a geometric characterization of c optimal designs for heteroscedastic regression we consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables the c optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest a geometric characterization of c optimal designs in this context is presented which generalizes the classical result of elfving 1952 for c optimal designs as in elfving s famous characterization c optimal designs can be described as representations of boundary points of a convex set however in the case where there appear parameters of interest in the variance the structure of the elfving set is dierent roughly speaking the elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space the c optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new elfving set the theory is illustrated in several examples including pharmacokinetic models with random eects

maximin d optimal designs for longitudinal mixed effects models in this article the optimal selection and allocation of time points in repeated measures experiments is considered d optimal cohort designs are computed numerically for the first and second degree polynomial models with random intercept random slope and first order autoregressive serial correlations because the optimal designs are locally optimal it is proposed to use a maximin criterion it is shown that for a large class of symmetric designs the smallest relative efficiency over the model parameter space is substantial

bayesian experimental design for nonlinear mixed effects models with application to hiv dynamics bayesian experimental design is investigated for bayesian analysis of nonlinear mixed effects models existence of the posterior risk for parameter estimation is shown when the same prior distribution is used for both design and inference existence of the preposterior risk for design is also proven if the prior distribution used in design is different from that used for inference sufficient conditions are established for existence of the preposterior risk for design a case study of design for an experiment in population hiv dynamics is provided

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

curvature adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models in pharmacokinetics compartment models are often used to describe the time course of blood concentration after the administration of a drug in this article we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non sequential design of sampling times of blood drug concentration data in individual pharmacokinetics the proposed optimal design criterion minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated by the curvature array therefore the proposed criterion considers the intrinsic and parameter effects nonlinearity underlying the compartment model and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small

determination of effective brain connectivity from activity correlations effective connectivity embodied in transfer functions is derived from symmetric network activity correlations under task free conditions via a recent causal spectral factorization method this generalizes previous covariance based analyses to include frequency dependencies and time delays results are verified against analytic solutions of equations that have reproduced many aspects of experimental brain dynamics and against cases of more complex connectivity robustness to noise is also demonstrated

on explicit wiener hopf factorization of 2 2 matrices in a vicinity of a given matrix as it is known the existence of the wienerhopf factorization for a given matrix is a well studied problem severe difficulties arise however when one needs to compute the factors approximately a

on optimal designs for nonlinear models a general and efficient algorithm finding optimal designs for nonlinear models is challenging in general although some recent results allow us to focus on a simple subclass of designs for most problems deriving a specific optimal design still mainly depends on numerical approaches there is need for a general and efficient algorithm that is more broadly applicable than the current state of the art methods we present a new algorithm that can be used to find optimal designs with respect to a broad class of optimality criteria when the model parameters or functions thereof are of interest and for both locally optimal and multistage design strategies we prove convergence to the optimal design and show in various examples that the new algorithm outperforms the current state of the art algorithms

a note on the de la garza phenomenon for locally optimal designs the celebrated de la garza phenomenon states that for a polynomial regression model of degree p 1 any optimal design can be based on at most p design points in a remarkable paper yang ann statist 38 2010 2499 2524 showed that this phenomenon exists in many locally optimal design problems for nonlinear models in the present note we present a different view point on these findings using results about moment theory and chebyshev systems in particular we show that this phenomenon occurs in an even larger class of models than considered so far

optimal design for additive partially nonlinear models we develop optimal design theory for additive partially nonlinear regression models showing that bayesian and standardized maximin d optimal designs can be found as the products of the corresponding optimal designs in one dimension a sufficient condition under which analogous results hold for ds optimality is derived to accommodate situations in which only a subset of the model parameters is of interest to facilitate prediction of the response at unobserved locations we prove similar results for q optimality in the class of all product designs the usefulness of this approach is demonstrated through an application from the automotive industry where optimal designs for least squares regression splines are determined and compared with designs commonly used in practice

the numerical solution for reaction diffusion combustion with fuel consumption this paper describes and studies the reaction between a gaseous oxidant and a solid porous fuel which allows for the consumption of the fuel j appl math 47 1991 33 here we consider the effects of including consumption of the solid fuel of a poisoning of a solid catalyst by the ongoing reaction we assume a single exothermic process first order in the concentration each of fuel and oxidant and with an arrhenious temperature dependence of the reaction rate constant the solid reactant has an infinite slab geometry of thickness 2l and is surround by a reservoir of constant oxidant concentration and a fixed ambient temperature further the reaction is assumed not to alter the integrity or pore structure of the solid we solved three coupled partial differential equations which describe this problem by using the finite difference scheme numerical solution of partial differential equations finite difference method clarendon press oxford 1978 which was first applied to a finite difference scheme by crank q j mech appl math 10 1957 220 j inst math appl 10 1872 19 using this method we can transform the partial differential equations to a system of linear algebraic equations int j eng sci 36 5 6 1998 625 appl math computat 113 2 3 2000 85 this system is solved with the dirichlet boundary conditions the effects of the frank kamenetskii parameter l and the lewis number l are examined

