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By modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories, and category-specific scalar effects, the ordered stereotype logit model is a flexible regression model for ordinal response variables. In this article, we propose a generalized estimating equations (GEE) approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices, which are not required to be correctly specified. A simulation study compares the performance of GEE estimators based on various working correlation matrices and working covariance matrices using local odds ratios. Estimation of the model is illustrated using a real-world dataset. The results from the simulation study suggest that GEE estimation of this model is feasible in medium-sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix. For low true correlations, the efficiency gains seem to be rather small and if the working covariance structure is too flexible, the corresponding estimator may even be less efficient compared with the GEE estimator assuming independence. Like for GEE estimators more generally, if the true correlations over time are high, then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence.

This work was supported by the Marsden Fund (Award Num-ber E2987-3648) administrated by the Royal Society of NewZealand and also supported by grant RTI2018-100927-J-I00 administrated by Ministerio de Ciencia e Innovación (MCI,Spain), by the Agencia Estatal de Investigación (AEI, Spain),and by the European Regional Development Fund FEDER(FEDER, UE). We are grateful to Dr Louise McMillan for helpful comments.

For a two-way contingency table, odds ratios are commonly used to describe the relationships between the row and column variables. In the ordinary case cells are mutually exclusive, that is each subject must fit into one and only one cell. However, in many surveys respondents may select more than one outcome category, commonly referred to as multiple responses. We discuss model-based and Mantel–Haenszel estimators of an assumed common odds ratio for several $$2\times c$$ tables, where the two rows refer to independent groups and the c columns to multiple responses, treating the multiple responses as an extension of the multinomial sampling model. We derive new dually consistent (co)variance estimators for the Mantel–Haenszel odds ratio estimators and show their performance in a simulation study and illustrate the estimators on a linguistic data set.

Many of the methods which deal with clustering in matrices of data are based on mathematical techniques such as distance-based algorithms or matrix decomposition and eigenvalues. In general, it is not possible to use statistical inferences or select the appropriateness of a model via information criteria with these techniques because there is no underlying probability model. This article summarizes some recent model-based methodologies for matrices of binary, count, and ordinal data, which are modelled under a unified statistical framework using finite mixtures to group the rows and/or columns. The model parameter can be constructed from a linear predictor of parameters and covariates through link functions. This likelihood-based one-mode and two-mode fuzzy clustering provides maximum likelihood estimation of parameters and the options of using likelihood information criteria for model comparison. Additionally, a Bayesian approach is presented in which the parameters and the number of clusters are estimated simultaneously from their joint posterior distribution. Visualization tools focused on ordinal data, the fuzziness of the clustering structures, and analogies of various standard plots used in the multivariate analysis are presented. Finally, a set of future extensions is enumerated.

This paper presents two new model-based goodness-of-fit tests for the ordered stereotype model applied to an ordinal response variable. The proposed tests are based on the Lipsitz test, which partitions the subjects into G groups following the popular Hosmer–Lemeshow test for binary data. The tests construct an alternative model where group effects are added into the null model. If the model fits the data well then the null model is correct, and there should be no group effects. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The two proposed tests use this new adjusted spacing. One test uses the form of the original ordered stereotype model, and the other uses an ordinary linear model. We demonstrate the performance of both tests under a variety of scenarios. Finally, the results of the application in three examples are presented.

Deciding on the best statistical method to apply when the response variable is ordinal is essential because the way the categories are ordered in the data is relevant as it could change the results of the analysis. Although the models for continuous variables have similarities to those for ordinal variables, this paper presents the advantages of the use of the ordering information on the outcomes with methods developed for modeling ordinal data such as the ordered stereotype model. The novelty of this article lies in showing the dangers of assigning equally spaced scores to ordered response categories in statistical analysis, which are illustrated with a simulation study and a case study. We propose a new way to use the score parameters, which incorporates the fitted spacing dictated by the data. Additionally, this article uses score parameter estimates in the ordered stereotype model to propose a new measure to calculate continuous medians in the raw data: the adjusted c-median. It benefits the general audience who can easily understand the median as a summary statistic. Supplementary materials for this article are available online.

