Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Max–min optimal discriminating designs for several statistical models

In the literature, different optimality criteria have been considered for model identification. Most of the proposals assume the normal distribution for the response variable and thus they provide optimality criteria for discriminating between regression models. In this paper, a max–min approach is followed to discriminate among competing statistical models (i.e., probability distribution families). More specifically, k different statistical models (plausible for the data) are embedded in a more general model, which includes them as particular cases. The proposed optimal design maximizes the minimum KL-efficiency to discriminate between each rival model and the extended one. An equivalence theorem is proved and an algorithm is derived from it, which is useful to compute max–min KL-efficiency designs. Finally, the algorithm is run on two illustrative examples.     

Optimal Latin hypercube designs for the Kullback–Leibler criterion

Space-filling designs are commonly used for selecting the input values of time-consuming computer codes. Computer experiment context implies two constraints on the design. First, the design points should be evenly spread throughout the experimental region. A space-filling criterion (for instance, the maximin distance) is used to build optimal designs. Second, the design should avoid replication when projecting the points onto a subset of input variables (non-collapsing). The Latin hypercube structure is often enforced to ensure good projective properties. In this paper, a space-filling criterion based on the Kullback–Leibler information is used to build a new class of Latin hypercube designs. The new designs are compared with several traditional optimal Latin hypercube designs and appear to perform well.     

Optimal design for the Bradley–Terry paired comparison model

Paired comparisons are a popular tool for questionnaires in psychological marketing research. The quality of the statistical analysis of the responses heavily depends on the design, i.e. the choice of the alternatives in the comparisons. In this paper we show that the structure of locally optimal designs changes substantially with the parameters in the underlying utility. This fact is illustrated by elementary examples, where the optimal designs can be completely characterized. As an alternative maximin efficient designs are proposed which perform well for all parameter settings. Research supported by grant Ho 1286 of the German Research Council (Deutsche Forschungsgemeinschaft).     

Optimum mixture designs for the log-logistic dose–response model with mixture of two similar compounds

The article studies the log-logistic class of dose–response bioassay models in the binomial set-up. The dose is identified by the potency adjusted mixing proportions of two similar compounds. Models for both absence and presence of interaction between the compounds have been considered. The aim is to investigate the D- and D_s-optimal mixture designs for the estimation of the full set of parameters or for the estimation of potency for a best guess of the parameter values. We also indicate how to find the optimal design to estimate the mixing proportions at which the probability of success attains a given value in the absence of the interaction effect.     

Locally optimal designs for some dose–response models with continuous endpoints

We consider the problem of constructing static (or non sequential), approximate optimal designs for a class of dose–response models with continuous outcomes. We obtain conditions for a design being D-optimal or c-optimal. The designs are locally optimal in that they depend on the model parameters. The efficiency studies show that these designs have high efficiency when the mis-specification of the initial values of model parameters is not severe. A case study indicates that using an optimal design may result in a significant saving of resources.     

Optimal designs for contingent response models with application to toxicity–efficacy studies

We describe a general family of contingent response models. These models have ternary outcomes constructed from two Bernoulli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. A bivariate extreme value distribution illustrates the model and optimal design results. To provide a motivating context, we call the two binary events that compose the contingent responses toxicity and efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response (toxicity, efficacy without toxicity, neither efficacy nor toxicity). The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. While optimal designs for contingent response models are numerically found, limiting optimal designs can be expressed in closed forms. In particular, in the family of four parameter bivariate location-scale models we study, as the marginal probability functions of toxicity and no efficacy diverge, limiting D optimal designs are shown to consist of a mixture of the D optimal designs for each failure (toxicity and no efficacy) univariately. Limiting designs are also obtained for the case of equal scale parameters.     

Optimal trajectory to the general equilibrium state of the unstable stochastic lattice gas of prey–predator model with pair-approximation

This paper focuses on the study of the optimal control problem of the unstable stochastic lattice gas of prey–predator model with pair-approximation. The Pontryagin minimum procedure is used to derive the optimal behaviour of the proposed system that makes it trajectory closer to its general equilibrium states with respect to a selected performance measure during a specified period of time T. Wide numerical simulations and illustrations are presented in graphs and tables.     

