TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS

STAT 512 2-Level Factorial Experiments: Regular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS Bottom Line: A regular fractional factorial design consists of the treatments in one block of a (regular) blocked full 2 f experiment Example: f = 5, s = 2 2 5 (32 possible treatments) 2 2 fraction containing 2 5 2 treatments satisfying: I = +ABC = ADE (= BCDE) Now the generating relation contains just + or for each word, not ±, because we are talking about only one block

STAT 512 2-Level Factorial Experiments: Regular Fractions 2 Example, continued: ABC = + A B C D E + + + + + + + + + + + + + + + + + + + + given A, ADE = We know already that we ve lost information on 3 effects (ABC, ADE, BCDE), corresponding to 3 degrees of freedom BUT, there are 2 5 1(mean) 3(confounded effects) = 28 MORE factorial effects, and only 8 observations (in an unreplicated experiment)... something else is missing

STAT 512 2-Level Factorial Experiments: Regular Fractions 3 Look, for example, at the BC interaction: A B C D E BC + + + + + + + + + + + + + + + + + + + + + + + + A = BC... But neither is always +1 or 1 and so would not be confounded with blocks in a full blocked design

STAT 512 2-Level Factorial Experiments: Regular Fractions 4 This is not a problem when we have all 4 blocks because: I = +ABC +A = +BC in two blocks I = ABC +A = BC in two blocks That is, A and BC are orthogonal in the full design In fact, in fractional factorials, ALL factorial effects are aliased in groups... ABC, ADE, and BCDE with I, and all other effects in other groups of size 4 The defining relation (or generating relation, identifying relation ) for this design is: I = +ABC = ADE = BCDE i.e. the relationship between the effects intentionally aliased with the intercept (and confounded with blocks in a full 2 f ). Recall that these are words or generators that stand for columns in the model matrix. We can use element-wise multiplication of these columns to identify the groups of aliased effects.

STAT 512 2-Level Factorial Experiments: Regular Fractions 5 Continued Example: Main effect A I = +ABC = ADE = BCDE (2 s 1 words) A = +AABC = AADE = ABCDE A = +BC = DE = ABCDE The A main effect is aliased with 2 s 1 other words We also say that A is aliased with BC, DE, and ABCDE. Note: Underlines are added to highlight the lowest-order effect aliased with A.

STAT 512 2-Level Factorial Experiments: Regular Fractions 6 You can verify this by looking at the columns from the model matrix for this set of treatments: A B C D E BC DE ABCDE + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

STAT 512 2-Level Factorial Experiments: Regular Fractions 7 Also from I = +ABC = ADE = BCDE: B = +AC = ABDE = CDE C = +AB = ACDE = BDE D = +ABCD = AE = BCE E = +ABCE = AD = BCD For an unreplicated experiment, N = 8 = uncorrected total d.f.: 1 for I and its aliases (correction for mean) 5 estimable strings containing main effects So, 2 more estimable strings... generate these by using any two effects that are not in the first 5 sets: BD = +ACD = ABE = CE BE = +ACE = ABD = CD

STAT 512 2-Level Factorial Experiments: Regular Fractions 8 Analysis The result is 8 estimable strings of effects, 7 of which don t include I (or µ) The estimate of α is really an estimate of a string of effects: E[ˆα] = α + (βγ) (δɛ) (αβγδɛ) Similarly for other main effects, but their alias strings each include only 1 two-factor interaction Given significance (or apparent significance via normal plot) of some collection of these strings, effects that are most likely real must be identified by other information expert knowledge, hierarchy or heredity rules, further experiments...

