Ordering factorial experiments

Abstract

In many practical experiments, both the level combinations of factors and the addition orders will affect the responses. However, virtually no construction methods have been provided for such experimental designs. This paper focuses on such experiments, introduces a new type of design called the ordering factorial design, and proposes the nominal main effect component-position model and interaction-main effect component-position model. To obtain efficient fractional designs, we provide some deterministic construction methods. The resulting designs are D-optimal, and the run sizes are much smaller than that of the full designs. Moreover, in some cases, some constructed designs are still D-optimal after reducing the number of components and factors.

1 Introduction

Factorial experiments as a traditional and universal method have been widely studied and applied in many scientific, agricultural and industrial investigations. In a factorial experiment, it is assumed that the response is only affected by the levels of factors. However, in some experiments, not only the level combinations of factors, but also the addition orders will affect the responses. For example, Von Gillhaussen et al. (2014) carried out a grassland experiment to investigate factors that may influence the productivity and functional composition of species-rich grassland communities. Three factors were considered in their experiment: density, sowing interval, and order of arrival of functionally different species groups. In their experiment, the sowing interval factor and the density factor had 2 and 3 levels respectively, and four arrival orders of legumes, forbs and grasses species groups were considered. They fitted a three-way layout model to analyse the biomass data by regarding four arrival orders as one 4-level factor. The modelling result showed that early sowing of legumes with high density and long sowing interval was helpful to aboveground biomass.

Experiments considering both the levels of factors and addition orders of components have been commonly used in bio-engineering and ecology, see e.g., Schäffer et al. (2019) and Wang et al. (2020). In this paper, we call such experiments as ordering factorial experiments. The objective of an ordering factorial experiment is to model the relationship between the response and the inputs more accurately, or to find the values of the inputs that optimize the response. There is a lack of research on the design and modeling of such experiments. Intuitively, we can construct designs and model the data like Von Gillhaussen et al. (2014), i.e., defining a qualitative factor to represent the addition orders whose levels correspond to all possible orders. However, when the number of factors or components is large, the full design is unaffordable due to the experimental cost, and the run size of the fractional design should be large enough to ensure that the model can be established since there are many parameters to be estimated. For example, if an experiment contains six 3-level factors while considering the addition orders of 5 components, the defined qualitative factor which represents the addition orders has $5!=120$ levels, and the run size of the full design is $120×36=87480$⁠. Therefore, it is necessary to investigate the construction of efficient fractional designs and provide effective models correspondingly.

An ordering factorial experiment can be regarded as a combination of two kinds of experiments, i.e., a factorial experiment and an order-of-addition (OofA) experiment, where the former one has been studied extensively and the latter one has received great attention in recent years. The factorial design is an essential part of design of experiments and various kinds of factorial designs have been proposed based on different requirements, such as the orthogonal array (OA) which aims to pursuing the combination orthogonality between levels of factors (Hedayat et al., 1999; Wu & Hamada, 2021), the uniform design and the space-filling design which tend to spread the design points in the design region evenly (Fang et al., 2006, 2018). Accordingly, there are several models for these designs, such as the main effect model, the polynomial regression model, and the Kriging model (Fang et al., 2006). When the number of levels is small, the OA and the main effect model are commonly used. It is easy to verify that the OA can enable the uncorrelated estimate of each factorial effect included in the main effect model. More details of the main effect model and the OA can be found in Wu and Hamada (2021) and Hedayat et al. (1999), respectively. The OofA experimental design introduced by Van Nostrand (1995) can be used to explore how the addition sequence of components affects the response. Recently, some OofA experimental designs and corresponding models have been proposed. Voelkel (2019) discussed the linear model and the criterion of fractional OofA experimental designs based on pair-wise ordering (PWO) factors. A case study of three sequential experiments was provided by Voelkel and Gallagher (2019) based on the linear PWO model. Peng et al. (2019) considered a (nonlinear) tapered model for OofA experiments, which combines the influence of not only the order of each pair of components but also the distance between the two components in each pair on the response. Lin and Peng (2019) and Mee (2020) extended the linear PWO model to the higher-order model. Zhao et al. (2021, 2022) explored the construction methods to obtain minimal-point designs and optimal designs under the PWO model, respectively. Yang et al. (2021) proposed the component-position (CP) model which can reflect the effect of any component at any position, and provided two construction methods for component orthogonal arrays (COAs) which are D-optimal designs under the CP model. Xiao and Xu (2021) proposed the mapping-based universal Kriging model which can be generalized to deal with heterogeneous variances in replicated experiments. Stokes and Xu (2022) proposed the position model based on the position matrix and introduced a novel construction method to obtain efficient fractional OofA experimental designs. Chen et al. (2022) provided a method to construct fractional OofA experimental designs with flexible run sizes without assuming any prespecified model.

