Computational aspects of experimental designs in multiple-group mixed models

Prus, Maryna; Filová, Lenka Article — Published Version Computational aspects of experimental designs in multiple-group mixed models Statistical Papers Provided in Cooperation with: Springer Nature Suggested Citation: Prus, Maryna; Filová, Lenka (2023) : Computational aspects of experimental designs in multiple-group mixed models, Statistical Papers, ISSN 1613-9798, Springer, Berlin, Heidelberg, Vol. 65, Iss. 2, pp. 865-886, https://doi.org/10.1007/s00362-023-01416-1 This Version is available at: https://hdl.handle.net/10419/306482 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Statistical Papers (2024) 65:865–886 https://doi.org/10.1007/s00362-023-01416-1 REGULAR ARTICLE Computational aspects of experimental designs in multiple-group mixed models Maryna Prus1,2 ·Lenka Filová3 Received: 12 April 2022 / Revised: 9 December 2022 / Published online: 13 March 2023 © The Author(s) 2023 Abstract We extend the equivariance and invariance conditions for construction of optimal designs to multiple-group mixed models and, hence, derive the support of optimal designs for firstand second-order models on a symmetric square. Moreover, we provide a tool for computation of Dand L-efficient exact designs in multiple-group mixed models by adapting the algorithm of Harman et al. (Appl Stoch Models Bus Ind, 32:3–17, 2016). We show that this algorithm can be used both for size-constrained problems and also in settings that require multiple resource constraints on the design, such as cost constraints or marginal constraints. Keywords Optimal design ·Exact design ·Random coefficient regression · Equivariance ·Invariance ·Resource constraints 1 Introduction The aim of this work is computation of highly efficient experimental designs in multiple-group random coefficient regression models. Analytical approach for determining optimal approximate designs for this type of models has been discussed, i.e., in Fedorov and Jones (2005), Schmelter (2007) and Prus (2022). In Fedorov and Jones (2005), optimal designs were obtained for specific regression functions. Schmelter (2007) proposed optimality conditions in the particular case of group-wise identical designs for commonly used linear and determinant criteria. In Prus (2022), equivalence theorems for the general form of multiple-group models have been formulated. BMaryna Prus [email protected] 1Institute of Crop Science, University of Hohenheim, Stuttgart, Germany 2Division of Statistics and Machine Learning, Linköping University, Linköping, Sweden 3Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Bratislava, Slovakia 123 866 M. Prus, L. Filová For computing optimal designs in mixed-effect models, several solutions are available. However, most of them focus on computing approximate designs for several traditional criteria, and we are not aware of any work that considers additional constraints besides the size of the design. Namely, Dumont et al. (2018) created an R package for computing designs in mixed-effects model that is predominantly focused on nonlinear models that are used in drug development. However, they only consider the D-optimality criterion without any additional constraints that may arise in the experiment (such as budget, material or other types of resources). The software package (Aliev et al. 2012) is aimed at computing approximate designs, mainly for mixed-effects models arising in pharmacokinetic applications. Finally, the software solution of Nyberg et al. (2012) seems to be the most versatile, as it admits user-defined criteria, including their Bayesian versions, but the focus is on approximate