Economic expansible-contractible sequential factorial designs for exploratory experiments

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Virginia Polytechnic Institute and State University


Sequential experimentation, especially for factorial treatment structures, becomes important when one or more of the following, conditions exist: observations become available quickly, observations are costly to obtain, experimental results need to be evaluated quickly, adjustments in experimental set-up may be desirable, a quick screening of the importance of various factors is important. The designs discussed in this study are suitable for these situations. Two approaches to sequential factorial experimentation are considered: one-run-at-a-time (ORAT) plans and one-block-at-a-time (OBAT) plans.

For 2ⁿ experiments, saturated non-orthogonal 2ᵥⁿ fractions to be carried out as ORAT plans are reported. In such ORAT plans, only one factor level is changed between any two successive runs. Such plans are useful and economical for situations in which it is costly to change simultaneously more than one factor level at a given time. The estimable effects and the alias structure after each run have been provided. Formulas for the estimates of main-effects and two-factor interactions have been derived. Such formulas can be used for assessing the significance of their estimates.

For 3m and 2ⁿ3m experiments, Webb's (1965) saturated non-orthogonal expansible-contractible <0, 1, 2> - 2ᵥⁿ designs have been generalized and new saturated non-orthogonal expansible-contractible 3ᵥm and 2ⁿ3ᵥm designs have been reported. Based on these 2ᵥⁿ, 3ᵥm and 2ⁿ3ᵥm designs, we have reported new OBAT 2ᵥⁿ, 3ᵥm and 2ⁿ3ᵥm plans which will eventually lead to the estimation of all main-effects and all two-factor interactions. The OBAT 2ⁿ, 3m and 2ⁿ3m plans have been constructed according to two strategies: Strategy I OBAT plans are carried out in blocks of very small sizes, i.e. 2 and 3, and factor effects are estimated one at a time whereas Strategy II OBAT plans involve larger block sizes where factors are assumed to fall into disjoint sets and each block investigates the effects of the factors of a particular set. Strategy I OBAT plans are appropriate when severe time trends in the response may be present. Formulas for estimates of main-effects and two-factor interactions at the various stages of strategy I OBAT 2ⁿ, 3m and 2ⁿ3m plans are reported.