D- and A-Optimal Screening Designs Jonathan Stallrich North Carolina State University Department of Statistics

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D- and A-Optimal Screening Designs

Jonathan Stallrich

North Carolina State University, Department of Statistics

Katherine Allen-Moyer

North Carolina State University, Department of Statistics

Bradley Jones

JMP Statistical Discovery Software LLC

November 1, 2022

Abstract

In practice, optimal screening designs for arbitrary run sizes are traditionally generated

using the D-criterion with factor settings ﬁxed at ±1, even when considering continu-

ous factors with levels in [−1,1]. This paper identiﬁes cases of undesirable estimation

variance properties for such D-optimal designs and argues that generally A-optimal de-

signs tend to push variances closer to their minimum possible value. New insights about

the behavior of the criteria are found through a study of their respective coordinate-

exchange formulas. The study conﬁrms the existence of D-optimal designs comprised

only of settings ±1 for both main eﬀect and interaction models for blocked and un-

blocked experiments. Scenarios are also identiﬁed for which arbitrary manipulation of

a coordinate between [−1,1] leads to inﬁnitely many D-optimal designs each having

diﬀerent variance properties. For the same conditions, the A-criterion is shown to have

a unique optimal coordinate value for improvement. We also compare Bayesian version

of the A- and D-criteria in how they balance minimization of estimation variance and

bias. Multiple examples of screening designs are considered for various models under

Bayesian and non-Bayesian versions of the A- and D-criteria.

Keywords: Bayesian optimal design; blocking; continuous exchange algorithm; D-optimality;

factorial experiments; minimum aliasing

arXiv:2210.13943v2 [stat.ME] 31 Oct 2022

1 Introduction

A screening experiment is an initial step in a sequential experimental procedure to un-

derstand and/or optimize a process dependent upon many controllable factors. Such ex-

periments are common in pharmaceuticals, agriculture, genetics, defense, and textiles (see

Dean and Lewis (2006) for a comprehensive overview of screening design methodology and

applications). The screening analysis aims to identify the few factors that drive most of

the process variation often according to a linear model comprised of main eﬀects, interac-

tion eﬀects, and, in the case of numeric factors, quadratic eﬀects (Jones and Nachtsheim,

2011a). Each eﬀect corresponds to one or more factors, and a factor is said to be active if

at least one of its corresponding eﬀects is large relative to the process noise; otherwise the

factor is said to be inert. Analyses under this class of models often follow eﬀect principles

of sparsity, hierarchy, and heredity (see Chapter 9 of Wu and Hamada (2009)), with the

primary goal of correctly classifying each factor as active or inert.

A screening design is represented by an n×kmatrix, Xd, with rows xT

i= (xi1, . . . , xik)

where xij represents the j-th factor’s setting for run i. To standardize screening designs

across applications, continuous factor settings are scaled so xij ∈[−1,1] while categorical

factor settings are often restricted to two levels, making xij =±1. We compare Xd’s based

on the statistical properties of the eﬀects’ least-squares estimators because their properties

are tractable, particularly their variances and potential biases. The goal then is to identify

an Xdthat minimizes the individual variances and biases of these eﬀect estimators.

Suppose the model is correctly speciﬁed and there are designs having unique least-

squares estimators for all eﬀects. Then these estimators are unbiased and designs may

be compared based on their estimation variances. A design having variances that are as

small as possible will improve one’s ability to correctly classify factors as active/inert. For

models comprised solely of main eﬀects and interactions, orthogonal designs have estimation

variances simultaneously equal to their minimum possible value across all designs. Such

designs exist only when nis a multiple of 4; for other nit is unclear which design will

have the best variance properties. Still, designs should be compared based on how close

their variances are to their respective minimum possible values. This approach requires

knowledge of the minimum values as well as some measure of closeness.

One approach for identifying minimum variances is to approximate them using the

theoretical value assuming an orthogonal design exists, but such values may be unattainable.

The c-criterion (Atkinson et al., 2007) may be used to identify the minimum variance for a

given eﬀect, but without any guarantee of the estimability of the other eﬀects of interest. To

remedy this estimability issue, Allen-Moyer and Stallrich (2022) proposed the cE-criterion

to calculate these minimum variances exactly. It is less clear how to measure the proximity

of a design’s variances to their cEvalues. The Pareto frontier approach by Lu et al. (2011) is

well-suited for this problem but can be cumbersome in practice. A more practical solution

is to evaluate and rank designs according to a single criterion that involves a scalar measure

of all the variances. Such criteria should be straightforward to evaluate and optimize, and

the resulting optimal designs should have variances close to their cEvalues. Diﬀerent forms

of the D- and A-criterion (see Section 2.1) are popular variance-based criteria employed in

the screening design literature and will be the focus of this paper.

Designs that optimize D- and A-criteria can coincide for some n, but this does not

mean the criteria equivalently summarize variances. Consider a screening problem with

n= 7 runs and k= 5 factors that assumes a main eﬀect model. It is well-known that there

always exists a D-optimal design comprised of xij =±1, even when xij ∈[−1,1] (Box and

Draper, 1971). While other D-optimal designs having xij ∈(−1,1) may exist, the screening

literature predominantly ﬁxes xij =±1 with no assumed degradation to the resulting

variances. For example, Jones et al. (2020a) found an A-optimal design with xij values of

±1 and 0 having smaller variances compared to D-optimal designs comprised of xij =±1

only. Figure 1 shows this A-optimal design, which has x14 =x15 = 0. Figure 1 also shows

the corresponding main eﬀect variances (in ascending order) of the A-optimal design and

two D-optimal designs comprised of xij =±1. The minimum possible variances assuming

an orthogonal design exists are 1/7=0.1429 and the minimum variances under the cE-

criterion from Allen-Moyer and Stallrich (2022) are 0.1459. Each of the A-optimal design’s

variances are equal to or smaller than the two competing D-optimal designs comprised of

±1.