To provide efficient fractional designs and models for the ordering factorial experiment, a new type of design is introduced, and two models under different assumptions are proposed in this paper. The optimality of the full design under these two new models are proved, and the least-square estimates and the variances of estimates with the full design are also given. Moreover, to obtain efficient fractional designs, sufficient conditions for some designs to be optimal designs are derived and some construction methods are provided.

The rest of this paper is organized as follows. Section 2 introduces some basic definitions and notation, and defined the new type of design and models. The properties of COAs and the estimability of these new models are also discussed in this section. In Section 3, we prove that the full design is D-optimal under these models, and obtain the least-square estimates and the variances of estimates with the full design. In Sections 4 and 5, some construction methods and illustrative examples are provided to obtain D-optimal fractional designs under the proposed models. Extensions to mixed-level experiments are discussed in Section 6. Section 7 gives some concluding remarks. All the proofs are deferred to the Supplementary Material.

2 Preliminaries

In this section, we provide a new property of COAs, and define the new type of designs and models for ordering factorial experiments. Some basic definitions and notation are introduced as follows.

A design is called a balanced design, if each level appears the same number of times in each column. Let m be the number of components, q be the number of factors and $s0,…,sq−1$ be the numbers of levels of q factors. Denote the m components as $0,…,m−1$⁠, the q factors as $0,…,q−1$ and the $si$ levels as $0,…,si−1$ for $i=0,…,q−1$⁠. Recall that the OA is commonly used as a kind of factorial experimental design. A design with n runs and q factors of $s0,…,sq−1$ levels is called an OA, denoted by $OA(n,q,∏i=0q−1si,t)$⁠, if each t-tuple occurs the same number of times in any $n×t$ submatrix. Let $OA(n,q,∏j=0l−1sjqj,t)$ with $q=∑j=0l−1qj$ be an OA where the elements of $qj$ columns are from the set ${0,…,sj−1}$ for $j=0,…,l−1$⁠. If $s0=⋯=sq−1=s$⁠, the OA can be simply written as $OA(n,q,s,t)$⁠. Hedayat et al. (1999) summarised various construction methods for OAs, some of which are based on the difference matrix. An $r×c$ matrix with entries from a finite abelian group $(A,+)$ is called a difference matrix, denoted by $D(r,c,s)$⁠, if the vector difference between any two distinct columns contains every element of $A$ equally often, where the cardinality of $A$ is s. In Section 4, difference matrices will be used to construct efficient fractional designs.

For OofA experiments, each row of an OofA experimental design is a permutation of ${0,…,m−1}$⁠. A $COA(n,m)$ is an OofA experimental design with n runs satisfying that for any two columns, each of the level combinations $(i,j)$ with $i≠j$ for $i,j=0,…,m−1$ appears the same times. Yang et al. (2021) proved that the COA always exists when m is a prime power and n is a multiple of $m(m−1)$⁠. It is easy to find that both COAs and OAs are balanced designs. Let $GF(m)$ be a Galois field of order m, where $m=(m′)r$⁠, r is an integer and $m′$ is a prime. For $r≥1$⁠, $GF(m)={a0+a1x+⋯+ar−1xr−1:a0,…,ar−1∈GF(m′)}={0,σ1,…,σm−1}$ and $GF(m′)={0,…,m′−1}$⁠. Then, given any vector $w=(w1,…,wm)$⁠, a $COA(m(m−1),m)$ can be constructed by Yang et al. (2021) as follows, where $w$ is a permutation of ${0,σ1,…,σm−1}$⁠. Let $D(0)=(σ1,…,σm−1)Tw$⁠, then

is a $COA(m(m−1),m)$⁠, where $D(0)⊕σ$ means that σ is added to each element in $D(0)$ with the addition defined on $GF(m)$⁠. Given m, Yang et al. (2021) showed that all the $m!$ different permutations can be partitioned into $(m−2)!$ mutually exclusive $COA(m(m−1),m)$’s.

We say that an OofA experimental design D for m components is collapsed to $m0$ components with $m0<m$⁠, if we only retain the first $m0$ components in D. For example, for $m=5$ and $m0=2$⁠, $(1,0)$ is obtained from $(3,1,4,0,2)$ after component collapsing. The following theorem shows a new property of COAs.