designs in nonlinear mixed-effect models and no additional constraints can be included. In our paper, we propose to use the algorithm of Harman et al. (2016), originally developed for computing D-efficient exact designs in the linear regression model with possibly multiple resource constraints, for the general form of multiple-group models. We also propose analytical solutions based on equiand invariance properties of optimal designs for several particular models. The paper has the following structure: In Sect. 2, we shortly introduce the multiple group model, the design problem and the optimality conditions that are subsequently used in Sect.3to show the equivariance and invariance properties of Dand L-optimal designs. In Sect.4, we show that the problem of computing optimal designs in the multiple-group mixed model can be reformulated as a problem of computing optimal designs with respect to a monotonous criterion function with resource constraints on weights, and, hence, a modification of a recent algorithm for computing resource constrained designs can be used to obtain efficient exact designs in our model. In Sects.5and 6, we compute the Dand IMSE-optimal designs in bilinear and quadratic models and show that we can easily solve problems with additional constraints that cannot be solved analytically. 2 Multiple-group RCR model 2.1 Model specification In multiple-group random coefficient regression model the h-th observation of the j-th observational unit in the i-th group is given by Yijh =F(i)(xih)βij +εijh,xih ∈Xi,i=1,...,s,j=1,...,ni, h=1,...,mi,(1) where niis the number of observational units in group i,miis the number of observations per unit in group i, observational settings xih come from some experimental region Xi. In this work we allow for multivariate (l-variate) response and F(i)denotes a group-specific (l×p) matrix of known regression functions in group i. In the particular 123 Computational aspects of experimental designs... 867 case of univariate response we deal with “classical” regression functions: F(i)=f (i), and l=1. Unit-specific random parameters βij =(βij1,...,β ijp)have unknown mean β0and given (p×p) covariance matrix Di,εijh denote observational errors with zero mean and non-singular (l×l) covariance matrix i. All observational errors and all random parameters are assumed to be uncorrelated. The covariance matrix of the best linear unbiased estimator for β0is given by Cov ˆ β0=s  i=1 ni(( ˜ F i˜ Fi)−1+Di)−1−1 ,(2) where ˜ Fi=(˜ F (i)(xi1),..., ˜ F (i)(ximi))for ˜ F(i)(xih)=−1/2 iF(i)(xih),h= 1,...,mi, and the symmetric positive definite matrix 1/2 iwith the property i= 1/2 i1/2 i. 2.2 Design criteria The experimental settings xi1,...,ximiin formula 1are not necessarily all distinct. We define an exact design in group ias ξi=xi1,...,xiki mi1,...,miki,(3) where xi1,...,xikiare the distinct support points in Xiwith the corresponding numbers of observations mi1,...,miki∈N,ki k=1mik =mi. For analytical purposes we also introduce approximate designs: ξi=xi1,...,xiki wi1,...,w iki, where wik ≥0 denotes the weight of observations at xik,k=1,...ki, and ki k=1wik =1. We will use the following notation for the moment (or information) matrix in group i: Mi(ξi)=mi ki  k=1 wik ˜ F(i)(xik)˜ F(i)(xik). (4) For exact designs we have wik =mik/miand Mi(ξi)=˜ F i˜ Fi, which follows from formula (4) and the definition of ˜ Fibelow formula (2). We will also use the notation ξfor the tuple of all group-designs ξi:ξ=(ξ1,...,ξ s). 123 868 M. Prus, L. Filová Further we extend the definition of the variance–covariance matrix (2) with respect to approximate designs: Covξ=s  i=1 niMi(ξi)−1+Di−1−1 .