Regina Dittrich

A paired comparison approach for the analysis of sets of Likert-scale responses.

This paper provides an alternative methodology for the analysis of a set of Likert responses measured on a common attitudinal scale when the primary focus of interest is on the relative importance of items in the set. The method makes fewer assumptions about the distribution of the responses than the more usual approaches such as comparisons of means, MANOVA or ordinal data methods. The approach transforms the Likert responses into paired comparison responses between the items. The complete multivariate pattern of responses thus produced can be analyzed by an appropriately reformulated paired comparison model. The dependency structure between item responses can also be modelled flexibly. The advantage of this approach is that sets of Likert responses can be analyzed simultaneously within the Generalized Linear Model framework, providing standard likelihood-based inference for model selection. This method is applied to a recent international survey on the importance of environmental problems.

Analysing partial ranks by using smoothed paired comparison methods: an investigation of value orientation in Europe

This paper introduces the paired comparison model as a suitable approach for the analysis of partially ranked data. For example, the Inglehart index, collected in international social surveys to examine shifts in post-materialistic values, generates such data on a set of attitude items. However, current analysis methods have failed to account for the complex shifts in individual item values, or to incorporate subject covariates. The paired comparison model is thus developed to allow for covariate subject effects at the individual level, and a reparameterization allows the inclusion of smooth non-linear effects of continuous covariates. The Inglehart index collected in the 1993 International Social Science Programme survey is analysed, and complex non-linear changes of item values with age, level of education and religion are identified. The model proposed provides a powerful tool for social scientists. Copyright 2002 Royal Statistical Society.

A log-linear approach for modelling ordinal paired comparison data on motives to start a PhD programme

The purpose of this paper is to propose an alternative log-linear representation of an adjacent categories (AC) paired comparison (PC) model. The AC model is well suited for modelling ordinal PC data by postulating a power relationship between the response category and the probability of preferring one object over another object. The model is applied to data collected on the motivation of Vienna students to start a doctoral programme of study.

Modeling heterogeneity in ranked responses by nonparametric maximum likelihood: How do Europeans get their scientific knowledge?

This paper is motivated by a Eurobarometer survey on science knowledge. As part of the survey, respondents were asked to rank sources of science information in order of importance. The official statistical analysis of these data however failed to use the complete ranking information. We instead propose a method which treats ranked data as a set of paired comparisons which places the problem in the standard framework of generalized linear models and also allows respondent covariates to be incorporated. An extension is proposed to allow for heterogeneity in the ranked responses. The resulting model uses a nonparametric formulation of the random effects structure, fitted using the EM algorithm. Each mass point is multivalued, with a parameter for each item. The resultant model is equivalent to a covariate latent class model, where the latent class profiles are provided by the mass point components and the covariates act on the class profiles. This provides an alternative interpretation of the fitted model. The approach is also suitable for paired comparison data.

Modelling dependencies in paired comparison data a log-linear approach

In many Bradley-Terry models a more or less explicit assumption is that all decisions of the judges are independent. An assumption which might be questionable at least for the decisions of a given judge. In paired comparison studies, a judge chooses among objects several times, and in such cases, judgements made by the same judge are likely to be dependent. A log-linear representation for the Bradley-Terry model will be developed, which takes into account dependencies between judgements. The modelling of the dependencies is embedded in the analysis of multiple binomial responses, which has the advantage of interpretability in terms of conditional odds ratios. Furthermore, the modelling is done in the framework of generalized linear models, thus parameter estimation and the assessment of goodness of fit can be obtained in the standard way by using e.g. GLIM or another standard software.

Modelling dependency in multivariate paired comparisons: A log-linear approach

A log-linear representation of the Bradley-Terry model is presented for multivariate paired comparison data, where judges are asked to compare pairs of objects on more than one attribute. By converting such data to multiple binomial responses, dependencies between the decisions of the judges as well as possible association structures between the attributes can be incorporated in the model, providing an advantage over parallel univariate analyses of individual attributes. The approach outlined gives parameters which can be interpreted as (conditional) log–odds and log–odds ratios. As the model is a generalised linear model, parameter estimation can use standard software and the GLM framework can be used to test hypotheses on these parameters.

Missing observations in paired comparison data

This paper considers the analysis of paired comparison experiments in the presence of missing responses. Various scenarios for how missing data might arise in paired comparisons are considered, and it is suggested that the most common types of missing data mechanism would be either missing completely at random or missing not at random. A new model is then proposed based on the paired comparison set of responses augmented by a set of missing data indicators for each comparison. Taking a sample selection approach, the proposed new method is based on the classical Bradley-Terry model for the response outcomes and a multinomial model for the missing indicators. Different models for the two missing data mechanisms—missing completely at random (MCAR) and missing not at random (MNAR)—are then discussed and a blockwise composite link formulation is used to construct the likelihood. Additionally, an extension to account for dependence between the paired comparison items is introduced. The methodology is illustrated by a survey paired comparison experiment on five distinct teaching qualities of teachers. We show that there is little evidence of a MNAR process in this dataset. A discussion on the sizes of problems that can be fitted using this approach concludes the paper.

A Mixture Model for Longitudinal Partially Ranked Data

This article discusses the use of mixture models in the analysis of longitudinal partially ranked data, where respondents, for example, choose only the preferred and second preferred out of a set of items. To model such data we convert it to a set of paired comparisons. Covariates can be incorporated into the model. We use a nonparametric mixture to account for unmeasured variability in individuals over time. The resulting multi-valued mass points can be interpreted as latent classes of the items. The work is illustrated by two questions on (post)materialism in three sweeps of the British Household Panel Survey.

Behaviour of the length test for medium sample sizes

In this note it is shown that even for relatively large sample sites the asymptotic distribution of the smoothed length as derived in Reschenhofer and Bomse (1991) should not be used for the determination of critical values. Therefore extended tables of critical values for both the 1% and 5% levels of significance generated by simulation are presented.

An Extended Model for Paired Comparisons

The aim of this paper is to present a log-linear formulation of an extended Bradley-Terry model for paired comparisons that allows for simultaneous modelling of ties, order effects, categorical subject-specific covariates as well as object-specific covariates.