Two-stage designs for identification and estimation of polynomial models
Introduction
Suppose that an experimenter is interested in fitting a regression model such that the mean response function E[y(x)] belongs to the class of functionswhere and flT(x)=[1,x,…,xl]. Hence, E[y(x)] is believed to follow a polynomial of order m, at the most. The random response variable Y(x) is observed at different values of the controllable variable x, and these observations are independent and normally distributed with the same variance σ2. An experimental design ξ is defined to be a probability measure with respect to x on a so-called design region X.
For the sake of simplicity, we shall restrict X to be the compact interval [−1,1]. The matrix defined byis called the information matrix of ξ under the model E[y(x)]=gl(x). If ξ has masses ξi only at the support points xi,i=1,…,n, and Nξi=ni are integers, then one can interpret ξ to be an experimental design wherein N uncorrelated observations are taken, with ni replications at each level xi of the controllable variable. The covariance matrix of the least-squares estimator for the parameter vector θl is proportional to the inverse of the information matrix Ml(ξ).
An approximate optimal design maximizes or minimizes some sensible functional based on the information matrix or its inverse, in the set Ξ of all probability measures. Some of the more widely used criteria are, for instance, the log determinant (D-criterion), the trace (A-criterion) of the inverse of the information matrix and the maximum variance of the mean response on a region of interest {x∈S} (G-criterion). See Fedorov (1972) or Pukelsheim (1993) for details. These classical criteria, however, assume that the true form of the model is known.
In this paper, we consider the dual problem of finding designs that give good powers in identifying the degree of a polynomial model up to order m and at the same time provide competitive parameter estimates of the model ultimately chosen. Hence, these designs should exhibit desirable robustness properties for estimation under different models as well as for model identification. We propose a two-stage strategy, which extends the ideas of Montepiedra and Yeh (1997) to the case when there are multiple rival polynomial regression models. In their paper, they extol the philosophy of the sequential approach since it allows the flexibility of revising the design depending on the information provided by the observations from the previous stages. They provided a methodology that specifically addresses the case when there are only two rival models.
We describe in Section 2 an extension of Montepiedra and Yeh's (1997) two-stage strategy to the case when the multiple rival models fall in the class of polynomials Fm. Examples are provided for the case when m=3. Comparisons with existing uni-stage designs are made with respect to both the standard relative efficiencies and a proposed extension of the concept of expected relative efficiencies introduced in Montepiedra and Yeh (1997).
There exist in the literature some optimality criteria which are designed to possess some robustness properties, i.e., they perform well (in the sense of providing good efficiencies) under different estimation and/or identification scenarios. They will be called uni-stage designs in this paper because all observations will be collected at the same time. Each of these criteria falls under one of three categories, based on the inference goal on which the design will be used: (i) estimation, (ii) model identification, or (iii) both (i) and (ii).
Läuter (1974) suggested taking the weighted average of some reasonable functional of the information matrix under different rival models, such that the weights βT=[β0,β1,…,βm] quantify some prior belief about the order of the polynomial model. Essentially, β constitutes a probability measure on {0,1,…,m}. For the D-criterion, the design ξM,β is defined to be optimal if it maximizes
Dette and Studden (1995) proposed the maximization of a weighted p-mean of the relative efficiencies in the different models and derived a complete analytical solution to this problem. Optimization criterion (2) turns out to be a special case of this problem.
Since the regression model is only known to belong to the class Fm, designs which can help identify the actual order of the polynomial are also desirable. For instance, if the researcher is to employ a stepwise selection procedure after the data are collected, he would be interested in testing H0:θll=0 vs. H1:θll≠0 at an appropriate step in the model selection process. The power to reject H0 if H1 is true depends on the design ξ through the quantity:where . Hence, Dette (1994) considered the maximization of the weighted p-meanwhere βT=(β1,…,βl) again denotes a set of priors and p is the order associated with the weighted mean. The corresponding D-criterion is obtained if one lets p approach 0 in the limit:and the design ξI,β is said to be optimal if it maximizes (4). Atkinson and Cox (1974) studied a special case of (3) and Dette (1995) proposed a maximin alternative for the case when p=−∞.
Dette (1993) suggested a mixture of criteria for model identification and parameter estimation:where β∈[0,1]. This criterion reflects the need for a design that provides good parameter estimates for the model gm(x) and at the same time gives good power for testing for θmm. It is clear from the formulation of the criterion that the optimal designs thus obtained will provide competitive efficiencies for estimation under model gm(x) and for testing for θmm. For more general related criteria, the reader is referred to Dette and Franke 2000, Dette and Franke 2001. In this paper, all our comparisons are limited to uni-stage design criteria strictly for robust estimation or strictly for robust testing.
Section snippets
The two-stage robust design
We now utilize the ideas introduced in Montepiedra and Yeh (1997) by extending their proposed methods to the case when several possible models are considered. Suppose that a fraction, f, of the total resources are allocated to the first stage of the experiment. Then the following two-stage procedure will be implemented:
Stage 1: Find the design that maximizes expression (4) over the set Ξ of all probability measures on X. This criterion is applicable assuming that the experimenter uses
A comparative study
We present comparisons under two settings: and . The former is used when there is non-informative prior, which is a common scenario. The latter provides an example when there is knowledge that the higher order model is more plausible.
We use relative efficiencies to assess robustness of the design proposed under different model scenarios. Using terminology in the literature, we define the relative D-efficiency of a design ξ for estimating under model gl(x) as
Expected relative efficiency
From the discussion in the previous section, we tend to emphasize the relative efficiencies of ξ2sl if the objective were estimation/identification of the model gl(x). This is sensible because if gl(x) were the correct model, then design ξ2sl will most likely be the final design used, with the hope that gl(x) will ultimately be selected and used for estimation. However, there is still some chance that the other two possible designs will be used. Thus, their relative efficiencies should not be
Discussion
This paper provides a strategy when an experiment can be performed in two stages, with no concern about the possible presence of a block effect induced by a restricted within-stage randomization scheme. The idea is that the first-stage design places full emphasis on model identification, and the second-stage design is determined by optimizing the precision of the model parameter estimates.
In 3 A comparative study, 4 Expected relative efficiency, the performance of the proposed strategy was
Acknowledgements
The authors thank two referees for their insightful comments and numerous suggestions which greatly improve the presentation of the paper.
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