 

Theorem 1 confirms a general case that the property of a COA also holds after component collapsing. In addition to the requirements for m and $m′$ in this theorem, there may be some special cases that make the conclusion tenable. For example, there are only two mutually exclusive $COA(12,4)$’s constructed by Equation (1), and it is easy to verify that these two COAs are also $COA(12,3)$’s after component collapsing.

For ordering factorial experiments, we need to define a new type of design.

 

Note that, Definition 1 provides a representation for the design of an ordering factorial experiment, but it does not require any extra properties of the design. For example, all n permutations of the last m columns could be equal, and the first q columns may not form a balanced design. The construction methods of efficient fractional designs and the corresponding properties will be discussed in Sections 4–6. Nevertheless, the following example shows an efficient OFD.

 

To analyse the experimental data, we propose some models for ordering factorial experiments. First, let us consider the model with only main effects. There are two parts in the model: one contains the effects of level combinations and the other contains the effects of addition orders. Consider the following model:

where y is the response, μ is the overall mean, $ai(k)=1$ if the factor i takes the level k and 0 otherwise, $αi(k)$ is the effect of the level k of the factor i, $bc(j)=1$ if the component c is used at position j and 0 otherwise, $βc(j)$ is the effect of the component c at the jth position, $ϵ∼N(0,σ2)$ is a random error and all errors are independent.

Model (2) implies the assumption that there is no interaction effect between the levels of factors and addition orders of components. However, in some experiments the interactions should be considered. For example, Wang et al. (2020) conducted a three-drug experiment, where two drugs were tested at two concentration levels while the third drug was kept at a low dose level, and addition orders of these three drugs were also taken into account. Thus, we also consider the model with the interaction terms. For simplicity, we only consider the case that the addition orders of the first m factors should also be designed in the experiment. Other cases can be handled in a similar manner. Now, the model with interaction terms is:

where $y,μ,ai(k),αi(k),bc(j),βc(j)$ and $ϵ$ have the same meanings as in Model (2), $γc(k,j)$ is the effect of level k of the component c appearing at the jth position.

Obviously, Models (2) and (3) are unestimable since the following constraints exist,

To make both models to be estimable, the following baseline constraints can be considered for them,

These constraints mean that we use the component 0, the position m and the level $s−1$ as references in the model. Under the above constraints, Models (2) and (3) become

respectively. We call these two estimable models as the nominal main effect component-position (MCP) model and nominal interaction-main effect component-position (I-MCP) model, which have $(m−1)2+q(s−1)+1$ and $(m−1)2s+q(s−1)+1$ parameters, respectively.

Here we only consider the interactions between the levels and positions of each factor whose addition order will affect the response, rather than the interactions between levels of all factors and addition orders of all components. Nevertheless, in some cases it can also be used to estimate the interaction effects between factors. For example, considering an experiment to study both addition orders and two concentration levels of three drugs. By setting $q=7$ and $m=3$⁠, and choosing an appropriate design, the interactions between levels of any two or three drugs can also be estimated with the I-MCP model, in which $α3(0),…,α6(0)$ are used to represent the interaction effects between levels of three drugs. Specifically, for the three-drug experiment in Wang et al. (2020), the term $α2(0)$ can reflect the interaction effect between levels of the first two drugs by setting $q=m=3$⁠, $s=2$ and $a2(0)=a0(0)a1(0)$⁠. And the stepwise regression implies that the interaction between the levels and positions of the second drug is significant, while the interaction between the levels of the first two drugs is estimable but insignificant.

3 The full OFD

Let $⊗$ be the Kronecker product and $1m$ be an $m×1$ vector with all elements 1. Obviously, the full $OFD(n,sq,m)$$Df$ can be obtained by $Df=(1m!⊗Ff,Of⊗1sq)$ with the run size $n=sq×m!$⁠, where $Ff$ is the full factorial design with $sq$ different level combinations and $Of$ is the full OofA experimental design with $m!$ distinct permutations of m components.

Intuitively, the full OFD should be an optimal design under both the MCP and I-MCP models. Suppose the model matrix of a design D is an $n×p$ matrix X, then the D-value of this design is defined as

Given the run size, a design is called a D-optimal design if its D-value achieves the maximum value. The following proposition confirms that the full OFD is a D-optimal design.

 

Define the D-efficiency of a design D as $Deff(D)=Dvalue(D)/Dvalue(Df).$ From Proposition 1, it can be seen that a design D is D-optimal if $Deff(D)=1$⁠.

