The person trade-off method and the transitivity principle: an example from preferences over age weighting

The person trade-off (PTO) is increasingly being used to elicit preferences in health. This paper explores the measurement properties of the PTO method in the context of a study about how members of the public prioritise between patients of different ages. In particular, it considers whether PTO responses satisfy the transitivity principle; that is, whether one PTO response can be inferred from two other PTO responses. The results suggest that very few responses to PTO questions satisfy cardinal transitivity condition. However, this study has produced results that suggest that cardinal transitivity will hold, on average, when respondents who fail to satisfy the ordinal transitivity condition have been excluded from the analysis. This suggests that future PTO studies should build in checks for ordinal transitivity.     

Classification to ordinal categories using a search partition methodology with an application in diabetes screening.

A method is proposed for classification to ordinal categories by applying the search partition analysis (SPAN) approach. It is suggested that SPAN be repeatedly applied to binary outcomes formed by collapsing adjacent categories of the ordinal scale. By a simple device, whereby successive binary partitions are constrained to be nested, a partition for classification to the ordinal states is obtained. The approach is applied to ordinal categories of glucose tolerance to discriminate between diabetes, impaired glucose tolerance and normal states. The results are compared with analysis by ordinal logistic regression and by classification trees.     

Are ordinal models useful for classification?

There is recent interest in classification procedures intended for use only when the response is ordinal. Ordinal response, however, is evident in the parameters estimated by either multinomial logistic or normal discriminant analyses, both of which classify either ordinal or non-ordinal responses. Further, there may be harm in applying ordinal models inappropriately and ample opportunity to assume mistakenly ordinality in real data sets. Therefore, it becomes important to ascertain whether there is benefit obtained in the appropriate application of ordinal models. This paper presents the results of a simulation study designed to compare classification accuracy of various models. We show that ordinal models classify less accurately than the multinomial logistic and normal discriminant procedures under a variety of circumstances. Until further studies become available, we presently conclude that ordinal models confer no advantage when the main purpose of the analysis is classification.     

Using DCE and ranking data to estimate cardinal values for health states for deriving a preference‐based single index from the sexual quality of life questionnaire

There is an increasing interest in using data derived from ordinal methods, particularly data derived from discrete choice experiments (DCEs), to estimate the cardinal values for health states to calculate quality adjusted life years (QALYs). Ordinal measurement strategies such as DCE may have considerable practical advantages over more conventional cardinal measurement techniques, e.g. time trade‐off (TTO), because they may not require such a high degree of abstract reasoning. However, there are a number of challenges to deriving the cardinal values for health states using ordinal data, including anchoring the values on the full health–dead scale used to calculate QALYs. This paper reports on a study that deals with these problems in the context of using two ordinal techniques, DCE and ranking, to derive the cardinal values for health states derived from a condition‐specific sexual health measure. The results were compared with values generated using a commonly used cardinal valuation technique, the TTO. This study raises some important issues about the use of ordinal data to produce cardinal health state valuations. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of clinical reasoning in the context of uncertainty: the effect of variability within the reference panel

The Script Concordance Test (SCT) assesses reasoning in the context of uncertainty. Because there is no single correct answer, scoring is based on the comparison of answers provided by examinees with those provided by members of a reference panel made up of experienced practitioners. The study aimed to assess the discriminatory power of the SCT based on the variability of the reference panel's answers. Items from a bank covering different family medicine domains were classified into 3 groups according to the degree of variability of answers provided by a pool of experienced doctors. A variability index (mean squared error) was used to select items in the low, moderate and high variability categories. A 102-item test (Cronbach's alpha 0.70), made up of 3 subtests of each category, was administered to 3 contrasting groups in family medicine: 157 clerkship students, 30 residents and 30 practising doctors. anova and effect size (ES) were used to quantify and test the discrimination power of the 3 subtests. The high variability subtest showed high effect size for discrimination between extreme groups (ES = 1.5; F = 16.3, P < 0.001), whereas the moderate variability subtest showed less effect size (ES = 0.56; F = 57, P = 0.041). The low variability subtest did not discriminate significantly (ES = 0.31; F = 2.9, P = 0.06). Variability of answers within the reference panel is a key component of the discriminatory power of the SCT. In accordance with theory, the presence of variability ensures discrimination between levels of clinical experience. These results imply important considerations for the construction of efficient SCTs.     

