Showing papers on "Ordinal regression published in 1996"

Ordinal methods for behavioral data analysis

Abstract: Contents: Why Ordinal Methods? Ordinal Correlation. Inferences About Ordinal Correlation. Predicting Ordinal Relations: Ordinal Analogues of Multiple Regression. Alternatives to Mean Comparisons. Extension of d to Correlated Data and Broader Applications.

Marginal Regression Models for Clustered Ordinal Measurements

Abstract: This article constructs statistical models for clustered ordinal measurements. We specify two regression models: one for the marginal means and one for the marginal pairwise global odds ratios. Of particular interest are problems in which the odds ratio regression is a focus. Simple assumptions about higher-order conditional moments give a quadratic exponential likelihood function with second-order estimating equations (GEE2) as score equations. But computational difficulty can arise for large clusters when both the mean response and the association between measures is of interest. First, we present GEE1 as an alternative estimation strategy. Second, we extend to repeated ordinal measurements the method developed by Carey et al. for binary observations that is based on alternating logistic regressions (ALR) for the marginal mean parameters and the pairwise log-odds ratio parameters. We study the efficiency of GEE1 and ALR relative to full maximum likelihood. We demonstrate the utility of our regr...

Answering Ordinal Questions with Ordinal Data Using Ordinal Statistics

Abstract: It is argued that ordinal statistical methods are often more appropriate than their more common counterparts for three types of reasons: Conclusions from them will be unaffected by monotonic transformation of the variables, they are statistically more robust when used appropriately, and they often correspond more closely to the goals of the investigator. Kendall's tau (Kendall, 1970) and its counterpart delta are recommended as having wide I applicability and good statistical behavior. It is recommended that they be estimated as population parameters and their standard errors estimated form the data. Ways in which they can then substitute for Pearson correlations and mean comparisons in a number of research contexts are suggested.

Ordinal measures for visual correspondence

Abstract: We present ordinal measures for establishing image correspondence. Linear correspondence measures like correlation and the sum of squared differences are known to be fragile. Ordinal measures, which are based on relative ordering of intensity values in windows, have demonstrable robustness to depth discontinuities, occlusion and noise. The relative ordering of intensity values in each window is represented by a rank permutation which is obtained by sorting the corresponding intensity data. By using a novel distance metric between the rank permutations, we arrive at ordinal correlation coefficients. These coefficients are independent of absolute intensity scale, i.e. they are normalized measures. Further, since rank permutations are invariant to monotone transformations of the intensity values, the coefficients are unaffected by nonlinear effects like gamma variation between images. We have developed a simple algorithm for their efficient implementation. Experiments suggest the superiority of ordinal measures over existing techniques under non-ideal conditions. Though we present ordinal measures in the context of stereo, they serve as a general tool for image matching that is applicable to other vision problems such as motion estimation and image registration.

Ordinal regression methodology for ROC curves derived from correlated data

Abstract: We present an approach for the analysis of correlated ROC data, using ordinal regression models in conjunction with generalized estimating equations. The approach applies to the analysis of degree-of-suspicion data derived from multiple interpretations of the same diagnostic study and from the examination of the same patients with multiple diagnostic modalities. The regression models make it possible to incorporate patient and reader characteristics into the analysis, without having to resort to stratification. We illustrate the potential of the approach with analysis of data from two studies in diagnostic oncology.

Goodness-of-fit Tests for Ordinal Response Regression Models

Abstract: SUMMARY In this paper, goodness-of-fit test statistics for ordinal regression models are proposed, which have approximate X2-distributions when the model has been correctly specified. The statistics proposed can be viewed as extensions of the Hosmer-Lemeshow statistic to ordinal categorical data and can be easily calculated by using existing statistical software for analysing ordinal response data The methods are illustrated by using data from an arthritis clinical trial comparing the drug auranofin and placebo therapy for the treatment of rheumatoid arthritis, in which the response is a self-assessment of arthritis, classified as poor, fair and good. The covariates of interest are age, gender, treatment and base-line response. A proportional odds model is fitted to the data, and the proposed goodness-of-fit statistics are applied to the fitted model. Also, the small sample properties of the proposed goodness-of-fit statistics are compared in a simulation study.

Generalized estimating equations for ordinal data: a note on working correlation structures.

Abstract: Generalized estimating equations (Liang, K. Y. and Zeger, S., 1986, Biometrika 73, 13-22) allow longitudinal or clustered data to be modeled with minimal assumptions about their dependence structures. Association structures for polytomous data have generally required the estimation of a large number of parameters. In many applications involving repeated categorical data, an ordinal structure is present. A range of association structures and computational methods for ordinal categorical data is described, based on the cumulative odds ratio, which allows much more parsimonious models. This permits the generalized estimating equation methodology to be used for smaller sets of ordinal data and with less effort expended on modeling associations. The method is illustrated on sets of ordinal data from medical studies.