(5) We generally search for the designs which minimize the variance-covariance matrix. Instead of the minimization of the matrix itself (which is in general not possible), we instead minimize suitable functions of this matrix which we call optimality criteria. We focus on the commonly used linear (L-) and determinant (D-) criteria for the estimation of the population parameters β0, which are given by φL(ξ)=tr ⎛ ⎝s  i=1 niMi(ξi)−1+Di−1−1 V⎞ ⎠,(6) where Vis some non-negative definite (p×p) matrix, and φD(ξ)=−ln det s  i=1 niMi(ξi)−1+Di−1,(7) respectively (see Prus 2022). (Note that matrix Mi(ξi)here differs from that in Prus Prus 2022 by constant mi.) Frequently used particular cases of the L-criterion are the cand A-criterion, which are of the form (6) with V=cc,c∈Rp, and V=Ip, where Ipis the p×pidentity matrix, respectively. Another frequently used linear criterion is the IMSE-criterion. For the estimation of the mean parameters β0in multiple-group model (1) we define this criterion as follows: φIMSE(ξ) = s  i=1 aitr Xi EF(i)(x)ˆ β0−F(i)(x)β0F(i)(x)ˆ β0−F(i)(x)β0νi(dx), (8) where νiis some suitable measure on the experimental region Xi(typically uniform on Xi) with νi(Xi)=1 and aiis a coefficient related to group i,s i=1ai=1. The coefficients a1,...,asmay depend on the group sizes or, alternatively, equal weight may be given to each group. IMSE-criterion (8) may be rewritten in form φIMSE(ξ)=tr Cov ˆ β0s  i=1 aiXi F(i)(x)F(i)(x)ν i(dx).(9) 123 Computational aspects of experimental designs... 869 Then we extend it for approximate designs by using the extended variancecovariance matrix (5) and we obtain the particular linear criterion with V= s i=1aiXiF(i)(x)F(i)(x)ν i(dx), which simplifies to V=X1F(1)(x)F(1)(x)ν 1(dx) if the regression matrices F(i), the experimental regions Xiand the weighting measures νiare the same among all groups: F(i)=F(1),Xi=X1and νi=ν1for i=1,...,s. 2.3 Optimality conditions The optimality conditions for the Land D-criteria are provided by the following theorems (see Prus 2022): Theorem 1 Approximate designs ξ∗=(ξ∗ 1,...,ξ∗ s)are L-optimal for estimation of the mean parameters β0iff mitr ⎧ ⎨ ⎩ ˜ F(i)(xi)⎡ ⎣Mi(ξ∗ i)−1Mi(ξ∗ i)−1+Di−1s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 V ·s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 Mi(ξ∗ i)−1+Di−1Mi(ξ∗ i)−1⎤ ⎦˜ F(i)(xi)⎫ ⎬ ⎭ ≤tr ⎧ ⎨ ⎩ Mi(ξ∗ i)−1Mi(ξ∗ i)−1+Di−1s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 V ·s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 Mi(ξ∗ i)−1+Di−1⎫ ⎬ ⎭ (10) for xi∈Xi,i =1,...,s. For support points of ξ∗ iequality holds in (10). Theorem 2 Approximate designs ξ∗=(ξ∗ 1,...,ξ∗ s)are D-optimal for estimation of the mean parameters β0iff mitr ⎧ ⎨ ⎩ ˜ F(i)(xi)⎡ ⎣Mi(ξ∗ i)−1Mi(ξ∗ i)−1+Di−1s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 ·Mi(ξ∗ i)−1+Di−1 Mi(ξ∗ i)−1˜ F(i)(xi) ≤tr ⎧ ⎨ ⎩ Mi(ξ∗ i)−1Mi(ξ∗ i)−1+Di−1s  r=1 nrMr(ξ∗ r)−1+Dr−1−1 ·Mi(ξ∗ i)−1+Di−1(11) for xi∈Xi,i =1,...,s. For support points of ξ∗ iequality holds in (11). 123 870 M. Prus, L. Filová Example 1 We consider the two-groups model of general form (1) with the regression functions F(i)(x)=(1,x), x∈Xi: Yijh =βij1+βij2xih +εijh,j=1,...,ni,h=1,...,mi,i=1,2,(12) on the design regions Xi=[0,1]. The covariance structures of random effects and observational errors are given by Di=diag(di1,di2)and i=1 for both groups. For this model the left hand sides of the optimality conditions (10) and (11) are parabolas with positive leading terms. Therefore, Dand L-optimal approximate group-designs have the form ξi=01 1−wi1wi1,(13) where wi1denotes the weight of observations in point 1 for the i-th group and may depend on the choice of the design criterion as well as on model parameters. The moment matrices are given by Mi(ξi)=mimi1 mi1mi1,(14) where mi1=wi1mi. Optimal designs for random intercept and random slope have been considered in more detail in Prus (2022). 