Abstract
Six categories of systematic 2n－(n－k) designs derivable from the full 2k factorial experiment by the interactions-main effects assignment are available for carrying out 2n－(n－k) factorial experiments sequentially run after the other such that main effects are protected against the linear/quadratic time trend and/or such that the number of factor level changes (i.e. cost) between the runs is minimal. Three of these six categories are of resolution at least III and three are of resolution at least IV. The three categories of designs within each resolution are: 1) minimum cost 2n－(n－k) designs, 2) minimum cost linear trend free 2n－(n－k) designs and 3) minimum cost linear and quadratic trend free 2n－(n－k) designs. This paper characterizes these six categories and documents their differences with regard to either time trend resistance of factor effects and/or the number of factor level changes. The paper introduces the last category of systematic 2n－(n－k) designs (i.e. the sixth) for the purpose of extending the design resolution from III into IV and also for raising the level of protection of main effects from the linear time trend into the quadratic, where a catalog of minimum cost linear and quadratic trend free 2n－(n－k) designs (of resolution at least IV) will be proposed. The paper provides for each design in any of the six categories: 1) the sequence of its runs in minimum number of factor level changes 2) the defining relation or its 2n－(n－k) alias structure and 3) the k independent generators needed for sequencing the 2n－(n－k) runs by the generalized foldover scheme. A comparison among these six categories of designs reveals that when the polynomial degree of the time trend increases from linear into quadratic and/or when the design’s resolution increases from III to IV, the number of factor level changes between the 2n－(n－k) runs increases. Also as the number of factors (i.e. n) increases, the design’s resolution decreases.
Highlights
Factorial experiments are experiments for studying the effect of several factors and their interactions on a particular outcome
In the case where a fractionated 2n − k experiment is to be carried out sequentially run after run and the n−k responses drift linearly with time and quadratically, it becomes necessary to sequence runs of this 2n − k experiment in minimum number of factor level changes such that main effects are orthogonal to the quadratic time trend as well as orthogonal to all non-negligible two-factor interactions
We describe the six categories of 16-run systematic 2 ( ) designs that can be constructed from the full 2 experiment by the interactions-main effects assignment, where each design of each category has its own: 1) run sequence in minimum number of factor level changes, 2) defining relation and 3) level in protection of main effects against the linear/quadratic time trend
Summary
Factorial experiments (confirmatory or exploratory) are experiments for studying the effect of several factors and their interactions on a particular outcome (i.e. response). Such experiments are either full or fractionated depending on whether three-factor and higher-order interactions are negligible or not. Many full or fractionated experiments are carried out sequentially (not randomly) one run or one block at a time. This experimentation scheme suffers sometimes from some drawbacks, where main effects and/or two-factor interactions may be contaminated by the time trend which might be present in the successive responses, biasing factor
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