3 blocks are too high (or too low) by amounts S, T, U, respectively. (These letters play the role of X in 2-block confounding). (R + S + T + U = 0 by definition) DOE Course

  21. Given the allocation of the 16 t.c.’s to the smaller blocks shown above, (lengthy) examination of all the 15 effects reveals that these unknown but constant (and systematic) block differences R, S, T, U, confound estimates AB, BCD, and ACD (# of estimates confounded at minimum = 1 fewer than # of blocks) but leave UNAFFECTED the 12 remaining estimates in the 24 design. This result is illustrated for ACD (a confounded effect) and D (a “clean” effect). DOE Course

  22. ACD D Sign of treatment block effect Sign of treatment block effect 1 a b ab c ac bc abc d ad bd abd cd acd bcd abcd - + - + + - + - + - + - - + - + -R +S -T +U +U -T +S -R +U -T +S -R -R +S -T +U - - - - - - - - + + + + + + + + -R -S -T -U -U -T -S -R +U +T +S +R +R +S +T +U DOE Course

  23. In estimating D, block differences cancel. In estimating ACD, block differences DO NOT cancel (the R’s, S’s, T’s, and U’s accumulate). In fact, we would estimate not ACD, but [ACD - R/2 + S/2 - T/2 + U/2] The ACD estimate is hopelessly confounded with block effects. DOE Course

  24. Summary • How to divide up the treatments to run in smaller blocks should not be done randomly • Blocking involves sacrifices to be made – losing one or more effects • In the next part, we will examine how to determine what effects are confounded. DOE Course

  25. Design and Analysis ofMulti-Factored Experiments Part II Determining what is confounded DOE Course

  26. We began this discussion of multiple confounding with 4 treatment combo’s allocated to each of the four smaller blocks. We then determined what effects were and were not confounded. Sensibly, this is ALWAYS REVERSED. The experimenter decides what effects he/she is willing to confound, then determines the treatments appropriate to each smaller block. (In our example, experimenter chose AB, BCD, ACD). DOE Course

  27. As a consequence of a theorem by Bernard, only two of the three effects can be chosen by the experimenter. The third is then determined by “MOD 2 multiplication”. Depending which two effects were selected, the third will be produced as follows: AB x BCD = AB2CD = ACD AB x ACD = A2BCD = BCD BCD x ACD = ABC2D2 = AB DOE Course

  28. Need to select with care: in 25 with 4 blocks, each of 8 t.c.’s, need to confound 3 effects: Choose ABCDE and ABCD. (consequence: E - a main effect) Better would be to confound more modestly: say - ABD, ACE, BCDE. (No Main Effects nor “2fi’s” lost). DOE Course

  29. Once effects to be confounded are selected, t.c.’s which go into each block are found as follows: Those t.c.’s with an even number of letters in common with all confounded effects go into one block (the principal block); t.c.’s for the remaining block(s) are determined by MOD - 2 multiplication of the principal block. DOE Course

  30. Example: 25 in 4 blocks of 8. Confounded: ABD, ACE, [BCDE] of the 32 t.c.’s: 1, a, b, ……………..abcde, the 8 with even # letters in common with all 3 terms (actually the first two alone is EQUIVALENT): DOE Course

  31. ABD, ACE, BCDE Prin. Block* 1, abc, bd, acd, abe, ce, ade, bcde a, bc, abd, cd, be, ace, de, abcde b, ac, d, abcd, ae, bce, abde, cde e, abce, bde, acde, ab, c, ad, bcd Mult. by a: Mult. by b: Mult. by e: any thus far “unused” t.c. * note: “invariance property” DOE Course

  32. Remember that we compute the 31 effects in the usual way. Only, ABD, ACE, BCDE are not “clean”. Consider from the 25 table of signs: DOE Course

  33. CONFOUNDED CLEAN ABD ACE BCDE AB D Block 1 (too high or low by R 1 abc bd acd abe ce ade bcde - - - - - - - - - - - - - - - - + + + + + + + + + + - - + + - - - - + + - - + + Block 2 (too high or low by S) a bc abd cd be ace de abcde + + + + + + + + + + + + + + + + + + + + + + + + - - + + - - + + - - + + - - + + Block 3 (too high or low by T b ac d abcd ae bce abde cde + + + + + + + + - - - - - - - - - - - - - - - - - - + + - - + + - - + + - - + + Block 4 (too high or low by U e abce bde acde ab c ad bcd - - - - - - - - + + + + + + + + - - - - - - - - + + - - + + - - - - + + - - + + DOE Course

  34. If the influence of the unknown block effect, R, is to be removed, it must be done in Block 1, for R appears only in Block 1. You can see when it cancels and when it doesn’t. (Similarly for S, T, U). DOE Course

  35. In general: (For 2k in 2r blocks) 2r number of smaller blocks 2r-1 number of confounded effects r number of confounded effects experimenter may choose 2r-1-r number of automatically confounded effects 2 4 8 16 1 3 7 15 1 2 3 4 0 1 4 11 DOE Course

  36. It may appear that there would be little interest in designs which confound as many as, say, 7 effects. Wrong! Recall that in a, say, 26, there are 63 =26-1 effects. Confounding 7 of 63 might well be tolerable. DOE Course

  37. Design and Analysis of Multi-Factored Experiments Part III Analysis of Blocked Experiments DOE Course

  38. Blocking a Replicated Design • This is the same scenario discussed previously • If there are n replicates of the design, then each replicate is a block • Each replicate is run in one of the blocks (time periods, batches of raw material, etc.) • Runs within the block are randomized DOE Course

  39. Blocking a Replicated Design Consider the example; k = 2 factors, n = 3 replicates This is the “usual” method for calculating a block sum of squares DOE Course

  40. ANOVA for the Blocked Design DOE Course

  41. Confounding in Blocks • Now consider the unreplicated case • Clearly the previous discussion does not apply, since there is only one replicate • This is a 24, n = 1 replicate DOE Course

  42. Example Suppose only 8 runs can be made from one batch of raw material DOE Course

  43. The Table of + & - Signs DOE Course

  44. ABCD is Confounded with Blocks Observations in block 1 are reduced by 20 units…this is the simulated “block effect” DOE Course

  45. Effect Estimates DOE Course

  46. The ANOVA The ABCD interaction (or the block effect) is not considered as part of the error term The rest of the analysis is unchanged DOE Course

  47. Summary • Better effects estimates can be made by doing a large experiments in blocks • Choice of effect to sacrifice must be made carefully – avoid losing main and 2 f.i.’s. • Luckily, most good software will do the blocking and subsequent analysis for you – but you must check to make sure that the effects you want estimated are not confounded with blocks. DOE Course

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