on the equivalence of constrained and compound optimal designs abstract constrained and compound optimal designs represent two well known methods for dealing with multiple objectives in optimal design as reflected by two functionals 1 and 2 on the space of information matrices a constrained optimal design is constructed by optimizing 2 subject to a constraint on 1 and a compound design is found by optimizing a weighted average of the functionals 1 1 2 0 1 we show that these two approaches to handling multiple objectives are equivalent

model robust test plans with applications in accelerated life testing we study a criterion based on a weighted asymptotic sample ratio asr that yields model robust test plans with relatively low asrs under different models we apply the criterion to accelerated life test planning in which interest lies in estimating life quantiles and there is uncertainty as to whether the weibull or the lognormal model best describes the lifetime distribution we use the criterion to determine optimum and best 4 2 1 allocation plans we compare the asrs of our test plans with the optimum and best 4 2 1 allocation test plans when the model is specified incorrectly test plans are evaluated in terms of asrs expected numbers of failures and extrapolation to use stress we illustrate the criterion in a practical situation

generalized simulated annealing for function optimization a generalized simulated annealing method has been developed and applied to the optimization of functions possibly constrained having many local extrema the method is illustrated in some difftcult pedagogical examples and used to solve a problem analyzed by bates technometrics 25 pp 373 376 1983 for which we identify an improved optimum the sensitivity of the solution to changes in the constraints and in other specifications of the problem is analyzed and discussed

generalized pointwise bias error bounds for response surface approximations this paper proposes a generalized pointwise bias error bounds estimation method for polynomial based response surface approximations when bias errors are substantial a relaxation parameter is introduced to account for inconsistencies between the data and the assumed true model the method is demonstrated with a polynomial example where the model is a quadratic polynomial while the true function is assumed to be cubic polynomial the effect of relaxation parameter is studied it is demonstrated that when bias errors dominate the bias error bounds characterize the actual error field better than the standard error the bias error bound estimates also help to identify regions in the design space where the accuracy of the response surface approximations is inadequate it is demonstrated that this information can be utilized for adaptive sampling in order to improve accuracy in such regions copyright 2005 john wiley sons ltd

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

on the performance of tests by wu and by hausman for detecting the ordinary least squares bias problem abstract we first consider the performance of the wu 1973 hausman 1978 w h specification error test as a test for the existence of ordinary least squares ols bias we discuss power properties of the test under alternative null hypotheses one of which has not previously been considered we next consider how the w h test performs as an indicator of the extent rather than the existence of an ols bias problem since this usage of the test seems common in applied studies finally monte carlo methods are used to evaluate wu s two step estimation procedure involving the w h test as a pretest

experimental designs for model discrimination abstract we present designs that perform well for several objectives simultaneously three different approaches are discussed to augment a given design in an optimal way to evaluate a mixture of the various criteria and to optimize one objective subject to achieving a prescribed efficiency level for the others our sample designs are for the situation of discriminating between a second and third degree polynomial fit under the d criterion and geometric mixtures of d criteria

limited information estimators and exogeneity tests for simultaneous probit models a two step maximum likelihood procedure is proposed for estimating simultaneous probit models and is compared to alternative limited information estimators conditions under which each estimator attains the cramer rao lower bound are obtained simple tests for exogeneity based on the new two step estimator are proposed and are shown to be asymptotically equivalent to one another and to have the same local asymptotic power as classical tests based on the limited information maximum likelihood estimator finite sample comparisons between the new and alternative estimators are presented based on some monte carlo evidence the performance of the proposed tests for exogeneity is also assessed

simulation designs for quadratic response surface models in the presence of model misspecification this article considers the selection of experimental designs for the estimation of second order response surface metamodels in a simulation environment rather than construct designs based on the premise that the postulated model exactly represents the simulated response as is the case in optimal design theory we assume that the estimation process may be biased by the presence of unfitted third order terms we therefore seek to specify experimental plans that address the bias due to possible model misspecification as well as traditional variance considerations the performance measure used for this fit protect scenario is box and draper s design criterion of average mean squared error of predicted response four important classes of response surface experimental plans are examined 1 central composite designs 2 box behnken plans 3 three level factorial experiments and 4 small central composite designs each design class is studied in conjunction with three pseudorandom number assignment str