designing dose finding studies with an active control for exponential families in a recent paper dette et al 2014 introduced optimal design problems for dose finding studies with an active control these authors concentrated on regression models with normal distributed errors with known variance and the problem of determining optimal designs for estimating the smallest dose which achieves the same treatment eect as the active control this paper discusses the problem of designing activecontrolled dose finding studies from a broader perspective in particular we consider a general class of optimality criteria and models arising from an exponential family which are frequently used analyzing count data we investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control optimal designs are constructed for several situations and the dierences arising from dierent distributional assumptions are investigated in detail in particular our results are applicable for constructing optimal experimental designs to analyze active controlled dose finding studies with discrete data and we illustrate the eciency of the new optimal designs with two

mean squared error of estimation or prediction under a general linear model abstract the problem considered is that of predicting a linear combination of the fixed and random effects of a mixed effects linear model more generally the problem considered is that of predicting an unobservable random variable from a set of observable random variables the best linear unbiased predictor depends on parameters which generally are unknown various exact or approximate expressions are given for the mean squared error mse of the predictor obtained by replacing the unknown parameters with estimates several estimators of the mse are investigated

optimal designs for comparing curves we consider the optimal design problem for a comparison of two regression curves which is used to establish the similarity between the dose response relationships of two groups an optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions optimal design theory equivalence theorems efficiency bounds is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly the results are illustrated in several examples modeling dose response relationships it is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models in particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used non optimal designs yields a reduction of the width of the confidence band by more than 50

robust optimum designs for transformation of the responses in a multivariate chemical kinetic model transformation of both sides of a nonlinear model is often necessary to obtain tractable error distributions but can have a dramatic effect on optimum designs for the parameters of the model this article develops methods for the design of experiments when the value of the power transformation parameter is not precisely known optimum designs for particular transformations together with the efficiency of these designs as the transformation varies are studied for a multivariate model it is shown that by a suitable choice of prior bayesian designs can be found that are robust to the correct transformation a typical equispaced design is evaluated and the properties of designs are studied under various departures from the assumptions

optimal two period repeated measurement designs with two or more treatments summary for two period repeated measurement designs for comparing t 2 treatments we discuss universally optimal designs when the number of subjects n is a multiple of t2 and some cases when n is a multiple of t we examine efficiencies of balanced designs and other symmetric designs relative to strongly balanced designs in addition efficiencies of the universally optimal designs obtained by hedayat zhao 1990 are examined under both fixed subject and random subject effect model assumptions

computing exact d optimal designs by mixed integer second order cone programming i 1 wiaia t i where ai i 1 s are known matrices with m rows in this paper we show that the criterion of d optimality is secondorder cone representable as a result the method of second order cone programming can be used to compute an approximate d optimal design with any system of linear constraints on the vector of weights more importantly the proposed characterization allows us to compute an exact d optimal design which is possible thanks to highquality branch and cut solvers specialized to solve mixed integer second order cone programming problems our results extend to the case of the criterion of dk optimality which measures the quality of w for the estimation of a linear parameter subsystem defined by a full rank coefficient matrixk we prove that some other widely used criteria are also secondorder cone representable for instance the criteria of a ak gand i optimality we present several numerical examples demonstrating the efficiency and general applicability of the proposed method we show that in many cases the mixed integer second order cone programming approach allows us to find a provably optimal exact design while the standard heuristics systematically miss the optimum

optimal designs for estimating individual coefficients in fourier regression models in the common trigonometric regression model we investigate the optimal design problem for the estimation of the individual coefficients where the explanatory variable varies in the interval a a 0 a le pi it is demonstrated that the structure of the optimal design depends sensitively on the size of the design space for many important cases optimal designs can be found explicitly where the complexity of the solution depends on the value of the parameter a and the order of the term for which the corresponding coefficient has to be estimated the main tool of our approach is the reduction of the problem for the trigonometric regression model to a design problem for a polynomial regression in particular we determine the optimal designs for estimating the parameters corresponding to the cosine terms explicitly if the design space is sufficiently small and prove that under this condition all optimal designs for estimating the parameters corresponding to the sine terms are supported at the same points

