On the Construction of Trend-Free Run Orders of Treatments

Abstract

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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.…”

Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

“…The designs resistant to nuisance effects are an extension of the designs orthogonal to the time trend (balanced for trend, or trend-free, cf. Cox [1951], Jacroux and Ray [1990]) to a more general class of models and treatment contrasts.The approach of first finding a design in a simpler model and then assuring that the information is retained in a finer model was used, e.g., in Schwabe [1996] and Kunert [1983].Schwabe [1996] studied optimal product designs, unlike the present paper, where optimal designs with non-product structure are provided too; Kunert [1983] studied exact designs in the case of universal optimality. Universal optimality, formulated by Kiefer [1975], means optimality for estimating a maximal system of orthonormal contrasts (which is a special case of the general system of contrasts that we consider), with respect to a wide range of criteria.For selected systems of treatment contrasts and a wide class of optimality criteria Φ, we calculate Φ-optimal treatment weights and thus obtain a class of Φ-optimal designs.…”

Designs for varietal experiments in the presence of trends

“…Jacroux and Saha Ray (1990) used a "fold over" technique described by Coster and Cheng (1988) to construct linear trend-free and higher order p-trend-free designs. Jacroux and Saha Ray (1992) showed methods to construct unequally replicated linear trend-free designs that are A-optimal or optimal by another criterion, and equally repli cated higher order p-trend-free designs that are .A.-optimal among the class of all designs that have the same n and v. A disadvantage of the designs constructed by Saha Ray (1990, 1992) is that the number of units tends to be large. In fact, the form of n must be n = uu"', where u and w are positive integers.…”

“…In showing so, we can use the fact that rrih, = 1 < /i < p-1. One final tool for the construction of p-trend-free sequences, essentially from Jacroux and Saha Ray (1992), is the combination of p-trend-free sequences.…”

“…For some interesting examples, see Joiner and Campbell (1976). Among relevant papers on multifactorial experiments in the presence of trends, including quantitative factors, or on run orders of factorial experiments are Atkinson and Donev (1996), Bailey, Cheng andKipnis (1992), Box (1952), Box and Hay (1953), Chatterjee and Mukerjee (1986), Cheng (1985, 1990, Cheng and Jacroux (1988), Cheng and Steinberg (1991, Coster and Cheng (1988), Daniel andWilcoxon (1966), Dickinson (1974), Draper andStoneman (1968), Hill (1960), Jacroux (1990Jacroux ( , 1993Jacroux ( , 1994ab, 1996, Saha Ray (1990b, 1991), John (1990), , Phillips (1964, 1968), Steinberg (1988), and Wang (1991, 1996.…”