3 Equiand invariance considerations for construction of optimal designs Equiand invariance of design criteria play an important role for determining optimal designs in fixed effects models (see e.g. Heiligers 1992 or Schwabe 1996, ch. 3). Prus and Schwabe (2016) investigated the related properties of designs, which are optimal for prediction of individual random parameters in single-group mixed effects models. Here we extend those results to multiple-group models. We consider a one-to-one transformation gof the experimental regions Xifor all i=1,...ssimultaneously with g(Xi)=Xg i. We assume the regression matrices F(i)to be defined on both Xiand Xg i. We also assume the existence of a non-singular p×pmatrix Qgsuch that ˜ F(i)(g(x)) =Qg˜ F(i)(x), ∀x∈Xi,i=1,...,s,(15) i.e. all ˜ F(i)are linearly equivariant with respect to the transformation g(see e.g. Schwabe 1996, ch. 3). We denote by ξg ithe following transformation of an approximate design ξi: ξg i=g(xi1),...,g(xiki) wi1,...,w iki,(16) 123 Computational aspects of experimental designs... 871 where the weight wik is the same for both ξiand ξg iand only the design points xik are transformed. Then we obtain the next property of the moment matrices: Mi(ξg i)=QgMi(ξi)Q g,i=1,...,s.(17) Further we use the notations D=(D1,...,Ds)and X=× s i=1Xifor the tuple of covariance matrices and the Cartesian product of the experimental regions, respectively, in all groups. For the covariance matrix (5) the following relation can be easily verified: Covξg(Dg)=Q− gCovξ(D)Q−1 g,(18) where ξg=(ξg 1,...,ξg s),Dg=(Dg 1,...,Dg s),Dg i=Q− gDiQ−1 gand Q− g= (Q g)−1. We use the notation Covξ(D)[instead of Covξas in formula (5)] to emphasize the dependence on the covariance matrices Dof random effects. Then the equivariance of the Dand L-criteria with respect to a transformation g can be established. Theorem 3 If the approximate designs ξ∗are D-optimal for the estimation of β0 on the experimental regions Xunder the dispersion matrices D, then the induced approximate designs ξgare D-optimal for the estimation of β0on the experimental regions Xg=× s i=1Xg iunder the induced dispersion matrices Dg. Proof From the definition of the D-criterion for the estimation of β0and formula (18) we obtain φD(ξg,Dg)=−2ln|det(Qg)|+φD(ξ,D), which proves the optimality of ξgon Xgfor ξoptimal on X. Theorem 4 If the approximate designs ξ∗are L-optimal for the estimation of β0 on the experimental regions Xunder the dispersion matrices Dwith respect to the transformation matrix V, then the induced approximate designs ξ∗gare L-optimal for the estimation of β0on the experimental regions Xgunder the induced dispersion matrices Dgwith respect to the induced transformation matrix Vg=QgVQ g. Proof Using formulas (6) and (18) it can be easily verified that φL(ξg,Dg,Vg)=φL(ξ,D,V), which proves the optimality of ξgon Xg. Corollary 1 The A-criterion for the estimation of β0is equivariant with respect to a transformation g if Qgis orthogonal, i.e.: QgQ g=Q gQg=Ip.(19) 123 872 M. Prus, L. Filová To verify the equivariance of the IMSE-criterion we assume, besides the transformed regression matrices ˜ F(i), the original regression matrices F(i)to be linearly equivariant with respect to the transformation g: F(i)(g(x)) =QgF(i)(x), ∀x∈Xi,i=1,...,s.