Ordinal response prediction using bootstrap aggregation,with application to a high‐throughput methylation data set

Many investigators conducting translational research are performing high‐throughput genomic experiments and then developing multigenic classifiers using the resulting high‐dimensional data set. In a large number of applications, the class to be predicted may be inherently ordinal. Examples of ordinal outcomes include tumor‐node‐metastasis (TNM) stage (I, II, III, IV); drug toxicity evaluated as none, mild, moderate, or severe; and response to treatment classified as complete response, partial response, stable disease, or progressive disease. While one can apply nominal response classification methods to ordinal response data, in doing so some information is lost that may improve the predictive performance of the classifier. This study examined the effectiveness of alternative ordinal splitting functions combined with bootstrap aggregation for classifying an ordinal response. We demonstrate that the ordinal impurity and ordered twoing methods have desirable properties for classifying ordinal response data and both perform well in comparison to other previously described methods. Developing a multigenic classifier is a common goal for microarray studies, and therefore application of the ordinal ensemble methods is demonstrated on a high‐throughput methylation data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Random-effect models for ordinal responses: application to self-reported disability among older persons

BACKGROUND: Longitudinal studies with ordinal repeated outcomes are now widespread in epidemiology and clinical research. The statistical analysis of these studies combines two difficulties: the choice of the best ordinal model and taking into account correlations for within-subject responses. METHODS: Random-effect models are of particular value in this context and we propose here a fitting strategy. The various ordinal models extended to the case of repeated responses are detailed. We explain how the choice of model constrains the random effect structure. Model selection criteria and goodness-of-fit measures are also presented. These issues are dealt with by using an example of self-reported disability in older women assessed annually over a period of seven years. RESULTS: The proportionality of the odds ratios was validated for the covariables "age" and "gait speed". In contrast the impact of the covariable "pain" differs according to the levels of disability. The restricted partial proportional odds model was found to have a goodness of fit equivalent to the full generalized ordered logit model while the stereotype model appeared to give poorer fit. CONCLUSIONS: The random-effects models presented in this paper allow taking into account the ordinal nature of the outcome in longitudinal studies. Furthermore the impact of the risk factors can be modeled according to the response levels. This approach can be useful for a better understanding of complex processes of evolution.     

Time‐varying effect models for ordinal responses with applications in substance abuse research

Ordinal responses are very common in longitudinal data collected from substance abuse research or other behavioral research. This study develops a new statistical model with free SAS macros that can be applied to characterize time‐varying effects on ordinal responses. Our simulation study shows that the ordinal‐scale time‐varying effects model has very low estimation bias and sometimes offers considerably better performance when fitting data with ordinal responses than a model that treats the response as continuous. Contrary to a common assumption that an ordinal scale with several levels can be treated as continuous, our results indicate that it is not so much the number of levels on the ordinal scale but rather the skewness of the distribution that makes a difference on relative performance of linear versus ordinal models. We use longitudinal data from a well‐known study on youth at high risk for substance abuse as a motivating example to demonstrate that the proposed model can characterize the time‐varying effect of negative peer influences on alcohol use in a way that is more consistent with the developmental theory and existing literature, in comparison with the linear time‐varying effect model. Copyright © 2014 John Wiley & Sons, Ltd.     

A comparison of methods for correlated ordinal measures with ophthalmic applications

For many clinical trials and epidemiologic investigations in the field of ophthalmology, paired ordinal data are often collected through the detailed grading of retinal photographs. One method for analysis of these data is the extension of the generalized estimating equation (GEE) methodology to multinomial data with cumulative link functions. Prior to the development of this advanced technique, however, ophthalmologists developed a method of combining the ordinal responses of both eyes of a patient into a single person-level response on a new ordinal scale. A relationship between the regression coefficients of these two methods is derived as a function of the correlation between eyes. We investigate the applicability of this result and the relationship of the standard errors in simulation experiments and in an example from the Wisconsin Epidemiologic Study of Diabetic Retinopathy.     

Analysis of repeated categorical data using generalized estimating equations

Moment methods for analysing repeated binary responses have been proposed by Liang and Zeger, and extended by Prentice and Zhao and Prentice. In these estimating equations, models are proposed for the correlation between the repeated binary responses. We extend Liang and Zeger's method to models for the correlation between repeated nominal or ordinal categorical responses; in particular, when the repeated responses are binary, our methods reduce to Liang and Zeger's method. Our method is illustrated with two datasets. One dataset contains repeated observations of self-assessment of arthritis, an ordered variable with three categories, collected during a randomized comparative study of alternative treatments of patients with rheumatoid arthritis. The second dataset is a longitudinal study of the health effects of air pollution, in which the repeated ordered multinomial response is the wheezing status (no wheeze, wheeze with cold, wheeze apart from cold) of a child at ages 9, 10, 11 and 12 years.     

Modelling ordinal responses from co-twin control studies

The co-twin control design has been widely used in studying the effects of environmental factors on the development of diseases. For binary outcomes that arise from co-twin control studies, the conditional likelihood method is commonly used. This approach, however, does not readily extend to ordinal response data because the standard conditional likelihood does not exist for cumulative logit or proportional odds models. In this paper, we investigate the applicability of the random-effects and GEE approaches in analysing ordinal response data from co-twin control studies. Using both approaches, we re-analyse data from a co-twin control study of the impact of military services during the Vietnam era on post-traumatic stress disorders (PTSD). The ordinal models have considerably increased power in detecting the effects of exposure when compared to the analyses using a dichotomized response. We discuss the interpretation of the estimates from GEE and random-effects models in the context of the twin data. © 1998 John Wiley & Sons, Ltd.     

