Bayesian sequential $\text{D}$-$\text{D}$ optimal model-robust designs

Abstract

Alphabetic optimal design theory assumes that the model for which the optimal design is derived is known. However in practice, this assumption may not be credible, as models are rarely known in advance. Therefore, optimal designs derived under the classical approach may be the best design but for the wrong assumed model. The Bayesian two-stage approach to design experiments for the general linear model when initial knowledge of the model is poor, is reviewed and extended. A Bayesian optimality procedure that works well under model uncertainty is used in the first stage and the second stage design is then generated from an optimality procedure that incorporates the improved model knowledge from the first stage. In this way, the Bayesian $\text{D}$-$\text{D}$ optimal model-robust design is developed. Results show that the Bayesian $\text{D}$-$\text{D}$ optimal designs are in general superior in performance to the classical one-stage $\text{D}$-optimal and the one-stage Bayesian $\text{D}$-optimal designs. The ratio of sample sizes for the two stages and the minimum sample size desirable in the first stage is also examined in a simulation study.

Introduction

In the context of the classical linear model y=Xβ+ε,y denotes the n×1 vector of observations and X is the extended design matrix of size n×p. β is the p×1 vector of regression coefficients and ε is the n×1 vector of random error terms assumed to be independent and identically normally distributed with zero mean and common variance $\text{σ}{}^2\text{I}{}_{\mathrm{n}}$. In the full rank model, X′X is non-singular and the least-squares estimate of β is $\text{β}\text{̂}\text{=(}\text{X}\text{′}\text{X}\text{)}{}^{\mathrm{-1}}\text{X}\text{′}\text{y}$ which is equivalent to the maximum likelihood estimator under normal errors. Also the variance–covariance matrix of $\text{β}\text{̂}$ is equal to $\text{cov}\text{(}\text{β}\text{̂}\text{)=σ}{}^2\text{(}\text{X}\text{′}\text{X}\text{)}{}^{\mathrm{-1}}$.

Classical design optimality theory involves selecting the rows of X so as to optimize some function of the Fisher's information matrix (X′X). The most popular measure of design criterion is the $\text{D}$-optimality criterion which minimizes the determinant of the variance–covariance matrix of the parameter estimates. As can be seen, by minimizing |(X′X)⁻¹| a lot of effort is concentrated on precise estimation of parameters of the model at the expense of checking whether the assumed model is appropriate. Therefore in the literature, criticisms on optimal design theory have centered around the fact that they are frequently quite sensitive to the form of the model used (see for example Box, 1982 or the historical review by Myers et al., 1989). It is thus fundamental that designs which are less dependent on the effect of model misspecification be developed. Box and Draper (1959) were the first authors to consider this issue in depth. They argued that a more appropriate criterion for such model-robust designs is one which uses the average mean squared error over the region of interest. Steinberg and Hunter (1984) give a nice overview with extensive references on these model-robust and model-sensitive designs. See also Chang and Notz (1996) for their good summary on model robust designs.

The Bayesian approach to design optimality has gained popularity among research workers over the recent years as a way to address this problem of strong model dependence of optimality criteria. The basic idea to Bayesian design optimality is to choose the design that maximizes posterior information about some or all of the β's conditional on the prior information available. The notion of a Bayesian optimality criterion for linear models was introduced by Covey-Crump and Silvey (1970). For general discussions on Bayesian designs see for example, Chaloner and Verdinelli (1995) and Clyde (2001).

Dette (1993) also developed Bayesian $\text{D}$-optimal and model-robust designs in linear regression models. The interesting work by DuMouchel and Jones (1994) illustrates a very elegant use of Bayesian methods to obtain designs which are more resistant to the bias caused by an incorrect model. Similar type of work has been done by Andere-Rendon et al. (1997) for mixture models where they show that the performance of their Bayesian designs are superior to standard $\text{D}$-optimal designs by producing smaller bias errors and improved coverage over the factor space.

Another strategy to develop robust designs found in the literature is the use of Bayesian sequential procedures. With this approach, it is possible to develop designs in two or more stages that lead to less dependence on model specification. The idea behind the sequential designs is intuitively appealing in the sense that the experimenter could have the opportunity to revise the design and model in the course of the experiment.

Box and Lucas (1959) were among the first authors to discuss designs in non-linear situations and the general notion of a sequential approach. Throughout the optimal design literature, the sequential approach has been studied and developed within the non-linear framework. This is natural as the classical alphabetic optimality criterion requires prior knowledge of the model parameters β due to the non-linearity of the problem. In this context, several two-stage designs have been developed. Abdelbasit and Plackett (1983) suggested a two-stage procedure to derive optimal designs for binary responses. Minkin (1987) generalized the idea of Abdelbasit and Plackett and proposed an improvement to their two-stage proposal. Letsinger (1995) developed two-stage designs for the logistic regression model. Sitter and Wu (1995) proposed two-stage designs for quantal response studies where there may be insufficient knowledge on a new therapeutic treatment or compound for the dose levels to be chosen properly. Myers et al. (1996) also proposed a two-stage procedure for the logistic regression that uses $\text{D}$-optimality in the first stage followed by Q-optimality in the second. Myers (1999) gives a good discussion on Bayesian and two-stage designs in his reflection paper on the current status and future directions of response surface methodology.

Development of two-stage designs for linear models has been scant in the literature. Neff (1996) developed Bayesian two-stage designs under model uncertainty for mean estimation models. Montepiedra and Yeh (1998) developed a two-stage strategy for the construction of $\text{D}$-optimal approximate designs for linear models. Lin et al. (2000) developed Bayesian two-stage $\text{D}$-$\text{D}$ optimal designs for mixture models.

In this paper, we review the approach of Neff (1996) for generating two-stage designs with reduced dependence on regressor specification. We propose some modifications in her algorithm to handle uncertainty in the prior distribution specifications using the approach proposed by Box and Meyer (1992) and also used by Lin et al. (2000). A few examples of two-stage designs are presented in Section 3 using the updated algorithm. In Section 4, the two-stage designs are compared relative to the classical one-stage designs and the approach of Neff (1996). Finally, in Section 5, a simulation study is conducted to investigate the allocation of sample sizes in the two stages and the desirable number of runs in the first stage.