(20) Then if the measure νiis transformed to its image νg i, we obtain Vg= s  i=1 aiXg i F(i)(x)F(i)(x)νg i(dx)=QgVQ g. Corollary 2 The I M SE-criterion for the estimation of β0is equivariant with respect to a transformation g if condition (20)is satisfied. Example 1(continued). We consider again the two-groups linear regression model (12)on Xi=[0,1]with diagonal covariance structure of random effects. For the IMSE-criterion we chose the uniform weighting νi=λ[0,1],i=1,2, where λ[c1,c2] denotes the Lebesgue measure on [c1,c2].Letξ∗ ibe D-, A-or I M SE-optimal groupdesigns of form (13)with the optimal weight of observations w∗ i1(which generally depends on the choice of the design criterion). Now we consider the linear transformation g(x)=ax,a>0, for which we obtain Qg=diag(1,a).Then the D-, A-orIMSE-optimal group-designs in model (12)on Xg i=[0,a]for Dg i=diag(di1,di2/a2)and νg i=1 aλ[0,a]are given by ξ∗ i g=0a 1−w∗ i1w∗ i1.(21) Same behavior of optimal designs has been established for the prediction of random effects in single-group model in Prus and Schwabe (2016). Further we consider a finite group G of transformations g :Xi→Xiof the experimental regions Xionto themselves for all i =1,...s simultaneously. We assume the equivariance condition (15)to be satisfied and the dispersion matrices to be invariant: Dg i=Di,for all g ∈G,i=1,...,s.For the linear criteria we additionally assume the invariance of the transformation matrices: Vg=V.Then the D-and Lcriteria are invariant with respect to all g ∈G and the following statement can be formulated: Theorem 5 If the approximate designs ξ∗are Dor L-optimal for the estimation of β0, then the symmetrized designs ¯ ξ∗=(¯ ξ∗ 1,...,¯ ξ∗ s)for ¯ ξ∗ i=1 #Gg∈Gξ∗ i gare also Dor L-optimal for the estimation of β0. Proof Let the designs ξ∗be D-optimal for the estimation of β0. Then it follows from Theorem 3and the invariance of the dispersion matrices that the induced designs ξ∗g are also D-optimal, i.e. φD(ξg∗,D)=φD(ξ∗,D), ∀g∈G. 123 Computational aspects of experimental designs... 879 Table 1 Exact D-efficient designs in bilinear model (26)on[−1,1]2with total numbers of observations in the groups given by m=(20,20,20)with di∈{−0.5,0,0.5} (d1,d2,d3)(−1,−1)(1,1)(−1,−1)(1,1)(−1,−1)(1,1) (0,0,0)555555 (0,0,0.5)464673 (0,0.5,0.5)467373 (0.5,0.5,0.5)646464 (0,0,−0.5)646437 (0,0.5,−0.5)557337 (−0.5,0.5,−0.5)378237 6 Quadratic regression on a symmetric interval Consider the two-groups model of the form (1) with the regression functions F(i)(x)= (1,x,x2),x∈Xi, and the design region Xi=[−1,1],i=1,2: Yijh =βij1+βij2xih +βij3x2 ih +εijh,j=1,...,ni,h=1,...,mi.(38) The covariance structures of random effects and observational errors are given by Di=diag(di1,di2,di3)and i=1 for both groups. The corresponding approximate optimal group-designs ξiare supported by three design points −1,0,1 (see Sect.3): ξ∗ i=−10 1 w∗ i11−2w∗ i1w∗ i1.(39) This result was heuristically confirmed to hold also for the exact designs: we discretized the design region into qpoints −1=x1<x2<···<xq=1 and confirmed that for all cases considered below, the support points are indeed −1,0 and 1. The exact Dand IMSE-efficient designs for this case and several particular m= (m1,m2)are given in Table 2in the Appendix. We can see that for the criterion of D-optimality and the values of the diagonal of the matrix Diequal to either (1,1,1)or (1,1,0), half of the measurements is in the point 0 and the remaining half is distributed equally among the points −1 and 1. For (d1,d2,d3)equal either to (0,1,1)or (1,0,0), the measurements are heavily concentrated in the support point 0, but the designs are still nonsingular. The case (1,0,1)shows opposite phenomenon with the measurements being concentrated in the points −1 and 1. For the remaining cases, the pattern is not so clear and the weights depend more on m, sometimes even resulting in singular designs. For the IMSE