on the de la garza phenomenon deriving optimal designs for nonlinear models is in general challenging one crucial step is to determine the number of support points needed current tools handle this on a case by case basis each combination of model optimality criterion and objective requires its own proof the celebrated de la garza phenomenon states that under a p 1 th degree polynomial regression model any optimal design can be based on at most p design points the minimum number of support points such that all parameters are estimable does this conclusion also hold for nonlinear models if the answer is yes it would be relatively easy to derive any optimal design analytically or numerically in this paper a novel approach is developed to address this question using this new approach it can be easily shown that the de la garza phenomenon exists for many commonly studied nonlinear models such as the emax model exponential model three and four parameter log linear models emax pk1 model as well as many classical polynomial regression models the proposed approach unifies and extends many well known results in the optimal design literature it has four advantages over current tools i it can be applied to many forms of nonlinear models to continuous or discrete data to data with homogeneous or nonhomogeneous errors ii it can be applied to any design region iii it can be applied to multiple stage optimal design and iv it can be easily implemented

a geometric characterization of optimal designs for regression models with correlated observations we consider the problem of optimal design of experiments for random effects models especially population models where a small number of correlated observations can be taken on each individual whereas the observations corresponding to different individuals are assumed to be uncorrelated we focus on c optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of elfving from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients the theory is demonstrated by finding optimal designs for a linear model with correlated observations and a non linear random effects population model which is commonly used in pharmaco kinetics

on the algorithmization of janashia lagvilava matrix spectral factorization method we consider three different ways of algorithmization of the janashia lagvilava spectral factorization method the first algorithm is faster than the second one however it is only suitable for matrices of low dimension the second algorithm on the other hand can be applied to matrices of substantially larger dimension the third algorithm is a superfast implementation of the method but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary all three algorithms fully utilize the advantage of the method which carries out spectral factorization of leading principal submatrices step by step the corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms

optimal weights for experimental designs on linearly independent support points an explicit formula is derived to compute the a optimal design weights on linearly independent regression vectors for the mean parameters in a linear model with homoscedastic variances the formula emerges as a special case of a general result which holds for a wide class of optimality criteria there are close links to iterative algorithms for computing optimal weights

d optimal input design for nonlinear fir type systems optimal input design is an important step of the identification process in order to reduce the model variance in this work a d optimal input design method for finite impulse response type nonlinear systems is presented the optimization of the determinant of the fisher information matrix is expressed as a convex optimization problem this problem is then solved using a dispersion based optimization scheme which is easy to implement and converges monotonically to the optimal solution without constraints the optimal design cannot be realized as a time sequence by imposing that the design should lie in the subspace described by a symmetric and non overlapping set a realizable design is found a graph based method is used in order to find a time sequence that realizes this optimal constrained design these methods are illustrated on a numerical example of which the results are thoroughly discussed additionally the computational speed of the algorithm is compared with the general convex optimizer cvx

an elementary proof of the polynomial matrix spectral factorization theorem a very simple and short proof of the polynomial matrix spectral factorization theorem on the unit circle as well as on the real line is presented which relies on elementary complex analysis and linear algebra

optimal design for goodness of fit of the michaelis menten enzyme kinetic function we construct efficient designs for the michaelis menten enzyme kinetic model capable of checking model assumptions an extended model called emax is also considered for this purpose this model is widely used in pharmacokinetics and reduces to the michaelis menten model for a specific choice of parameter settings our strategy is to find efficient designs for estimating the parameters in the emax model and at the same time test the validity of the michaelis menten model against the emax model by maximizing a minimum of the d or d1 efficiencies taken over a range of values for the nonlinear parameters in particular we show that such designs are a efficient for estimating parameters in the emax model b about 70 efficient for estimating parameters in the michaelis menten model c efficient for testing the michaelis menten model against the emax model and d robust with respect to misspecification of the unknown parameters in the nonlinear model

bayesian t optimal discriminating designs the problem of constructing bayesian optimal discriminating designs for a class of regression models with respect to the t optimality criterion introduced by atkinson and fedorov 1975a is considered it is demonstrated that the discretization of the integral with respect to the prior distribution leads to locally t optimal discriminating design problems with a large number of model comparisons current methodology for the numerical construction of discrimination designs can only deal with a few comparisons but the discretization of the bayesian prior easily yields to discrimination design problems for more than 100 competing models a new efficient method is developed to deal with problems of this type it combines some features of the classical exchange type algorithm with the gradient methods convergence is proved and it is demonstrated that the new method can find bayesian optimal discriminating designs in situations where all currently available procedures fail

