Optimal Designs for Generalized Linear Models

Abstract

Generalized linear models with group effects are commonly used in scientific studies. However, there appear to be no results for selecting optimal designs. In this paper, we identify the structure of locally optimal designs, provide a general strategy to determine the design points and the corresponding weights for optimal designs, and present theoretical results for the special case of D-optimality. The results can be applied to many commonly studied models, including the logistic, probit, and loglinear models. The design region can be restricted or unrestricted, and the results can also be applied for a multi-stage approach.

Optimal designs for 2k factorial experiments with binary response

“…For results on optimal designs with continuous factors in the GLM setup see, for example, Stufken and Yang (2012).…”

“…Khuri, Mukherjee, Sinha, and Ghosh (2006) provided a systematic study of the optimal design problem in the GLM setup and recently there has been an upsurge in research in both theory and computation of optimal designs. Russell et al (2009), Majumdar (2008, 2009), Stufken (2009), Yang, Zhang, andHuang (2011), Stufken and Yang (2012) are some of the papers that developed theory, while Woods et al (2006), Steinberg (2006, 2008), Waterhouse et al (2008), Woods and van de Ven (2011) focused on developing efficient numerical techniques for obtaining optimal designs under generalized linear models. Our focus is on optimal designs for GLMs with qualitative factors.…”

“…GLMs have been widely used for modeling binary response. Stufken and Yang (2012) noted that "the study of optimal designs for experiments that plan to use a GLM is however not nearly as well developed (see also Khuri, Mukherjee, Sinha, and Ghosh (2006)), and tends to be much more difficult than the corresponding and better studied problem for the special case of linear models." For optimal designs under GLMs, there are four different approaches proposed in the literature to handle the dependence of the design optimality criterion on the unknown parameters: the local optimality approach of Chernoff (1953) in which the parameters are replaced by assumed values; the Bayesian approach (Chaloner and Verdinelli (1995)) that incorporates prior belief on unknown parameters; the maximin approach that maximizes the minimum efficiency over a range of values of the unknown parameters (see Pronzato and Walter (1988) and Imhof (2001)); the sequential approach where the design and parameter estimates are updated in an iterative way (see Ford, Titterington, and Kitsos (1989)).…”

A fast algorithm for constructing efficient event-related functional magnetic resonance imaging designs

“…We follow Maus et al [27] to search over the obtained locally optimal designs for a design maximizing F * M m2 . Instead of evaluating the F * M m2 -values of all the 1798 locally optimal designs, we consider a much more efficient two-stage procedure to approximate the result; a similar procedure has also been considered in other studies [30]. At the first stage, a design maximizing F * M m2 is selected among a subclass of locally optimal designs, each being optimal for a φ-value over a coarse grid (0.1 × 0.1) on Ω.…”

Designs for Generalized Linear Models

“…For example, with b blocks, a fixed-effects design would require the specification of b − 1 block effects. See Stufken and Yang (2012a) for locally optimal designs with fixed group effects and a single variable.…”