Order-restricted tests for stratified comparisons of binomial proportions

Abstract: The data set presented relates a binomial response to ordered levels of an explanatory variable, representing doses of a drug, with data collected at several centers. A study goal is to test independence of the response and the ordinal factor, assuming under the alternative only that the binomial parameter is a monotonically increasing function of the ordinal predictor. We present two likelihood-ratio tests that are sensitive to order-restricted alternatives. Simulating the exact distributions of the test statistics yields nearly exact P-values. We also discuss related analyses for comparing two groups on an ordinal response, and we propose a test that is sensitive to a stochastic ordering alternative.

A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses.

Abstract: The mixed effects model for binary responses due to Conaway (1990, A Random Effects Model for Binary Data) is extended to accommodate ordinal responses in general and discrete time survival data with ordinal responses in particular. Given a multinomial likelihood, cumulative complementary log-log link function, and log-gamma random effects distribution, the resulting marginal likelihood has a closed form. As a result, a Newton-Raphson estimation procedure is feasible without resorting to numerical, approximation-based, or Monte Carlo integration techniques. The parameters in the model have a proportional hazards interpretation in terms of multivariate discrete time data with ordinal responses. Using data from a psychological example, the proposed method is compared with other mixed effects approaches as well as population-averaged models.

Computing Min and Max Scorings for Two-Sample Ordinal Data.

Abstract: : Ordinal response variables often occur in practice. For example, in clinical trials a subject's response to a drug regime might be categorized as negative, none, fair, or good. There are several common approaches to analyzing two-sample ordinal response data. These procedures applied to the same data can lead to contradictory conclusions. In an attempt to reconcile contradictory results and provide guidance to the practitioner, Kimledorf, Sampson and Whitaker (1992) propose an alternative approach. They find the scores which when assigned to the levels of the ordinal response variable maximize a two-sample test statistic and the scores that minimize that same statistic. Since many of the two-sample statistics are related by monotonic transformations, these extreme scores are in fact extreme scores for several test statistics. Both minimized and maximized test statistics falling into the rejection region clearly indicate a difference between the two populations or treatments. On the other hand if neither of the two extreme statistics fall in the rejection region then no matter what scores are used there will be no significant difference in the two populations. In this paper we review the KsW procedure and its implementation in SAS software.

Efficient Estimation of Ordered Probit Models

Abstract: This article discusses a model in which both the dependent variable and the explanatory variables are ordinal and have an arbitrary number of categories. Assuming joint normality of the underlying continuous latent variables, we compare estimation based on the joint distribution to estimation based on the conditional distribution. Because the explanatory variables are not weakly exogenous in this model, the latter approach implies a loss in efficiency that can be substantial in many cases, as shown in detail for the special case of trichotomous data with symmetric thresholds. Therefore, latent variables underlying the observed ordinal variables should always be considered to be jointly endogenous; that is, the joint distribution should be considered.

Population‐averaged and cluster‐specific models for clustered ordinal response data

Abstract: We compare population-averaged and cluster-specific models for clustered ordinal data. We consider generalized estimating equations and constrained equations maximum likelihood estimation of population-averaged cumulative logit regression models, and mixed effects estimation of cluster-specific cumulative logit regression models. A previously reported relationship between population-averaged and cluster-specific parameters for the binary logistic link appears to hold for analogous parameters under the cumulative logit link. We address these issues in the context of data from two cross-over clinical trials.

Interval Censoring and Marginal Analysis in Ordinal Regression

Abstract: This article develops a methodology for regression analysis of ordinal response data subject to interval censoring. This work is motivated by the need to analyze data from multiple studies in toxicological risk assessment. Responses are scored on an ordinal severity scale, but not all responses can be scored completely. For instance, in a mortality study, information on nonfatal but adverse outcomes may be missing. In order to address possible within-study correlations, we develop a generalized estimating approach to the problem, with appropriate adjustments to uncertainty statements. We develop expressions relating parameters of the implied marginal model to the parameters of a conditional model with random effects, and, in a special case, we note an interesting equivalence between conditional and marginal modeling of ordinal responses. We illustrate the methodology in an analysis of a toxicological database.

Effect of dropouts in a longitudinal study: an application of a repeated ordinal model.

Abstract: An analysis is presented of a longitudinal study of fluvoxamine, an antidepressant drug, with ordinal responses, regressed on a combination of discrete and continuous covariates and with a substantial proportion of dropouts. Classical methods, such as weighted least squares (SAS procedure CATMOD) and logistic regression, are not suitable for the analysis of such data. Instead, we illustrate how a recently introduced model can be used to solve most of the problems posed. The method is likelihood-based and is an extension of the bivariate Dale model to an arbitrary number of outcomes. The method is suitable for several types of designs commonly employed in clinical trials.

Regression analysis of forest damage by marginal models for correlated ordinal responses

Abstract: Studies on forest damage generally cannot be carried out by common regression models, for two main reasons: Firstly, the response variable, damage state of trees, is usually observed in ordered categories. Secondly, responses are often correlated, either serially, as in a longitudinal study, or spatially, as in the application of this paper, where neighbourhood interactions exist between damage states of spruces determined from aerial pictures. Thus so-called marginal regression models for ordinal responses, taking into account dependence among observations, are appropriate for correct inference. To this end we extend the binary models of Liang and Zeger (1986) and develop an ordinal GEEI model, based on parametrizing association by global cross-ratios. The methods are applied to data from a survey conducted in Southern Germany. Due to the survey design, responses must be assumed to be spatially correlated. The results show that the proposed ordinal marginal regression models provide appropriate tools for analysing the influence of covariates, that characterize the stand, on the damage state of spruce.

