When performing an experiment, the observed responses are often influenced by a temporal trend possibly due to aging of material, learning effects, equipment wear-out, or warm-up effects. The construction of run orders that are optimally balanced for time trend effects usually relies on the incorporation of a parametric representation of the time dependence. Using a parametric approach works very well as long as the unknown time dependence is properly specified or overspecified. However, for complicated temporal trends of unknown periodicity, or when the design size is small compared to the complexity of the response model, a parametric approach may lead to underspecification of the true time trend. Serious problems of bias can result. In this paper we show that, contrary to a fully parametric approach with an underfitted time trend, modeling the time trend nonparametrically is very attractive in terms of both bias and precision of the parameter estimators. An algorithm is presented for the construction of optimal run orders when kernel smoothing is used to model the temporal trend. An industrial example illustrates the practical utility of the proposed design methodology.
Key Words: Design of Experiments, Optimal Design, Simulated Annealing, Trends.
By LIEVEN TACK and MARTINA VANDEBROEK Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven, Belgium
PERFORMING experiments in a time sequence often creates time order dependence in the observed responses. Incorporation of time trend effects results in the mathematical model
where f (×) is the p × 1 vector representing the polynomial expansion of × for the response model, g(t) is the q × 1 vector representing the polynomial expansion for the time trend, expressed as a function of time t [-1, 1], is the p × 1 vector of important parameters, and β is the q × 1 vector of parameters of the polynomial time trend. As is common in the literature on the construction or the existence of trend-free designs, the error terms are assumed to be independently distributed with expectation zero and constant variance . In order to preserve uncorrelated errors, each factor is assumed to be independently reset for each run. In this paper, no interaction e.ects between x and t are considered, an assumption which holds true in many practical situations. For n observations, it is convenient to rewrite the model in Equation (1) as
where y is an n-dimensional vector of observed responses, responses, and F and G are the n × p and the n × q design matrices, respectively. When primary interest is in the precision of the parameter estimators, the construction of run orders that are optimally balanced for time trends is based on the maximization of the information on the important parameters in , whereas the parameters in β modeling the time trend are treated as nuisance parameters. The resulting run order is called the -optimal run order , and the value of the optimality criterion equals
In the absence of time trend effects, the -optimal design maximizes the determinant of the information matrix, i.e., = |F´F|. Atkinson and Donev (1996) compared the -optimal and the -optimal design through the trend factor
The power 1/p ensures that the trend factor has the
dimension of the variance. This means, for instance,
that a design with trend factor 0.5 has to be
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