criterion, we get results identical to D-optimality if the diagonal of Diis (1,1,1). For the rest of the cases, the situation is more varied and we refer reader to Table 2for details. From practical point of view, it may not be desirable to only have three support points for each group. Therefore, additional constraints on the design were suggested, 123 880 M. Prus, L. Filová where it is prescribed that, for each group, maximum one half of the measurements can be taken at −1,0 or 1. Formally, these constraints can be written in the form A(1)w≤b(1)(see Sect.4for details), where A(1)=cq0 q 0 qcq∈R2×2q,b(1)=m1/2 m2/2,(40) where cq=(1,0,...,0,1,0,...,0,1)∈Rqwith 1 on the positions corresponding to the points −1,0,1. The Dand IMSE-efficient designs for the discretization Xi={−1,−0.8,...,0.8,1}of the interval [−1,1]with the step 0.2 (i.e. q=11) are given in Tables 3and 4. Note that for both criteria, the tendency is to distribute the measurements as close as possible to the original support points −1,0 and 1. Another type of constraint that is often used in practical situations, is the cost constraint: this is natural, for example, in clinical trials, where taking a measurement at a point xconsumes a certain number of time, personal or material resources and the total cost of the experiment is limited. In our case, let the measurement at the point xcost |x|+0.1 units, and, for group j, let the maximum admissible cost be mj/4. This leads to adding the constraints A(2)w≤b(2)with the following A(2),b(2)to the problem (36): A(2)=uq+0.11 q0 0uq+0.11 q∈R2×2q,b(2)=1 4m1 m2,(41) where uq=(1,0.8,...,0.8,1). Again, we computed Dand IMSE-efficient designs with respect to this constraint for the discretization Xi={−1,−0.8,...,0.8,1}. Now, the designs are supported on −1,0 and 1, but, compared to the unconstrained designs, much more measurements are made at the point 0, which is ’cheap’: the results are summarized in Table 5. Finally, it is also feasible and possible to consider both types of constraints together, resulting in A(3)w≤b(3)with A(3)=A(1) A(2)∈R4×4q,b(3)=b(1) b(2).(42) The resulting Dand IMSE-efficient designs for this constraint are given in Tables 6and 7. Note that in some cases, the additional constraints on the designs were saturated for a number of measurements that is lower than the maximum attainable number of measurements given by (m1,m2). This is demonstrated in a more detailed way in Fig.2, where we again consider the D-efficient design with (m1,m2)=(20,40) and the cost constraints (41), but now the cost b(2)can vary between 0 and mifor the i-th group. All the designs are supported in the points −1,0 and 1 and the figure shows that when the maximum allowed cost is too low, the total maximum number of measurements is (sometimes significantly) lower than the corresponding mi. 123 Computational aspects of experimental designs... 881 Fig. 2 The numbers of measurements in the point 0 (full line), −1 (dashed line) and 1 (dot-dashed line) for the second group in the D-efficient design in model (38) with (m1,m2)=(20,40)and constraints of the type (41) with the maximum cost b(2)in the second group varying between 0 and 40 7 Discussion In the paper, we have considered equiand invariance properties of approximate optimal designs in multiple-group mixed models. We have used these properties to fix the support points and, consequently, to reduce the number of unknown variables in firstand second-order models on a symmetric square. As we currently have no universal computational tool for approximate designs, these results can be used to determine optimal designs analytically in a few isolated and easy cases, as shown in the examples in