likelihood inference for mixtures geometrical and other constructions of monotone step length algorithms summary this paper considers algorithms for the maximum likelihood estimator of the mixing distribution of a family of parametric densities specific interest is devoted to the monotonicity of the step length of the solution in 2 we introduce the idea of estimating the area above the second derivative curve and in theorem 1 we relate area overestimation to the monotonicity of the associated algorithm in 3 we apply theorem 1 to the monotonicity analysis of well known algorithms as well as to the construction of a new class of monotone step length choices which is particularly simple in the mixture setting numerical comparisons and refinements in keeping the monotonic step length from becoming too conservative are given theorem 3 characterizes conditions under which the newton raphson step is monotonic and if these conditions fail to hold states that the regula falsi step is monotonic

digital simulation of thermal reactions abstract simulations of the equation for thermal expansion of a reacting gas have been carried out exploring both the possible steady states and time marching solutions the critical frank kamenetskii parameter cr has been evaluated to seven decimal places for the slab cylinder and spherical geometries and the role of the critical activation parameter was explored it was found that there exist one or more mathematical steady states for any if 0 the curves for steady temperature at the center of the geometry plotted against tending to a straight line at large critical values of the values above which this plot has a single solution for a given have been computed to eight decimals time marching simulations showed that the crank nicolson method applied consistently produces very accurate results compared with the implementation in which the nonlinear term is rendered explicit where for a given there are several mathematical steady states a time march usually settles on the lowest such state if it settles at all regardless of where the simulation is started within the possible limits the mathematical multiple steady states are not attained by time marching simulations and are also physically unlikely

characterization of c l and ϕk optimal designs for a class of non linear multiple regression models optimal designs for multiple regression models are determined we consider a general class of non linear models including proportional hazards models with different censoring schemes the poisson and the negative binomial model for these models we provide a complete characterization of c optimal designs for all vectors c in the case of a single covariate for multiple regression with an arbitrary number of covariates c optimal designs for certain vectors c are derived analytically using some general results on the structure of optimal designs for multiple regression we determine l and k optimal designs for models with an arbitrary number of covariates

design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models bioequivalence or interaction trials are commonly studied in crossover design and can be analysed by nonlinear mixed effects models as an alternative to noncompartmental approach we propose an extension of the population fisher information matrix in nonlinear mixed effects models to design crossover pharmacokinetic trials using a linearisation of the model around the random effect expectation including within subject variability and discrete covariates fixed or changing between periods we use the expected standard errors of treatment effect to compute the power for the wald test of comparison or equivalence and the number of subjects needed for a given power we perform various simulations mimicking crossover two period trials to show the relevance of these developments we then apply these developments to design a crossover pharmacokinetic study of amoxicillin in piglets and implement them in the new version 3 2 of the r function pfim

optimal designs for frequentist model averaging we consider the problem of designing experiments for estimating a target parameter in regression analysis when there is uncertainty about the parametric form of the regression function a new optimality criterion is proposed that chooses the experimental design to minimize the asymptotic mean squared error of the frequentist model averaging estimate necessary conditions for the optimal solution of a locally and bayesian optimal design problem are established the results are illustrated in several examples and it is demonstrated that bayesian optimal designs can yield a reduction of the mean squared error of the model averaging estimator by up to 45

optimal designs for quantile regression models despite their importance optimal designs for quantile regression models have not been developed so far in this article we investigate the d optimal design problem for nonlinear quantile regression analysis we provide a necessary condition to check the optimality of a given design and use it to determine bounds for the number of support points of locally d optimal designs the results are illustrated determining locally bayesian and standardized maximin d optimal designs for quantile regression analysis in the michaelis menten and emax model which are widely used in such important fields as toxicology pharmacokinetics and dose response modeling

optimal designs for dose finding studies with an active control summary dose finding studies often compare several doses of a new compound with a marketed standard treatment as an active control in the past however research has focused mostly on experimental designs for placebo controlled dose finding studies to the best of our knowledge optimal designs for dose finding studies with an active control have not been considered so far as the statistical analysis for an active controlled dose finding study can be formulated in terms of a mixture of two regression models the related design problem is different from what has been investigated before in the literature we present a rigorous approach to the problem of determining optimal designs for estimating the smallest dose achieving the same treatment effect as the active control we determine explicitly the locally optimal designs for a broad class of models employed in such studies we also discuss robust design strategies and determine related bayesian and standardized minimax optimal designs we illustrate the results by investigating alternative designs for a clinical trial which has recently appeared in a consulting project of one of the authors