As it turns out, the A-optimal design in Figure 1 is also D-optimal despite having some

xij = 0. In fact, changing either x14 or x15 to any value in [−1,1] produces yet another

D-optimal design but with equal or larger variances than the A-optimal design. The A-

optimal design in this case, however, is unique. The existence of inﬁnitely many D-optimal

designs, each with equal or larger variances relative to the A-optimal design, is cause for

concern about utilizing the D-criterion to rank screening designs. In this example, the

11100

-1 -1 1 -1 1

-1 1 -1 -1 1

1 -1 -1 -1 -1

-1 -1 1 1 -1

1 -1 -1 1 1

-1 1 -1 1 -1

Figure 1: (Left) n= 7, k = 5, A-optimal design. (Right) Main eﬀect variances (in ascending

order) for A- and D-optimal designs. The design “D-optimal 1,1” replaces x14 and x15 of

left design with 1. Design “D-optimal −1,1” is similarly deﬁned. The minimum possible

variances assuming an orthogonal design would each be 1/7=0.1429 and the minimum

variances under the cE-criterion from Allen-Moyer and Stallrich (2022) are 0.1459.

A-criterion was better able to diﬀerentiate designs in terms of their ability to minimize the

main eﬀect variances simultaneously.

This is not to say D-optimal designs are less valuable than A-optimal designs. Such

designs have been used with great success in practice and the relative diﬀerences of the

variances in Figure 1 do not appear large. Whether these diﬀerences impact the analysis

depends on the ratio of the true main eﬀect, denoted βj, and the process variance, σ2. When

performing a two-sided t-test for the null hypothesis βj= 0, the associated noncentrality

parameter will be βj/σ divided by the square root of the variances shown in Figure 1.

When βj/σ is large, slight diﬀerences in the variances will not aﬀect the noncentrality

parameter, and hence will not aﬀect power of the tests. The diﬀerences in variances will

have a signiﬁcant impact as βj/σ gets smaller. For example, suppose βj/σ = 1 and we

perform a t-test for β1= 0 with signiﬁcance level α= 0.05. The power for this test under

the D-optimal design with x14 =x15 = 1 is 0.6355 while for the A-optimal design it is

0.7135. Without any prior knowledge of the βj/σ, it is important then to ﬁnd a design that

decreases the individual variances as much as possible.

Based on the eﬀect principles, it is common to ﬁt a main eﬀect model even though

interactions and/or quadratic eﬀects may be active. The least-squares estimators for the

main eﬀect model may then become biased. Rather than try to estimate all potentially

important eﬀects, one can quantify the bias of the estimators and identify a design that

simultaneously reduces estimation variance and bias. Let βbe the vector of the largest

collection of eﬀects that may be important and hence captures the true model. Partition

βinto β1and β2where β1are eﬀects we believe are most likely to be important and

correspond to the eﬀects in the ﬁtted model, and β2are the remaining eﬀects that are

potentially important but ignored in the ﬁtted model. The possible bias from estimating

β1under the ﬁtted model when the true model includes all βis Aβ2where Ais the

design’s so-called alias matrix. DuMouchel and Jones (1994) construct designs under model

uncertainty by assigning a prior distribution to β1and β2, and ranking designs according

to the D-criterion applied to β’s posterior covariance matrix. While Bayesian D-optimal

designs have shown an ability to balance minimizing bias and variance, the possible ﬂaws

of the D-criterion pointed out earlier are still concerning. Better designs may then be

found with a Bayesian A-criterion, which has not received much attention in the screening

literature.

This paper makes two important contributions that build a strong case for construct-

ing screening designs under diﬀerent forms of the A-criterion. The ﬁrst contribution is a

comparison of the behavior of the D- and A-criteria in response to manipulating a single

coordinate of a given design. Our investigation not only provides insights into the criteria’s

coordinate exchange algorithms, a popular design construction algorithm, but also estab-

lishes the existence of D-optimal designs with xij =±1 for models including main eﬀects

and/or interactions, as well as nuisance eﬀects, such as block eﬀects. We are only aware of

such a result for main eﬀect models with an intercept. We also identify cases in which the

D-criterion is invariant to any possible coordinate exchange, meaning the D-criterion con-

siders all such designs as having equal value despite potentially having diﬀerent variances.

For such cases, we show that the A-criterion has a unique optimal coordinate exchange. Our

second contribution is the promotion of a weighted Bayesian A-criterion for constructing

designs that balance bias and variance minimization. We compare new screening designs

generated under coordinate-exchange algorithms for common factorial models and show

the Bayesian A-optimal designs have more appealing variance and bias properties than

Bayesian D-optimal designs.

The paper is organized as follows. Section 2 reviews traditional and current screening

models and criteria. Section 3 investigates the behavior of the D- and A-criteria following

coordinate exchanges to an existing design for models including nuisance eﬀects. It also

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D-andA-OptimalScreeningDesignsJonathanStallrichNorthCarolinaStateUniversity,DepartmentofStatisticsKatherineAllen-MoyerNorthCarolinaStateUniversity,DepartmentofStatisticsBradleyJonesJMPStatisticalDiscoverySoftwareLLCNovember1,2022AbstractInpractice,optimalscreeningdesignsforarbitraryrunsizesaretradit...

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