Calculating Ordinal Regression Models in SAS and S‐Plus

Although a number of regression models for ordinal responses have been proposed, these models are not widely known and applied in epidemiology and biomedical research. Overviews of these models are either highly technical or consider only a small part of this class of models so that it is difficult to understand the features of the models and to recognize important relations between them. In this paper we give an overview of logistic regression models for ordinal data based upon cumulative and conditional probabilities. We show how the most popular ordinal regression models, namely the proportional odds model and the continuation ratio model, are embedded in the framework of generalized linear models. We describe the characteristics and interpretations of these models and show how the calculations can be performed by means of SAS and S‐Plus. We illustrate and compare the methods by applying them to data of a study investigating the effect of several risk factors on diabetic retinopathy. A special aspect is the violation of the usual assumption of equal slopes which makes the correct application of standard models impossible. We show how to use extensions of the standard models to work adequately with this situation.     

Sequential ordinal modeling with applications to survival data

This paper considers the class of sequential ordinal models in relation to other models for ordinal response data. Markov chain Monte Carlo (MCMC) algorithms, based on the approach of Albert and Chib (1993, Journal of the American Statistical Association 88, 669-679), are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several nonnested models, such as the sequential model, the cumulative ordinal model, and Weibull and log-logistic models.     

Ordinal response regression models in ecology

Abstract. Although ordinal data are not rare in ecology, ecological studies have, until now, seriously neglected the use of specific ordinal regression models. Here, we present three models – the Proportional Odds, the Continuation Ratio and the Stereotype models – that can be successfully applied to ordinal data. Their differences and respective fields of application are discussed. Finally, as an example of application, PO models are used to predict spatial abundance of plant species in a Geographical Information System. It shows that ordinal models give as good a result as binary logistic models for predicting presence‐absence, but are additionally able to predict abundance satisfactorily.     

An Application of Logistic Models to the Analysis of Ordinal Responses

Logistic probability models—models linear in the log odds of the outcome event—have found extensive application in modelling of unordered categorical responses. This paper illustrates some extensions of logistic models to the modelling of probabilities of ordinal responses. The extensions arise naturally from discrete probability models for the conditional distribution of the ordinal response, as well as from linear modelling of the log odds of response. Methods of estimation and examination of fit developed for the binary logistic model extend in a straightforward manner to the ordinal models. The models and methods are illustrated in an analysis of the dependence of chronic obstructive respiratory disease prevalence on smoking and age.     

Bayesian multilocus association mapping on ordinal and censored traits and its application to the analysis of genetic variation among Oryza sativa L. germplasms

Association mapping can be a powerful tool for detecting quantitative trait loci (QTLs) without requiring line-crossing experiments. We previously proposed a Bayesian approach for simultaneously mapping multiple QTLs by a regression method that directly incorporates estimates of the population structure. In the present study, we extended our method to analyze ordinal and censored traits, since both types of traits are common in the evaluation of germplasm collections. Ordinal-probit and tobit models were employed to analyze ordinal and censored traits, respectively. In both models, we postulated the existence of a latent continuous variable associated with the observable data, and we used a Markov-chain Monte Carlo algorithm to sample the latent variable and determine the model parameters. We evaluated the efficiency of our approach by using simulated- and real-trait analyses of a rice germplasm collection. Simulation analyses based on real marker data showed that our models could reduce both false-positive and false-negative rates in detecting QTLs to reasonable levels. Simulation analyses based on highly polymorphic marker data, which were generated by coalescent simulations, showed that our models could be applied to genotype data based on highly polymorphic marker systems, like simple sequence repeats. For the real traits, we analyzed heading date as a censored trait and amylose content and the shape of milled rice grains as ordinal traits. We found significant markers that may be linked to previously reported QTLs. Our approach will be useful for whole-genome association mapping of ordinal and censored traits in rice germplasm collections.     

Log-Linear Models for Change in Manifest Categorical Variables

Applied developmental scientists engaged in the evaluation of community-based programs, in tests of therapeutic effectiveness, or in developmental research on special populations (e.g., youth gang members, street children, or refugees) must often conduct individual or contextual analyses that involve multiple categorizations over time. This article discusses models for analysis of change in cross classifications of nominal or ordinal variables. Examples of such models include Clogg, Eliason, and Grego's (1990) change-persistence models and quasi-independence models for nominal or ordinal variables. A definition of trends in nominal level variables is provided. A general model is specified, and two groups of log-linear models are presented as special cases of this model. The first allows researchers to test specific hypotheses concerning, for example, general trends and trends in subtables. The second group extends application of log-linear modeling of mobility tables to rectangular tables. Examples use data from research on change in schizophrenics and from pubertal development.     