integer valued minimax robust designs for estimation and extrapolation in heteroscedastic approximately linear models abstract we present our findings on a new approach to robust regression design this approach differs from previous investigations into this area in three respects the use of a finite design space the use of simulated annealing to carry out the numerical minimization problems and in our search for integer valued rather than continuous designs we present designs for the situation in which the response is thought to be approximately polynomial we also discuss the cases of approximate first and second order multiple regression in each case we allow for possible heteroscedasticity and also obtain minimax regression weights the results are extended to cover extrapolation of the regression response to regions outside of the design space a case study involving dose response experimentation is undertaken the optimal robust designs which protect against bias as well as variance can be roughly described as being obtained from the classical variance minimizing designs by replacing replicates with clust

graphical tools for assessing the sensitivity of response surface designs to model misspecification when comparing different designs for an experiment optimality criteria and other measures often depend on the correctness of the assumed model in this article we develop and illustrate an approach for comparing designs given the potential effect of bias due to an underspecified model we illustrate this approach using graphical summaries of the expected mean squared error emse that allow assessment of the robustness of designs to model misspecification for response surface designs in cuboidal regions the excellent performance of the central composite designs when the quadratic model is correct is tempered when bias is present the box behnken designs demonstrate superiority in the presence of missing cubic terms for a screening design scenario the amount of bias present leads to different conclusions as to which design is best

efficient designs with minimal aliasing for some experimenters a disadvantage of the standard optimal design approach is that it does not consider explicitly the aliasing of specified model terms with terms that are potentially important but are not included in the model for example when constructing an optimal design for a first order model aliasing of main effects and interactions is not considered this can lead to designs that are optimal for estimation of the primary effects of interest yet have undesirable aliasing structures in this article we construct exact designs that minimize the squared norm of the alias matrix subject to constraints on design efficiency we demonstrate use of the method for the construction of screening and response surface designs

response surface approximations for structural optimization response surface methodology can be used to construct global and midrange approximations to functions in structural optimization since structural optimization requires expensive function evaluations it is important to construct accurate function approximations so that rapid convergence may be achieved in this paper techniques to find the region of interest containing the optimal design and techniques for finding more accurate approximations are reviewed and investigated aspects considered are experimental design techniques the selection of the best regression equation intermediate response functions and the location and size of the region of interest standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size while the selection of a larger number of points is not necessarily cost effective in a further attempt to improve efficiency different regression models were investigated the results indicate that the use of the two methods investigated does not significantly improve the results finding an accurate global approximation is challenging and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space 1998 john wiley sons ltd

minimax robust designs and weights for approximately specified regression models with heteroscedastic errors abstract this article addresses the problem of constructing designs for regression models in the presence of both possible heteroscedasticity and an approximately and possibly incorrectly specified response function working with very general models for both types of departure from the classical assumptions i exhibit minimax designs and correspondingly optimal weights simulation studies and a case study accompanying the theoretical results lead to the conclusions that the robust designs yield substantial gains over some common competitors in the presence of realistic departures that are sufficiently mild so as to be generally undetectable by common test procedures specifically i exhibit solutions to the following problems p1 for ordinary least squares determine a design to minimize the maximum value of the integrated mean squared error imse of the fitted values with the maximum being evaluated over both types of departure p2 for weighted least squares determine both weights and a design to m

constrained d and d_1 optimal designs for polynomial regression in the common polynomial regression model of degree m we consider the problem of determining the d and d 1 optimal designs subject to certain constraints for the d 1 efficiencies in the models of degree m j m j 1 m k m j 0 k 0 given we present a complete solution of these problems which on the one hand allow a fast computation of the constrained optimal designs and on the other hand give an answer to the question of the existence of a design satisfying all constraints our approach is based on a combination of general equivalence theory with the theory of canonical moments in the case of equal bounds for the d 1 efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials

a connection between uniformity and aberration in regular fractions of two level factorials summary we show a link between the two apparently unrelated areas of uniformity and minimum aberration with reference to regular fractions of two level factorials we derive an expression for the centred l2 discrepancy measure for uniformity in terms of the word length pattern this result indicates in particular excellent behaviour of minimum aberration designs with regard to uniformity and provides further justification for the popular criterion of minimum aberration

robust regression designs when the design space consists of finitely many points we consider the nearly linear regression problem when the assumed first degree model is contaminated by some smal l constant and the design space consists of finitely many points symmetrically distributed on the interval 2 2 under the usual squared loss function for the estimation of the slope and the intercept and with the use of the least squares estimators the problem is to find the designs which are optimal in the sense of minimizing the maximum risk among symmetric designs the results turn out to be quite different from those obtained by li and notz 1981 in a setup that is similar except that the design space is the whole interval 1 2 1 2 in many cases the optimal solution has a support containing more than two points

the application of the annealing algorithm to the construction of exact optimal designs for linear regression models this article reports the application of the annealing algorithm to the construction of exact d i and g optimal designs for polynomial regression of degree 5 on the interval 1 l and for the second order model in two factors on the design space 1 l 1 1 details of the perturbation scheme and the annealing schedules used are given and the method of implementation is illustrated by means of a simple example the algorithm is assessed by comparing its performance in terms of computer time and effkiency with the modified fedorov procedure and it is shown to be particularly effective in finding g optimal designs the salient features of the exact designs constructed in this study are also summarized

some theory for constructing minimum aberration fractional factorial designs summary minimum aberration is the most established criterion for selecting a regular fractional factorial design of maximum resolution minimum aberration designs for n runs and n 2 m n factors have previously been constructed using the novel idea of complementary designs in this paper an alternative method of construction is developed by relating the wordlength pattern of designs to the so called confounding between experimental runs this allows minimum aberration designs to be constructed for n runs and 5n 16 m n 2 factors as well as for n 2 m n

near minimal matrix profiles and wavefronts for testing nodal resequencing algorithms the speeds of algorithms that are specifically designed to solve sparse matrix equations depend on the ordering of the unknowns because it is difficult to know what a good ordering is many resequencing algorithms have been developed to reorder the equations in a manner that minimizes the execution time of the solver being used there is no theoretical way of evaluating resequencing algorithms but four widely used algorithms cuthill mckee gibbs poole stockmeyer levy gibbs king have been compared with one another on the basis of their performance on a set of benchmark test problems this paper reports what we believe to be are minimal or near minimal matrix profiles and wavefronts for the benchmark problems comparisons of the minimal results with those produced by the widely used resequencing algorithms show that they produce profiles typically a few tens of per cent greater than minimal but 50 per cent to 100 per cent greater on two problem types the algorithm that produced the near minimal results used a simulated annealing technique and is far too slow for general use

model robust two level designs using coordinate exchange algorithms and a maximin criterion we propose a candidate list free exchange algorithm that facilitates construction of exact model robust two level experiment designs in particular we investigate two model spaces previously considered in the literature the first assumes that all main effects and an unknown subset of two factor interactions are active but that the experimenter knows the number of active interactions the second assumes that an unknown subset of the main effects and all associated two factor interactions are active previous literature uses two criteria for design construction first maximize the number of estimable models then differentiate between designs equivalent in estimability by choosing the design with the highest average efficiency we adopt a similar strategy but 1 do not impose orthogonality or factor level balance constraints resulting in generally equal or larger numbers of estimable models and 2 use a flexible secondary criterion that maximizes the minimum efficiency we provide results fo

k circulant supersaturated designs a class of supersaturated designs called k circulant designs is explored these designs are constructed from cyclic generators by cycling k elements at a time the class of designs includes many es2 optimal designs some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity generators for the most efficient designs are listed and projection properties of some of the designs are explored we also illustrate that some k circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions

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

pitfalls of using a single criterion for selecting experimental designs for surrogate construction a good experimental design ed is essential to simultaneously reduce the effect of noise and bias errors however most eds cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria we use multiple criteria to assess the performance of different popular eds we demonstrate that these eds offer different trade offs and that use of a single criterion is indeed risky in addition we show that popular eds such as latin hypercube sampling lhs and d optimal designs often leave large regions of the design space unsampled even for moderate dimensions we discuss a few possible strategies to combine multiple criteria and illustrate them with examples we show that complementary criteria e g bias handling criterion for variance based designs and vice versa can be combined to improve the performance of eds we demonstrate improvements in the trade offs between noise and bias error by combining a model based criterion like the d optimality criterion and a geometry based criterion like lhs next we demonstrate that selecting an ed from three candidate eds using a suitable error based criterion helped eliminate potentially poor designs finally we show benefits from combining the multiple criteria based strategies that is generation of multiple eds using the d optimality and lhs criteria and selecting one design using a pointwise bias error criterion the encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail copyright 2007 john wiley sons ltd

a graphical method for finding maximin efficiency designs we consider the problem of designing an experiment when there are two competing optimality criteria designs that maximize the minimum efficiencies under the two criteria are proposed along with a graphical method for finding these maximin designs

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

robust designs for polynomial regression by maximizing a minimum of d and d1 efficiencies in the common polynomial regression of degree m we determine the design which maximizes the minimum of the d efficiency in the model of degree m and the d 1 efficiencies in the models of degree m j m k j k 0 given the resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness of fit of the assumed model of degree m by testing the highest coefficients in the polynomials of degree m j m k our approach is based on a combination of the theory of canonical moments and general equivalence theory for minimax optimality criteria the optimal designs can be explicitly characterized by evaluating certain associated orthogonal polynomials

model robust linear optimal designs motivated by a pioblem concerning the estimation of uranium content in calibration standards a new design criterion is developed that generalizes linear optimality to a situation in which a priori the exact form of the regression model need not be known an equivalence theorem is given some properties of the new designs are delineated and iterative methods for design construction are presented finally by using methods developed within optimal designs are computed in a number of situations including that of the motivating example

another look at first order saturated designs the p efficient designs the problem of constructing first order saturated designs that are optimal in some sense has received a great deal of attention in the literature since these saturated designs are frequently used in screening situations the focus will be on the potential projective models rather than the full model this article discusses some practical concerns in choosing a design and presents some first order saturated designs having two desirable properties near equal occurrence and near orthogonality these saturated designs are shown to be reasonably efficient for estimating the parameters of projective submodels and thus are called p efficient designs comparisons with the efficiency of d optimal designs are given for designs for all n from 3 to 30

optimal designs for identifying the degree of a polynomial regression if an experimenter wants to determine the degree of a polynomial regression on the basis of a sample of observations anderson showed that the following method is optimal starting with the highest specified degree the procedure is to test in sequence whether the coefficients are 0 in this paper optimal designs for anderson s procedure are determined explicitly the optimal design maximizes the minimum power of a given set of alternatives

a mean squared error criterion for the design of experiments in estimating a response surface over a design region of interest mean squared error can arise both from sampling variance and bias introduced by model inadequacy the criterion adopted here for experimental design attempts to protect against bias resulting from a large class of deviations from the assumed model two algorithms are proposed for the construction of either approximate or exact designs and an example is given

some robust type d optimal designs in polynomial regression abstract consider a polynomial regression situation on an interval and also a robustness type formulation of stigler s a technique involving canonical moments is discussed and some explicit solutions are given

approximate model spaces for model robust experiment design optimal designs depend upon a prespecified model form a popular and effective model robust alternative is to design with respect to a set of models instead of just one however model spaces associated with experiments of interest are often prohibitively large and so algorithmically generated designs are infeasible here we present a simple method that largely eliminates this problem by choosing a small set of models that approximates the full set and finding designs that are explicitly robust for this small set we build our procedure on a restricted columnwise pairwise algorithm and explore its effectiveness for two model spaces in the literature for smaller full model spaces we find that the designs constructed with the new method compare favorably with robust designs that use the full model space with construction times reduced by orders of magnitude we also construct designs that heretofore have been unobtainable due to the size of their model spaces supplementary material available online