on the efficiency of two stage response adaptive designs in this paper we investigate the efficiency of response adaptive locally optimum designs we focus on two stage adaptive designs where after the first stage the accrued data are used to determine a locally optimum design for the second stage on the basis of an explicit expansion of the information matrix we compare the variance of the maximum likelihood estimates obtained from a two stage adaptive design and a fixed design without adaptation for several one parameter models we provide explicit expressions for the relative efficiency of these two designs which is seen to depend sensitively on the statistical problem under investigation in particular we show that in non linear regression models with moderate or large variances the first stage sample size of an adaptive design should be chosen sufficiently large in order to address variability in the interim parameter estimates these findings support the results of recent simulation studies conducted to compare adaptive designs in more complex situations we finally present an application to a real clinical dose finding trial aiming at the estimation of the smallest dose achieving a certain percentage of the maximum treatment effect by using a three parameteremax model copyright 2012 john wiley sons ltd

optimal designs for active controlled dose finding trials with efficacy toxicity outcomes author s schorning kirsten dette holger kettelhake katrin wong weng kee bretz frank abstract we derive optimal designs to estimate efficacy and toxicity in active controlled dose finding trials whenthe bivariate continuous outcomes are described using nonlinear regression models we determine upperbounds on the required number of different doses and provide conditions under which the boundary pointsof the design space are included in the optimal design we provide an analytical description of minimallysupported optimal designs and show that they do not depend on the correlation between the bivariateoutcomes

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

combined algorithm to compute d optimal designs algorithmic techniques for computing optimal designs continue being a need in the optimal experimental design field the increasing interest in finding the optimal experimental conditions makes that new methods are demanded for more complex frameworks showing more realistic situations numerical techniques are often the unique viable option for computing these designs since to tackle analytically the problem results impracticable in most of cases wynn fedorov algorithm multiplicative algorithm and their modifications are the more frequently used methods in the literature for computing d optimal designs however they are not always suitable and efficient to compute optimal designs since their procedures are very limited by the computational requirements a new algorithm to obtain d optimal designs is proposed in this paper it is based on combining suitable strategies followed by the traditional algorithms a proof of the convergence is provided and several numerical examples are presented to illustrate its improved results

on the number of support points of maximin and bayesian optimal designs we consider maximin and bayesian d optimal designs for nonlinear regression models the maximin criterion requires the specification of a region for the nonlinear parameters in the model while the bayesian optimality criterion assumes that a prior for these parameters is available on interval parameter spaces it was observed empirically by many authors that an increase of uncertainty in the prior information i e a larger range for the parameter space in the maximin criterion or a larger variance of the prior in the bayesian criterion yields a larger number of support points of the corresponding optimal designs in this paper we present analytic tools which are used to prove this phenomenon in concrete situations the proposed methodology can be used to explain many empirically observed results in the literature moreover it explains why maximin d optimal designs are usually supported at more points than bayesian d optimal designs

a maximum likelihood estimation method for random coefficient regression models summary a method for estimating the distribution of the parameters of a random coefficient regression model is proposed this distribution accounting for interindividual variability is assumed to lie in a wide class of probability distributions rather than in a given parametric class estimation is based on observations from a sample of individuals and likelihood is the estimation criterion experimental designs may be different among individuals allowing the method to apply to routinely collected data the problem has strong connections with the theory of optimum design of experiments conditions are given under which the problem has a unique solution which then corresponds to a discrete distribution a simple pharmacokinetic model involving two parameters is used as an example these parameters have a bimodal distribution as statistical specification moreover only one observation is available per individual thus the method applies even when the model is not identifiable

optimal designs for the emax log linear and exponential models we derive locally d and edp optimal designs for the exponential log linear and three parameter emax models for each model the locally d and ed p optimal designs are supported at the same set of points while the corresponding weights are different this indicates that for a given model d optimal designs are efficient for estimating the smallest dose that achieves 100p of the maximum effect in the observed dose range conversely ed p optimal designs also yield good d efficiencies we illustrate the results using several examples and demonstrate that locally d and ed p optimal designs for the emax log linear and exponential models are relatively robust with respect to misspecification of the model parameters

optimal designs for longitudinal and functional data summary we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited by determining the optimal locations in time where measurements should be taken as for all optimal designs some prior information is needed to implement the optimal designs proposed we demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements where for each subject one has available a small random number of repeated measurements that are randomly located on the domain a second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data we construct optimal designs by targeting two criteria optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses again from a few optimally placed measurements the optimal designs proposed address the need for sparse data collection when planning longitudinal studies by taking advantage of the close connections between longitudinal and functional data analysis we demonstrate in simulations that the designs perform considerably better than randomly chosen design points and include a motivating data example from the baltimore longitudinal study of aging the designs are shown to have an asymptotic optimality property

