Semifolding 2 k-p Designs

Transcription

1 Semifolding 2 k-p Designs Robert W. Mee and Marta Peralta Department of Statistics University of Tennessee Knoxville, TN rmee@utk.edu Abstract This article addresses the varied possibilities for following a two-level fractional factorial with another fractional factorial half the size of the original experiment. While follow-up fractions of the same size as an original experiment are common practice, in many situations a smaller follow-up experiment will suffice. Peter John coined the term semifolding to describe using half of a foldover design. Existing literature does include brief mention and examples of semifolding but no thorough development of this followup strategy. After a quick examination of the estimation details for semifolding the design, we focus on following 16-run fractions with a semifold design of eight runs. Two such examples are considered - one where the initial fraction is resolution, the other resolution III. A general result is proven for semifolding foldover designs in two blocks is also recommended k p 2 designs. Conducting full Key words: blocking, foldover, irregular fraction, sequential experiment. Submitted July 1997; revised June

2 1. INTRODUCTION Daniel (1962) discusses a wide range of possible follow-up designs that are smaller than an original two-level fractional factorial experiment. One suggestion by Daniel involved augmenting a 8 4 experiment with eight follow-up runs, in order to estimate the seven two-factor interactions involving one of the factors. This particular strategy is named semifolding in Barnett, Czitrom, John and Leon (1997) since it involves running half of a foldover design. Barnett et al. (1997) presented a case study where an initial 6 2 experiment with centerpoints indicated the presence of two or three main effects, plus at least four different two-factor interactions. In situations such as this where several contrasts of aliased two-factor interactions have large estimates, some follow-up design is needed. For this particular application, experimentation was costly, so only eight runs of a foldover fraction were performed. The original quarter fraction ( 6 2 ) plus the follow-up experiment together form a 3/8ths fraction of the 2 6 factorial. Although this irregular fraction is not an orthogonal design, it does permit estimation of the six main effects and as many as twelve of the fifteen two-factor interactions. Completing the foldover fraction would make the design orthogonal (i.e., the combined 32 runs form a 6 1 ) and so improve the precision of each estimated coefficient, but this would not permit estimation of any more two-factor interactions. Most design of experiment texts that include 2 k-p series designs discuss strategies for adding a second fraction from the same family (e.g., Box and Draper 1987, p. 156ff., Montgomery 1997, p. 412ff.). These strategies include: 2

3 For resolution III designs, increase the resolution by reversing the signs of all factors For resolution III or, estimate all k-1 two-factor interactions for one factor by reversing that factor. As in Montgomery and Runger (1996), we use foldover design to refer to any new 2 k-p fraction obtained by reversing signs of one or more factors. While easy to construct and analyze, foldover designs following a k-p 2 design are generally degree of freedom inefficient, i.e., they provide relatively few additional estimable effects of interest. In fact, for k-p 2 designs of size n, adding a second fraction of size n from the same family provides fewer than n/2 additional degrees of freedom (df) for two-factor interactions in each of the following cases: the foldover is obtained by reversing a single factor - since k-1 or fewer additional any any two-factor interactions can be estimated by the foldover, and n 2k for all resolution designs; k-p 2 design of size n 32, regardless of how the foldover is obtained (this was verified by exhaustive search); k-p 2 design with only even-length words, since only n/2-1 df are associated with effects of even length; k k<12 for n = 64, or in general, < n, since the number of df for aliased two factor 2 interactions cannot exceed half the number of two-factor interactions. 3

4 The bounds on k for n=64 and larger designs can surely be strengthened. The smallest k we know for which a foldover provides n/2 or more additional df for two-factor interactions is 16, specifically design in Chen, Sun and Wu (1993). To summarize this discussion regarding foldovers of 2 designs, while they are easy to construct and provide an orthogonal design when combined with the original the number of additional two-factor interactions that can be estimated will, with few exceptions, be less than half the size of the follow-up design. These designs are worthy of consideration if increasing the precision of estimates is important, but they are poor choices for follow-up when the chief objective is estimating more effects. Though only half the size of a foldover, semifold designs following a estimation of just as many two-factor interactions. k-p k-p 2, k-p 2 experiment generally support If three or more factorial effects are aliased following an initial fraction, at most two effects can be estimated following the addition of either n runs obtained by foldover or n/2 runs by semifolding. What alternative follow-up procedures exist for estimating all aliased effects? Possibilities include using a sequence of follow-up fractions from the same family (Addelman 1969), a sequence from different but equivalent families (Pajak 1972, Pajak and Addelman 1975), or adding one or more fractions with different resolution (Daniel 1962, Pajak 1972). Each follow-up strategy cited above involves combining regular fractions. Even greater flexibility for follow-up designs is possible if an algorithmic approach based on an optimal design (Mitchell 1974) or Bayesian criterion (Meyer, Steinberg and Box 1996, and the discussion that follows) is used. D-optimal designs have the well-known 4