geometric isomorphism and minimum aberration for factorial designs with quantitative factors factorial designs have broad applications in agricultural engineering and scientific studies in constructing and studying properties of factorial designs traditional design theory treats all factors as nominal however this is not appropriate for experiments that involve quantitative factors for designs with quantitative factors level permutation of one or more factors in a design matrix could result in different geometric structures and thus different design properties in this paper indicator functions are introduced to represent factorial designs a polynomial form of indicator functions is used to characterize the geometric structure of those designs geometric isomorphism is defined for classifying designs with quantitative factors based on indicator functions a new aberration criteria is proposed and some minimum aberration designs are presented

nonregular designs with desirable projection properties in many industrial applications the experimenter is interested in estimating some of the main effects and two factor interactions in this article we rank two level orthogonal designs based on the number of estimable models containing a subset of main effects and their associated two factor interactions by ranking designs in this way the experimenter can directly assess the usefulness of the experimental plan for the purpose in mind we apply the new ranking criterion to the class of all non isomorphic two level orthogonal designs with 16 and 20 runs and introduce a computationally efficient algorithm based on two theoretical results which will aid in finding designs with larger run sizes catalogs of useful designs with 16 20 24 and 28 runs are presented

some guidelines for constructing exact d optimal designs on convex design spaces some problems unique to the construction of n point d optimal designs on convex design spaces are considered multiple point augmentation and exchange algorithms are shown to be more costly and less efficient than the analogous single point procedures moreover some recommendations for improving single point methods are given finally empirical evidence is found that supports the galil and kiefer option for constructing initial designs and powell s optimization method for design augmentation

minimum aberration and model robustness for two level fractional factorial designs summary the performance of minimum aberration two level fractional factorial designs is studied under two criteria of model robustness simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived it is also shown that a minimum aberration design of resolution ilil or higher maximizes the number of two factor interactions which are not aliases of main effects and subject to that condition minimizes the sum of squares of the sizes of alias sets of two factor interactions this roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform it follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here examples are given to show that minimum aberration designs are indeed highly efficient

generalized minimum aberration for asymmetrical fractional factorial designs by studying treatment contrasts and anova models we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs the criterion is independent of the choice of treatment contrasts and thus model free it works for symmetrical and asymmetrical designs regular and nonregular designs in particular it reduces to the minimum aberration criterion for regular designs and the minimum g 2 aberration criterion for two level nonregular designs in addition by exploring the connection between factorial design theory and coding theory we develop a complementary design theory for general symmetrical designs which covers many existing results as special cases

score tests for zero covariances in recursive linear models for grouped or censored data abstract the score test statistic for testing whether an error covariance is zero is derived for a normal linear recursive model for fully observed censored or grouped data the test which is obtained by regarding non zero error covariances as arising from correlated random parameter variation is shown to be closely related to the information matrix test it turns out that the statistic which is asymptotically n 0 1 under the null examines the sample covariance of appropriately defined residuals

optimal designs for a class of polynomials of odd or even degree prior y y1 v y d where the numbers yj correspond to the models of degree 2j 1 2j for j 1 r for a special class of priors optimal designs of a very simple structure are calculated generalizing the d1 optimal design for polynomial regression of degree 2r 1 2r the support of these designs splits up in three sets and the masses of the optimal design at the support points of every set are all equal the results are derived in a general context using the theory of canonical moments and continued fractions some applications are given to the d optimal design problem for polynomial regression with vanishing coefficients of odd or even powers

a general criterion for factorial designs under model uncertainty motivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design we show that the qb criterion which aims to improve the estimation in as many models as possible by incorporating experimenters prior knowledge along with an approximation to the as criterion is more general and has a better statistical interpretation than many standard criteria the generalization and application of the criterion to different types of designs are presented the relationships between qb and other criteria for different situations are explored it is shown that the e s2 criterion is a special case of qb and several aberration type criteria are limiting cases of our criterion so that qb provides a bridge between alphabetic optimality and aberration the two case studies illustrate the potential benefits of the qb criterion r programs for calculating qb are available online as supplemental materials

