On the de la Garza Phenomenon

arXiv:0912.3861v1 [stat.ME] 21 Dec 2009 On the de la Garza Phenomenon Min Yang University of Missouri Abstract Deriving optimal designs for nonlinear models is in general challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p −1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many clas- sical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or non-homogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design; and (iv) it can be easily implemented. KEY WORDS: Locally optimal; Loewner ordering; Support points. 1 Introduction The usefulness and popularity of nonlinear models have spurred a large literature on data analysis, but research on design selection has not kept pace. One complication in studying optimal designs for nonlinear models is that information matrices and optimal designs depend on unknown parameters. A common approach to solve this dilemma is to use locally optimal designs, which are based on one’s best guess of the unknown parameters. Research sponsored by NSF grants DMS-0707013 and DMS-0748409 1 While a good guess may not always be available, this approach remains of value to obtain benchmarks for all designs (Ford, Torsney, and Wu, 1992). In fact, most available results are under the context of locally optimal designs. (Hereafter, the word “locally” is omitted for simplicity.) There is a vast literature on identifying good designs for a wide variety of linear models, but the problem is much more difficult and not nearly as well understood for nonlinear models. Relevant references will be provided in later sections in this paper. In the field of optimal designs, there exist no general approaches for identifying good designs for nonlinear models. There are three main reasons for this significant research gap. First, in nonlinear models the mathematics tends to become more difficult, which makes proving optimality of designs a more intricate problem. Current available tools are mainly based on the geometric approach by Elfving (1952) or the equivalence approach by Kiefer and Wolfowitz (1960). This typically means that results can only be obtained on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. It is not feasible to derive a general solution. Second, while linear models are all of the form E(y) = Xβ, there is no simple canonical form for nonlinear models. Coupled with the first challenge, this means it is very difficult to establish unifying and overarching results for nonlinear models. Again, this means that individual consideration is typically needed for different models, different optimality criteria, and different objectives. Third, when considering the important practical problem of multi-stage experiments, the search for optimal designs becomes even more complicated because one needs to add design points on top of an existing design. Is there a practical way to overcome these challenges and derive a general approach for finding optimal designs for nonlinear models? One feasible strategy is to identify a subclass of designs with a simple format, so that one can restrict considerations to this subclass for any optimality problem. With a simple format, it would be relatively easy to derive an optimal design, analytically or numerically. To make this strategy meaningful, the number of support points for designs in the subclass should be as small as possible. By Carath´eodory’s theorem, we can always restrict our consideration to at most p(p+1)/2 design points (where p is the number of parameters). On the other hand, if we want all parameters to be estimable, the minimum number of support points should be at least p. Thus, the ideal situation is that the designs in the subclass have no more than p points. This reminds one of de la Garza (1954)’s result, which was discussed in detail by Pukelsheim (2006) under the concept of “admissib

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