TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING. Upper-case letters are associated with factors, or regressors of factorial effects, e.g.
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1 STAT Level Factorial Experiments: Blocking 1 TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING Some Traditional Notation: Upper-case letters are associated with factors, or regressors of factorial effects, e.g. ABC x 1 x 2 x 3 The treatment combination associated with µ is sometimes designated by listing lower-case letters associated with factors set to level 2, e.g. ac factors A and C at level 2, others at level 1 abcd factors A-D at level 2, others at level 1 (1) all factors at level 1 So upper-case letters are associated with columns, and lower-case letters with rows, of X.
2 STAT Level Factorial Experiments: Blocking 2 Blocks of Size 2 f (i.e. complete block designs) Put one full unreplicated factorial experiment in each block e.g. CBD for f = 2: (1) (1) (1) a b a b... a b all 2 f treatments in each block ab ab ab r blocks
3 STAT Level Factorial Experiments: Blocking 3 Effects Models: { y mijl... = µ + ρ m + (α i +...) + ɛ mijl... (overparameterized) µ + ρ m + (±α ±...) + ɛ mijl... (full rank) i = 1, 2, et cetera; m = 1, 2, 3,...r Assuming no block-by-treatment interaction, source df sum-of-squares blocks r 1 m 2f (ȳ m ȳ ) 2 treatments 2 f 1 ijl... r(ȳ ijl... ȳ ) 2 residual (2 f 1)(r 1) difference corr d total r2 f 1 mijl... (y mijl... ȳ ) 2
4 STAT Level Factorial Experiments: Blocking 4 Residual would be block-by-treatment interaction if that had been included in the model Treatments can be decomposed into 1-df components for each effect, e.g. N ˆα 2 just as in unblocked case leave out µ... this is taken out as correction factor e.g. f = 2, SST = N ˆα 2 + N ˆβ 2 + N (αβ) 2, df=3
5 STAT Level Factorial Experiments: Blocking 5 Blocks of Size 2 f 1 Put one half-replicate one-half of all treatments in each block Start with a single, full, unreplicated design... divide into two blocks of size 2 f 1 so that each treatment appears exactly once This leads to: source df blocks 1 (two blocks) treatments 2 f 2 (???) residual 0 (no replication) c.t. 2 f 1 (N 1) Doesn t accommodate 2 f 1 treatment d.f., even without any residual d.f.
6 STAT Level Factorial Experiments: Blocking 6 Need to give up one of the treatment degrees of freedom. For regular blocks, do this by intentionally confounding one of the factorial effects with blocks Arrange pairs of blocks so that, for a selected effect ( ), ( ) is always + in one block ( ) is always in the other block Can t estimate ( ) now because it is confounded with the block difference, so we generally want to select an effect that is: least likely to be important or interesting most likely to be zero The highest-order interaction is often used
7 STAT Level Factorial Experiments: Blocking 7 Example: 2 3, confounding (αβγ) with blocks treatment I A B C AB AC BC ABC block (1) a b c ab ac bc abc
8 STAT Level Factorial Experiments: Blocking 8 Notes: (1) a ab ac b c not a BIBD (or any other design we ve studied so far) bc abc COULD have used another effect to split treatments into 2 blocks, e.g. AB rather than ABC, if this had made sense Over the entire design, all factorial effects are orthogonal, so all except ABC are orthogonal to ABC+block Estimates and SS s for other factorial effects are unchanged Without replication, there is no pure error e.g. use normal plots omitting ˆµ AND block + (αβγ)