5 characteristic of being optimal for the model specified, but possibly quite poor otherwise. For this reason, we would favor computer-aided design approaches which take into account multiple criteria; Steinberg and Hunter (1984, p. 128) articulate well this point regarding optimal design. By contrast to D-optimal algorithms, Bayesian approaches to augmenting designs explicitly take into account model uncertainty. The user specifies a set of candidate models, with prior probabilities for each. The approach presented by Meyer, Steinberg and Box (1996) (MSB) for augmenting 2 k-p designs considers only saturated models, since their objective was to identify which factors have an effect on the response. By contrast, the objective in our article is to identify which interactions are important. Chipman and Hamada (1996) (CH) advocate an effect-based approach by allowing a richer family of models than just saturated models, and illustrate how the follow-up design selected depends on the family of models considered. The fact that Bayesian approaches are sensitive to prior assumptions regarding possible models is to be expected. See further comparison of these Bayesian approaches in the Conclusion. Semifolding designs have the following advantages to designs produced by D- optimal or Bayesian approaches. Semifold follow-up designs: are simple to construct, requiring no software to generate produce an irregular [(3/2)2 k-p ] design for which the analysis is well understood, can, if necessary, be followed by the remaining n/2 foldover treatment combinations to complete a 2 k-p+1 design, correlating a single factorial effect with blocks. We recommend semifolding following resolution designs where several contrasts of aliased two-factor interactions are clearly present, and one needs to identify which 5

6 interactions are in fact responsible. When the precision of the initial experiment is adequate, semifolding is a frugal strategy for estimating more two-factor interactions. When more precision is needed, full foldover designs should be considered, although generally in two blocks. (A different rationale for semifolding resolution III designs is explained later.) We describe semifolding for an initial details for this case, the smallest of design in Section 2. By presenting k-p 2 designs, we illustrate the typical block diagonal structure of the normal equations after semifolding and the resulting correlation of estimates. We generalize the notion of semifolding in Section 3, using the resolution example of Barnett et al. (1997) as motivation. Beginning in Section 3, and especially in Section 4, we emphasize follow-up designs for initial 16-run experiments, since we expect these applications to be among the most frequently encountered. Alternative eight-run follow-up experiments are discussed in great detail for the 6 2, 7 3, and 8 4 designs in Sections , respectively. The rationale for choosing a particular semifold design and the analysis of data are illustrated in Section 4.4 using a k=7 factor example from Buckner et al. (1997). Semifold follow-up designs are also recommended for larger resolution designs. Section 5 presents an overview of full foldover and semifold designs and proves that, for a single factor foldover of any k-p 2 design, a semifold fraction selected using any main effect provides as many estimable two-factor interactions as does the full foldover. Section 6.1 proposes the related idea of conducting a full foldover design in two blocks. This strategy envisions applications where semifolding is pursued with the 6

7 recognition that the remaining half of the foldover design may be run at a later time. The remainder of Section 6 considers semifolding or blocking foldover designs following a Resolution III experiment. We do not recommend semifolding primarily as a strategy for estimating more effects after a k p 2 III. Rather, Section 6.2 presents semifolding as the first step in a confirmation strategy following resolution III experiments. If the results of the semifold fraction are not as expected, the remaining half of the foldover design can be performed. We follow this discussion with a resolution III example in Section 6.3. Section 7 summarizes our results and briefly mentions several extensions. While the range of possible semifolding follow-up designs has never been carefully expounded, the idea has repeatedly arisen in the literature. In addition to Daniel (1962) and Barnett et al. (1997), see John (1966), and Steinberg and Hunter (1984, p. 88) SEMIFOLDING THE SIMPLEST CASE: 2 The eight-run fraction with defining relation I=ABCD is the smallest design. To fold over this design is to experiment at the eight treatment combinations for which I=-ABCD. Treating the two sets of eight runs as separate blocks, one can estimate all factorial effects, except the four-factor interaction that is confounded with blocks. To semifold the initial k-p fraction is to experiment at only four of the I=-ABCD treatment combinations. The semifolding choice given in Table 1 adds the four treatment combinations with I = -ABCD = A= -BCD. By symmetry of the original design, the normal equations described below would be similar if one added the four I = -ABCD treatment combinations with the same level for any of the four factors. 7

8 If we assume that three-factor interactions and higher are negligible, and that any block effect is additive, then the regression model is: Y = β β Block Block + β A A + β B B + β C C + β D D + β AB AB + β AC AC + β AD AD + β BC BC + β BD BD + β CD CD + ε were the β s are unknown coefficients, 1 is a vector of 1 s, Block, A,..., D are vectors defined using the coding in Table 1, and ε is a vector of independently distributed random variables, each with mean zero and variance σ 2. We refer to coefficients such as β A and β AB as main effects and two-factor interactions, respectively. The normal equations are: 12 ˆ β 0-4 ˆ β Block - 4 ˆ β A -4 ˆ β ˆ β Block - 4 ˆ β A -4 ˆ β 0-4 ˆ β Block + 12 ˆ β A 12 ˆ β B - 4 ˆ β AB + 4 ˆ β CD -4 ˆ β B +12 ˆ β AB + 4 ˆ β CD 4 ˆ β B + 4 ˆ β AB + 12 ˆ β CD = T = [Block] = [A] = [B] = [AB] = [CD] etc. where T denotes the sum of the 12 observations 1 Y, [Block] = (Block) Y, [A] = A Y, etc. The block diagonal structure of normal equations for such irregular fractions has been documented previously (see Addelman 1961). For semifolding applications, the non-zero off-diagonal elements of X X will be ±2 k-p-1, the size of the follow-up design. These twelve linear equations may be grouped into four disjoint sets of three. As a result, ˆ β 0 and ˆ β Block are correlated only with each other and with the main effect ˆ β A, since A is 8