Sect.3. However, from practical point of view, it is more important to be able to compute efficient exact designs, possibly even with some additional constraints given by the experimental conditions. We have shown a modified version of the algorithm of Harman et al. (2016) is a useful tool for such computations, even in the cases where there are several nontrivial constraints on the design. In the models considered here, covariance matrix of random effects is assumed to be known. A natural question that arises while reading this work is how to perform in the situation where no prior knowledge about variances and covariances is available. In this case an estimation can be used. However, the quality of obtained designs depends on the accuracy of the estimation. For some particular structures of the covariance matrix it may happen that optimal designs turn out to be independent on the variance parameters (consider, for example, compound symmetry structure in Prus and Piepho 2021). Acknowledgements This research was partially supported by Grant SCHW 531/16 of the German Research Foundation (DFG) and by Grant Number 1/0341/19 and 1/0362/22 from the Slovak Scientific Grant Agency (VEGA). A part of the research has been performed as the second author was visiting Comenius University of Bratislava in framework of the postdoc program (PKZ 91672803) of German Academic Exchange Service (DAAD). The authors are grateful to Radoslav Harman for fruitful discussions. Discussions with Rainer Schwabe on design criteria in multiple group-models were also very helpful. Funding Open Access funding enabled and organized by Projekt DEAL. 123 882 M. Prus, L. Filová Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 Computational aspects of experimental designs... 883 Appendix See Tables 2,3,4,5,6and 7. Table 2 Exact D-andIMSE-efficient designs in quadratic model on the interval [−1,1]with respect to the numbers of observations mifor Di=diag(d1,d2,d3),i=1,2 criterion D IMSE (d1,d2,d3)m1m2−10 1 −10 1 −10 1−10 1 (1,1,1)20805 1052040205 105204020 50 50 12 25 13 13 25 12 13 25 12 12 25 13 80 20 20 40 20 5 10 5 20 10 20 5 10 5 40 160 10 20 10 40 80 40 10 20 10 40 80 40 100 100 25 50 25 25 50 25 25 50 25 25 50 25 160 40 40 80 40 10 20 10 40 80 40 10 20 10 (1,1,0)20 80 5 10 5 20 40 20 9 2 9 15 49 16 50 50 12 25 13 13 25 12 13 25 12 12 25 13 80 20 20 40 20 5 10 5 15 49 16 9 2 9 40 160 10 20 10 40 80 40 19 2 19 31 98 31 100 100 25 50 25 25 50 25 25 50 25 25 50 25 160 40 40 80 40 10 20 10 31 98 31 19 2 19 (0,1,1)20 80 2 16 2 1 78 1 10 0 10 13 54 13 50501 4811 48112 2612132413 80201 7812 16213 5413100 10 40 160 2 36 2 1 158 1 20 0 20 27 106 27 1001001 9811 98125 5025255025 160 40 1 158 1 2 36 2 27 106 27 20 0 20 (1,0,1)20808 4 8401 394 124253025 50 50 25 1 24 24 1 25 13 25 12 12 25 13 80 20 39 1 40 8 4 8 25 30 25 4 12 4 40 160 18 4 18 80 1 79 8 25 7 49 61 50 100 100 50 1 49 49 1 50 25 50 25 25 50 25 160 40 80 1 79 18 4 18 49 61 50 8 25 7 (1,0,0)20802 1621 7811 190262826 50501 4811 48113 2512122513 80201 7812 16226 28261 190 401602 3621 15811 381535453 1001001 9811 98125 5025255025 160 40 1 158 1 2 36 2 53 54 53 1 38 1 123 884 M. Prus, L. Filová Table 2 continued criterion D IMSE (d1,d2,d3)m1m2−10 1 −101−101−10 1 (0,1,0)20 80 10 0 10 19 42 19 10 0 10 16 48 16 50 50 12 26 12 13 24 13 25 12 13 13 25 12 80 20 19 42 19 10 0 10 16 48 16 10 0 10 40 160 18 4 18 38 84 38 20 0 20 32 96 32 100 100 25 50 25 25 50 25 25 50 25 25 50 25 160 40 38 84 38 18 4 18 32 96 32 20 0 20 (0,0,1)20 80 9 2 9 19 42 19 10 0 10 17 46 17 50 50 12 26 12 13 24 13 12 26 12 13 24 13 80 20 19 42 19 9 2 9 17 46 17 10 0 10 40 160 18 4 18 38 84 38 20 0 20 35 91 34 100 100 25 50 25 25 50 25 25 50 25 