Models for Ordinal Agreement Data

Statistical models can be used to describe the probabilistic structure underlying cross‐classified agreement data. This article explains how models for ordinal agreement data can be understood in terms of an association component and an agreement component. The association component accounts for the positive association typically present in ordinal ratings of two observers. The agreement component specifies a model for the diagonal cells of the cross‐classified ratings. Several models for ordinal agreement data proposed in the literature are special cases of this approach. A new log‐linear model for agreement data that can also be understood in terms of the two components is presented and illustrated using data from a case‐control study of coronary heart disease.     

Power analysis for trend in ordinal cover classes: implications for long‐term vegetation monitoring

Question: We provide a method to calculate the power of ordinal regression models for detecting temporal trends in plant abundance measured as ordinal cover classes. Does power depend on the shape of the unobserved (latent) distribution of percentage cover? How do cover class schemes that differ in the number of categories affect power? Methods: We simulated cover class data by “cutting‐up” a continuous logit‐beta distributed variable using 7‐point and 15‐point cover classification schemes. We used Monte Carlo simulation to estimate power for detecting trends with two ordinal models, proportional odds logistic regression (POM) and logistic regression with cover classes re‐binned into two categories, a model we term an assessment point model (APM). We include a model fit to the logit‐transformed percentage cover data for comparison, which is a latent model. Results: The POM had equal or higher power compared to the APM and latent model, but power varied in complex ways as a function of the assumed latent beta distribution. We discovered that if the latent distribution is skewed, a cover class scheme with more categories might yield higher power to detect trend. Conclusions: Our power analysis method maintains the connection between the observed ordinal cover classes and the unmeasured (latent) percentage cover variable, allowing for a biologically meaningful trend to be defined on the percentage cover scale. Both the shape of the latent beta distribution and the alternative hypothesis should be considered carefully when determining sample size requirements for long‐term vegetation monitoring using cover class measurements.     

Regression modeling of ordinal data with nonzero baselines

This paper develops regression models for ordinal data with nonzero control response probabilities. The models are especially useful in dose-response studies where the spontaneous or natural response rate is nonnegligible and the dosage is logarithmic. These models generalize Abbott's formula, which has been commonly used to model binary data with nonzero background observations. We describe a biologically plausible latent structure and develop an EM algorithm for fitting the models. The EM algorithm can be implemented using standard software for ordinal regression. A toxicology data set where the proposed model fits the data but a more conventional model fails is used to illustrate the methodology.     

Generalized semiparametrically structured ordinal models

Semiparametrically structured models are defined as a class of models for which the predictors may contain parametric parts, additive parts of covariates with an unspecified functional form, and interactions which are described as varying coefficients. In the case of an ordinal response the complexity of the predictor is determined by different sorts of effects. Global effects and category-specific effects are distinguished; the latter allow the effect to vary across response categories. A general framework is developed in which global as well as category-specific effects may have unspecified functional form. The framework extends various existing methods of modeling ordinal responses. The Wilkinson-Rogers notation is extended to incorporate smooth model parts and varying coefficient terms, the latter being important for the smooth specification of category-specific effects.     

A penalized latent class model for ordinal data

Latent class models provide a useful framework for clustering observations based on several features. Application of latent class methodology to correlated, high-dimensional ordinal data poses many challenges. Unconstrained analyses may not result in an estimable model. Thus, information contained in ordinal variables may not be fully exploited by researchers. We develop a penalized latent class model to facilitate analysis of high-dimensional ordinal data. By stabilizing maximum likelihood estimation, we are able to fit an ordinal latent class model that would otherwise not be identifiable without application of strict constraints. We illustrate our methodology in a study of schwannoma, a peripheral nerve sheath tumor, that included 3 clinical subtypes and 23 ordinal histological measures.     

