1. L. M. LyeDOE Course1 Design and Analysis of Multi-Factored Experiments Part I Experiments in smaller blocks

2. L. M. LyeDOE Course2 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 –Unreplicated designs

3. L. M. LyeDOE Course3 Confounding In an unreplicated 2 k there are 2 k 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.

4. L. M. LyeDOE Course4 (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

5. L. M. LyeDOE Course5 Consider 1, a, b, ab, c, ac, bc, abc, M 1 a b ab c ac bc abc TM 1 ab c abc a b ac bc TM 1 ab ac bc a b c abc T Which is preferable? Why? Does it matter? 123

6. L. M. LyeDOE Course6 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?

7. L. M. LyeDOE Course7 Again consider M 1 ab ac bc a b c abc T Usual estimate: A= (1/4)[-1+a-b+ab-c+ac-bc+abc] NOW BECOMES

8. L. M. LyeDOE Course8 = (usual estimate) [x’s cancel out] Usual ABC = Usual estimate - x

9. L. M. LyeDOE Course9 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.

10. L. M. LyeDOE Course10 M 1 a b ab c ac bc abc TM 1 ab c abc a b ac bc TM 1 ab ac bc a b c abc T Confounded Effects: Only COnly ABOnly ABC

11. L. M. LyeDOE Course11 M 1 a b ac ab c bc abc T Confounded Effects: B, C, AB, AC (4 out of 7, instead of 1 out of 7)

12. L. M. LyeDOE Course12 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.

13. L. M. LyeDOE Course13 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.

14. L. M. LyeDOE Course14 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 2 k, 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 2 7, we give up 1 out of 127 effects- perhaps, ABCDEFG)

15. L. M. LyeDOE Course15 2 3 with 4 replications: Partial Confounding 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 Confound ABC Confound AB Confound AC Confound BC

16. L. M. LyeDOE Course16 Can estimate A, B, C from all 4 replications (32 “units of reliability”) AB from Repl.1, 3, 4 ACfrom1, 2, 4 BCfrom1, 2, 3 ABCfrom2, 3, 4 24 “units of reliability”

17. L. M. LyeDOE Course17 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

18. L. M. LyeDOE Course18 Only 2 weighing mechanisms are available, each able to handle (only) 4 t.c.’s. The 2 3 is replicated twice: 1 ab ac bc a b c abc 1 ab c abc a b ac bc Confound ABCConfound AB Machine 1 Machine 2Machine 1 Machine 2 A, B, C, AC, BC, “clean” in both replications. AB from repl. ; ABC from repl. 1 2 12

19. L. M. LyeDOE Course19 Multiple Confounding Further blocking: (more than 2 blocks) 1 cd abd abc a acd bd bc b bcd ad ac c d abcd ab 3421 2 4 = 16 t.c.’s Example: RSTU

20. L. M. LyeDOE Course20 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)

21. L. M. LyeDOE Course21 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 2 4 design. This result is illustrated for ACD (a confounded effect) and D (a “clean” effect).

22. L. M. LyeDOE Course22 1 a b ab c ac bc abc d ad bd abd cd acd bcd abcd -+-++-+-+-+--+-+-+-++-+-+-+--+-+ Sign of treatment -R +S -T +U -T +S -R +U -T +S -R +S -T +U block effect --------++++++++--------++++++++ Sign of treatment -R -S -T -U -T -S -R +U +T +S +R +S +T +U block effect ACD D

23. L. M. LyeDOE Course23 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.

24. L. M. LyeDOE Course24 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.

25. L. M. LyeDOE Course25 Design and Analysis of Multi-Factored Experiments Part II Determining what is confounded

26. L. M. LyeDOE Course26 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).

27. L. M. LyeDOE Course27 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 = AB 2 CD= ACD AB x ACD = A 2 BCD = BCD BCD x ACD = ABC 2 D 2 = AB

28. L. M. LyeDOE Course28 Need to select with care: in 2 5 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).

29. L. M. LyeDOE Course29 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.

30. L. M. LyeDOE Course30 Example: 2 5 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):

31. L. M. LyeDOE Course31 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 ABD, ACE, BCDE Prin. Block * Mult. by a: Mult. by b: Mult. by e: any thus far “unused” t.c. * note: “invariance property”