9 STAT Level Factorial Experiments: Blocking 9 Now, suppose we can afford to apply each treatment r > 1 times, but must still use blocks of size 2 f 1 One option is to copy the previous pattern r times: (1) ab ac bc... a b c abc r blocks r blocks... ABC is confounded with the difference between the two groups of blocks Let m index block... not replicate... so there are 2r blocks:
10 STAT Level Factorial Experiments: Blocking 10 source df sum-of-squares blk s 2r 1 m 2f 1 (ȳ m ȳ ) 2 trt s 2 f 2 N ( ) 2 (omit (αβγ)) resid (2 f 2)(r 1) difference c.t. r2 f 1 mijl... (y mijl... ȳ ) 2 In the last sum, not all possible combinations of index values appear since not all treatments (i, j,...) appear in each block (m) For r = 1, there are no d.f. for residual under the full model (of course), but an error estimate could come from d.f. in treatments corresponding to terms omitted from the model (and not confounded with blocks)
11 STAT Level Factorial Experiments: Blocking 11 If blocks can be regarded as random, this experiment can also be analyzed as a split-plot design, with levels of ABC compared between blocks, and other effects compared within blocks: stratum source df sum-of-squares w.p. ABC 1 2 N (αβγ) blk s ABC 2(r 1) resid. from prev. blk SS c.t. 2r 1 prev. blk SS s.p. other trt s 2 f 2 N 2 ( ) (omit (αβγ)) resid (2 f 2)(r 1) difference c.t. r2 f 1 mijl... (y mijl... ȳ ) 2
12 STAT Level Factorial Experiments: Blocking 12 Last 3 lines of the ANOVA table above are the same as with the fixed-block analysis. We get information about block variation by first eliminating any systematic difference due to ABC, i.e. looking at blocks ABC. In contrast, there is NO information about ABC blocks : blocks SS = 2 f 1 m (ȳ m... ȳ... ) 2 r blocks including (1),ab,ac,bc... ABC= r blocks including a,b,c,abc... ABC=+ as with split-plot design assigning r dams to diet 1, r dams to diet 2...
13 STAT Level Factorial Experiments: Blocking 13 But what if you: don t want to assume random blocks need to make inferences about all factorial effects are required to use blocks of size 2 f 1 Partial confounding (don t confound the same factorial effect with each pair of blocks): (1) ab ac bc a b c abc (1) bc a abc b c ab ac ABC conf. BC conf. (-) conf. r
14 STAT Level Factorial Experiments: Blocking 14 Estimates and SS s for non-confounded effects are computed as in the unblocked case, e.g. N ˆα 2 Estimates and SS s for confounded effects are computed from replicates in which they aren t confounded, e.g. 2 f (r 1) (αβγ) 2, with the estimate computed from data in replicates 2-through-r only source df blocks 2r 1 unconfounded effects confounded effects resid 2 f 1 r c.t. 2 f r 1 r (2 f 2)(r 1) 1 *: 1 did not appear before because the single confounded effect was not recovered... now all effects are estimated, some using partial data
15 STAT Level Factorial Experiments: Blocking 15 Blocks of Size 2 f 2 Put one quarter replicate 1/4 of all treatments in each block Accomplish this by further splitting of 2 f 1 -size blocks, by selecting a second factorial effect to confound with blocks This results in a group of 4 blocks in which each treatment is applied only once. Continuing previous f = 3 example where ABC was chosen to generate the first split, now add BC for the second. (Now we won t be able to estimate the factorial effect (βγ) associated with this contrast either.)
16 STAT Level Factorial Experiments: Blocking 16 treatment I A B C ABC BC block ( ) a b c ab ac bc abc Or, (-) a b ab bc abc c ac
17 STAT Level Factorial Experiments: Blocking 17 So in the unreplicated case, we have 8 observations 7=8-1 d.f. after correction for the mean 4=7-3 d.f. after accounting for 4 blocks But, there are 5 factorial effects not used in splitting : A, B, C, AB, AC... but not BC, ABC What else can t be estimated within blocks?