basis of an improved hybrid node renumbering algorithm for matrix bandwidth reduction abstract a new node renumbering algorithm for the reduction of matrix bandwidth is presented the algorithm is a hybrid of existing algorithms and original methods this new hybrid algorithm is compared against the commonly used gibbs poole stockmeyer technique gps and the direct reduction hybrid algorithm presented by armstrong nsas using example problems compiled by everstine the result showed a net overall improvement against these two highly efficient reduction schemes in fact two of the test cases were observed to exhibit the smallest bandwidth ever obtained and six other cases tied for the best yet bandwidths found

efficient rounding of approximate designs summary discretization methods to round an approximate design into an exact design for a given sample size n are discussed there is a unique method called efficient rounding which has the smallest loss of efficiency under a wide family of optimality criteria the efficient rounding method is a multiplier method of apportionment which otherwise is known as the method of john quincy adams or the method of smallest divisors

simple and effective bandwidth reduction algorithm an automatic algorithm is presented for reducing the bandwidth of symmetric interconnectivity matrices the technique takes an arbitrary input scheme and reorders the nodes so as to reduce the bandwidth a large number of examples is presented to illustrate the reliability and cost effectiveness of the method

two level designs to estimate all main effects and two factor interactions we study the design of two level experiments with n runs and n factors large enough to estimate the interaction model which contains all the main effects and all the two factor interactions yet an effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two factor interactions orthogonal arrays oas favor main effect estimation however complete enumeration becomes infeasible for cases relevant for practitioners we develop a partial enumeration procedure for these cases and we establish upper bounds on the d efficiency for the interaction model based on arrays that have not been generated by the partial enumeration we also propose an optimal design procedure that favors main effect estimation designs created with this procedure have smaller d efficiencies for the interaction model than d optimal designs but standard errors for the main effects in this model are improved generated oas for 7 10 factors and 32 72 runs are smaller or have a higher d efficiency than the smallest oas from the literature designs obtained with the new optimal design procedure or strength 3 oas which have main effects that are not correlated with two factor interactions are recommended if main effects unbiased by possible two factor interactions are of primary interest d optimal designs are recommended if interactions are of primary interest supplementary materials for this article are available online 2017 american statistical association and the american society for quality

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

the weight functions generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the jacobi and the gegenbauer polynomials abstract in the introduction the main object of the paper namely the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the jacoby and the gegenbauer polynomials is briefly discussed section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind section 3 which deals with the application to the case of the jacobi polynomials p n z n n starts with the construction of the orthogonality relation and the weight function 1 of the associated polynomials for arbitrary gt 1 the denominator of the fraction representing 1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use attention is also paid to the associated functions of the second kind and some of their properties next in a subsection 1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of pairs when 1 1 n gt 1 1 1 gt0 and 1 0 gt 1 in another subsection the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas in the final subsection the moments of the weight function 1 with respect to 0 1 and 1 are expressed in terms of the corresponding moments of the weight function of the jacobi polynomials both in finite determinantal form and by means of a recurrence relation in section 4 which concerns the application to the case of the gegenbauer polynomials c v n z n n the advantages of a new definition of the associated polynomials of the second kind are at first expounded then follow the orthogonality relation and the weight function 1 for arbitrary v gt 1 2 next 1 is calculated in terms of elementary functions for all integer and half odd integer values of v exceeding 1 2 in the former case a remarkable simplification occurs in the denominator of the fraction representing 1 after that follow three subsections devoted respectively to the functions of the second kind associated with the gegenbauer polynomials two different generating functions for the associated polynomials and related formulas and the moments of the weight function 1 the paper ends with two appendices in which certain types of sums are reduced to a one term expression

