Easier analysis and better reporting : modelling ordinal data in mathematics education research

This article presents an examination of the use of Rasch modelling in a major research project, 'Improving Middle Years Mathematics and Science' (IMYMS). It is unarguable that it is important to take students' perceptions, or views, into account when planning learning and teaching for them. The IMYMS student perceptions survey is an attempt to make visible these student viewpoints, and report them in a way that is accessible to teachers and researchers involved in the project. The project involves four clusters of schools from urban and regions of Victoria to investigate the role of mathematics and science knowledge and subject cultures in mediating change processes in the middle years of schooling. There are five secondary and twenty-eight primary schools. The project has generated both qualitative and quantitative data, with much of the qualitative data being ordinal in nature. Reporting the results of analyses for a range of audiences necessitates careful, well-designed report formats. Some useful new report formats based on Rasch modeling -the Modified Variable Map, the Ordinal Map, the Threshold Map, and the Annotated Ordinal Map - are illustrated using data from the IMYMS project. The Rasch analysis and the derived reporting formats avoid the pitfalls that exist when working with ordinal data and provide insights into the respondents' views about their experiences in schools unavailable by other approaches.

Ordinal Boltzmann machines for collaborative filtering

Collaborative filtering is an effective recommendation technique wherein the preference of an individual can potentially be predicted based on preferences of other members. Early algorithms often relied on the strong locality in the preference data, that is, it is enough to predict preference of a user on a particular item based on a small subset of other users with similar tastes or of other items with similar properties. More recently, dimensionality reduction techniques have proved to be equally competitive, and these are based on the co-occurrence patterns rather than locality. This paper explores and extends a probabilistic model known as Boltzmann Machine for collaborative filtering tasks. It seamlessly integrates both the similarity and cooccurrence in a principled manner. In particular, we study parameterisation options to deal with the ordinal nature of the preferences, and propose a joint modelling of both the user-based and item-based processes. Experiments on moderate and large-scale movie recommendation show that our framework rivals existing well-known methods.

Cumulative restricted Boltzmann machines for ordinal matrix data analysis

Ordinal data is omnipresent in almost all multiuser-generated feedback - questionnaires, preferences etc. This paper investigates modelling of ordinal data with Gaussian restricted Boltzmann machines (RBMs). In particular, we present the model architecture, learning and inference procedures for both vector-variate and matrix-variate ordinal data. We show that our model is able to capture latent opinion profile of citizens around the world, and is competitive against state-of-art collaborative filtering techniques on large-scale public datasets. The model thus has the potential to extend application of RBMs to diverse domains such as recommendation systems, product reviews and expert assessments.

Stabilized sparse ordinal regression for medical risk stratification

The recent wide adoption of electronic medical records (EMRs) presents great opportunities and challenges for data mining. The EMR data are largely temporal, often noisy, irregular and high dimensional. This paper constructs a novel ordinal regression framework for predicting medical risk stratification from EMR. First, a conceptual view of EMR as a temporal image is constructed to extract a diverse set of features. Second, ordinal modeling is applied for predicting cumulative or progressive risk. The challenges are building a transparent predictive model that works with a large number of weakly predictive features, and at the same time, is stable against resampling variations. Our solution employs sparsity methods that are stabilized through domain-specific feature interaction networks. We introduces two indices that measure the model stability against data resampling. Feature networks are used to generate two multivariate Gaussian priors with sparse precision matrices (the Laplacian and Random Walk). We apply the framework on a large short-term suicide risk prediction problem and demonstrate that our methods outperform clinicians to a large margin, discover suicide risk factors that conform with mental health knowledge, and produce models with enhanced stability. © 2014 Springer-Verlag London.

A two-stage dynamic group decision making method for processing ordinal information

In group decision making (GDM) problems, ordinal data provide a convenient way of articulating preferences from decision makers (DMs). A number of GDM models have been proposed to aggregate such kind of preferences in the literature. However, most of the GDM models that handle ordinal preferences suffer from two drawbacks: (1) it is difficult for the GDM models to manage conflicting opinions, especially with a large number of DMs; and (2) the relationships between the preferences provided by the DMs are neglected, and all DMs are assumed to be of equal importance, therefore causing the aggregated collective preference not an ideal representative of the group's decision. In order to overcome these problems, a two-stage dynamic group decision making method for aggregating ordinal preferences is proposed in this paper. The method consists of two main processes: (i) a data cleansing process, which aims to reduce the influence of conflicting opinions pertaining to the collective decision prior to the aggregation process; as such an effective solution for undertaking large-scale GDM problems is formulated; and (ii) a support degree oriented consensus-reaching process, where the collective preference is aggregated by using the Power Average (PA) operator; as such, the relationships of the arguments being aggregated are taken into consideration (i.e., allowing the values being aggregated to support each other). A new support function for the PA operator to deal with ordinal information is defined based on the dominance-based rough set approach. The proposed GDM model is compared with the models presented by Herrera-Viedma et al. An application related to controlling the degradation of the hydrographic basin of a river in Brazil is evaluated. The results demonstrate the usefulness of the proposed method in handling GDM problems with ordinal information.

Ordinal random fields for recommender systems

Recommender Systems heavily rely on numerical preferences, whereas the importance of ordinal preferences has only been recognised in recent works of Ordinal Matrix Factorisation (OMF). Although the OMF can effectively exploit ordinal properties, it captures only the higher-order interactions among users and items, without considering the localised interactions properly. This paper employs Markov Random Fields (MRF) to investigate the localised interactions, and proposes a unified model called Ordinal Random Fields (ORF) to take advantages of both the representational power of the MRF and the ease of modelling ordinal preferences by the OMF. Experimental result on public datasets demonstrates that the proposed ORF model can capture both types of interactions, resulting in improved recommendation accuracy.