Sequential monitoring of clinical trials with correlated responses

Abstract: SUMMARY This paper demonstrates how the alpha-spending method of Lan & DeMets (1983) can be applied to the generalised estimating equations regression model for correlated data proposed by Liang & Zeger (1986). Under large-sample conditions, the sequential regression parameters are shown to have an independent increments structure, conditional on the amount of Type I error allocated at each interim analysis. We propose and evaluate surrogates for the information fraction, which determines this allocation of Type I error. Data from the Early Treatment Diabetic Retinopathy Study are used to illustrate the proposed methods for ordered polytomous outcomes.

Measures of Variation for Ordinal Data as Functions of the Cumulative Distribution

Abstract: A new measure of ordinal variation, the LSQ, is developed using a geometric representation involving the cumulative distribution function. Connections among it and previously suggested measures, the LOV, IOV, and COV, are clarified. This geometric perspective helps demonstrate that all these statistics measure the distance between the observed cumulative distribution and that corresponding to the maximally dispersed distribution, given the sample size and the number of categories for the ordinal variable. From this perspective, it is clear that none of these measures relies on supra-ordinal assumptions concerning intercategory distances. Recent questions concerning scale invariance and unreasonable values for these measures are also clarified.

The Analysis of Longitudinal Ordinal Response Data in Continuous Time

Abstract: A simple Markov model is developed for assessing the predictive effect of time-dependent covariates on an intermittently observed ordinal response in continuous time. This is accomplished by reparameterizing an ergodic intensity matrix in terms of its equilibrium distribution and a parametrically independent component that assesses the rate of movement between ordinal categories. The effect of covariates on the equilibrium distribution can then be modeled using any link appropriate for ordinal data. A robust maximum likelihood estimator based on this model that is consistent and asymptotically normal is constructed. Practical data analysis issues are discussed, and a simple diagnostic tool for assessing model adequacy is developed. The utility of these methods is demonstrated with several analyses of visual acuity data, including a comparison analysis based on generalized estimating equation (GEE) methods.

Presentation of ordinal regression analysis on the original scale

Abstract: Frequently, ordinal measurement scales are constructed either by coarse measurement of interval or ratio scales, or by assigning numeric scores to ordinal categories. When data on such scales are analyzed using the ordinal regression techniques of McCullagh (1980, Journal of the Royal Statistics Society, Series B 42, 109-142), inference is usually performed, and results presented, on the scale of the linear predictor. This can be unsatisfactory from the point of view of the applied scientist familiar with the original scale definition. It is demonstrated how ordinal regression results can be presented on the original ordinal score scale.

Maximum likelihood assessment of clinical trials based on an ordered categorical response

Abstract: A unified maximum likelihood (ML) methodology is developed for assessing simultaneously both the statistical significance of treatment effects and the model fit when the response variable contains ordered categories. In general, for any treatment by response table it is shown that the better the model fit the more significant [he [realmenl effect. The paper begins by examining the fit for different logit model extensions to data derived under the assumption of an underlying bivariate normal. It is shown that the “parallel log-odds’’ model (I) based on adjacent odds often provides a parsimonious description of the data which fits better than the ‘proportional odds” model (2) based on cumulative odds. Using additional data from a published clinical study, the fit and descriptive utility of more general models (3,4) using an extended ML algorithm (5) are also examined. The paper concludes with some extensions to multi-way tables, where one or more covariates are present.

Univariate analysis of dichotomous or ordinal data from twin pairs: A simulation study comparing structural equation modeling and logistic regression

Abstract: The univariate analysis of categorical twin data can be performed using either structural equation modeling (SEM) or logistic regression. This paper presents a comparison between these two methods using a simulation study. Dichotomous and ordinal (three category) twin data are simulated under two different sample sizes (1,000 and 2,000 twin pairs) and according to different additive genetic and common environmental models of phenotypic variation. The two methods are found to be generally comparable in their ability to detect a "correct" model under the specifications of the simulation. Both methods lack power to detect the right model for dichotomous data when the additive genetic effect is low (between 10 and 20%) or medium (between 30 and 40%); the ordinal data simulations produce similar results except for the additive genetic model with medium or high heritability. Neither method could adequately detect a correct model that included a modest common environmental effect (20%) even when the additive genetic effect was large and the sample size included 2,000 twin pairs. The SEM method was found to have better power than logistic regression when there is a medium (30%) or high (50%) additive genetic effect and a modest common environmental effect. Conversely, logistic regression performed better than SEM in correctly detecting additive genetic effects with simulated ordinal data (for both 1,000 and 2,000 pairs) that did not contain modest common environmental effects; in this case the SEM method incorrectly detected a common environmental effect that was not present.