pointwise bias error bounds and min max design for response surface approximations bounds on response surface approximation errors as a result of model inadequacy bias error are presented and a design of experiments minimizing the maximal bias error is proposed the bias error bounds are considered as a tool to identify locations in the design space where the accuracy of the approximation fitted on a given design of experiments might be poor two approaches to characterize the bias error assume that the functional form of the true model is known and seek at each point in design space worst case bounds on the absolute error the first approach is implemented before data generation this data independent error bound can easily be implemented in a search for a design of experiments that minimize the bias error bound as it requires very little computation the second approach is to be used posterior to the data generation and provides tightened error bound consistent with the data this data dependent error bound requires the solution of two linear programming problems at each point the data independent error bound for design of experiments of two variable examples is demonstrated randomly generated polynomials in two variables are then used to validate the data dependent bias error bound distribution although the two approaches are used in conjunction in the given examples the data independent error bound design of experiment is not a prerequisite for the application of the data dependent error bounds in search for the high bias error regions

the coordinate exchange algorithm for constructing exact optimal experimental designs we describe the cyclic coordinate exchange algorithm for constructing d optimal and linear optimal experimental designs the algorithm uses a variant of the gauss southwell cyclic coordinate descent algorithm within the k exchange algorithm to achieve substantive reductions in required computing among its advantages are the following candidate sets which grow exponentially in the number of factors need not be explicitly constructed or enumerated convex design spaces or mixed convex by discrete design spaces are handled directly without the need for sophisticated nonlinear programming routines or candidate set adjustment for design problems having 10 or more factors the reductions in execution time are typically two or more orders of magnitude when compared to standard candidateset based procedures such as k exchange yet the designs produced exhibit no loss of efficiency

model robust response surface designs scaling two level factorials summary response surface methods use simple graduating functions to study the relationship between an experimental response variable and a set of continuous explanatory variables in designing a response surface study an experimenter must decide how far apart to set the levels of each factor i e how to scale the design a good choice should be sensitive to the fact that the graduating function is only an approximation to the true response function a bayesian model is proposed that makes explicit assumptions about the inadequacy of an assumed model and a design criterion based on the model leads to reasonable choices of scale for two level factorial designs the choice of scale is found to be insensitive to the prior distributions in the model

robust minimax designs for multiple linear regression abstract we establish an extension to the case of multiple regression of a result on minimax simple regression designs due to p huber designs are found which are minimax with respect to integrated mean squared error as the true response function varies over an l 2 neighbourhood of 1 a p dimensional plane or 2 a bivariate surface with possible interactions between the regressors

accelerated life test plans robust to misspecification of the stress life relationship this article discusses methodology for deriving the asymptotic distribution of maximum likelihood estimators of model parameters in accelerated life tests alts when the form of the stress life relationship is misspecified if the practitioner can describe possible departures from an assumed alt model often linear in the appropriately transformed accelerating variable then the results here can be used to obtain alt plans that provide protection against potential bias and loss in estimation efficiency general results are derived for log location scale distributions and time censored data whereas specific results are presented for the lognormal and weibull distributions in a practical alt situation the methodology is applied to derive robust test plans based on minimizing asymptotic bias asymptotic variance or asymptotic mean squared error test plans are evaluated in terms of bias efficiency extrapolation to use level and expected number failing

projective three level main effects designs robust to model uncertainty this paper is concerned with designing experiments under the assumption that an analysis strategy is used that considers interactions in addition to main effects a criterion which averages an approximation to as efficiency over lower dimensional projections of the design is introduced to compare designs a columnwise design procedure is used to construct three level designs for six factors in 18 runs the projection efficiencies of these designs are explored and compared with designs obtained from the l 18 orthogonal array results are also given for designs with 14 and 17 runs

bayesian inspired minimum aberration two and four level designs summary motivated by a bayesian framework we propose a new minimum aberration type criterion for designing experiments with two and four level factors the bayesian approach helps in overcoming the ad hoc nature of effect ordering in the existing minimum aberration type criteria the approach is also capable of distinguishing between qualitative and quantitative factors numerous examples are given to demonstrate its advantages

