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Design and Analysis of Multi-Factored Experiments Part I Experiments in smaller blocks L. M. Lye DOE Course 1 Design of Engineering Experiments Blocking & Confounding in the 2k • Blocking is a technique for dealing with controllable nuisance variables • Two cases are considered – Replicated designs – Unreplicated designs L. M. Lye DOE Course 2 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. L. M. Lye DOE Course 3 (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 L. M. Lye DOE Course 4 Consider 1, a, b, ab, c, ac, bc, abc, 1 M 1 a b ab T c ac bc abc 2 M 1 ab c abc T a b ac bc 3 M 1 ab ac bc T a b c abc Which is preferable? Why? Does it matter? L. M. Lye DOE Course 5 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? L. M. Lye DOE Course 6 Again consider M 1 ab ac bc T a b c abc Usual estimate: A= (1/4)[-1+a-b+ab-c+ac-bc+abc] NOW BECOMES L. M. Lye DOE Course 7 1 (1 x ) a b (ab x ) 4 c (ac x ) (bc x ) abc = (usual estimate) [x’s cancel out] 1 Usual ABC 1 a b ab c ac bc abc 4 1 (1 x ) a b (ab x ) 4 c (ac x) (bc x) abc = Usual estimate - x L. M. Lye DOE Course 8 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. L. M. Lye DOE Course 9 M T M T M T 1 a b ab c ac bc abc 1 ab c abc a b ac bc 1 ab ac bc a b c abc Confounded Effects: Only C L. M. Lye Only AB DOE Course Only ABC 10 Confounded Effects: M 1 a b ac B, C, AB, AC T ab c bc abc (4 out of 7, instead of 1 out of 7) L. M. Lye DOE Course 11 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. L. M. Lye DOE Course 12 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. L. M. Lye DOE Course 13 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 effectsperhaps, ABCDEFG) L. M. Lye DOE Course 14 Partial Confounding 23 with 4 replications: Confound ABC Confound Confound AB AC 1 a 1 a 1 a 1 b ab b ab b b ab a ab ac c c ac ac c bc c bc abc abc bc abc bc abc ac L. M. Lye DOE Course Confound BC 15 Can estimate A, B, C from all 4 replications (32 “units of reliability”) AB AC BC ABC L. M. Lye from Repl. from from from 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 DOE Course 24 “units of reliability” 16 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 L. M. Lye DOE Course 17 Only 2 weighing mechanisms are available, each able to handle (only) 4 t.c.’s. The 23 is replicated twice: Confound ABC 1 Confound AB 2 Machine 1 Machine 2Machine 1 Machine 2 1 a 1 a ab b ab b ac c c ac bc abc abc bc A, B, C, AC, BC, “clean” in both replications. AB from repl. 1 ; ABC from repl. 2 L. M. Lye DOE Course 18 Multiple Confounding Further blocking: (more than 2 blocks) 24 = 16 t.c.’s Example: L. M. Lye 1 2 3 4 1 a b c cd acd bcd d abd bd ad abcd abc bc ac ab R S T U DOE Course 19 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) L. M. Lye DOE Course 20 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). L. M. Lye DOE Course 21 ACD 1 a b ab c ac bc abc d ad bd abd cd acd bcd abcd L. M. Lye D Sign of treatment block effect Sign of treatment block effect + + + + + + + + -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 22 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. L. M. Lye DOE Course 23 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. L. M. Lye DOE Course 24 Design and Analysis of Multi-Factored Experiments Part II Determining what is confounded L. M. Lye DOE Course 25 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). L. M. Lye DOE Course 26 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 AB x ACD = A2BCD BCD x ACD = ABC2D2 L. M. Lye DOE Course = ACD = BCD = AB 27 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). L. M. Lye DOE Course 28 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. L. M. Lye DOE Course 29 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): L. M. Lye DOE Course 30 ABD, ACE, BCDE Prin. Block* 1, abc, bd, acd, abe, ce, ade, bcde Mult. by a: a, bc, abd, cd, be, ace, de, abcde Mult. by b: b, ac, d, abcd, ae, bce, abde, cde Mult. by e: e, abce, bde, acde, ab, c, ad, bcd any thus far “unused” t.c. L. M. Lye * note: “invariance property” DOE Course 31 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: L. M. Lye DOE Course 32 CONFOUNDED L. M. Lye ABD ACE CLEAN 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 33 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). L. M. Lye DOE Course 34 In general: (For 2k in 2r blocks) 2r number of smaller blocks 2r-1 r 2r-1-r number number of confounded number of effects of confounded experimenter automatically effects may choose confounded effects 2 4 8 1 3 7 1 2 3 0 1 4 16 15 4 11 L. M. Lye DOE Course 35 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. L. M. Lye DOE Course 36 Design and Analysis of Multi-Factored Experiments Part III Analysis of Blocked Experiments L. M. Lye DOE Course 37 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 L. M. Lye DOE Course 38 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 L. M. Lye Bi2 y...2 12 i 1 4 6.50 3 SS Blocks DOE Course 39 ANOVA for the Blocked Design L. M. Lye DOE Course 40 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 L. M. Lye DOE Course 41 Example Suppose only 8 runs can be made from one batch of raw material L. M. Lye DOE Course 42 The Table of + & - Signs L. M. Lye DOE Course 43 ABCD is Confounded with Blocks Observations in block 1 are reduced by 20 units…this is the simulated “block effect” L. M. Lye DOE Course 44 Effect Estimates L. M. Lye DOE Course 45 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 L. M. Lye DOE Course 46 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. L. M. Lye DOE Course 47 Design and Analysis of Multi-Factored Experiments Part IV Analysis with Blocking : More examples L. M. Lye DOE Course 48 Analysis of 2k 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. L. M. Lye DOE Course 49 Example 1 Consider a 24 experiment in two blocks with effect ABCD confounded. Using the method discussed, the two blocks are as follows with the responses given. Block 1 Block 2 L. M. Lye (1) = 3 a=7 ab = 7 b=5 ac =6 c=6 bc = 8 d=4 ad = 10 abc = 6 bd = 4 bcd = 7 cd = 8 acd = 9 abcd = 9 abd = 12 DOE Course 50 DESIGN-EASE Pl ot Y A: B: C: D: Half Normal plot A B C D 99 97 A Half Normal % probability 95 90 AC 85 D 80 AD 70 60 40 20 0 0.00 0.66 1.31 1.97 2.63 |Effect| L. M. Lye DOE Course 51 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Source Block (ABCD) Model A B C D AB AC AD BC BD CD Error (3 f.i.’s terms) Cor Total Sum of Squares 0.063 80.63 27.56 1.56 3.06 14.06 0.063 22.56 10.56 0.56 0.56 0.063 4.25 84.94 DF 1 10 1 1 1 1 1 1 1 1 1 1 4 15 Mean Square 0.063 8.06 27.56 1.56 3.06 14.06 0.063 22.56 10.56 0.56 0.56 0.063 1.06 F Value 7.59 25.94 1.47 2.88 13.24 0.059 21.24 9.94 0.53 0.53 0.059 The Model F-value of 7.59 implies the model is significant. There is only a 3.29% chance that a "Model F-Value" this large could occur due to noise. L. M. Lye DOE Course 52 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 R2 = 0.917 L. M. Lye DOE Course 53 DESIGN-EASE Pl ot Y Normal plot of residuals 99 95 Normal % probability 90 80 70 50 30 20 10 5 1 -1.79 -0.89 0.00 0.89 1.79 Studentized Res iduals L. M. Lye DOE Course 54 DESIGN-EASE Pl ot Interaction Graph Y C 12 X = A: A Y = C: C Y 9.75 C- -1.000 C+ 1.000 Actual Factors B: B = 0.00 D: D = 0.00 7.5 5.25 3 -1.00 -0.50 0.00 0.50 1.00 A L. M. Lye DOE Course 55 DESIGN-EASE Pl ot Interaction Graph Y D 12 X = A: A Y = D: D Y 9.75 D- -1.000 D+ 1.000 Actual Factors B: B = 0.00 C: C = 0.00 7.5 5.25 3 -1.00 -0.50 0.00 0.50 1.00 A L. M. Lye DOE Course 56 Example 2 Consider a 25 experiment that were conducted in 4 blocks. Effects ABCD, BCDE, and AE are confounded with blocks. L. M. Lye DOE Course 57 ANOVA Table L. M. Lye DOE Course 58 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 L. M. Lye DOE Course 59