Experimental designs when there are one or more factor constraints

In response surface methodology, designs of orders one or two are often needed such that some or all the factor levels satisfy one or more linear constraints. A method is discussed for obtaining such designs by projection of a standard design onto the constraint hyperplane. It is shown that a projected design obtained from a rotatable design is also rotatable, and for a rotatable design that is also orthogonal (in particular any orthogonal first-order design) a least squares analysis carried out on the generating design supplies a least squares solution for the constrained design subject to the constraints. Some useful properties of the generating design, such as orthogonal blocking and fractionation are retained in the projected design. Some second-order mixture designs generated by two-level factorials are discussed.     

A NEW MEASURE OF ROTATABILITY OF RESPONSE SURFACE DESIGNS

This paper proposes a new measure of rotatability of a response surface design. It is based on a comparison between the design moments of a given design and the design moments of a rotatable design closest to the given design in the least squares sense. This measure is easier to calculate than Khur?s measure. An alternative easier non-geometrical way of deriving Khuri's measure is also presented. Both measures are calculated to illustrate some designs considered by Khuri.     

Bayesian Geostatistical Design

Abstract.  This paper describes the use of model-based geostatistics for choosing the set of sampling locations, collectively called the design, to be used in a geostatistical analysis. Two types of design situation are considered. These are retrospective design, which concerns the addition of sampling locations to, or deletion of locations from, an existing design, and prospective design, which consists of choosing positions for a new set of sampling locations. We propose a Bayesian design criterion which focuses on the goal of efficient spatial prediction whilst allowing for the fact that model parameter values are unknown. The results show that in this situation a wide range of inter-point distances should be included in the design, and the widely used regular design is often not the best choice.     

Recent progresses in outcome-dependent sampling with failure time data

An outcome-dependent sampling (ODS) design is a retrospective sampling scheme where one observes the primary exposure variables with a probability that depends on the observed value of the outcome variable. When the outcome of interest is failure time, the observed data are often censored. By allowing the selection of the supplemental samples depends on whether the event of interest happens or not and oversampling subjects from the most informative regions, ODS design for the time-to-event data can reduce the cost of the study and improve the efficiency. We review recent progresses and advances in research on ODS designs with failure time data. This includes researches on ODS related designs like case–cohort design, generalized case–cohort design, stratified case–cohort design, general failure-time ODS design, length-biased sampling design and interval sampling design.     

Highly Efficient Designs for Logistic Models with Categorical Variables

Optimal designs for logistic models generally require prior information about the values of the regression parameters. However, experimenters usually do not have full knowledge of these parameters. We propose a design that is D-optimal on a restricted design region. This design assigns an equal weight to design points that contain more information and ignores those design points that contain less information about the regression parameters. The design can be constructed in practice by means of the rank order of the outcome variances. A numerical study compares the proposed design with the D-optimal and completely balanced designs in terms of efficiency.     

Optimal parametric design with applications to pharmacokinetic and pharmacodynamic trials

This paper considers optimal parametric designs, i.e. designs represented by probability measures determined by a set of parameters, for nonlinear models and illustrates their use in designs for pharmacokinetic (PK) and pharmacokinetic/pharmacodynamic (PK/PD) trials. For some practical problems, such as designs for modelling PK/PD relationship, this is often the only feasible type of design, as the design points follow a PK model and cannot be directly controlled. Even for ordinary design problems the parametric designs have some advantages over the traditional designs, which often have too few design points for model checking and may not be robust to model and parameter misspecifications. We first describe methods and algorithms to construct the parametric design for ordinary nonlinear design problems and show that the parametric designs are robust to parameter misspecification and have good power for model discrimination. Then we extend this design method to construct optimal repeated measurement designs for nonlinear mixed models. We also use this parametric design for modelling a PK/PD relationship and propose a simulation based algorithm. The application of parametric designs is illustrated with a three-parameter open one-compartment PK model for the ordinary design and repeated measurement design, and an Emax model for the phamacokinetic/pharmacodynamic trial design.     