9 the factor with unequal replication. Each other main effect is correlated with two interaction coefficients: its interaction with A and the alias of that interaction in the initial experiment: ˆ β B with ˆ β AB and ˆ β CD ; ˆ β C with ˆ β AC and ˆ β BD ; ˆ β D with ˆ β AD and ˆ β BC. The solution to the normal equations is ˆ β 0 = {2T + [Block] + [A]}/16 ˆ β Block = {T + 2[Block] + [A]}/16 ˆ β A = {T + [Block] + 2[A]}/16 ˆ β B = {2[B] + [AB] - [CD]}/16 ˆ β AB = {[B] + 2[AB] - [CD]}/16 ˆ β CD = {-[B] - [AB] + 2[CD]}/16 The variance for each estimator is σ 2 /8. Correlations among estimators in the same subset are ±0.5. etc. Similar results hold for semifolding other k-p 2 designs. If we fit a saturated model, we have 2 k-p-1 sets of three estimates, each with variance σ 2 /2 k-p. Whenever we fit less than a saturated model, some coefficients will be correlated with only one coefficient, or with none. Coefficients correlated in pairs will have variance (3/4)σ 2 /2 k-p ; coefficients uncorrelated in all others will have variance (2/3)σ 2 /2 k-p. For details, see Addelman (1961, p. 489). Following semifolding, the D-efficiency and A-efficiency will be 84% (=2 4/3 /3) and %, respectively, for saturated models, and slightly larger, otherwise. 9

10 3. ALL POSSIBLE SEMIFOLD FRACTIONS In both Daniel (1962) and Barnett et al. (1997), semifolding has been applied by folding over the design on one factor and then selecting the new treatment combinations where that factor is at the same level. For example, Daniel (1962) semifolds the 8 4 design by folding over on factor A and choosing the eight new treatment combinations with the low level for A. Barnett et al. (1997) semifold a 6 2 experiment by folding over on A and choosing the treatment combinations with high A. Notice, however, that semifolding actually involves two distinct steps: 1. Choosing a foldover fraction 2. Selecting half of these treatment combinations to be run It is advantageous to view these two steps separately, since the motivation for choosing a foldover fraction is distinct from the reasons for choosing which half of that fraction to run (see Section 3.2). We now discuss these possibilities in the context of the Barnett et al. (1997) example. The six factors in an etch uniformity experiment were: revolutions per minute (A), pre-etch total flow (B), pre-etch vapor flow (C), etch total flow (D), etch vapor flow (E), and amount of oxide etched (F). The initial 6 2 experiment was generated by defining E=ABC and F=BCD. For the response ln(uniformity) the estimated regression coefficients are reported in Table 2. The standard error for each regression coefficient is.027 (= root mean square error/ 16, based on 2 degrees of freedom). Barnett et al. (1997) entertained the possibility of eight two-factor interactions corresponding to the four largest interaction estimates. Only the DF interaction was ruled out a priori. 10

11 Since this initial experiment is a one-quarter fraction, there are three possible foldover fractions: 2: reverse the sign of B or C: E=-ABC and F=-BCD 2 : reverse the sign of A or E: E=-ABC and F= BCD 2 : reverse the sign of D or F: E= ABC and F=-BCD. Given the engineering judgment that DF is negligible, one can easily verify from Table 2 that folding over on A or E enables estimating all eight interactions considered of interest after the initial experiment. Barnett et al. (1997) semifold by taking the half of fraction 2 with A at the high level. Although folding over on A is equivalent to folding over on E, clearly semifolding can involve different subsets of these 16 treatment combinations. The most obvious alternative half-fractions of the 2 foldover design (E=-ABC, F=BCD) are: A= -1 A=+1 (Barnett et al. s semifold design) E= -1 E=+1 (Our alternative recommendation) Given these four possibilities, we propose E=+1 rather than A=+1, since factor E has a large negative estimate, while A has no apparent main effect. Choosing E=+1 thus ensures that our follow-up experiment is located at a desirable level for this important factor. Additional subset possibilities, as well as other reasons for choosing a half-fraction will be discussed later in Section 3.2. If engineering judgment had ruled out other interactions (e.g., BC rather than DF), we might have considered folding over on D or F, and then choosing the half fraction 11

12 based on E=+1. In this foldover design, the foldover and subset choices are based on different factors. How would using this half of foldover 2 compare with our recommendation above? We explore the details in Section 4. However, first we summarize the steps for choosing a semifold design. 3.1 Choosing the Foldover Fraction A 2 k-p design is a (1/2) p fraction. Thus, there are 2 p -1 possible foldover fractions obtained by changing the sign of one or more of the design generators. For the 16 run, resolution designs in Section 4, the possibilities are: 6 2 : reverse one factor (3 distinct fractions); 7 3 : reverse one factor (7 distinct fractions); 8 4 : reverse one factor (8 distinct fractions) or reverse a pair of factors (7 distinct fractions). The choice of a foldover fraction determines which effects can be separated from their aliases. The 6 2 design confounds the 15 two-factor interactions in six pairs plus one set of three. Each full foldover fraction separates four pairs of confounded two-factor interactions and one other interaction from the set of three. The 7 3 design confounds the 21 two-factor interactions in sets of three. Each full foldover fraction separates a twofactor interaction from six of these sets. The 8 4 design confounds the 28 two-factor interactions in sets of four. Folding over on one factor separates a two-factor interaction from each of the seven sets. Folding over on two factors breaks six chains in half; for details, see Montgomery and Runger (1996). 12