complete classes of designs for nonlinear regression models and principal representations of moment spaces in a recent paper yang and stufken ann statist 40 2012a 1665 1685 gave su cient conditions for complete classes of designsfor nonlinear regression models in this note we demonstrate thatthere is an alternative way to validate this result our main argumentutilizes the fact that boundary points of moment spaces generated bychebyshev systems possess unique representations

optimal designs for dose response models with restricted design spaces in dose response studies the dose range is often restricted due to concerns over drug toxicity and or efficacy we present restricted and unrestricted interval locally optimal designs with respect to a very general class of optimality criteria for estimating the underlying dose response curve the underlying curve belongs to a diversified set of link functions suitable for the dose response studies and having a common canonical form these include the fundamental binary response models the logit and the probit as well as the skewed versions of these models the results are illustrated through the re design of a dose ranging trial conducted at the merck research laboratories zeng and zhu 1997 this work is a generalization of the results of dai and zhu 2002 in terms of the design interval the underlying dose response curve and the optimality criterion

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

information matrix and d optimal design with gaussian inputs for wiener model identification we present a closed form expression for the fischer s information matrix associated with the identification of wiener models in the derivation we assume that the input signal is gaussian the analysis allows the linear sub system in the wiener model to have a generic rational transfer function of arbitrary order it also allows the static nonlinearity of the wiener model to be a polynomial of arbitrary degree in addition we show how this analysis can be used to design tractable algorithms for d optimal input design the idea is further extended to design optimal inputs consisting of a sequence of gaussian signals with different mean values and variances by combining gaussian inputs with different means we can tune the amplitude distribution of the input to achieve the best identification accuracy in d optimal sense the analytical results are also illustrated with some numerical simulations

exact optimum coin bias in efron s randomization procedure efron s biased coin design is a restricted randomization procedure that has very favorable balancing properties yet it is fully randomized in that subjects are always randomized to one of two treatments with a probability less than 1 the parameter of interest is the bias p of the coin which can range from 0 5 to 1 in this note we propose a compound optimization strategy that selects p based on a subjected weighting of the relative importance of the two fundamental criteria of interest for restricted randomization mechanisms namely balance between the treatment assignments and allocation randomness we use exact and asymptotic distributional properties of efron s coin to find the optimal p under compound criteria involving imbalance variability expected imbalance selection bias and accidental bias for both small moderate trials and large samples

bayesian optimal design via interacting particle systems we propose a new stochastic algorithm for bayesian optimal design in nonlinear and high dimensional contexts following peter muller we solve an optimization problem by exploring the expected utility surface through markov chain monte carlo simulations the optimal design is the mode of this surface considered a probability distribution our algorithm relies on a particle method to efficiently explore high dimensional multimodal surfaces with simulated annealing to concentrate the samples near the modes we first test the method on an optimal allocation problem for which the explicit solution is available to compare its efficiency with a simpler algorithm we then apply our method to a challenging medical case study in which an optimal protocol treatment needs to be determined for this case we propose a formalization of the problem in the framework of bayesian decision theory taking into account physicians knowledge and motivations we also briefly review further improvements and alternatives

compound optimal allocation for individual and collective ethics in binary clinical trials in recent years several authors have investigated response adaptive allocation rules for comparative clinical trials in order to favour at each stage of the trial the treatment that appears to be best in this paper we define admissible allocations namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion reflecting the inferential properties of the experiment and the proportion of patients assigned to the best treatment or treatments we survey existing designs from this viewpoint we also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria with weights that depend on the actual treatment effects this compound criterion leads to a locally optimal allocation that can be targeted by some response adaptive randomization rule the paper mainly deals with the case of two treatments but the suggested methodology is shown to extend to more than two

kiefer ordering of simplex designs for second degree mixture models with four or more ingredients for mixture models on the simplex we discuss the improvement of a given design in terms of increasing symmetry as well as obtaining a larger moment matrix under the loewner ordering the two criteria together define the kiefer design ordering for the second degree mixture model we show that the set of weighted centroid designs constitutes a convex complete class for the kiefer ordering for four ingredients the class is minimal complete of essential importance for the derivation is a certain moment polytope which is studied in detail 1 introduction many practical problems are associated with the investigation of mixture ingredients of m factors assumed to influence the response only through the proportions in which they are blended together the definitive text cornell 1990 lists numerous examples and provides a thorough discussion of both theory and practice early seminal work was done by scheff e 1958 1963 in which he suggested and analyzed canonical model forms when the regression function for the expected response is apolynomia l of degree one two or three

