Two-stage designs for identification and estimation of polynomial models

Abstract

A sequential approach to experimentation allows the investigator to adjust the design according to data obtained from earlier stages, as well as prioritize design objectives at each stage of the study. In this paper, we develop a two-stage design strategy for model identification and parameter estimation when the relationship between the response and the predictor is known to belong to a class of polynomial models. This approach is an extension of Montepiedra and Yeh's (Comm. Statist. Simulations Comput. 27 (1997) 377) two-stage design when there are only two candidate models. The two-stage strategy is compared to robust uni-stage designs for the case when the model is known to be polynomial up to the third order (cubic model). A generalization of the concept of expected relative efficiency is also provided.

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 functions

$$\text{F}{}_{\mathrm{m}}\text{={g}{}_{\mathrm{l}}\text{:}\text{g}{}_{\mathrm{l}}\text{(x)=θ}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{f}{}_{\mathrm{l}}\text{(x),}\text{l=0,1,…,m},}$$

where $\text{θ}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{=[θ}{}_{\mathrm{0l}}\text{,θ}{}_{\mathrm{1l}}\text{,…,θ}{}_{\mathrm{ll}}\text{]∈}\text{R}{}^{\mathrm{l+1}}$ and f_l^T(x)=[1,x,…,x^l]. 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 by

$$\text{M}{}_{\mathrm{l}}\text{(ξ)=}\text{∫}\text{−1}\text{1}\text{f}{}_{\mathrm{l}}\text{(x)f}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{(x)ξ(}\text{d}\text{x)}$$

is called the information matrix of ξ under the model E[y(x)]=g_l(x). If ξ has masses ξ_i only at the support points x_i,i=1,…,n, and Nξ_i=n_i are integers, then one can interpret ξ to be an experimental design wherein N uncorrelated observations are taken, with n_i replications at each level x_i of the controllable variable. The covariance matrix of the least-squares estimator for the parameter vector θ_l is proportional to the inverse of the information matrix M_l(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 F_m. 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 β^T=[β₀,β₁,…,β_m] 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 ξ_M,β is defined to be optimal if it maximizes

$$\text{Ψ}{}_{\mathrm{\beta }}\text{(ξ)=}\text{∑}\text{l=0}\text{m}\text{β}{}_{\mathrm{l}}\text{l+1}\text{log}\text{|M}{}_{\mathrm{l}}\text{(ξ)|.}$$

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 F_m, 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:θ_ll=0 vs. H₁:θ_ll≠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:

$$\text{δ}{}_{\mathrm{ll}}\text{(ξ)=}\text{|M}{}_{\mathrm{l}}\text{(ξ)|}\text{|M}{}_{\mathrm{l-1}}\text{(ξ)|}\text{=(b}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{M}{}_{\mathrm{l}}{}^{\mathrm{-1}}\text{(ξ)b}{}_{\mathrm{l}}\text{)}{}^{\mathrm{-1}}\text{,}$$

where $\text{b}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{=[0,0,…,0,1]∈}\text{R}{}^{\mathrm{l+1}}$. Hence, Dette (1994) considered the maximization of the weighted p-mean

$$\text{Φ}{}_{\mathrm{p,\beta }}{}^{\mathrm{b}}\text{(ξ)=}\text{∑}\text{l=1}\text{m}\text{β}{}_{\mathrm{l}}\text{(b}{}_{\mathrm{l}}{}^{\mathrm{<mtext>T</mtext>}}\text{M}{}_{\mathrm{l}}{}^{\mathrm{-1}}\text{b}{}_{\mathrm{l}}\text{)}{}^{\mathrm{-p}}{}^{\mathrm{1/p}}\text{,}$$

where β^T=(β₁,…,β_l) 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:

$$\text{Φ}{}_{\mathrm{\beta }}\text{(ξ)=}\text{∑}\text{l=1}\text{m}\text{β}{}_{\mathrm{l}}\text{ln}\text{δ}{}_{\mathrm{ll}}\text{(ξ)}$$

and the design ξ_I,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:

$$\text{Ψ}{}_{\mathrm{\beta }}\text{(ξ)=(1−β)}\text{log}\text{|M}{}_{\mathrm{m}}\text{(ξ)|}\text{|M}{}_{\mathrm{m-1}}\text{(ξ)|}\text{+}\text{β}\text{m+1}\text{log}\text{|M}{}_{\mathrm{m}}\text{(ξ)|,}$$

where β∈[0,1]. This criterion reflects the need for a design that provides good parameter estimates for the model g_m(x) and at the same time gives good power for testing for θ_mm. It is clear from the formulation of the criterion that the optimal designs thus obtained will provide competitive efficiencies for estimation under model g_m(x) and for testing for θ_mm. 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.