13 When only half of a foldover fraction is performed, the choice of the full foldover fraction is still the primary determinant of which effects can be estimated. Obviously, the semifold fraction cannot support estimation of any more effects than could be estimated by the full foldover fraction. However, as we will see in Section 4, eight-run semifolding can be used to estimate all of the two-factor interactions that can be estimated with a full foldover design of 16 runs. 3.2 Choosing a Half-Fraction of the Foldover Design 16 For a given 16 run foldover design, there are = 12,870 distinct subsets of 8 eight treatment combinations. Although we disregard most of these possibilities, we do consider the thirty possible subsets defined by choosing the eight treatment combinations with a common sign for one of the 15 orthogonal factorial effect contrasts. Four issues deserve consideration in choosing a semifold design, with only the first two customarily addressed in foldover discussions: 1) Which additional effects can be estimated? 2) What precision is obtained for the estimates? 3) Is there a desired level for one of the factors? 4) Is one factor more difficult to change? Regarding 3), choosing the preferred level will reduce the possible ill-effect of lack-of-fit. For example, suppose β ABC 0 but this term is omitted from the model we fit. For the semifold example of Section 2, it turns out that ˆ β BC depends only on the eight observations with A=1. Consequently, ˆ β BC estimates the BC interaction at A=1, i.e., β BC 13

14 + β ABC. One also has higher precision for predicting the response at the semifold runs than at the foldover runs with A=-1. Thus, if one expects to make inferences only at high A, one has better precision and less potential for lack-of-fit by semifolding with A=+1. Regarding 4), an experiment that holds this factor constant will be easier to run. Fixing this hard to change factor may also reduce the random error, since re-setting this factor likely entails more variation than keeping it fixed (refer to Appendix C). 3.3 Terminology and Notation Semifolding for the 6 2 fraction involves 30 3 = 90 possibilities (30 subsets for each of three foldover fractions), while there are 210 and 450 distinct possibilities for the 7 3 and 8 4 fractional factorial designs. To clearly distinguish the possibilities, we will describe a semifold design in the following way: fold over on ; subset on. For Barnett et al. s (1997) example, fold over on E; subset on E+ is our tentative recommendation, while under different a priori information on interactions we might suggest foldover on F; subset on E+. Abbreviating fold over with fo and subset with ss, we will denote this alternative semifold as fo=f; ss=e+. The next section summarizes the possibilities for the resolution 16-run designs for k = 6, 7, and 8 factors. 4. SEMIFOLDING RESOLUTION SIXTEEN-RUN DESIGNS The defining relations for the 6 2, 7 3, and 8 4 designs contain words only of length four (or eight). When followed with a semifold design, the 16+8=24 runs combine to be a 3/8th, 3/16th or 3/32nd fraction, respectively, of the full factorial. If a coefficient 14

15 is uncorrelated with other estimates, its variance is σ 2 /24. However, coefficient estimates are generally correlated with either one or two other coefficients, due to off-diagonal elements of the X X matrix of ±8. If only a pair of coefficients is correlated, the correlation between the two effects will be ±1/3, and the variance for the estimated coefficients will be σ 2 /( /24) = σ 2 / However, as we saw in Section 2, most coefficients will be correlated in sets of three and have the larger variance σ 2 /16, the same precision as estimable coefficients in the original 16-run orthogonal design. 4.1 Semifolding Following a 6 2 Experiment We use the defining relation I=ABCE=BCDF=ADEF. Note that A and E always appear together, as do B and C (D and F). Without loss of generality, we consider only the foldover design fo=a. We found that, due to symmetries in the defining relation, the 30 possible subset half-fractions correspond to only five different cases (refer to Table 3). When subsetting on a main effect, i.e., cases 6.1 and 6.2, the combined 24 runs permit estimation of 12 two-factor interactions - nine clear of aliasing and the remaining three each aliased with one two-factor interaction - just as in the case of a full foldover fraction. However, the correlation of effects is different, depending on which factor is used to define the subset. In case 6.1, each main effect is correlated with two other effects. In particular, the subset factor (say, E) is correlated with the intercept and with blocks. Each other main effect is correlated with its interaction with E and with the 15

16 previous alias of that interaction. For example, BE is aliased with AC in the original 6 2. Thus, with case 6.1 with ss=e+, BE is correlated with B and with AC. Recall that when three effects are correlated together, their variances are σ Only two estimates have higher precision, those corresponding to the aliased pairs of interactions that remain intact from the original fraction (refer to Table 4). If either of these aliased pairs had a large estimate, we likely would not have chosen this foldover fraction. For example, for the Barnett et al. (1997) experiment, we consider using fo=a (or E) in part because BD=CF and CD=BF appear insignificant. For case 6.2, where the foldover and subset are not obtained using the same factor, two main effects and six interactions can be estimated with greater precision (refer to Table 4 again). This is because the two alias chains that remain intact (AC=BE and AB=CE) involve the subset factor E, and thus are correlated with main effects (BE with B, CE with C). This is a potential advantage, though for the Barnett et al. (1997) experiment, the estimates with increased precision under semifolding option 6.2 are insignificant based on the initial 6 2. Instead of subsetting on a main effect (cases 6.1 and 6.2), we consider the option of subsetting on a two-factor interaction (cases 6.3 and 6.4). With case 6.3 (6.4), we can estimate three (two) fewer interactions than with a semifold design which subsets on a main effect. Appendix A provides a simple indication of why subsetting on a two-factor interaction permits estimation of fewer two-factor interactions. Cases 6.3 and 6.4 are clearly inferior and will not be considered further. 16