optimal discriminating designs for several competing regression models the problem of constructing optimal designs for a class of regression models is considered we investigate a version of the tp optimality criterion as introduced by atkinson and fedorov 1975b and demonstrate that optimal designs with respect to this type of criteria can be obtained by solving nonlinear vector valued approximation problems we provide a characterization of the best approximations in this context and use these results to develop an efficient algorithm for the determination of the optimal discriminating designs the results are illustrated by fnumerical examples

hopscotch method the numerical solution of the frank kamenetskii partial differential equation numerical solutions to the frank kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme we observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term numerical solutions obtained via matlab mathematica and the crank nicolson implicit scheme are employed as a means of comparison to gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the lie group method the numerical solution is also observed to converge to a well known steady state solution a linear stability analysis is performed to validate the stability of the results obtained

k optimal design via semidefinite programming and entropy optimization in this paper we consider the problem of optimal design of experiments a two step inference strategy is proposed the first step consists in minimizing the condition number of the so called information matrix this step can be turned into a semidefinite programming problem the second step is more classical and it entails the minimization of a convex integral functional under linear constraints this step is formulated in some infinite dimensional space and is solved by means of a dual approach numerical simulations will show the relevance of our approach

design in nonlinear mixed effects models optimization using the fedorov wynn algorithm and power of the wald test for binary covariates we extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi exponential model we first show the relevance of the predicted standard errors ses given by the computation of the population fisher information matrix using the r function pfim in comparison to those computed with the stochastic approximation expectation maximization algorithm implemented in the monolix software we then highlight the usefulness of the fedorov wynn fw algorithm for designs optimization compared to the simplex algorithm from the predicted se of pfim we compute the predicted power of the wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power using the fw algorithm we investigate the influence of the design on the power and show that for optimized designs with the same total number of samples the power increases when the number of subjects increases and the number of samples per subject decreases a simulation study is also performed with the nlme function of r to confirm this result and show the relevance of the predicted powers compared to those observed by simulation

multi objective optimal designs in comparative clinical trials with covariates the reinforced doubly adaptive biased coin design the present paper deals with the problem of allocating patients to two competing treatments in the presence of covariates or prognostic factors in order to achieve a good trade off among ethical concerns inferential precision and randomness in the treatment allocations in particular we suggest a multipurpose design methodology that combines efficiency and ethical gain when the linear homoscedastic model with both treatment covariate interactions and interactions among covariates is adopted the ensuing compound optimal allocations of the treatments depend on the covariates and their distribution on the population of interest as well as on the unknown parameters of the model therefore we introduce the reinforced doubly adaptive biased coin design namely a general class of covariate adjusted response adaptive procedures that includes both continuous and discontinuous randomization functions aimed to target any desired allocation proportion the properties of this proposal are described both theoretically and through simulations

monotonic convergence of a general algorithm for computing optimal designs 1 a general class of algorithms optimal experimental design approximate theory is a well developed area and we refer to kiefer 1974 silvey 1980 p zm n 1986 and pukelsheim 1993 for a general introduction and basic results we consider computational aspects of optimal designs focusing on a finite design space x x xn suppose the probability density or mass function of the response is specified as p y x where 0i 9m t is the parameter of interest let a denote the m m expected fisher information matrix from a unit assigned to x with the j k entry the expectation is with respect to p y x

exact and optimum tests in unbalanced split plot designs under mixed and random models abstract unbalanced split plot designs are considered where the whole plot treatments could be replicated an unequal number of times in the design but each split plot treatment occurs exactly once in every whole plot analysis of such designs is provided when the various effects could be fixed or random in each case valid exact tests and optimum invariant tests whenever they exist are derived for testing the significance of the various effects in some cases optimum tests exist only under additional conditions on the design and these conditions are specified for the various setups

optimal discrimination designs we consider the problem of constructing optimal designs for model discrimination between competing regression models various new properties of optimal designs with respect to the popular t optimality criterion are derived which in many circumstances allow an explicit determination of t optimal designs it is also demonstrated that in nested linear models the number of support points of t optimal designs is usually too small to estimate all parameters in the extended model in many cases t optimal designs are usually not unique and we give a characterization of all t optimal designs finally t optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study

optimum experimental designs for properties of a compartmental model summary three properties of interest in bioavailability studies using compartmental models are the area under the concentration curve the maximum concentration and the time to maximum concentration methods are described for finding designs that minimize the variance of the estimates of these quantities in such a model these methods use prior information both prior estimates and prior distributions are used the designs for an open one compartment model are compared with the corresponding d optimum design for all parameters and also with designs that minimize the sum of the scaled variances of the individual properties