Next, using the full OFD to fit the MCP model or the I-MCP model, we can obtain the least-square estimates of the parameters in the model and the corresponding variances.

  

4 Optimal fractional OFDs under the MCP model

Even though the full OFD is D-optimal under the proposed models, it is generally unaffordable due to the limitation of experimental cost, since the run size of the full OFD increases rapidly with the increases of $q,s$ and m. It is necessary to construct a design with a high D-efficiency and a small run size. In this section, we provide the construction methods of D-optimal fractional designs under the MCP model.

First, we give some sufficient conditions for a design to be a D-optimal OFD under the MCP model, which can be used to propose the construction methods.

 

Now, we provide two construction methods for D-optimal OFDs with different run sizes under the MCP model. It can be proved that all the resulting designs have the properties in Lemma 1, and consequently are D-optimal designs under the MCP model.

If m is a prime power and there exists a difference matrix $D(n1,q,s)$⁠, then through the following construction method, we can obtain an $OFD(n,sq,m)$⁠, where $n≥sm(m−1)$ is a multiple of the least common multiple of $m(m−1)$ and $sn1$⁠.

 

Note that the juxtaposition order of V does not influence the optimality of the resulting design under the MCP model, which can be seen in the proof of Theorem 4. The following example shows the procedure of constructing an $OFD(36,35,4)$ by Construction 1.

 

Let $[a,b]$ be the least common multiple of a and b. In general, the resulting design of Construction 1 has the following property.

 

Theorem 4 shows that designs obtained by Construction 1 are D-optimal. Furthermore, one may consider component collapsing and dropping factors when only the effects of some components and factors are assumed to be significant.

 

Corollary 1 shows that the resulting design of Construction 1 is also D-optimal with component collapsing in many cases. Thus, the constructed designs are still useful under the effect sparsity principle.

We next provide another construction method to obtain some OFDs with new run sizes. If there exists an $OA(n2,q,s,2)$⁠, where $n2$ is a multiple of $m−1$ and m is a prime power, then with the following construction method, we can obtain an $OFD(mn2,sq,m)$⁠.

 

Note that the juxtaposition order of U does not influence the optimality of the resulting design under the MCP model, which can be seen in the proof of Theorem 5. Here is an example to illustrate the procedure of Construction 2.

 

The following theorem ensures that the resulting design of Construction 2 is a D-optimal design under the MCP model.

 

With the two proposed construction methods, we can obtain D-optimal OFDs under the MCP model with different run sizes. Table 3 collects some resulting designs of the proposed construction methods with $m≤5,s≤4$ and $n<50$⁠. In this table, m is the number of components, s is the number of levels of each factor, n is the run size of the constructed OFD, $n1$ and $n2$ are the parameters in Constructions 1 and 2 respectively, N is the run size of the full OFD, and $qmax$ is the maximum number of factors that can be arranged given $n,m$ and s. See Online Supplementary Material, Table S4 of the Supplementary Material for more results with $n<100$⁠. Recall that these designs are all D-optimal designs under the MCP model while their run sizes are much smaller than that of the full designs.

5 Optimal fractional OFDs under the I-MCP model

From Section 3, we know that the full OFD is also a D-optimal design under the I-MCP model. However, the full OFD will be unaffordable when $q,s$ or m is large, which impels us to construct efficient fractional designs.

The following theorem provides some sufficient conditions for a design to be a D-optimal OFD under the I-MCP model.

 

It is easy to find that the properties in Lemma 1 also hold for the OFDs possessing the properties in Lemma 2. Thus, the designs satisfying the above properties are also D-optimal OFDs under the MCP model. According to the sufficient conditions, two construction methods for D-optimal OFDs under the I-MCP model can be provided as follows.

  

Note that the run sizes of the resulting designs of Constructions 3 and 4 are the same, but the resulting designs are different. Define that two designs are different if they cannot be obtained from each other through row permutations. If there exist $km(m−1)$ different $OA(n2,q,s,2)$’s with no duplicate rows in each design and between any two designs, and k mutually exclusive $COA(m(m−1),m)$’s, then there is no duplicate rows in the first q columns of the resulting design of Construction 3, and the last m columns are $n2$ replications of a $COA(km(m−1),m)$⁠. For the resulting design of Construction 4, the first q columns are $m(m−1)$ replications of an $OA(kn2,q,s,2)$ with no duplicate rows, and there is no duplicate rows in the last m columns if U consists of $kn2$ different COAs with no duplicate rows in each design and between any two designs. Thus, if we have a prior information that the response is affected by levels of factors and may be influenced by the addition orders of components, we prefer the resulting designs of Construction 3. Otherwise, the resulting designs of Construction 4 are more recommended.