Sequential designs for a Poisson regression model

The locally D-optimal design for the two-variable interaction Poisson regression model is not robust to parameter misspecification. To deal with this issue, a sequential design approach is investigated. Simulation is used to evaluate the sequential design. It is found that the sequential design is very efficient and quite robust to parameter misspecification on both unrestricted and restricted design regions.     

Non-linear design problem in a chemical kinetic model with non-constant error variance

For a locally optimum non-linear design problem for a chemical kinetic model, we investigate the influence of the dispersion structure of the random observation errors on the design and its efficiency. We find that there are two kinds of design determined by the model parameters and the error variance function: “interior” designs, and “boundary” designs that depend also on the design range. We give an exact criterion for determining which kind of design will arise and we illustrate the qualitative difference between the two kinds of design in terms of the design locus and the equivalence theorem. We tabulate quantitative details of the designs for a range of parameter values.     

Improving The Efficiency of The Case-Crossover Design

The case-crossover design has been used by many researchers to study the transient effect of an exposure on the risk of a rare outcome. In a case-crossover design, only cases are sampled and each case will act as his/her own control. The time of failure acts as the case and non failure times act as the controls. Case-crossover designs have frequently been used to study the effect of environmental exposures on rare diseases or mortality. Time trends and seasonal confounding may be present in environmental studies and thus need to be controlled for by the sampling design. Several sampling methods are available for this purpose. In time-stratified sampling, disjoint strata of equal size are formed and the control times within the case stratum are used for comparison. The random semi-symmetric sampling design randomly selects a control time for comparison from two possible control times. The fixed semi-symmetric sampling design is a modified version of the random semi-symmetric sampling design that removes the random selection. Simulations show that the fixed semi-symmetric sampling design improves the variance of the random semi-symmetric sampling estimator by at least 35% for the exposures we studied. We derive expressions for the asymptotic variance of risk estimators for these designs, and show, that while the designs are not theoretically equivalent, in many realistic situations, the random semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size two and the fixed semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size three.     

The plan of enrichment designs for dealing with high placebo response

To deal with high placebo response in clinical trials for psychiatric and other diseases, different enrichment designs, such as the sequential parallel design, two‐way enriched design, and sequential enriched design, have been proposed and implemented recently. Depending on the historical trial information and the trial sponsors' resources, detailed design elements are needed for determining which design to adopt. To assist in making more suitable decisions, we perform evaluations for selecting required design elements in terms of power optimization and sample size planning. We also discuss the implementation of the interim analysis related to its applicability.     

Evaluate the relative efficiency of a targeted clinical trial design to an untargeted design under the issue of cost

In this article, we evaluate the relative cost of a targeted design to an untargeted design for a randomized clinical trial comparing a new treatment to a control. We observed that when the ratio of screen cost per person and drug cost per person is very small, say 0.008, the trial cost for conducting a targeted design might be fewer than that for conducting an untargeted design. When the difference between the treatment effect for the targeted patients and the treatment effect for the untargeted patients is increased, the savings in trial cost for the targeted design would be increased significantly. The greater sensitivity and specificity for the screen tool will lead to greater savings in trial cost for the targeted design. The relative cost of a targeted design to an untargeted design is associated with the treatment effect difference between targeted and untargeted patients, the proportion of targeted patients in the population, the screen tool performance, and the ratio of drug cost per person and screen cost per person.     

On the optimality of the generalized room squares

Using a model given by Archbold and Johnson (1958) it is shown that the generalised Room square design is universally optimal for the estimation of the treatment effects. Further, if the variety-block design in a generalised Room square design is a BIBD, this design is also universally optimal for the estimation of the varietal effects.     

