It is sometimes necessary to perform two-level factorial designs in blocks of size two. We investigate what combinations of designs in different pairing arrangements permit estimation of all main effects and two-factor interactions. The confounding patterns of selected combination designs which combine several 2^{k}designs in different pairings are presented for k = 2, 3, 4, and 5. We seek "best" combination designs which provide the most pure (within block) estimates of main effects and two-factor interactions under each confounding pattern given. Some "best" sequential combinations of 2^{k}designs with various choices of initial arrangements are also given. The popular mirror-image pairing design is not the best initial arrangement, but is valuable for providing pure estimates of all the main effects. Extensions to 2^{k-p}fractional factorial designs are illustrated. An example illustrates the use of a follow-up inter-block analysis.Key Words: Engineering Process Control, Kalman Filtering, Linear Quadratic Control, Setup Adjustment, Stochastic Approximation.

*By* **YUYUN JESSIE YANG and NORMAN R. DRAPER, University of Wisconsin, Madison, WI 53706**

**Introduction**

IN many experimental situations, it is desirable to group sets of experimental runs together in blocks. The block size is governed by many considerations, and represents, in most experiments, the number of runs that can be made without worrying (much) about variation caused by factors not being studied speciffcally in the experiment. Often, a block is some natural interval of time (e.g., a week, a day, or a work shift), of space (an oven, a greenhouse, a work bench, or a reactor), of personnel (a research worker or a research team), and so on. An excellent discussion of some of the practical considerations that dictate the need to block experimental designs is given by Rosenbaum (1999, p. 127).

Consider a product that can be made in di.erent ways by varying a set of input factors, each with two levels. In making boots, for example, variable factors that may be considered are the type of leather in the uppers, stiffness of the leather uppers, type of sole/heel cushioning, type of insoles, thickness of insoles, .exibility of sole, padded or thin tongue, overall weight of boot, Velcro or laced closure, and so forth. Thus, in making boots, one could perform a two level factorial design that employed every combination of such levels (a 2^{k} factorial design) or perhaps a subset of these combinations (a 2^{k-p} fractional factorial design). If each boot of every pair were made to the same specification and if the boots were worn and used by testers, the boot results would be perfectly confounded with the testers. However, if the two boots of a pair looked similar, but had some manufacturing differences, a valid within block comparison would be possible. When only one factor is being examined, we have the case of the simple comparative experiment described by Box, Hunter, and Hunter (1978, pp. 97–102). When more than two factors are involved, any allocation of the runs of a 2^{k} or 2^{k-p} design into pairs will involve confounding some effects of possible interest with the blocks. Thus, more than one performance of the design will be necessary. Box, Hunter, and Hunter (1978, p. 341) discussed an example of this type, in which a 23 design is blocked into pairs in four separate ways to give a total of 4(23) = 32 runs in 16 blocks. In each of the four portions of the design, different effects are confounded with blocks, so that an overall balance is achieved, and all main effects and interactions are estimable. Is it necessary to perform so many runs? Draper and Guttman (1997) showed that for a 2^{k} design, *k*2^{k} runs are needed to estimate all main effects and interactions. In this article, we con.ne our interest to estimating only main effects and two-factor interactions, while assuming that all interactions between three or more factors are zero. Using the notation *x*. for an interaction between *x* factors for *x* ≥ 2, we can write this assumption as "≥ 3fi = 0." As might be anticipated, this can be done with more economical designs containing fewer blocks. We start by assuming that block effects are additive, representing changes in overall level only, and that there are no block-factor interactions. In a later section we also discuss the inter-block analysis of our designs, in which it is assumed that block e.ects are random variables.

Because blocks of size two are very restrictive, any 2* ^{k}* design must be run several times in various blocking con.gurations in order to estimate all main effects and 2fis. Consider, for example, the so-called "mirror image" or "foldover" pairs of runs, in which the levels of the factors are completely reversed in the second run of the pair. These pairs are commonly used in blocks of size two, but only main effects (and not 2fis) can be estimated in this way. Other pairs of blocked runs are thus needed to estimate the 2fis. Moreover, in certain applications where 2

In blocking entire 2^{k} or 2^{k-p} designs in blocks of size two, we pair runs using conventional ideas of blocking generators. Note that we are not examining the most general situation here. A much wider problem would be to form all the possible pairs that could be chosen from 2^{k} runs (there are 2^{k}(2^{k} - 1)/2 such pairs), and then to consider how to add pairs one at a time sequentially to form a useful design. (For *k* = 3, we would select from 28 pairs, for example.) We suggest that a design chosen in this more general way would not be an improvement over the designs we choose by using blocking generators, simply because it is necessary to build certain symmetries to estimate the effects. Moreover, such designs might not be resolvable, that is, permit division into sets of blocks, each set of which contains an entire 23 design within it. However, we have not investigated these wider issues.

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