18 STAT Level Factorial Experiments: Blocking 18 Within each block: ABC = x 1 x 2 x 3 is constant BC = x 2 x 3 is constant But x 1 x 2 x 3 x 2 x 3 = x 1... if both x 1 x 2 x 3 and x 2 x 3 are constant within blocks, then x 1 must be also So, A = x 1 = ABC BC is also constant within each block Symbolically, within: block 1, I = ABC = +BC = A block 2, I = +ABC = +BC = +A block 3, I = +ABC = BC = A block 4, I = ABC = BC = +A By choosing to confound ABC and BC with blocks, we also confound their generalized interaction, A. (This is a bad design choice for most purposes since a main effect is not estimable.) Summarized by the generating relation I = ± ABC = ± BC = ±A
19 STAT Level Factorial Experiments: Blocking 19 Usually better: I = ± AB = ± BC = ± AC (-) abc c ab a bc b ac Now, replicate each of these blocks r times so that each treatment is applied to r units in the the overall experiment (of 4r blocks of size 2) For fixed block effects: source d.f. blocks 4r 1 treatments 2 f 4 residual (2 f 4)(r 1) c.t. 2 f r 1
20 STAT Level Factorial Experiments: Blocking 20 For random block effects: stratum source d.f. sums-of-squares w.p. AB,AC,BC 3 2 N (αβ) +... blocks AB,AC,BC 4r 4 c.t. 4r 1 (fixed blocks ) s.p. other fact. effects 2 f 4 (as before) residual (2 f 4)(r 1) (as before) c.t. 2 f r 1
21 STAT Level Factorial Experiments: Blocking 21 Partial confounding also works here Example: 2 4 2, 2 replicates Rep 1: I = ± ABC = ± BCD ( = ± AD ) Rep 2: I = ± ABCD = ± BC ( = ± AD ) ABC and BCD estimates and SS from Rep 2 only ABCD and BC estimates and SS from Rep 1 only AD is confounded with blocks in both replicates; no information available df(resid) = 31 (c.t.) 7 (blocks) 4 (ABC,BCD,ABCD,BC) 10 ( others except for AD) = 10 If confounded effects had been entirely different in each rep: df(resid) = 31 (c.t.) 7 (blocks) 6 (all confounded effects) 9 (others) = 9
22 STAT Level Factorial Experiments: Blocking 22 Blocks of size 2 f s, s < f Have already taken care of s = 1, 2 Ideas can be made more general by sequentially splitting blocks (s times), and realizing that new generalized interactions are also confounded Example : 2 6 = 64 treatments 2 4 = 16 blocks = 4 observations per block
23 STAT Level Factorial Experiments: Blocking 23 Think in terms of the effects we select to confound with blocks with each split: I = ± ABF 1st split = ± ACF (= ± BC ) 2nd split = ± BDF (= ± AD = ± ABCD = ±CDF ) 3rd split = ± DEF (= ± ABDE = ± ACDE = ± BCDEF 4th split = ± BE = ± AEF = ± ABCEF = ± CE) Generalized interactions added at each stage are between the new independent effect, and all previously implicated effects (independent or GI s) 2 s 1 effects or words, in all, are confounded with blocks (all possible symbolic products formed with s independent effects)
24 STAT Level Factorial Experiments: Blocking 24 Note: We cannot pick a previously identified GI as a new independent effect. Why? because these are ALREADY constant-within-blocks What do these blocks look like? The block containing treatment ( ) has + signs attached to each word in the generating relation: I = +ABF = +ACF = +BDF = +DEF A B F C D E We don t need to consider the generlized interactions here because their signs are determined by those of the independent words.
25 STAT Level Factorial Experiments: Blocking 25 All other blocks can be formed by reversing entire columns from this set, e.g. A and B: A B F C D E I = +ABF = ACF = BDF = +DEF Signs on ACF and BDF are reversed because they each have one of A or B Signs on ABF and DEF are not reversed because they each have both of A and B... likewise with generalized interactions
26 STAT Level Factorial Experiments: Blocking 26 Principal block: Reverse all, so that run set at the low level for each factor will be included A B F C D E I = ABF = ACF = BDF = DEF In general, the generating relation for the principal block assigns /+ to words of odd/even length
27 STAT Level Factorial Experiments: Blocking 27 For fixed block effects: source d.f. sum of squares blocks 2 s r 1 2 f s (ȳ block ȳ) 2 treatments 2 f 2 s 2 N ( ) except confounded residual difference c.t. 2 f r 1
28 STAT Level Factorial Experiments: Blocking 28 For random block effects: stratum source d.f. w.p. confounded effects 2 s 1 blocks conf effects 2 s (r 1) c.t. 2 s r 1 (split-plot blocks ) s.p. other fact. effects 2 f 2 s (as before) residual difference (as before) c.t. 2 f r 1 As with 2 f 2, partial confounding can also be used For example: r = 5, ABC in confounding pattern for replicates 1 and 2, estimate from replicates 3-5 only... some effects may be confounded in more replicates than other effects
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