25 50 25 160 40 38 84 38 18 4 18 34 91 35 20 0 20 Table 3 Exact D-efficient designs in quadratic model on the interval [−1,1]with constraints given by 40 and numbers of observations mifor Digiven by 37 with di=0.5, i=1,2 m1m2−1−0.2 0 0.2 1 −1−0.2 0 0.2 1 20 40 4 5 2 5 4 8 10 4 10 8 40 20 8 10 4 10 8 4 5 2 5 4 25 100 5 7 2 6 5 21 26 9 24 20 100 25 21 27 9 23 20 5 6 2 7 5 Table 4 Exact IMSE-efficient designs in quadratic model on the interval [−1,1]with constraints given by (40) and numbers of observations mifor Digiven by (37) with di=0.5, i=1,2 m1m2−1−0.2 0 0.2 1 −1−0.2 0 0.2 1 20 40 5 5 0 5 5 9 10 1 10 10 40 20 9 10 1 10 10 5 5 0 5 5 25 100 6 6 0 7 6 23 26 3 24 24 100 25 24 25 3 25 23 6 6 0 7 6 123 Computational aspects of experimental designs... 885 Table 5 Exact D- (left) and IMSE-efficient (right) designs in quadratic model on the interval [−1,1]with constraints given by (41) and numbers of observations mifor Digiven by (37) with di=0.5, i=1,2 (m1,m2)−10 1−10 1−10 1−10 1 (20,40) 2 1713 3432 6 24 233 (40,20) 3 34 3 2 17 1 4 23 3 2 6 2 (25,100) 3 14 3 8 74 8 3 14 3 9 52 9 (100,25) 8 74 8 3 14 3 8 63 9 3 14 3 Table 6 Exact D-efficient designs in quadratic model on the interval [−1,1]with constraints given by (42)and numbers of observations mifor Digiven by (37) with di=0.5, i=1,2 (m1,m2)−1−0.2 0 0.2 1 −1−0.2 0 0.2 1 (20,40) 2 3 7 0 1 3 3 14 3 3 (40,20) 3 0 13 3 4 2 0 6 0 2 (25,100) 3 5 7 1 2 8 7 34 6 8 (100,25) 8 7 34 6 8 2 4 8 5 2 Table 7 Exact IMSE-efficient designs in quadratic model on the interval [−1,1]with constraints given by (42) and numbers of observations mifor Digiven by (37) with di=0.5, i=1,2 (m1,m2)−1−0.2 0 0.2 1 −1−0.2 0 0.2 1 (20,40) 2 0 6 0 2 4 0 12 3 3 (40,20) 4 0 12 0 4 2 0 6 0 2 (25,100) 3 1 7 2 3 8 2 38 0 12 (100,25) 9 1 35 0 11 3 2 7 1 3 References Aliev A, Fedorov V, Leonov S, McHugh B, Magee M (2012) Pkstamp library for constructing optimal population designs for pk/pd studies. Commun Stat Simul Comput 41(6):717–729 Cook RD, Thibodeau L (1980) Marginally restricted D-optimal designs. J Am Stat Assoc 75(370):366–371 Dumont C, Lestini G, Nagard HL, Mentré F, Comets E, Nguyen TT (2018) PFIM 4.0, an extended R program for design evaluation and optimization in nonlinear mixed-effect models. Comput Methods Programs Biomed 156:217–229 Fedorov V, Jones B (2005) The design of multicentre trials. Stat Methods Med Res 14:205–248 Galil Z, Kiefer J (1977) Comparison of designs for quadratic regression on cubes. J Stat Plan Inference 1:121–132 Harman R, Bachratá A, Filová L (2016) Construction of efficient experimental designs under multiple resource constraints. Appl Stoch Models Bus Ind 32:3–17 Heiligers B (1992) Admissible experimental designs in multiple polynomial regression. J Stat Plan Inference 31:219–233 Nyberg J, Ueckert S, Strömberg EA, Hennig S, Karlsson MO, Hooker AC (2012) Poped: an extended, parallelized, nonlinear mixed effects models optimal design tool. Comput Methods Programs Biomed 108(2):789–805 Prus M (2022) Equivalence theorems for multiple-design problems with application in mixed models. J Stat Plan Inference 217:153–164 Prus M, Piepho H-P (2021) Optimizing the allocation of trials to sub-regions in multi-environment crop variety testing. J Agric Biol Environ Stat 26:267–288 123 886 M. Prus, L. Filová Prus M, Schwabe R (2016) Optimal designs for the prediction of individual parameters in hierarchical models. J R Stat Soc B 78:175–191 Schmelter T (2007) Considerations on group-wise identical designs for linear mixed models. J Stat Plan Inference 137:4003–4010 Schwabe R (1996) Optimum designs for multi-factor models. Springer, New York Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123