The Construction of Trend-Free Run Orders of Two-Level Factorial Designs

Abstract

In experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence, the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed; however, randomization may lead to an undesirable run order. For example, in a factory experiment, there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied, or there may be equipment wear-out. In either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur, and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend. Joiner and Campbell (1976) gave some more specific examples as to when trend effects can occur in sequential experiments. Therefore, it is important to consider systematic designs in which the estimates for factor effects of interest are trend-resistant. This article constructs run orders of 2" complete and 2 " -p fractional factorial designs in which the estimates of main effects and two-factor interactions are orthogonal to some polynomial trends. It is shown that in the standard order of a complete 2" design, any k-factor interaction contrast is orthogonal to a (k -1)-degree polynomial trend. Thus by designating some high-order interaction contrasts in the standard ordering as the desired effects or interaction contrasts, a run order can be derived in which these desired effects are orthogonal to high-degree polynomial trends. Symmetric block designs are shown to be useful for constructing such trend-free run orders of 2" complete factorials. Run orders of fractional factorial designs are also considered. The concept of local trend resistance is defined, and it is shown that trend-free run orders also have the property of local trend resistance.

Sequenciamento sistemático de experimentos fatoriais como alternativa à ordem aleatória

“…Although the studies conducted by Draper and Stoneman (1968), Dickinson (1974) and Joiner and Campbell (1976) already considered linear trend effects through maximum time count, only in the late twentieth century these criteria were actually used to develop factorial designs, especially in the studies by Cheng and Jacroux (1988), Bailey et al (1992) and Atkinson & Donev (1996). Coster & Cheng (1988), Jacroux (1994), Githinji & Jacroux (1998), and Tsao & Liu (2008), in turn, approached the issue either based on time count or on the number of factor level changes.…”

“…Cheng & Jacroux (1988) presented mathematical evidences that the randomization is inadequate for experiments subjected to linear trend effects. They suggested developing plans that are robust to time effects in order to obtain null covariance between variables and time.…”

“…Cheng (1985) developed the generalized foldover scheme in order to build sequences of experiments robust to the linear trend effects of full factorial and fractional factorial designs. Subsequently, Cheng & Jacroux (1988) generalize the scheme by minimizing the biases in the estimates of the main effects as well as in the estimates of double interactions in 2 k experiments. Tack & Vandebroek (2004) innovated the research in the field when they simultaneously studied the robustness and cost of two-factorial orthogonal and semi-orthogonal experiments.…”

Trend-free Repeated Measurement Designs

“…However, one may use suitable designs in the presence of trends to avoid the complications of analysis of covariance and increase design efficiencies. Some studies in this direction have been made by Atkinson and Donev (1996), Box (1952), Box and Hay (1953), Bailey, Cheng and Kipnis (1992), Bradley and Yeh (1985), Bradley and Odeh (1988), Chai and Majumdar (1993), Cheng (1985), Cheng and Jacroux (1988), Coster (1993), Coster and Cheng (1988), Cox (1951Cox ( ,1952Cox ( ,1958, Daniel (1976, chapter 15), Daniel and wilcoxon (1966), Githinji and Jacroux (1998), Hill (1960), Jaroux (1993Jaroux ( , 1994, Jacroux and Ray (1990), Joiner and Campbell (1976), Lin and Dean (1991), Mukerjee and Sengupta (1994), Ogilvie(1963), Philips (1964Philips ( ,1968aPhilips ( ,1968b, Prescott (1981), Stufken (1988), Taylor (1967), Yeh and Bradley (1983), Yeh, Bradley and Notz (1985), Whittinghill (1989) and Wilkie (1987). In the present paper, we consider two families of repeated measurement designs (RMDs) and show that efficient/optimal trend-free RMDs exist.…”

Characterization of Six Categories of Systematic 2<sup><i>n</i>－(<i>n</i>－<i>k</i>)</sup> Fractional Factorial Designs

“…For systematic complete 2 n factorial experiments, there are four main algorithms for sequencing their 2 n runs to overcome either of the above mentioned two problems. These algorithms are due to [1][2][3][4]. [5] has conducted a comparison among these algorithms and documented their differences according to three criteria: 1) which algorithm produces run orders in less number of factor level changes, 2) which algorithm produces run orders with more linear/quadratic time trend free main effects and 3) which run order of an algorithm can be generated by another algorithm using either the generalized foldover scheme or the interactions-main effects assignment.…”