17 The final option to consider is case 6.5. This case is similar to case 6.2 in that the same effects are estimable and two main effects are estimated with greater precision. For case 6.5 with fo=a (or E), then the A and E main effects, the AE interaction, and the three alias pairs have variance σ However, 6.5 does not permit the advantage of selecting only treatment combinations with a preferred level for one factor. The one situation where case 6.5 is relevant is for blocking the foldover fraction. Since the same effects can be estimated using, e.g., fo=a; ss=abd+ as with the full foldover fo=a, it will sometimes be useful first to run the semifold design {fo=a; ss=abd+}, analyze the results and decide whether to complete the foldover design. This sacrifices precision for ABD, but not for any main effects or estimable two-factor interactions. More will be said about blocking in Section Semifolding Following a 7 3 Experiment We use the 7 3 design with generators E=ABC, F=BCD, and G=ACD. The defining relation for this one-eighth fraction is: I=ABCE=BCDF=ADEF=ACDG=BDEG=ABFG=CEFG. The remaining seven fractions each correspond to folding over on a different factor. Without loss of generality, we choose fo=a. We now consider the 30 possible subsets for this foldover fraction as categorized in Table 5. This 7 3 design aliases the 21 two-factor interactions in sets of three. Any foldover design separates one interaction from six of these sets of three. For example, via fo=a all two-factor interactions involving A are clear of other two-factor interactions. 17

18 Similarly, adding only eight treatment combinations from fo=a by subsetting via any main effect (i.e., semifolding cases 7.1 and 7.2) permits estimation of the same effects. With either 24 or 32 runs we estimate seven main effects, all interactions involving the foldover factor, and the remaining seven chains of aliased effects. Under case 7.1, the unbroken chain of two-factor interactions is uncorrelated with all other effects; all other effects are correlated in sets of three. Under case 7.2, four effects are correlated with just one other and so have variance σ ; all remaining effects are correlated in sets of three effects. The example in Section 4.4 is of type 7.2. Although differences exist among the alternative semifolding fractions represented under cases 7.1 and 7.2, generally the choices should be made based on Criteria 1) and 3) from Section 3.2: what factor s interactions are of greatest interest? This determines fo. which factor s level is most desirable? This defines ss. What about subsetting using a two-factor interaction? Again we find this illadvised. Semifolding via Case 7.3 or 7.4 provides a total of 10 df for two-factor interactions, rather than the 13 permitted by subsetting on a main effect or the three-factor interaction ABD (i.e., Case 7.5). As with Case 6.5, subsetting based on a three-factor interaction would be recommended for blocking the foldover fraction. 4.3 Semifolding Following a 8 4 Experiment We use the 8 4 design with generators E=ABC, F=BCD, G=ACD, and H=ABD. The defining relation for this one-sixteenth fraction is: I=ABCE=BCDF=ADEF=ACDG=BDEG=ABFG=CEFG=ABDH 18

19 =CDEH=ACFH=BEFH=BCGH=AEGH=DFGH=ABCDEFGH. Eight of the remaining fractions correspond to folding over on a single factor. The other seven are obtained by folding over a pair of factors. Without loss of generality, we choose fo=a and fo=ab. We now consider the 30 possible subsets for each of these foldover fractions (see Table 6). Folding over on a single factor breaks each chain of four aliased two-factor interactions; the seven interactions involving the factor defining the foldover are now estimable clear of other two-factor interactions. The same is accomplished if we semifold by subsetting on any main effect (case 8.1). The result is a 24 run design in two blocks. A saturated model would include a block effect, the eight main effects and 14 two-factor interactions. These 23 effects and the intercept are correlated in eight sets of three effects. Thus, for the saturated model, every effect is estimated with the same precision. If instead we subset using a two-factor interaction (Case 8.2), we gain only three additional df for two-factor interactions, as opposed to seven more via Case 8.1. The only merit to Case 8.2 is that each main effect is correlated only with one other main effect, so that its precision will increase to σ If we were folding over the design primarily to increase the precision of main effects, we might semifold subsetting on a two-factor interaction that was deemed inactive. A decision may then be made after 24 treatment combinations whether it is needed to complete the foldover. Folding over on two factors (e.g., fo =AB) separates the alias chains as follows: AG = BF = CD = EH AH = BD = CF = EG AG = BF and CD = EH AH = BD and CF = EG 19

