G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm

Abstract

Hierarchical linear models are widely used in many research disciplines and estimation issues for such models are generally well addressed. Design issues are relatively much less discussed for hierarchical linear models but there is an increasing interest as these models grow in popularity. This paper discusses the G-optimality for predicting individual parameters in such models and establishes an equivalence theorem for confirming the G-optimality of an approximate design. Because the criterion is non-differentiable and requires solving multiple nested optimization problems, it is much harder to find and study G-optimal designs analytically. We propose a nature-inspired meta-heuristic algorithm called competitive swarm optimizer (CSO) to generate G-optimal designs for linear mixed models with different means and covariance structures. We further demonstrate that CSO is flexible and generally effective for finding the widely used locally D-optimal designs for nonlinear models with multiple interacting factors and some of the random effects are correlated. Our numerical results for a few examples suggest that G and D-optimal designs may be equivalent and we establish that D and G-optimal designs for hierarchical linear models are equivalent when the models have only a random intercept only. The challenging mathematical question of whether their equivalence applies more generally to other hierarchical models remains elusive.

Particle swarm optimization for searching efficient experimental designs: A review

“…Most recently, a variant of CSO called CSO with mutated agent (CSO-MA) was also used to find G-optimal designs for various hierarchical models with random effects (X. Liu et al, 2021;Zhang et al, 2020).…”

“…For example, quantum PSO, DE, ICA and a modified CSO algorithm have been recently used to newly find various types of optimal designs for generalized linear models, like the logistic and the negative binomial regression models with several interacting factors, with and without random effects (Lukemire et al, 2019; Masoudi et al, 2019; Stokes et al, 2020; Xu et al, 2019). Most recently, a variant of CSO called CSO with mutated agent (CSO‐MA) was also used to find G ‐optimal designs for various hierarchical models with random effects (X. Liu et al, 2021; Zhang et al, 2020).…”

“…Extensions to more complicated cases, such as, when errors are not normally distributed or are correlated or the model is hierarchical, are possible and can be found in some of the references cited herein; see for example, R.‐B. Chen et al (2020); Zhang et al (2020), and X. Liu et al (2021).…”

Equivalence theorems for c and DA‐optimality for linear mixed effects models with applications to multitreatment group assignments in health care

“…Others include Chen et al (2015) and Masoudi et al (2017), who found standardized maximin optimal designs under a nondifferentiable criterion for a few enzyme kinetic models that required solving multiple nested optimization problems over different spaces. In addition, Masoudi et al (2019) applied ICA and found Bayesian optimal designs and, both Zhang et al (2020) and Liu et al (2021) modified the CSO algorithm and found optimal designs for different types of regression models with random effects. A most recent application using PSO and Genetic Algorithm is described in Ushijima et al (2021), where they found efficient designs for groundwater models after the complex models were simplified using a Galerkin method and a Proper Orthogonal Decomposition.…”

Robust Optimum Life-Testing Plans under Progressive Type-I Interval Censoring Schemes with Cost Constraint

“…There are few tuning parameters required of the algorithm and the knowledge of good solutions is retained by all particles, and particles in the swarm share information between, which makes the algorithm easily escape from local minima and converge at a fast rate. Recently, different versions of PSO have been used to solve all kinds of optimal design problems (see Chen et al [34], Zhou et al [35], and Liu et al [36]). In the following we give a summary of the PSO algorithm for completeness.…”