optimal supersaturated designs we consider screening experiments where an investigator wishes to study many factors using fewer observations our focus is on experiments with two level factors and a main effects model with intercept since the number of parameters is larger than the number of observations traditional methods of inference and design are unavailable in 1959 box suggested the use of supersaturated designs and in 1962 booth and cox introduced measures for efficiency of these designs including e s2 which is the average of squares of the off diagonal entries of the information matrix ignoring the intercept for a design to be e s2 optimal the main effect of every factor must be orthogonal to the intercept factors are balanced and among all designs that satisfy this condition it should minimize e s2 this is a natural approach since it identifies the most nearly orthogonal design and orthogonal designs enjoy many desirable properties including efficient parameter estimation factor balance in an e s2 optimal d

the equivalence of constrained and weighted designs in multiple objective design problems abstract several competing objectives may be relevant in the design of an experiment the competing objectives may not be easy to characterize in a single optimality criterion one approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria an alternative approach has been to optimize one criterion subject to constraints on the other criteria an equivalence theorem is presented for the bayesian constrained design problem equivalence theorems are essential in verifying optimality of proposed designs especially when as in most nonlinear design problems numerical optimization is required this theorem is used to show that the results of cook and wong on the equivalence of the weighted and constrained problems apply much more generally the results are applied to bayesian nonlinear design problems with several objectives

a comparison of sequential and non sequential designs for discrimination between nested regression models summary classical regression analysis is usually performed in two steps in a first step an appro priate model is identified to describe the data generating process and in a second step statistical inference is performed in the identified model in this paper we investigate a sequential and a non sequential design strategy which take into account these different goals of the analysis for a class of nested models it is demonstrated that non sequential designs usually identify the correct model with a higher probability than sequential methods although non sequential designs can never be guaranteed to achieve the best possible efficiency in the correct model it is demonstrated by means of a simulation study that for realistic sample sizes the efficiencies of the non sequential designs for the estimation of the parameters in the correct model are at least as high as the corresponding efficiencies of the sequential methods

robust optimal extrapolation designs summary we study robustness properties of optimal extrapolation designs at a single point under various model assumptions a curious result is the efficiency of the optimal extrapolation design for a polynomial of degree m is the same whether the true underlying model is a polynomial of degree k or m k k 1 2 m 1 in addition the loss in efficiency of the optimal extrapolation designs is between 40 50 when we overestimate the degree of the polynomial model and the extrapolated point is far from the design space we propose a new class of optimality criteria for extrapolation the new optimal extrapolation designs are shown to be more efficient and robust to the regression functions they enjoy good power for discriminating among polynomial models and if the extrapolated point is sufficiently far from the design space they coincide with the optimal discrimination designs studied in dette 1994

a general theory of minimum aberration and its applications minimum aberration is an increasingly popular criterion for comparing and assessing fractional factorial designs and few would question its importance and usefulness nowadays in the past decade or so a great deal of work has been done on minimum aberration and its various extensions this paper develops a general theory of minimum aberration based on a sound statistical principle our theory provides a unified framework for minimum aberration and further extends the existing work in the area more importantly the theory offers a systematic method that enables experimenters to derive their own aberration criteria our general theory also brings together two seemingly separate research areas one on minimum aberration designs and the other on designs with requirement sets to facilitate the design construction we develop a complementary design theory for quite a general class of aberration criteria as an immediate application we present some construction results on a weak version of this class of criteria

principal representations and canonical moment sequences for distributions on an interval abstract an initial expository introduction to moment space geometry and canonical moment sequences for distributions on the unit interval leads to the exposure of previously unknown but striking symmetries in stieltjes transforms for these distributions when the transforms are viewed as moment functions the n th and the n 1 st canonical moments corresponding to a moment point interior to the space of the first n moments are shown to depend in a very simple way on the canonical intervals determined by the two principal representations of the moment point for example the n th canonical moment is just the sum of the lengths of the upper canonical intervals

exogeneity tests in a truncated structural equation abstract two bayesian and one non bayesian test statistics are derived to test the hypothesis that stochastic regressors in a truncated structural equation are independent of the error term the truncation is caused by an indicator function sampling experiments are conducted to examine sizes and powers of the test statistics and to compare them with those of the mroz test the tests are also applied to a labor supply equation for married women to see whether the wage variable is exogenous the michigan panel study of income dynamics data for 1976 are used