STAT 512 2-Level Factorial Experiments: Regular Fractions 9 Comparison of Fractions: Resolution Design resolution focuses on the shortest-length word in the defining relation. Suppose: I = + AB (lowest-order effect) = +... Then: A = + B =... i.e. can t resolve main effects Suppose: I = + ABC (lowest-order effect) = +... Then: A = + BC =... i.e. o.k. for estimating all main effects cleanly if there are no two-factor interactions Suppose: I = + ABCD (lowest-order effect) = +... Then: A = + BCD =... i.e. o.k. for estimating all main effects cleanly if there are no three-factor interactions Then: AB = + CD =... i.e. can t resolve two-factor interactions

STAT 512 2-Level Factorial Experiments: Regular Fractions 10 Suppose: I = + ABCDE (lowest-order effect) = +... Then: A = + BCDE =... i.e. o.k. for estimating all main effects cleanly if there are no four-factor interactions Then: AB = + CDE =... i.e. o.k. for estimating all two-factor interactions if there are no three-factor interaction The worst case of aliasing lower-order effects with higher-order effects is determined by the lowest-order effect aliased with I, i.e. the shortest word in the defining relation The length of this shortest word (i.e. number of letters involved) is called the resolution of the design, often denoted by a roman numeral The bigger, the better.

STAT 512 2-Level Factorial Experiments: Regular Fractions 11 Summary: In a design of resolution R, no O-order effect is aliased with any effect of order less than R O Res. III: main effects aren t aliased with main effects Res. IV: main effects aren t aliased with other main effects or two-factor interactions; but two-factor interactions are aliased with other two-factor interaction Res. V: main effects aren t aliased with other main effects, two-factor interactions, or three-factor interactions; two-factor interactions aren t aliased with other two-factor interactions... these are the classes of regular fractional factorials that are most commonly used in practice

STAT 512 2-Level Factorial Experiments: Regular Fractions 12 Examples: 2 5 I = +ABCDE is a 2 5 1 V I = +ABCD is a 2 5 1 IV, would usually be considered worse I = +ABC = BDE ( = ACDE) is a 2 (5 2) III, of less resolution, but also a smaller design Note: () s are used to emphaize that ACDE is actually implied by ABC and BDE... a more compact notation could omit this with no loss of information.

STAT 512 2-Level Factorial Experiments: Regular Fractions 13 Comparing Fractions of Equal Resolution: Aberration Example: 2 7 2 IV I = +ABCD = +DEFG = +ABCEFG I = +ABCD = +CDEFG = +ABEFG The second design has fewer pairs of aliased 2-factor interactions, less aberration Goal is to find a design of: 1. maximum resolution (maximum length of shortest word), and among these 2. minimum aberration (minimum number of shortest words)

STAT 512 2-Level Factorial Experiments: Regular Fractions 14 These two criteria can be combined by looking at a list of word lengths, the word length pattern, for each candidate design: design length of words 1 2 3 4 5 6 7 I=ABCD=DEFG=ABCEFG (0 0 0 2 0 1 0) Res IV I=ABCD=CDEFG=ABEFG (0 0 0 1 2 0 0) Res IV, min ab. I=ABC=DEFG=ABCDEFG (0 0 1 1 0 0 1) Res III Look for vectors with (1.) the largest number of leading zero elements (max resolution), and from among these (2.) the smallest first non-zero element (min aberrration).

STAT 512 2-Level Factorial Experiments: Regular Fractions 15 Blocking Regular Fractional Factorial Designs As with full factorial experiments, but now realizing that the effects we chose to estimate or confound with blocks are really strings of aliased effects Previous 2 5 2 example, 8 estimable strings: Defining Relation: I = +ABC = ADE (= BCDE) I = I +ABC ADE BCDE A = A +BC DE ABCDE B = B +AC ABDE CDE C = C +AB ACDE BDE D = D +ABCD AE BCE E = E +ABCE AD BCD BD = BD +ACD ABE CE BE = BE +ACE ABD CD Without blocking, the last 7 of these are associated with the 7 d.f. that would be available for treatments

STAT 512 2-Level Factorial Experiments: Regular Fractions 16 Original fractional factorial: ad ae b bde c cde abcd abce To divide into 2 blocks (of size 4), we must confound one of the effect strings with blocks Pick, e.g., BD (split using BD column in design matrix, so BD+ACD-ABE-CE is actually confounded with blocks) source df ae bde ad b blk 1 c abcd cde abce trt 6 c.t. 7