Bounding the resampling risk for sequential Monte Carlo implementation of hypothesis tests

Sequential designs can be used to save computation time in implementing Monte Carlo hypothesis tests. The motivation is to stop resampling if the early resamples provide enough information on the significance of the p-value of the original Monte Carlo test. In this paper, we consider a sequential design called the B-value design proposed by Lan and Wittes and construct the sequential design bounding the resampling risk, the probability that the accept/reject decision is different from the decision from complete enumeration. For the B-value design whose exact implementation can be done by using the algorithm proposed in Fay, Kim and Hachey, we first compare the expected resample size for different designs with comparable resampling risk. We show that the B-value design has considerable savings in expected resample size compared to a fixed resample or simple curtailed design, and comparable expected resample size to the iterative push out design of Fay and Follmann. The B-value design is more practical than the iterative push out design in that it is tractable even for small values of resampling risk, which was a challenge with the iterative push out design. We also propose an approximate B-value design that can be constructed without using a specially developed software and provides analytic insights on the choice of parameter values in constructing the exact B-value design.     

On the performance of the nearest proportional to size sampling design

Assuming a super-population model the expected variance of the generalized difference estimator (Basu,1971) based on the nearest proportional to size sampling design introduced by Gabler(1987) is shown to be less than that of the same estimator based on an arbitrary sampling design from which the former design is realized. The former strategy is also shown to fare better than an unbiased ratio-cum-generalized difference estimator based on the nearest proportional to size sampling design in the sense of having less expected design variance under the same model.     

The exact D-optimal first order saturated design with 17 observations

The exact D-optimal first order saturated design with 17 observations is given. The upper bound of the determinant of the information matrix is established and a design attaining this value is constructed. The information matrix is proved to be unique and the optimal design contains the B.I.B. design (16, 16, 6, 6, 2).     

Quantitative comparison of randomization designs in sequential clinical trials based on treatment balance and allocation randomness

To evaluate the performance of randomization designs under various parameter settings and trial sample sizes, and identify optimal designs with respect to both treatment imbalance and allocation randomness, we evaluate 260 design scenarios from 14 randomization designs under 15 sample sizes range from 10 to 300, using three measures for imbalance and three measures for randomness. The maximum absolute imbalance and the correct guess (CG) probability are selected to assess the trade-off performance of each randomization design. As measured by the maximum absolute imbalance and the CG probability, we found that performances of the 14 randomization designs are located in a closed region with the upper boundary (worst case) given by Efron's biased coin design (BCD) and the lower boundary (best case) from the Soares and Wu's big stick design (BSD). Designs close to the lower boundary provide a smaller imbalance and a higher randomness than designs close to the upper boundary. Our research suggested that optimization of randomization design is possible based on quantified evaluation of imbalance and randomness. Based on the maximum imbalance and CG probability, the BSD, Chen's biased coin design with imbalance tolerance method, and Chen's Ehrenfest urn design perform better than popularly used permuted block design, EBCD, and Wei's urn design.     

A measure of slope-rotatability for second order response surface experimental designs

In the design of experiments for estimating the slope of a response surface, slope-rotatability is a desirable property. In this paper, a measure is introduced that enables us to assess the degree of slope-rotatability for a given response surface design. The measure takes the value zero if and only if the design is slope-rotatable, and becomes larger as the design deviates from a slope-rotatable design. Examples of applying this measure to some response surface designs are also given.     

Some new lower bounds to various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this article, some new lower bounds of various discrepancies of combined designs, such as the centered, symmetric, and wrap-around L₂-discrepancies, are obtained, which can be used as a better benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Bayesian optimal design for changepoint problems

We investigate Bayesian optimal designs for changepoint problems. We find robust optimal designs which allow for arbitrary distributions before and after the change, arbitrary prior densities on the parameters before and after the change, and any log‐concave prior density on the changepoint. We define a new design measure for Bayesian optimal design problems as a means of finding the optimal design. Our results apply to any design criterion function concave in the design measure. We illustrate our results by finding the optimal design in a problem motivated by a previous clinical trial. The Canadian Journal of Statistics 37: 495–513; 2009 © 2009 Statistical Society of Canada     

The nearest proportional to size sampling design

The construction of a proportional to size sampling design nearest to a given design is introduced. The realization of the new design can be easily made with the help of the original one.