Random effects in censored ordinal regression: latent structure and Bayesian approach

This paper discusses random effects in censored ordinal regression and presents a Gibbs sampling approach to fit the regression model. A latent structure and its corresponding Bayesian formulation are introduced to effectively deal with heterogeneous and censored ordinal observations. This work is motivated by the need to analyze interval-censored ordinal data from multiple studies in toxicological risk assessment. Application of our methodology to the data offers further support to the conclusions developed earlier using GEE methods yet provides additional insight into the uncertainty levels of the risk estimates.     

Modeling different behaviors in disclosing risk perception

In many fields, people are requested to express their level of awareness about some risk (mainly associated with health, environment, energy, etc.) by selecting a category in an ordered scale. We propose a model for such ordinal data by taking into account that the observed response does not necessarily reflect the respondent's true opinion since the final answer can be inaccurate or completely random. The proposed model hypothesizes three behaviors in the process of answering: accurate interviewees express their risk perception exactly, uncertain ones randomly select the response according to the uniform distribution, and inaccurate interviewees make evaluation errors but with high probability they choose a rating close to the true one. Statistical inference for the proposed models is addressed without assuming that the model, to be fitted, is correctly specified. Two real case studies on the awareness of geo-hydrological risk and work-related stress risk are considered using the proposed methodology.     

Incorporating Inter‐item Correlations in Item Response Data Analysis

This paper concerns with the analysis of item response data, which are usually measured on a rating scale and are therefore ordinal. These study items tended to be highly inter‐correlated. Rasch models, which convert ordinal categorical scales into linear measurements, are widely used in ordinal data analysis. In this paper, we improve the current methodology in order to incorporate inter‐item correlations. We have advocated the latent variable approach for this purpose, in combination with generalized estimating equations to estimate the Rasch model parameters. The data on a study of families of lung cancer patients demonstrate the utility of our methods.     

Estimating time-to-event from longitudinal ordinal data using random-effects Markov models: application to multiple sclerosis progression

Longitudinal ordinal data are common in many scientific studies, including those of multiple sclerosis (MS), and are frequently modeled using Markov dependency. Several authors have proposed random-effects Markov models to account for heterogeneity in the population. In this paper, we go one step further and study prediction based on random-effects Markov models. In particular, we show how to calculate the probabilities of future events and confidence intervals for those probabilities, given observed data on the ordinal outcome and a set of covariates, and how to update them over time. We discuss the usefulness of depicting these probabilities for visualization and interpretation of model results and illustrate our method using data from a phase III clinical trial that evaluated the utility of interferon beta-1a (trademark Avonex) to MS patients of type relapsing-remitting.     

Group sequential methods for an ordinal logistic random-effects model under misspecification

Interim analyses in clinical trials are planned for ethical as well as economic reasons. General results have been published in the literature that allow the use of standard group sequential methodology if one uses an efficient test statistic, e.g., when Wald-type statistics are used in random-effects models for ordinal longitudinal data. These models often assume that the random effects are normally distributed. However, this is not always the case. We will show that, when the random-effects distribution is misspecified in ordinal regression models, the joint distribution of the test statistics over the different interim analyses is still a multivariate normal distribution, but a sandwich-type correction to the covariance matrix is needed in order to obtain the correct covariance matrix. The independent increment structure is also investigated. A bias in estimation will occur due to the misspecification. However, we will also show that the treatment effect estimate will be unbiased under the null hypothesis, thus maintaining the type I error. Extensive simulations based on a toenail dermatophyte onychomycosis trial are used to illustrate our results.     

The control of progression order in a captive herd of sable antelope (Hippotragus niger)

A small herd of captive sable antelope (Hippotragus niger) was observed repeatedly while progressing through a gate connecting a pasture to a corral. The ordinal position of each herd member was recorded during each progression. These data were analyzed quantitatively with respect to three models of progression order control and one of random ordering. An end point-control model, stating that the oldest female of a progression occupies the first position while the remainder of the animals randomly distribute themselves within the progression, accounts for the data. These results are related to both protection and resource learning theories of progression order. The terminology used in the study of leadership and ordinal progressions is discussed as well.     

