Ulrike Graßhoff

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.

A catalogue of designs for partial profiles in paired comparison experiments with three groups of factors

A common strategy for avoiding information overload in multi-factor paired comparison experiments is to employ pairs of options which have different levels for only some of the factors in a study. For the practically important case where the factors fall into three groups such that all factors within a group have the same number of levels and where one is only interested in estimating the main effects, a comprehensive catalogue of D-optimal approximate designs is presented. These optimal designs use at most three different types of pairs and have a block diagonal information matrix.

Optimal Design for Count Data with Binary Predictors in Item Response Theory

The Rasch Poisson counts model (RPCM) allows for the analysis of mental speed which represents a basic component of human intelligence. An extended version of the RPCM, which incorporates covariates in order to explain the difficulty, provides a means for modern rule-based item generation. After a short introduction to the extended RPCM we develop locally D-optimal calibration designs for this model. To this end the RPCM is embedded in a particular generalized linear model. Finally, the robustness of the derived designs is investigated.

Efficient Designs for Paired Comparisons with a Polynomial Factor

In psychological research paired comparisons, which demand judges to evaluate the trade-off between two alternatives, have been shown to yield valid estimates of the judges’ preferences. For this situation we present optimal and efficient designs in a response surface setting where the alternatives are modelled by a polynomial.

Optimal Designs for Linear Logistic Test Models

An important class of models within item response theory are Linear Logistic Test Models (LLTM). These models provide a means for rule-based item generation in educational and psychological testing based upon cognitive theories. After a short introduction into the LLTM, optimal designs for the LLTM will be developed with respect to the item calibration step assuming that persons’ abilities are known. Therefore, the LLTM is embedded in a particular generalized linear model. Finally, future developments are outlined.

Poisson Model with Three Binary Predictors: When are Saturated Designs Optimal?

In this paper, Poisson regression models with three binary predictors are considered. These models are applied to rule-based tasks in educational and psychological testing. To efficiently estimate the parameters of these models locally D-optimal designs will be derived. Eight out of all 70 possible saturated designs are proved to be locally D-optimal in the case of active effects. Two further saturated designs which are the classical fractional factorial designs turn out to be locally D-optimal for vanishing effects.

A Comparison of Efficient Designs for Choices Between Two Options

Optimal designs for choice experiments with choice sets of size two are frequently derived under the assumption that all model parameters in a multinomial logit model are equal to zero. In this case, optimal designs for linear paired comparisons are also optimal for the choice model. It is shown that the methods for constructing linear paired comparison designs often require a considerably smaller number of choice sets when the parameters of primary interest are main effects.

Optimal Design for the Rasch Poisson-Gamma Model

Many tests, measuring human intelligence, yield count data. Often, these data can be analyzed by the Rasch Poisson counts model which incorporates parameters representing the ability of the respondents and the difficulty of the items. In a generalized version, the so-called Rasch Poisson-Gamma counts model, the ability parameter is specified as random with an underlying Gamma distribution. We will develop locally D-optimal calibration designs for an extended version of this model which includes two binary covariates in order to explain the difficulty of an item.

Efficient Paired Comparison Designs for Utility Elicitation

In applications data are often available from the comparison of two alternatives rather than the direct valuation of a single object on its own. Experiments have to be designed for these paired comparisons in a different way than for standard situations. In this note we deal with a problem arising from organizational psychology and present an application of design considerations to the estimation of utility functions.