Experimental investigation of consumer price evaluations

We develop a procedure to collect experimental choice data for estimating consumer preferences with a special focus on consumer price evaluations. For this purpose we employ a heteroskedastic mixed logit model that measures the effect of the way prices are specified on the variance of choice. Our procedure is based on optimal design ideas from the statistics literature and on some algorithms for constructing choice designs published in marketing journals. In an empirical application on mobile phone preferences we find evidence that the way prices are specified significantly affects the variance of choice. In a simulation study we show that our design is significantly more efficient than randomly generated designs., which can be regarded as equivalent to most commonly used experimental designs in the literature.heterogeneity;Bayesian design;demand;quasi-random;task complexity

A geometric characterization of cc-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 [Ann. Math. Statist. 23 (1952) 255--262] 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 different. 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 effects.Comment: Published in at http://dx.doi.org/10.1214/09-AOS708 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal exact designs of experiments via Mixed Integer Nonlinear Programming

Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studyingtheir properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariablyapplicable to the particular problem only.We propose a systematic approach to construct optimal exact designs by incorporatingthe Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. Asexamples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global orlocal optimizers. Our examples include design problems with constraints on the locations or the number of replicates at theoptimal design points

Exploiting correlation in the construction of D-optimal response surface designs.

Cost considerations and difficulties in performing completely randomized experiments often dictate the necessity to run response surface experiments in a bi-randomization format. The resulting compound symmetric error structure not only affects estimation and inference procedures but it also has severe consequences for the optimality of the designs used. Fir this reason, it should be taken into account explicitly when constructing the design. In this paper, an exchange algorithm for constructing D-optimal bi-randomization designs is developed and the resulting designs are analyzed. Finally, the concept of bi-randomization experiments is refined, yielding very efficient designs, which, in many cases, outperform D-optimal completely randomized experiments.Structure;

Model-robust experimental designs for the fractional polynomial response surface models

Fractional polynomial response surface models are polynomial models whose powers are restricted to a small predefined set of rational numbers. Very often these models can give a good a fit to the data and much more plausible behavior between design points than the polynomial models. In this paper, we propose a one-stage and two-stage design strategy for obtaining designs under model uncertainty for these nonlinear class of models.Keywords: nonlinear, model-robust, lack of fit, nesting strategy, support points, locally optimal designs

Efficient D-optimal designs under multiplicative heteroscedasticity.

In optimum design theory designs are constructed that maximize the information on the unknown parameters of the response function. The major part deals with designs optimal for response function estimation under the assumption of homoscedasticity. In this paper, optimal designs are derived in case of multiplicative heteroscedasticity for either response function estimation or response and variance function estimation by using a Bayesian approach. The efficiencies of Bayesian designs derived with various priors are compared to those of the classic designs with respect to various variance functions. The results show that any prior knowledge about the sign of the variance function parameters leads to designs that are considerably more efficient than the classic ones based on homoscedastic assumptions.Optimal;

Trend-resistant and cost-efficient cross-over designs for mixed models.

A mixed model approach is used to construct optimal cross-over designs. In a cross-over experiment the same subject is tested at different points in time. Consider as an example an experiment to investigate the influence of physical attributes of the work environment such as luminance, ambient temperature and relative humidity on human performance of acceptance inspection in quality assurance. In a mixed model context, the subject effects are assumed to be independent and normally distributed. Besides the induction of correlated observations within the same inspector, the mixed model approach also enables one to specify the covariance structure of the inspection data. Here, several covariance structures are considered either depending on the time variable or not. Unfortunately, a serious drawback of the inspection experiment is that the results may be influenced by an unknown time trend because of inspector fatigue due to monotony of the inspection task. In other circumstances, time trend effects can be caused by learning effects of the test subjects in behavioural and life sciences, heating or aging of material in prototype experiments, etc. An algorithm is presented to construct cross-over designs that are optimally balanced for time trend effects. The costs for using the subjects and for altering the factor levels between consecutive observations can also be taken into account. A number of examples illustrate utility of the outlined design methodology.Optimal; Models; Model;

Optimal designs for random effect models with correlated errors with applications in population pharmacokinetics

We consider the problem of constructing optimal designs for population pharmacokinetics which use random effect models. It is common practice in the design of experiments in such studies to assume uncorrelated errors for each subject. In the present paper a new approach is introduced to determine efficient designs for nonlinear least squares estimation which addresses the problem of correlation between observations corresponding to the same subject. We use asymptotic arguments to derive optimal design densities, and the designs for finite sample sizes are constructed from the quantiles of the corresponding optimal distribution function. It is demonstrated that compared to the optimal exact designs, whose determination is a hard numerical problem, these designs are very efficient. Alternatively, the designs derived from asymptotic theory could be used as starting designs for the numerical computation of exact optimal designs. Several examples of linear and nonlinear models are presented in order to illustrate the methodology. In particular, it is demonstrated that naively chosen equally spaced designs may lead to less accurate estimation.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS324 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Model-robust and model-sensitive designs.

Abstract: The main drawback of the optimal design approach is that it assumes the statistical model is known. In this paper, a new approach to reduce the dependency on the assumed model is proposed. The approach takes into account the model uncertainty by incorporating the bias in the design criterion and the ability to test for lack-of-fit. Several new designs are derived in the paper and they are compared to the alternatives available from the literature.A-optimality; Bias; D-optimality; Lack-of-fit; Model-discrimination; Model-robustness;

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. --Model discrimination,optimal design,T-optimality,Ds-optimality,nonlinear approximation