Heritability and phenotypic variation of canine hip dysplasia radiographic traits in a cohort of Australian German shepherd dogs

Canine Hip Dysplasia (CHD) is a common, painful and debilitating orthopaedic disorder of dogs with a partly genetic, multifactorial aetiology. Worldwide, potential breeding dogs are evaluated for CHD using radiographically based screening schemes such as the nine ordinally-scored British Veterinary Association Hip Traits (BVAHTs). The effectiveness of selective breeding based on screening results requires that a significant proportion of the phenotypic variation is caused by the presence of favourable alleles segregating in the population. This proportion, heritability, was measured in a cohort of 13,124 Australian German Shepherd Dogs born between 1976 and 2005, displaying phenotypic variation for BVAHTs, using ordinal, linear and binary mixed models fitted by a Restricted Maximum Likelihood method. Heritability estimates for the nine BVAHTs ranged from 0.14-0.24 (ordinal models), 0.14-0.25 (linear models) and 0.12-0.40 (binary models). Heritability for the summed BVAHT phenotype was 0.30 ± 0.02. The presence of heritable variation demonstrates that selection based on BVAHTs has the potential to improve BVAHT scores in the population. Assuming a genetic correlation between BVAHT scores and CHD-related pain and dysfunction, the welfare of Australian German Shepherds can be improved by continuing to consider BVAHT scores in the selection of breeding dogs, but that as heritability values are only moderate in magnitude the accuracy, and effectiveness, of selection could be improved by the use of Estimated Breeding Values in preference to solely phenotype based selection of breeding animals.     

Deep transformation models for functional outcome prediction after acute ischemic stroke

In many medical applications, interpretable models with high prediction performance are sought. Often, those models are required to handle semistructured data like tabular and image data. We show how to apply deep transformation models (DTMs) for distributional regression that fulfill these requirements. DTMs allow the data analyst to specify (deep) neural networks for different input modalities making them applicable to various research questions. Like statistical models, DTMs can provide interpretable effect estimates while achieving the state-of-the-art prediction performance of deep neural networks. In addition, the construction of ensembles of DTMs that retain model structure and interpretability allows quantifying epistemic and aleatoric uncertainty. In this study, we compare several DTMs, including baseline-adjusted models, trained on a semistructured data set of 407 stroke patients with the aim to predict ordinal functional outcome three months after stroke. We follow statistical principles of model-building to achieve an adequate trade-off between interpretability and flexibility while assessing the relative importance of the involved data modalities. We evaluate the models for an ordinal and dichotomized version of the outcome as used in clinical practice. We show that both tabular clinical and brain imaging data are useful for functional outcome prediction, whereas models based on tabular data only outperform those based on imaging data only. There is no substantial evidence for improved prediction when combining both data modalities. Overall, we highlight that DTMs provide a powerful, interpretable approach to analyzing semistructured data and that they have the potential to support clinical decision-making.     

Estimated Breeding Values for Canine Hip Dysplasia Radiographic Traits in a Cohort of Australian German Shepherd Dogs

Canine hip dysplasia (CHD) is a serious and common musculoskeletal disease of pedigree dogs and therefore represents both an important welfare concern and an imperative breeding priority. The typical heritability estimates for radiographic CHD traits suggest that the accuracy of breeding dog selection could be substantially improved by the use of estimated breeding values (EBVs) in place of selection based on phenotypes of individuals. The British Veterinary Association/Kennel Club scoring method is a complex measure composed of nine bilateral ordinal traits, intended to evaluate both early and late dysplastic changes. However, the ordinal nature of the traits may represent a technical challenge for calculation of EBVs using linear methods. The purpose of the current study was to calculate EBVs of British Veterinary Association/Kennel Club traits in the Australian population of German Shepherd Dogs, using linear (both as individual traits and a summed phenotype), binary and ordinal methods to determine the optimal method for EBV calculation. Ordinal EBVs correlated well with linear EBVs (r = 0.90–0.99) and somewhat well with EBVs for the sum of the individual traits (r = 0.58–0.92). Correlation of ordinal and binary EBVs varied widely (r = 0.24–0.99) depending on the trait and cut-point considered. The ordinal EBVs have increased accuracy (0.48–0.69) of selection compared with accuracies from individual phenotype-based selection (0.40–0.52). Despite the high correlations between linear and ordinal EBVs, the underlying relationship between EBVs calculated by the two methods was not always linear, leading us to suggest that ordinal models should be used wherever possible. As the population of German Shepherd Dogs which was studied was purportedly under selection for the traits studied, we examined the EBVs for evidence of a genetic trend in these traits and found substantial genetic improvement over time. This study suggests the use of ordinal EBVs could increase the rate of genetic improvement in this population.