20 AE = BC = DF = GH AF = BG = CH = DE AD = BH = CG = EF AC = BE = DG = FH AB = FG = DH = CE AE = BC and DF = GH AF = BG and CH = DE AD = BH and CG = EF AC = BE and DG = FH <no change> Thus, this full foldover permits estimation of only six new two-factor interactions, one fewer than with, e.g., fo =A. One rationale for this choice would be that the likely twofactor interactions do not all involve a single factor, and rather than leave some of these aliased with two other interactions, we prefer to allow aliasing of no more than one interaction with ones considered likely. Taking half of this foldover design by subsetting on any main effect (Case 8.3) permits estimation of these thirteen interaction contrasts and the main effects. If we subset on, e.g., C, then AG=BF and CD=EH will be correlated with D. Analogous results hold for each of the other broken alias chains. The subsetting factor (here C) will be correlated with the intercept and block effect. The unbroken alias chain (involving CE) is correlated with only one effect, the main effect E. The remaining cases, 8.4 and 8.5, permit estimation of only three additional df for two-factor interactions (for a total of ten). As with case 8.2, when we subset using a twofactor interaction, the main effects are each correlated with only one other main effect and so are estimable with higher precision. Subsetting on a main effect (case 8.1 or 8.3) seems preferred except for the purpose of blocking. 4.4 A 7 3 Example 20

21 Buckner et al. (1997) report experimentation to characterize the uniformity capabilities of a tungsten deposition tool used in the manufacture of integrated circuits. The levels were carefully chosen for the seven factors: temperature (A), pressure (B), [partial pressure of H 2 ] 1/2 (C), WF 6 flow (D), argon flow (E), backside H 2 /Ar ratio (F), and backside total flow (G). The initial experiment was a 7 3 design plus three center points spaced at the beginning, middle, and end of the sequence of nineteen runs. Uniformity of tungsten thickness was determined using 49 sheet resistance measurements across the surface of each wafer. We analyze the uniformity data using the coefficient of variation (CV) as our response. The CV s ranged from a low of 2.14% at the design center to a high of 7.44%. Given this superior performance at the center (the nominal process parameters recommended by the supplier), there seemed that little could be learned. While wrestling to understand the curvature indicated by the center points, the experiment team discovered that all eight wafers at high hydrogen pressure (C=+1) had thicker tungsten at the center, while the wafers at C=-1 were all thicker at the edge. This was by far the most pronounced uniformity effect and it explained the success at the center point without resorting to a model with pure quadratic effects. Clearly, we would prefer to analyze the individual data rather than the CV. However, given that we are limited to the data as summarized by Buckner et al. (1997), we have no alternative. (We do not consider Buckner et al. s ad hoc response of signed CV well advised since there are no observed CV s near zero; with no observations between and 2.14, 37% in the middle of the range for signed CV is empty.) 21

22 Initially we analyze the data using a model with seven main effects and all possible two-factor interactions. Table 7 summarizes the results. Five interaction contrasts are notable, with three of these more than twice the magnitude of the largest main effect. Since each two-factor interaction is confounded with two others, we can only conjecture that the three largest interactions are AC, BC, and CF, and not their aliases. If this conjecture is true, the preferred levels for A, B, and F depend on the level of C, since these factors have profoundly different effects on uniformity at C=-1 vs. C=+1. Only for factor G does it appear that a single level is preferred, regardless of the levels of the other factors. The interpretation above supposes that the interactions named above involving C are in fact present and not other interactions aliased with these. To verify that this is the case, one would fold over the initial design by reversing the level of factor C. This was in fact done; Buckner et al. (1997) report a follow-up 7 3 experiment, fo=c plus three more center points. While agreeing with the choice of folding over the design on C, we suggest that semifolding would have been a reasonable option. Given the adequate precision of the initial design, a half fraction of fo=c may be all that is needed to verify which effects are present and to identify desired treatment combinations. How should this subset of eight runs from fo=c be chosen? Certainly not using C, since neither extreme level of C is preferred. If fact, we are interested in determining whether low coefficient of variation can be obtained by judicious choices for A, B, D, and F at either extreme level for C. Since we are likely interested in both the low and high 22

23 levels for the factors with strong interactions with C, we choose ss=g- as the semifold half, since ˆ β G =.254 is the largest of the main effect estimates and larger than ˆ β CG. Table 8 gives the results for the combined analysis of the initial 7 3 plus the semifold design fo=c, ss=g-. Consistent with our expectations, the AC, BC, CD and CF interactions are statistically significant, while BE=DG, AE=DF, BF=AG, and EG=BD are not significant. Unexpected, however, is the appearance of a mild AB or FG interaction. We conclude that at high hydrogen pressure (C=+), F=1 and A=G=-1 are desired, whereas F=-1 and B=-1 appear beneficial if hydrogen pressure is low. At these combinations, we expect results with a coefficient of variation only 0.7% higher than at the design center. Changes in the other factors appear relatively unimportant, once the levels are specified as above. k p 5. FOLDOVER AND SEMIFOLDS OF LARGER 2 DESIGNS k p Semifolding may also be used with larger 2 designs. Indeed, the preference for semifolding (or some smaller fraction) versus a full foldover design would seem greater for larger designs, since the savings in the number of runs is greater. We prove the following general result: k p For any regular 2 design and any factors X and Y, the full foldover design fo = X and the semifold design fo = X, ss=y+ (or ss=y-) permit estimation of the same estimable functions of two-factor interactions, assuming that three-factor and higher order interactions are negligible. Appendix B proves this result first for X=Y and then for X Y. 23

