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Statistics and Computing (2021) 31:70

https://doi.org/10.1007/s11222-021-10046-2

Optimal design of multifactor experiments via grid exploration

Radoslav Harman1

·Lenka Filová1

·Samuel Rosa1

Received: 10 April 2021 / Accepted: 22 August 2021 / Published online: 13 September 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021

Abstract

We propose an algorithm for computing efﬁcient approximate experimental designs that can be applied in the case of very

large grid-like design spaces. Such a design space typically corresponds to the set of all combinations of multiple genuinely

discrete factors or densely discretized continuous factors. The proposed algorithm alternates between two key steps: (1) the

construction of exploration sets composed of star-shaped components and separate, highly informative design points and (2)

the application of a conventional method for computing optimal approximate designs on medium-sized design spaces. For a

given design, the star-shaped components are constructed by selecting all points that differ in at most one coordinate from

some support point of the design. Because of the reliance on these star sets, we call our algorithm the galaxy exploration

method (GEX). We demonstrate that GEX signiﬁcantly outperforms several state-of-the-art algorithms when applied to D-

optimal design problems for linear, generalized linear and nonlinear regression models with continuous and mixed factors.

Importantly, we provide a free R code that permits direct veriﬁcation of the numerical results and allows researchers to easily

compute optimal or nearly optimal experimental designs for their own statistical models.

Keywords Optimal design ·Multifactor experiments ·Regression models ·Generalized linear models ·Algorithms

Mathematics Subject Classiﬁcation 62K05 ·90C59

1 Introduction

The usual aim of the so-called “optimal” design of experi-

ments is to perform experimental trials in a way that enables

efﬁcient estimation of the unknown parameters of an underly-

ing statistical model (see, e.g., Fedorov 1972; Pázman 1986;

Pukelsheim 2006; Atkinson et al. 2007; Goos and Jones

2011; Pronzato and Pázman 2013). The literature provides

optimal designs in analytical forms for many speciﬁc sit-

uations; for a given practical problem at hand, however,

analytical results are often unavailable. In such a case, it

is usually possible to compute an optimal or nearly optimal

design numerically (e.g., Chapter 4 in Fedorov 1972, Chap-

ter 5 in Pázman 1986, Chapter 12 in Atkinson et al. 2007,

and Chapter 9 in Pronzato and Pázman 2013).

In this paper, we propose a simple algorithm for solving

one of the most common optimal design problems: com-

BRadoslav Harman

harman@fmph.uniba.sk

1Department of Applied Mathematics and Statistics, Faculty of

Mathematics, Physics and Informatics, Comenius University

in Bratislava, Bratislava, Slovakia

puting efﬁcient approximate designs for experiments with

uncorrelated observations and several independent factors.

The proposed algorithm employs a speciﬁc strategy to adap-

tively explore the grid of factor-level combinations without

the need to enumerate all elements of the grid. The key idea

of this algorithm is to form exploration sets composed of star-

like subsets and other strategically selected points; therefore,

we refer to this algorithm as the “galaxy” exploration method

(GEX).

If the set of all combinations of factor levels is ﬁnite and

not too large, it is possible to use many available efﬁcient

and provably convergent algorithms to compute an optimal

design (e.g., those of Fedorov 1972; Atwood 1973; Silvey

et al. 1978; Böhning 1986; Vandenberghe et al. 1998;Uci´nski

and Patan 2007;Yu2011; Sagnol 2011; Yang et al. 2013;Har-

man et al. 2020). However, in the case of multiple factors,

each with many levels, the number of factor-level combina-

tions is often much larger than the applicability limit of these

methods.

The main advantage of GEX is that it can be used to solve

problems with an extensive number of combinations of fac-

tor levels, e.g., 1015 (5 factors, each with 1000 levels), and

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