Semi-bayesian D-optimal designs and estimation procedures for mean and variance functions.

Semi-Bayesian D-optimal designs for fitting mean and variance functions are derived for some prior distributions on the variance function parameters. The impact of the mean of the prior and of the uncertainty about this mean is analyzed. Simulation studies are performed to investigate whether the choice of design has a substantial impact on the efficiency of the mean and the variance function parameter estimation and whether the D-optimality criterion is appropriate irrespective of the method applied to estimate the variance function parameters.Functions;

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;

The D-optimal design of blocked and split-plot experiments with mixture components.

So far, the optimal design of blocked and split-plot experiments involving mixture components has received scant attention. In this paper, an easy method to construct efficient blocked mixture experiments in the presence of fixed and/or random blocks is presented. The method can be used when qualitative variables are involved in a mixture experiment as well. It is also shown that orthogonally blocked mixture experiments are highly inefficient compared to D-optimal designs. Finally, the design of a split-plot mixture experiment with process variables is discussed.Design; Fixed and random blocks; Minimum support design; Mixture experiment; Optimal; Optimal design; Orthogonal blocking; Process variables; Processes; Qualitative variables; Split-plot experiment; Variables;

Outperforming completely randomized designs.

Bi-randomization designs have become increasingly popular in industry because some of the factors under investigation are often hard-to-change. It is well-known that the resulting compound symmetric error structure not only affects estimation and inference procedures but also the efficiency of the experimental designs used. In this paper, the use of bi-randomization designs is shown to outperform completely randomized designs in terms of D-efficiency. This result suggests that bi-randomization designs should be considered as an alternative to completely randomized designs even if all experimental factors are easy-to-change.Optimal;

Estimating the intercept in an orthogonally blocked experiment when the block effects are random.

Abstract: For an orthogonally blocked experiment, Khuri (1992) has shown that the ordinary least squares estimator and the generalized least squares estimator of the factor effects in a response surface model with random block effects coincide. However, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. When the block sizes are homogeneous, ordinary and generalized least squares provide an identical estimate for the intercept.Effects;

The optimal design of an experiment with blocks of size two for quadratic regression on one variable.

Exact D-optimal designs are derived for an optometry experiment for the estimation of a quadratic polynomial in one explanatory variable. Two observations are made for each subject participating in the experiment, such that each subject serves as a block of two possibly correlated observations. The exact D-optimal designs are compared to the best possible three-level designs and to the continuous D-optimal designs.Optimal;

The importance of attribute interactions in conjoint choice design and modeling.

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for the attribute interaction models. In this paper, we present Bayesian D-optimal interaction-effects designs for the multinomial logit models and compare their predictive performances with those of main-effects designs. We show that in situations where a researcher is not sure whether or not the attribute interaction effects are present, incorporating interaction effects into both design stage and model estimation stage is most robust against misspecification of the underlying model for making precise predictions.Bayesian; Choice; Interaction effects; Experimental design; Predictions; Multinomial logit;

Individually adapted sequential Bayesian designs for conjoint choice experiments.

In this paper, we propose an efficient individually adapted sequential Bayesian approach for constructing conjoint choice experiments. It uses Bayesian updating, a Bayesian analysis and a Bayesian design criterion for generating choice-set-designs for each individual respondent based on previous answers of that particular respondent. The proposed design approach is compared with two non-adaptive design approaches (the average customization design proposed by Arora and Huber 2001 and the nearly orthogonal design constructed with Sawtooth software) under various degree of response error and respondent heterogeneity. The simulation study shows that the individually adapted sequential Bayesian approach leads to designs which are robust not only to respondent heterogeneity but also to response error. It turns out that the proposed method outperforms the benchmark methods in all scenarios that we have looked at. In particular, for conditions with high response error (the responses from a respondent can hardly provide proper information about the individual-level parameter and is therefore very challenging for individually adapted choice designs), our approach leads to substantially improvement not only in the precision of the parameter estimates but also in the predictive accuracy when the respondent heterogeneity is large. The new method therefore overcomes the limitation of the recently proposed adaptive polyhedral choice-based question design approach by Toubia et al. (2004), whose method performs well only when the response error is low. Furthermore, our study provides compelling evidence that adapting each respondent's choice sets based on the previous responses of that particular respondent in a Bayesian framework enables one to capture more information for the individual- level parameters and therefore also on the population-level parameters. It is shown that it is substantially better to employ the adaptive approach when the response heterogeneity is high.Adaptive Bayesian design; Conjoint choice experiments; Respondent heterogeneity; Response error;

Comparing different sampling schemes for approximating the integrals involved in the semi-Bayesian optimal design of choice experiments.

In conjoint choice experiments, the semi-Bayesian D-optimality criterion is often used to compute efficient designs. The traditional way to compute this criterion which involves multi-dimensional integrals over the prior distribution is to use Pseudo-Monte Carlo samples. However, other sampling approaches are available. Examples are the Quasi-Monte Carlo approach (randomized Halton sequences, modified Latin hypercube sampling and extensible shifted lattice points with Baker's transformation), the Gaussian-Hermite quadrature approach and a method using spherical-radial transformations. Not much is known in general about which sampling scheme performs best in constructing efficient choice designs. In this study, we compare the performance of these approaches under various scenarios. We try to identify the most efficient sampling scheme for each situation.Conjoint choice design; Pseudo-Monte Carlo; Quasi-Monte Carlo; Gaussian-Hermite quadrature; Spherical-radial transformation;

Optimal designs for rating-based conjoint experiments.

The scope of conjoint experiments on which we focus embraces those experiments in which each of the respondents receives a different set of profiles to rate. Carefully designing these experiments involves determining how many and which profiles each respondent has to rate and how many respondents are needed. To that end, the set of profiles offered to a respondent is viewed as a separate block in the design and a respondent effect is incorporated in the model, representing the fact that profile ratings from the same respondent are correlated. Optimal conjoint designs are then obtained by means of an adapted version of the algorithm of Goos and Vandebroek (2004). For various instances, we compute the optimal conjoint designs and provide some practical recommendations.Conjoint analysis; D-Optimality; Design; Model; Optimal; Optimal block design; Rating-based conjoint experiments; Recommendations;