32. L. M. LyeDOE Course32 Remember that we compute the 31 effects in the usual way.Only, ABD, ACE, BCDE are not “clean”. Consider from the 2 5 table of signs:

33. L. M. LyeDOE Course33 1 abc bd acd abe ce ade bcde ---------------- ---------------- ++++++++++++++++ ++--++--++--++-- --++--++--++--++ Block 1 (too high or low by R a bc abd cd be ace de abcde ++++++++++++++++ ++++++++++++++++ ++++++++++++++++ --++--++--++--++ --++--++--++--++ Block 2 (too high or low by S) b ac d abcd ae bce abde cde ++++++++++++++++ ---------------- ---------------- --++--++--++--++ --++--++--++--++ Block 3 (too high or low by T e abce bde acde ab c ad bcd ---------------- ++++++++++++++++ ---------------- ++--++--++--++-- --++--++--++--++ Block 4 (too high or low by U ABDACEABD CONFOUNDEDCLEAN BCDE

34. L. M. LyeDOE Course34 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).

35. L. M. LyeDOE Course35 In general: (For 2 k in 2 r blocks) 2 r number of smaller blocks 2 r -1 number of confounded effects r number of confounded effects experimenter may choose 2 r -1-r number of automatically confounded effects 2 4 8 16 1 3 7 15 12341234 0 1 4 11

36. L. M. LyeDOE Course36 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, 2 6, there are 63 =2 6 -1 effects. Confounding 7 of 63 might well be tolerable.

37. L. M. LyeDOE Course37 Design and Analysis of Multi-Factored Experiments Part III Analysis of Blocked Experiments

38. L. M. LyeDOE Course38 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

39. L. M. LyeDOE Course39 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

40. L. M. LyeDOE Course40 ANOVA for the Blocked Design

41. L. M. LyeDOE Course41 Confounding in Blocks Now consider the unreplicated case Clearly the previous discussion does not apply, since there is only one replicate This is a 2 4, n = 1 replicate

42. L. M. LyeDOE Course42 Example Suppose only 8 runs can be made from one batch of raw material

43. L. M. LyeDOE Course43 The Table of + & - Signs

44. L. M. LyeDOE Course44 ABCD is Confounded with Blocks Observations in block 1 are reduced by 20 units…this is the simulated “block effect”

45. L. M. LyeDOE Course45 Effect Estimates

46. L. M. LyeDOE Course46 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

47. L. M. LyeDOE Course47 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.

48. L. M. LyeDOE Course48 Design and Analysis of Multi-Factored Experiments Part IV Analysis with Blocking : More examples

49. L. M. LyeDOE Course49 Analysis of 2 k factorial experiments with blocking Method for obtaining estimates of effects and sum-squares is exactly the same as without blocking. The only difference is in the ANOVA table. An additional line for variation due to “Blocks” must be added.

50. L. M. LyeDOE Course50 Example 1 Consider a 2 4 experiment in two blocks with effect ABCD confounded. Using the method discussed, the two blocks are as follows with the responses given. Block 1Block 2 (1) = 3a = 7 ab = 7b = 5 ac =6c = 6 bc = 8d = 4 ad = 10abc = 6 bd = 4bcd = 7 cd = 8acd = 9 abcd = 9abd = 12

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53. L. M. LyeDOE Course53 Regression Equation Effects and sum-squares are obtained by Yate’s algorithm in the usual way. Final Equation in Terms of Coded Factors: Y = 6.94 + 1.31 A + 0.44 C +0.94 D - 1.19 AC + 0.81 AD R 2 = 0.917

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57. L. M. LyeDOE Course57 Example 2 Consider a 2 5 experiment that were conducted in 4 blocks. Effects ABCD, BCDE, and AE are confounded with blocks.

58. L. M. LyeDOE Course58 ANOVA Table

59. L. M. LyeDOE Course59 Summary ANOVA table with blocking has an extra line – SS due to Blocking Other steps are the same as without blocking Examples shown here were done using Design-Ease Fractional design uses similar concepts are blocking – next topic