STAT 512 2-Level Factorial Experiments: Regular Fractions 17 We COULD split a second time by confounding a second effect string, say BE BUT, this would involve the generalized interaction also: BD BE = DE; but DE also has aliases: A = A +BC DE ABCDE SO, A isn t estimable, even with aliases. Still, we COULD... source df bde ae b ad blk 3 c abcd cde abce trt 4 c.t. 7 * associated with B, C, D, and E

STAT 512 2-Level Factorial Experiments: Regular Fractions 18 Recombining Fractions Suppose a 2 5 2 has been completed: I = +ABC = ADE (= BCDE) no blocking 8 estimable strings: I, A through E, BD, BE Results were interesting, but indicate more (or more complex) effects of factors than was expected... now we want to expand the study augment the design Can convert this 1/4 fraction to a 1/2 fraction by adding any one of the other three 1/4 fractions based on the same defining relation (but different signs). For example:

STAT 512 2-Level Factorial Experiments: Regular Fractions 19 1st 1/4 fraction: I = +ABC = ADE = ( BCDE) 2nd 1/4 fraction: I = ABC = ADE = (+BCDE) 1/2 fraction : I = ADE Note: Estimable strings are now half as long (i.e. aliased groups are now half the size) as before and there are twice as many of them (counting I ) Note: Selecting a different 2nd 1/4 fraction results in a different augmented design. For example: 1st 1/4 fraction: I = +ABC = ADE = ( BCDE) 2nd 1/4 fraction: I = ABC = +ADE = ( BCDE) 1/2 fraction : I = BCDE This one is of greater resolution, and so would ordinarily be preferred

STAT 512 2-Level Factorial Experiments: Regular Fractions 20 Generally: Start with a 2 f s : I = W 1 = W 2 =... W s ( = W 1 W 2... ) 2 s 1 words (other than I) in all Add another 2 f s : I = W 1 = W 2 =... W s ( = W 1 W 2... ) - s on any combination of independent generators 2 s 1 of all words will have s... how would you show this? Together, 2 f s+1 : I = all independent words and G.I.s for which sign didn t change

STAT 512 2-Level Factorial Experiments: Regular Fractions 21 Example: 2 6 3 I = +ABC = +CDE ( = +ABDE) = ADF ( = BCDF = ACEF = BEF) Would be good to eliminate all words of length 3 here... this would increase resolution to IV W 1 = +ABC W 2 = +CDE W 3 = ADF W 1 W 2 W 3 = BEF If signs on W 1, W 2 and W 3 are changed, the sign on W 1 W 2 W 3 would also change, so add: I = ABC = CDE ( = +ABDE) = +ADF ( = BCDF = ACEF = +BEF)

STAT 512 2-Level Factorial Experiments: Regular Fractions 22 Result: 2 6 2 I = +ABDE = BCDF (= ACEF) Could subsequently expand this 1/4 fraction to a 1/2 fraction the same way, e.g. add Result: 2 6 1 I = ABDE = BCDF (= +ACEF) I = BCDF But note: we COULD have had a resolution VI 1/2 fraction if we had begun by selecting the best 2 6 1, e.g.: I = +ABCDEF Can you find another 2 6 3 III doubled twice? that yields a 26 1 V or 2 6 1 V I when

STAT 512 2-Level Factorial Experiments: Regular Fractions 23 Fold-Over Designs Recall from the discussion of blocked factorial designs: Given 1 block of runs, you can construct another by REVERSING SIGNS on a selected set of factors This leads to two practical techniques for augmenting a Resolution III fractional factorial design, based on the analysis of the data Example: 2 6 3 III I = +ABC = +CDE ( = +ABDE ) = ADF ( = BCDF = ACEF = BEF ) A = +BC..., AB = +C..., et cetera