This paper presents a new goodness-of-fit test for an ordered stereotype model used for an ordinal response variable. The proposed test is based on the well-known Hosmer-Lemeshow test and its version for the proportional odds regression model. The latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The proposed test takes the use of this new adjusted spacing to partition data. A simulation study shows good performance of the proposed test under a variety of scenarios. Finally, the results of the application in two examples are presented. Copyright © 2016 John Wiley & Sons, Ltd.

Medical research frequently focuses on the relationship between quality of life (QoL) and survival time of subjects. QoL may be one of the most important factors that could be used to predict survival, making it worth identifying factors that jointly affect survival and QoL. We propose a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. Several popular ordinal models are considered and compared in the item response component, while the Cox proportional hazards model is used in the survival component. We estimate the baseline hazard function and model parameters simultaneously, through a profile likelihood approach. We illustrate the method using an example from a clinical study.

Dimension reduction is a preprocessing step in many classification tasks, but reducing dimensionality and finding the optimal set of features or attributes are challenging because of the big search space and interactions between attributes. This paper proposes a new dimension reduction method by using a statistical variable grouping method that groups similar attributes into a group by considering interaction between attributes and using particle swarm optimisation as a search technique to adopt the discovered statistical grouping information to search optimal attribute subsets. Two types of approaches are developed, where the first aims to select one attribute from each group to reduce the dimensionality, and the second allows the selection of multiple attributes from one group to further improve the classification performance. Experiments on ten datasets of varying difficulties show that all the two approaches can successfully address dimension reduction tasks to decrease the number of attributes, and achieve the similar of better classification performance. The first approach selects a smaller number of attributes than the second approach while the second approach achieves better classification performance. The proposed new algorithms outperform other recent dimension reduction algorithms in terms of the classification performance, or further reduce the number of attributes while maintaining the classification performance.

The work in this paper introduces finite mixture models that can be used to simultaneously cluster the rows and columns of two-mode ordinal categorical response data, such as those resulting from Likert scale responses. We use the popular proportional odds parameterisation and propose models which provide insights into major patterns in the data. Model-fitting is performed using the EM algorithm, and a fuzzy allocation of rows and columns to corresponding clusters is obtained. The clustering ability of the models is evaluated in a simulation study and demonstrated using two real data sets. Electronic supplementary material The online version of this article (doi:10.1007/s11336-016-9503-3) contains supplementary material, which is available to authorized users.

Outcome-dependent sampling designs are commonly used in economics, market research and epidemiological studies. Case-control sampling design is a classic example of outcome-dependent sampling, where exposure information is collected on subjects conditional on their disease status. In many situations, the outcome under consideration may have multiple categories instead of a simple dichotomization. For example, in a case-control study, there may be disease sub-classification among the “cases” based on progression of the disease, or in terms of other histological and morphological characteristics of the disease. In this note, we investigate the issue of fitting prospective multivariate generalized linear models to such multiple-category outcome data, ignoring the retrospective nature of the sampling design. We first provide a set of necessary and sufficient conditions for the link functions that will allow for equivalence of prospective and retrospective inference for the parameters of interest. We show that for categorical outcomes, prospective–retrospective equivalence does not hold beyond the generalized multinomial logit link. We then derive an approximate expression for the bias incurred when link functions outside this class are used. Most popular models for ordinal response fall outside the multiplicative intercept class and one should be cautious while performing a naive prospective analysis of such data as the bias could be substantial. We illustrate the extent of bias through a real data example, based on the ongoing Prostate, Lung, Colorectal and Ovarian (PLCO) cancer screening trial by the National Cancer Institute. The simulations based on the real study illustrate that the bias approximations work well in practice.