24 It is not always true that semifolding on a main effect succeeds in the same way when the foldover is obtained by reversing two or more factors. For example, with the and designs in Montgomery (1997, pp. 690, 694), semifold designs of the form fo = XY, ss=z permit estimation of the same effects as the full foldover fo = XY for some but not all of the choices of the subset factor Z. For all possible 32-run, Resolution designs [as enumerated in Chen, Sun, and Wu s (1993) Table 3], we have verified that semifolding for some choice of Z permits estimation of as many two factor interactions as any full foldover. For larger resolution designs, there are a vast number of alternative foldover fractions. Often one must reverse three factors to maximize the number of df for estimating two-factor interactions. Such foldover designs may not be the most desirable ones for other objectives, e.g., isolating all two-factor interactions involving a specific factor. Thus we do not attempt to enumerate the possibilities. When faced with the need to split in two each of several chains of aliased two-factor interactions, a practitioner should begin by carefully choosing a foldover fraction that best separates the interactions of interest. It may then be determined whether subsetting this foldover design by choosing the preferred level of one factor permits the same estimable functions of twofactor interactions as the full foldover. Of course, if a foldover design provides n/2 or more additional df for two-factor interactions (e.g., the design cited in Section 1), no design with n/2 observations will be able to support estimating so many effects. However, such cases only occur for large k. 24

25 In some cases, fewer than 2 k-p-1 runs will suffice. For example, following up the and designs in Montgomery (1997, pp. 686, 688) with only four and eight treatment combinations, respectively, from a foldover design permits estimation of as many two-factor interactions as does a full 32-run foldover, although with much lower precision. 6. BLOCKING AND CONFIRMATION 6.1 Blocking for a Resolution Foldover As described in Section 4, adding eight runs to a 16-run resolution experiment can provide for estimation of as many as seven more two-factor interactions. If the remaining half of the foldover design follows this semifolding experiment, one again has a regular fraction. Although these final eight runs may not increase the number of twofactor interactions that can be estimated, they will increase the precision of estimates substantially. If the error variance is the same for the follow-up experiments as for the original experiment, the final eight runs decrease the variances for effect estimates as much as 50%, due to the combined benefit of increasing the sample size and removing the correlation among estimates. In general, effects with variance σ 2 2 k-p after semifolding will have variances reduced to σ 2 2 k-p+1 by completing the foldover. The only effect for which the precision is not increased by the final set of runs is the one used to block the foldover fraction. Even for applications where a follow-up design equal in size to the original experiment is planned, the notion of running that follow-up experiment in two blocks of size 2 k-p-1 may be useful. The results of Sections 2-5 indicate the useful information that 25

26 may be obtained once the first half of the follow-up experiment is complete. Furthermore, blocking provides some protection against unforeseen complications. If equipment failure or other difficulties prevent the completion of one block of foldover runs, the other block plus the original experiment can be analyzed and, if necessary, the missing block re-done. 6.2 Confirmation via Semifolding Resolution III Designs Commonly, when a resolution III experiment is performed, only a few of the effect estimates are deemed both statistically and practically significant. The experimenter makes a judgment as to which factors are active and which are not. It is also common practice then to ignore the inactive factors and project the design into the few dimensions believed to be active. However, the more inactive factors ignored, the more tentative the results. It is wise to remember Cuthbert Daniel s (1962, p. 417) advice at this point: It is my own practice to recommend great caution in dropping factors. The simplification so produced is sometimes illusory. In accord with Daniel s advice, we recommend that confirmation runs be performed to verify the validity of one s assessment of active vs. inactive factors. We believe that semifolding will be useful in many such cases. Our basic strategy is as follows: 1) Following the conclusion of a successful resolution III experiment, identify a suitable foldover fraction that would increase the resolution of the design. 2) Choose a half fraction of this foldover design to use as confirmation of the original experiment s supposed interpretation. 26

27 3) If the confirmatory runs support the tentative model describing the original experiment, base one s conclusions and subsequent action on that model; otherwise, complete the foldover design. Our advice for selecting the confirmation runs is: Choose a foldover fraction and subset that permit separation of aliased effects most open to debate. Choose a subset that includes several of the most desirable treatment combinations. If the expected good results are obtained, this is direct support for the model in the vicinity of interest. If possible, subset the foldover design based on an effect for which no more information is needed (since that effect will be confounded with blocks for the follow-up runs). An example that takes into account these considerations follows. 6.3 A 9 5 III Example Hsieh and Goodwin (1986) describe a successful experiment with 16 treatment combinations conducted jointly by Chrysler Motors and its supplier of grille panels, Eagle Picher. The objective was to reduce the occurrence of pops - surface imperfections in the grilles that become visible after the panels are shipped to Chrysler and painted. There were nine factors: mold pressure (A), mold temperature (B), mold cycle (C), cutting pattern (D), priming (E), viscosity (F), weight (G), material thickening process (H), and glass type (J). These were investigated using a 9 5 III design with generators E=-ABCD, F=-CD, G=ACD, H=-BC and J=ABC. Though a better 9 5 III design exists, Hsieh and 27