STAT 512 2-Level Factorial Experiments: Regular Fractions 24 Suppose analysis suggests that factor A is potentially important, and we want more information on this factor. REVERSE THE SIGN FOR ONLY FACTOR A in the augmenting fraction: I = ABC = +CDE ( = ABDE ) = +ADF ( = BCDF = +ACEF = BEF ) Together: I = +CDE = BCDF ( = BEF ) no A s A = +ACDE = ABCDF = ABEF AB = +ABCDE = ACDF = AEF each alias has 4 letters each alias has 3 letters The A main effect and all two-factor interactions involving A are estimable if there are no interactions of order 3 or more (that is, the doubled design is Res V for factor A only) Still Res III for all other factors

STAT 512 2-Level Factorial Experiments: Regular Fractions 25 Suppose analysis suggests that ALL factors are potentially interesting, and we want more information on the entire system. REVERSE THE SIGNS ON ALL LETTERS in the augmenting fraction: I = ABC = CDE ( = +ABDE ) = +ADF ( = BCDF = ACEF = +BEF ) Together: I = +ABDE = BCDE ( = ACEF ) no odd-length (3, esp.) words A = +BDE = ABCDE = CEF each alias has 3 letters All main effects are estimable if there are no interactions of order 3 or more (Res IV for all factors) What happens in intermediate cases, where signs are changed on a SUBSET of factors?

STAT 512 2-Level Factorial Experiments: Regular Fractions 26 Practical Reality of Experimenting in Stages Operating conditions or raw material may change The 2 s fractions may need to be treated as blocks Example (again): 2 5 2 : I = +ABC = ADE ( = BCDE) block 1 8 estimable strings of 4 effects each 2 5 2 : I = ABC = +ADE ( = BCDE) block 2 same 8 strings of effects, but different signs within strings Together: I = BCDE 16 strings of 2 effects each one of them is ABC = ABC ADE THE WORDS THAT CHANGED SIGNS this is the contrast confounded with blocks

STAT 512 2-Level Factorial Experiments: Regular Fractions 27 Example: Begin with 2 6 3 (block 1): I = +ABC = +CDE ( = +ABDE ) = ADF ( = BCDF = ACEF = BEF ) Estimable strings contain 8 effects at this point. Add 2 6 3 (block 2): I = ABC = CDE ( = +ABDE ) = +ADF ( = BCDF = ACEF = +BEF )

STAT 512 2-Level Factorial Experiments: Regular Fractions 28 Result 2 6 2 : I = +ABDE = BCDF ( = ACEF ) ABC + CDE ADF BEF is confounded with blocks Now estimable strings contain 4 effects each. Add 2 6 2 (blocks 3 and 4): Result 2 6 1 : I = +ABDE = +BCDF ( = +ACEF ) I = +ABDE BCDF ACEF is confounded with blocks +ABC + CDE is confounded with blocks ADF BEF is confounded with blocks

STAT 512 2-Level Factorial Experiments: Regular Fractions 29 Add 2 6 1 (blocks 5 through 8): I = ABDE Result 2 6 : +ABDE is confounded with blocks BCDF is confounded with blocks ACEF is confounded with blocks...

STAT 512 2-Level Factorial Experiments: Regular Fractions 30 Summary of Example: ABC CDE (ABDE) ADF (BCDF ACEF BEF) block 1 + + + 8 runs block 2 + + + 8 runs accumulated + 16 runs block 3 + + + + + + + 8 runs block 4 + + + 8 runs accumulated + 32 runs block 5 + + + 8 runs block 6 + + + 8 runs block 7 + + + 8 runs block 8 + + + 8 runs accumulated 64 runs

STAT 512 2-Level Factorial Experiments: Regular Fractions 31 Alternatively, it may make more sense to think of this as four blocks of increasing size, with a new block added at each doubling : ABC CDE (ABDE) ADF (BCDF ACEF BEF) block 1 + + + 8 runs block 2 + + + 8 runs block 3 + + + 16 runs block 4 32 runs 1 1 2 1 3 3 1 Group 1: aliased, but estimable from data in blocks 3 and 4 Group 2: not estimable... confounded throughout with blocks Group 3: aliased, but estimable from data in block 4 Note: In most cases, this isn t done. Blocks of equal size: make operation simpler are generally more consistend with an assumption of equal experimental control (and noise ) within blocks