locally optimal designs for errors in variables models we consider the construction of optimal designs for nonlinear regression models when there are measurement errors in the covariates corresponding approximate design theory is developed for maximum likelihood and least squares estimation with the latter leading to nonconcave optimization problems analytical characterizations of the locally d optimal saturated designs are provided for the michaelis menten e rm max and exponential regression models through concrete applications we illustrate how measurement errors in the covariates affect the optimal choice of design and show that the locally d optimal saturated designs are highly efficient for relatively small misspecifications of the parameter values

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

efficient algorithms for optimal designs with correlated observations in pharmacokinetics and dose finding studies summary random effects models are widely used in population pharmacokinetics and dose finding studies however when more than one observation is taken per patient the presence of correlated observations due to shared random effects and possibly residual serial correlation usually makes the explicit determination of optimal designs difficult in this article we introduce a class of multiplicative algorithms to be able to handle correlated data and thus allow numerical calculation of optimal experimental designs in such situations in particular we demonstrate its application in a concrete example of a crossover dose finding trial as well as in a typical population pharmacokinetics example additionally we derive a lower bound for the efficiency of any given design in this context which allows us on the one hand to monitor the progress of the algorithm and on the other hand to investigate the efficiency of a given design without knowing the optimal one finally we extend the methodology such that it can be used to determine optimal designs if there exist some requirements regarding the minimal number of treatments for several in some cases all experimental conditions

optimal designs for the rasch model in this paper optimal designs will be derived for estimating the ability parameters of the rasch model when difficulty parameters are known it is well established that a design is locally d optimal if the ability and difficulty coincide but locally optimal designs require that the ability parameters to be estimated are known to attenuate this very restrictive assumption prior knowledge on the ability parameter may be incorporated within a bayesian approach several symmetric weight distributions e g uniform normal and logistic distributions will be considered furthermore maximin efficient designs are developed where the minimal efficiency is maximized over a specified range of ability parameters

on the first degree fejer riesz factorization and its applications to x a x 1 a q abstract given a laurent polynomial with matrix coefficients that is positive semi definite over the unit circle in the complex plane the fejer riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint this paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation x a x 1 a q in particular the nonlinear equation can be recast as a linear sylvester equation subject to unitary constraint the sylvester equation is readily obtainable from hermitian eigenvalue computation the unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate free newton iteration for high precision calculation this approach offers a complete parametrization of all solutions and in contrast to most existent algorithms makes it possible to find all solutions if so desired

a new method of matrix spectral factorization a new algorithm of matrix spectral factorization is proposed which can be applied to compute an approximate spectral factor of any positive definite matrix function which satisfies the paley wiener condition

a new approach to optimal design for linear models with correlated observations we consider the problem of designing experiments for regression in the presence of correlated observations with the location model as the main example for a fixed correlation structure approximate optimal designs are determined explicitly and it is demonstrated that under the model assumptions made by bickel and herzberg 1979 for the determination of asymptotic optimal design the designs derived in this article converge weakly to the measures obtained by these authors we also compare the asymptotic optimal design concepts of sacks and ylvisaker 1966 1968 and bickel and herzberg 1979 and point out some inconsistencies of the latter finally we combine the best features of both concepts to develop a new approach for the design of experiments for correlated observations and it is demonstrated that the resulting design problems are related to the logarithmic potential theory

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

optimal designs for dose response curves with common parameters a common problem in phase ii clinical trials is the comparison of dose response curves corresponding to different treatment groups if the effect of the dose level is described by parametric regression models and the treatments differ in the administration frequency but not in the sort of drug a reasonable assumption is that the regression models for the different treatments share common parameters this paper develops optimal design theory for the comparison of different regression models with common parameters we derive upper bounds on the number of support points of admissible designs and explicit expressions for d optimal designs are derived for frequently used dose response models with a common location parameter if the location and scale parameter in the different models coincide minimally supported designs are determined and sufficient conditions for their optimality in the class of all designs derived the results are illustrated in a dose finding study comparing monthly and weekly administration

robust t optimal discriminating designs this paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the t optimality criterion introduced by atkinson and fedorov 1975a t optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification as a solution of this problem we propose a bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the t optimality criterion it is shown that the corresponding bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria for the problem of discriminating between two polynomial regression models which differ in the degree by two the robust t optimal discriminating designs can be found explicitly the results are illustrated in several examples ams subject classification 62k05