Moreover, when $s=2$⁠, Construction 3 can be modified to improve the orthogonality of the factorial part without increasing the run size as follows. In Step 2, let $V(t)=(V1(t)T,…,Vm(m−1)(t)T)T$ for $t=1,…,k$ and $V=(V(1)T,…,$$V(k)T)T$⁠, where $Vi(t)$ is an $OA(n2,q,s,2)$ and $Vi+m(m−1)/2(t)$ is obtained by exchanging the two levels of $Vi(t)$ for $i=1,…,m(m−1)/2$ and $t=1,…,k$⁠. Then, the corresponding V is an $OA(n,q,s,3)$⁠. Now, the resulting OFD is also a D-optimal design under the I-MCP model, and the factorial part is an OA of strength 3, which means that there are more distinct level combinations in the factorial part of the resulting design when projected to any three dimensions.

Here is an example for constructing two $OFD(96,27,4)$’s with the proposed two construction methods.

 

The following theorem ensures that all the resulting designs of Constructions 3 and 4 are D-optimal OFDs under both the I-MCP and MCP models.

 

The following corollary indicates that the resulting designs of Constructions 3 and 4 still perform well under the effect sparsity principle.

 

Online Supplementary Material, Table S5 of the Supplementary Material collects some resulting OFDs of Constructions 3 and 4 with $m≤5,s≤4$ and $n<300$⁠. Recall that these designs are all D-optimal designs under both the MCP and I-MCP models while the run sizes are much smaller than that of the full designs. Moreover, all the resulting designs are still D-optimal designs under both the MCP and I-MCP models when reducing the number of components from m to $m′$⁠, where $m′=2$ or $(m′,m)=(3,4)$ or m is a power of a prime $m′$⁠.

6 Mixed-level OFDs

In the previous sections, we have discussed ordering factorial experiments with the same number of levels for all factors. However, the various factors may have different number of levels in some practical experiments. Thus, the models and designs for ordering factorial experiments with mixed-level factors need to be studied. The following definition generalizes the OFD to allow the factors to have different levels.

 

Table 4 shows a mixed-level OFD with 48 runs, which can be used to arrange an experiment of four 2-level factors, one 3-level factor and four components whose addition orders may affect the response.

Let $si$ be the number of levels of factor i for $i=0,…,q−1$⁠. For mixed-level ordering factorial experiments, the estimable MCP and I-MCP models become

respectively. There are $(m−1)2+∑i=0q−1(si−1)+1$ and $(m−1)∑i=1m−1si+∑i=0q−1(si−1)+1$ parameters in these two models, respectively. Now, the full mixed-level OFD can be obtained by $Df=(1m!⊗Ff,Of⊗1∏i=0q−1si)$ with the run size $n=m!∏i=0q−1si$⁠, where $Ff$ is the full factorial design with $∏i=0q−1si$ different level combinations and $Of$ is the full OofA experimental design with $m!$ distinct permutations of m components. The following proposition ensures the optimality of the full mixed-level OFD.

 

Let X and $XF$ be the model matrices of a fractional OFD and the full OFD respectively. From Proposition 2 we know that, if $XTX/n=XFTXF/N$ then this fractional OFD is D-optimal, where n and N are the run sizes of this fractional OFD and the full OFD respectively. Thus, some sufficient conditions for a design to be a D-optimal mixed-level OFD under the MCP and I-MCP models can be given as follows respectively.

  

It is easy to find that the properties in Lemma 3 also hold for the mixed-level OFDs possessing the properties in Lemma 4. Thus, the designs satisfying the properties in Lemma 4 are also D-optimal mixed-level OFDs under the MCP model. Besides, Constructions 2 to 4 can be extended to obtain D-optimal fractional mixed-level OFDs easily. For Construction 2, let V be an $OA(n2,q,∏i=0l−1siqi,2)$⁠, then the resulting design is a D-optimal $OFD(n,∏i=0l−1siqi,m)$ under the MCP model. Similarly, Constructions 3 and 4 can be used to obtain $OFD(n,∏i=0l−1siqi,m)$’s under both the I-MCP and MCP models by replacing the $OA(n2,q,s,2)$’s with $OA(n2,q,∏i=0l−1siqi,2)$’s. Online Supplementary Material, Tables S31 and S32 of the Supplementary Material collect some resulting mixed-level OFDs of the extended construction methods with $n≤100$ and $n≤300$⁠, respectively. It should be noted that, all designs in these two tables are D-optimal designs under the MCP model, while the run sizes are much smaller than that of the full designs. Moreover, designs in Online Supplementary Material, Table S32 are also D-optimal under the I-MCP model. And all the resulting designs are still D-optimal designs when reducing the number of components from m to $m′$⁠, where $m′=2$ or $(m′,m)=(3,4)$ or m is a power of a prime $m′$⁠.