28 Goodwin s choice does permit estimation of the five interactions reported to be of particular interest - AB, AC, AD, BF, FH - plus the CE=GH=FI contrast, assuming that other interactions are negligible. Hsieh and Goodwin (1986) report two counts on the number of pops at each of the sixteen treatment combinations and analyze the data as if a 32-run completely randomized design was performed. In the absence of information to the contrary, we will treat the results as true replication. Since the resulting pure-error mean square is larger than the sums of squares for at least half of the fifteen factorial effects, the replicates do not seriously underrepresent the error variation present. The response variable Y was the number of pops. Montgomery (1997, p. 434) suggests using Y, or some other transformation suitable for Poisson data. While the square root is a clear improvement over simply using the observed counts, the data exhibit more than Poisson variation. Our analysis suggests that Y 1/4 is a better variance-stabilizing transformation. Table 9 summarizes the results for Y 1/4. Our analysis clearly recognizes two main effects and one of the five interactions of interest. High mold pressure and priming method 2 each decrease the average number of pops. The viscosity*material thickening process interaction (FH) is also significant; since these two main effects are not significant, we tentatively conclude that which thickening process is optimal depends on the level of viscosity. (The case study does not clarify whether viscosity here is controllable or an uncontrollable characteristic of the incoming raw material.) Using the raw count Y as the response, Hsieh and Goodwin (1986) find evidence for a third main effect C (with a preference for high C) and two more interactions: AC 28

29 and BF. Using Y 1/2 as the response, AC and CE interactions appear significant, in addition to the three effects we found using Y 1/4. We are inclined to discount the evidence for these extra effects. However, the possibility certainly exists that other interactions besides FH are important. An eight-run follow-up experiment might be used to accomplish several goals simultaneously: obtain data at the conditions believed to be optimal based on our parsimonious model (A=E=1 and F=-H) investigate the possibility of other interactions effects if the confirmation runs indicate that more data are warranted, be prepared to complete a foldover design that would increase the resolution to. We now examine the possibilities. For the Hsieh and Goodwin 9 5 III design, only one of the 31 foldover fractions will increase the resolution. That new fraction may be obtained by reversing the levels for all nine columns, i.e., fo=abcdefghj. If we restrict our choice to this foldover, we can separate main effects from their aliased two-factor interactions but six sets of three aliased two-factor interactions remain inseparable. Our only remaining decision is which subset to choose. To guide this choice, we obtain predicted values at each of the foldover treatment combinations in this foldover fraction using the alternative models suggested by fitting Y 1/4 and Y 1/2 - see Table 10. We prefer to include treatment combinations expected to be lowest for the number of pops, especially if there is disagreement between the two models. Of the 15 possible contrasts to use, A and E are the only two that enable us to include (exclude) treatment combinations with the lowest (highest) expected number of pops. Recall that these were the only main effects in our models. Both A=1+ 29

30 and E=1+ subsets include the treatment combinations 1-4 in Table 10 (and exclude treatment combinations 13-16). Semifolding based on ss=e+ produces a design which clears A, C, F and H from their aliased two-factor interactions. Since these factors appear in likely interactions (and it is possible that their main effects are concealed by the presence of their aliases in the initial design), we recommend this subset. If we used ss=a+, the four clear main effects would be B, C, D, and E. Given the results of the initial 16 treatment combinations, we prefer extra information on F and H rather than B and C. Thus, by running treatment combinations 1-8 in Table 10, we will be able to estimate a total of 22 effects, with four main effects of interest clear of two-factor interactions. If the results for these eight trials indicate more effects than anticipated, we may follow with treatment combinations 9-16, to increase the precision of all estimates (except E) and clear all other main effects of aliasing with two-factor interactions. Foldovers of 7. CONCLUSION k-p 2 designs are degree of freedom inefficient, in that the number of additional two-factor interactions that can be estimated is typically less than half the additional df n-1. So when the initial experiment provides adequate precision, we recommend the frugal follow-up strategy of judiciously choosing half of a foldover fraction. For foldover designs obtained by reversing a single factor to augment a k-p 2, one can estimate the same functions of two-factor interactions with n/2 runs, subsetting on any main effect. This result was given in Section 5 and proven in Appendix B. For foldover designs obtained by reversing two or more factors, a weaker result was obtained by inspection. For any n=16 or 32 run resolution design, semifold designs exist which 30

31 provide as many df for two-factor interactions as does any foldover design obtained by reversing two or more factors. For foldovers reversing two or more factors, which main effect is chosen for subsetting can affect the number of estimable two-factor interactions. For n 64, foldover designs exist (obtained by reversing several factors) that provide n/2 or more additional df for two-factor interactions. Obviously, semifolding cannot add more than n/2 1 df. Following a k-p 2III design, semifolding is recommended to check one s tentative conclusions and to investigate treatment combinations of interest. A detailed example was given in Section 6.3. Even for cases where a foldover follow-up design is considered in order to increase precision as well as to estimate more effects, we recommend that one consider arranging the follow-up treatment combinations in two blocks. What alternatives exist to semifolding? Strategies based on optimal design criteria do provide a flexible means to select follow-up runs. D-optimal algorithms are widely available and they permit selection of designs for which, if desired, all df are utilized to estimate additional effects of interest. One example where even semifolding is df-inefficient is following a 9 4 [design in Chen, Sun and Wu (1993) or Montgomery (1997, p. 691)]. The original design produces 21 df for two-factor interactions - eight clear and 13 aliased. Semifolding or a full foldover breaks at most nine of the 13 alias chains. By contrast, a D-optimal algorithm can select 16 additional runs that enable estimation of all 36 two-factor interactions. Although this D-optimal augmentation produces numerous correlations > 0.8, it is fully efficient in terms of df. In spite of the versatility of D-optimal algorithms, one should not select a followup design based on this single criterion. Supported by Steinberg and Hunter s comments 31