Optimal generalized case–cohort sampling design under the additive hazard model

Generalized case–cohort designs have been proved to be a cost-effective way to enhance effectiveness in large epidemiological cohort. In generalized case–cohort design, we first select a subcohort from the underlying cohort by simple random sampling, and then sample a subset of the failures in the remaining subjects. In this article, we propose the inference procedure for the unknown regression parameters in the additive hazards model and develop an optimal sample size allocations to achieve maximum power at a given budget in generalized case–cohort design. The finite sample performance of the proposed method is evaluated through simulation studies. The proposed method is applied to a real data set from the National Wilm's Tumor Study Group.     

A Bayesian benchmarking of the Scott–Smith model for small areas

When the finite population ‘totals’ are estimated for individual areas, they do not necessarily add up to the known ‘total’ for all areas. Benchmarking (BM) is a technique used to ensure that the totals for all areas match the grand total, which can be obtained from an independent source. BM is desirable to practitioners of survey sampling. BM shifts the small-area estimators to accommodate the constraint. In doing so, it can provide increased precision to the small-area estimators of the finite population means or totals. The Scott–Smith model is used to benchmark the finite population means of small areas. This is a one-way random effects model for a superpopulation, and it is computationally convenient to use a Bayesian approach. We illustrate our method by estimating body mass index using data in the third National Health and Nutrition Examination Survey. Several properties of the benchmarked small-area estimators are obtained using a simulation study.     

A Mann–Whitney type effect measure of interaction for factorial designs

We propose a measure for interaction for factorial designs that is formulated in terms of a probability similar to the effect size of the Mann–Whitney test. It is shown how asymptotic confidence intervals can be obtained for the effect size and how a statistical test can be constructed. We further show how the test is related to the test proposed by Bhapkar and Gore [Sankhya A, 36:261–272 (1974)]. The results of a simulation study indicate that the test has good power properties and illustrate when the asymptotic approximations are adequate. The effect size is demonstrated on an example dataset.     

Tests for the variance parameter in the Fay–Herriot model

The Fay–Herriot model is a linear mixed model that plays a relevant role in small area estimation (SAE). Under the SAE set-up, tools for selecting an adequate model are required. Applied statisticians are often interested on deciding if it is worthwhile to use a mixed effect model instead of a simpler fixed-effect model. This problem is not standard because under the null hypothesis the random effect variance is on the boundary of the parameter space. The likelihood ratio test and the residual likelihood ratio test are proposed and their finite sample distributions are derived. Finally, we analyse their behaviour under simulated scenarios and we also apply them to real data.     

Bayesian inference for the Birnbaum–Saunders nonlinear regression model

We develop a Bayesian analysis for the class of Birnbaum–Saunders nonlinear regression models introduced by Lemonte and Cordeiro (Comput Stat Data Anal 53:4441–4452, 2009). This regression model, which is based on the Birnbaum–Saunders distribution (Birnbaum and Saunders in J Appl Probab 6:319–327, 1969a), has been used successfully to model fatigue failure times. We have considered a Bayesian analysis under a normal-gamma prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. Two empirical applications are considered in order to illustrate the developed procedures.     

Testing for random effect in the Fuller–Battese model

We consider the Fuller–Battese model where random effects are allowed to be from non-normal universes. The asymptotic distribution of the F-statistic in this model is derived as the number of groups tends to infinity (is large) and sample size from any group is either fixed or large. The result is used to establish an approximate test for the significance of the random effect variance component. Robustness of the established approximate test is given.     

Testing for the partial Koziol–Green model with covariates

In this paper we consider some non-parametric goodness-of-fit statistics for testing the partial Koziol–Green regression model. In this model, the response at a given covariate value is subject to random right censoring by two independent censoring times. One of these censoring times is informative in the sense that its survival function is some power of the survival function of the response. The goodness-of-fit statistics are based on an underlying empirical process for which large sample theory is obtained.     

A class of saturated row–column designs

A new class of row–column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The (m,s)-criterion is used to select optimal designs. It turns out that all (m,s)-optimal designs are binary. Square (m,s)-optimal designs are constructed and they are treatment-connected. Thus, all treatment contrasts are estimable regardless of the row and column effects.     

Optimal row–column design for three treatments

The A-optimality problem is solved for three treatments in a row–column layout. Depending on the numbers of rows and columns, the requirements for optimality can be decidedly counterintuitive: replication numbers need not be as equal as possible, and trace of the information matrix need not be maximal. General rules for comparing 3×3$3\times 3$ information matrices for their A-behavior are also developed, and the A-optimality problem is also solved for three treatments in simple block designs.