7 Concluding remarks

In this paper, we focus on experiments considering both addition orders and levels of factors, and call such experiments as ordering factorial experiments. The contributions of this paper are as follows. First, we find that a COA may also be a COA for a smaller number of components in some special cases. Secondly, a new type of design called the ordering factorial design (OFD) is introduced, and we propose two new types of models called the MCP and I-MCP models according to the cases whether there are interactions between the levels of factors and addition orders. Thirdly, we prove that the full OFDs are D-optimal designs under the proposed two models, and calculate the D-values of the full OFDs. Besides, since the full designs are generally unaffordable, sufficient conditions for some designs to be D-optimal fractional OFDs under both models are given, and four construction methods for such D-optimal designs are also provided. Run sizes of the resulting D-optimal designs are much smaller than that of the full designs. Moreover, some resulting designs are still D-optimal designs after reducing the number of components in some special cases, which implies that they are still useful under the effect sparsity principle. Finally, the designs and models for mixed-level ordering factorial experiments are also discussed.

Under the MCP and I-MCP models, the proposed construction methods require that the number of components, m, is a prime power, but this requirement may not be satisfied in some practical experiments. Thus, we may need other methods to construct D-optimal fractional OFDs for any integer m. Another research issue is to find the necessary and sufficient conditions for a design to be a D-optimal OFD under the proposed two models, which may be helpful to explore the construction methods for D-optimal fractional OFDs with more flexible run sizes.

As mentioned in Section 1, there are two common models for OofA experiments: the PWO model and CP model. The former is based on relative orderings among the components (Van Nostrand, 1995; Voelkel, 2019), and the latter is based on the absolute positions of the components (Yang et al., 2021). In this paper, both the MCP and I-MCP models are based on the CP model. In the existing literature, Voelkel (2019) and Mee (2020) mentioned searching optimal designs from a candidate set based on the Kronecker product of the full factorial design and the full OofA experimental design, where the linear PWO model is used. And Voelkel (2019) provided three OofA_OAs with 24 runs which can also be used for ordering factorial experiments. Table 5 shows the D-efficiencies of these OofA_OAs and OFDs with the similar run sizes under the MCP and MPWO models, where the MPWO model is obtained by replacing the CP model in the MCP model with the linear PWO model. In Table 5, $OFD(24,24,4)$ is obtained by deleting the 5th to 12th columns from the $OFD(24,212,4)$ shown in Online Supplementary Material, Table S10, $OFD(20,23,5)$ is shown Online Supplementary Material, Table S8, and $OFD(40,24,5)$ is obtained by deleting the 5th to 20th columns from the $OFD(40,220,5)$ shown in Online Supplementary Material, Table S14. Besides, let $(F*,O)$ be an OFD with more columns than needed, and F be the factorial part with required number of columns. We find that selecting the columns from $F*$ does not affect the D-efficiency of the resulting OFD under the MPWO model, but permuting the columns of O may lead to a higher D-efficiency. By permuting the last m columns of $(F,O)$⁠, we obtain the OFD with the highest D-efficiency under the MPWO model shown in Table 5. It can be seen that, the OofA_OAs given by Voelkel (2019) have the same D-efficiencies under the MPWO model as the full designs, while some D-optimal OFDs under the MCP model may fail to be used to fit a MPWO model. Thus, it is necessary to consider other models for ordering factorial experiments and construct optimal designs under those models.

Acknowledgments

The authors thank Editor Qiwei Yao, and two referees for their valuable comments and suggestions. The first two authors contributed equally to this work. Min-Qian Liu is the corresponding author.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12131001, 11871288 and 12226343), and the National Ten Thousand Talents Program of China.

Data availability

The data underlying this article are available in the article and in its online supplementary material.

Supplementary material

Supplementary material are available at Journal of the Royal Statistical Society: Series B online.

References

Author notes