Ordered multiple-class ROC analysis with continuous measurements

Receiver operating characteristic (ROC) curves have been useful in two-group classification problems. In three- and multiple-class diagnostic problems, an ROC surface or hyper-surface can be constructed. The volume under these surfaces can be used for inference using bootstrap techniques or U-statistics theory. In this article, ROC surfaces and hyper-surfaces are defined and their behaviour and utility in multi-group classification problems is investigated. The formulation of the problem is equivalent to what has previously been proposed in the general multi-category classification problem but the definition of ROC surfaces here is less complex and addresses directly the narrower problem of ordered categories in the three-class and, by extension, the multi-class problem applied to continuous and ordinal data. Non-parametric manipulation of both continuous and discrete test data and comparison between two diagnostic tests applied to the same subjects are considered. A three-group classification example in the context of HIV neurological disease is presented and the results are discussed.     

An imputation strategy for incomplete longitudinal ordinal data

A new quasi-imputation strategy for correlated ordinal responses is proposed by borrowing ideas from random number generation. The essential idea is collapsing ordinal levels to binary ones and converting correlated binary outcomes to multivariate normal outcomes in a sensible way so that re-conversion to the binary and then ordinal scale, after conducting multiple imputation, yields the original marginal distributions and correlations. This conversion process ensures that the correlations are transformed reasonably, which in turn allows us to take advantage of well-developed imputation techniques for Gaussian outcomes. We use the phrase 'quasi' because the original observations are not guaranteed to be preserved. We present an application using a data set from psychiatric research. We conclude that the proposed method may be a promising tool for handling incomplete longitudinal or clustered ordinal outcomes.     

Longitudinal quality of life data: a comparison of continuous and ordinal approaches

Purpose

In cancer clinical trials, health-related quality of life (HRQoL) is a major outcome measure. It is generally assessed at specified time intervals by filling out a questionnaire with ordered response categories. Despite recent advances in the statistical methodology for handling ordinal longitudinal outcome data, most users keep treating HRQoL scales as continuous rather than ordinal variables regardless of the number of categories. The purpose of this study was to compare the results of analyzing HRQoL longitudinal data under both approaches, continuous and ordinal.

Methods

The EORTC QLQ-C30 scores of two EORTC randomized brain cancer clinical trials (26951 and 26981) were analyzed using the two approaches. In the 26951 trial, a total of 368 patients were randomly assigned to receive either radiotherapy (RT) or the same RT plus procarbazine, CCNU, and vincristine. In the 26981 trial, 573 patients were randomly allocated to RT or RT plus temozolomide. Comparison of the two treatment arms was done using methods for longitudinal quantitative and longitudinal ordinal data. Both statistical methods were adapted to account for missing data and compared in terms of statistical significance of the results (p values) but also with respect to data interpretation.

Results

Three scales, i.e., appetite loss, insomnia, and drowsiness, presenting four response categories ("Not at all", "A little", "Quite a bite", and "Very much") were analyzed in each trial. Both statistical methods (continuous and ordinal) showed statistically significant differences between the two treatments, not only globally but also at the same assessment time points. The magnitude of the p values, however, varied at some time points and was less pronounced in the ordinal approach.

Conclusions

The analysis of the two clinical trials showed that treating the HRQoL scales by a quantitative or an ordinal method did not make much difference as far as statistical significance was concerned. The interpretation of results, however, was easier under the ordinal approach. Treatment effects may be more meaningful when expressed in terms of odds ratios than as mean values, particularly when the number categories is small.     

Measuring inequality in self-reported health-discussion of a recently suggested approach using Finnish data

Health surveys often include a general question on self-assessed health (SAH), usually measured on an ordinal scale with three to five response categories, from 'very poor' or 'poor' to 'very good' or 'excellent'. This paper assesses the scaling of responses on the SAH question. It compares alternative procedures designed to impose cardinality on the ordinal responses. These include OLS, ordered probit and interval regression approaches. The cardinal measures of health are used to compute and decompose concentration indices for income-related inequality in health. Results are provided using Finnish data on 15D and the SAH questions. Further evidence emerges for the internal validity of a method used in a pioneering study by van Doorslaer and Jones which was based on Canadian data on the McMaster Health Utility Index Mark III (HUI) and SAH. The study validates the conclusions drawn by van Doorslaer and Jones. It confirms that the interval regression approach is superior to OLS and ordered probit regression in assessing health inequality. However, regarding the choice of scaling instrument, it is concluded that the scaling of SAH categories and, consequently, the measured degree of inequality, are sensitive to characteristics of the chosen scaling instrument.     

Misclassification among methods used for multiple group discrimination--the effects of distributional properties

Methods of multiple group discriminant analysis have not been fully studied with respect to classification into more than two populations when the covariate distributions are normal or non-normal. The present study examines the classification performance of several multiple discrimination methods under a variety of simulated continuous normal and non-normal covariate distributions. The methods include polychotomous logistic regression, multiple group linear discriminant analysis, kernel density estimation, and rank transformations of the data as input into the linear function. The parameters of interest were distance among populations, configuration of population mean vectors (collinear or forming the vertices of a regular simplex), skewness, kurtosis and bimodality. Simulation of the last three parameters was by log-normal, sinh-1 normal and a two-component mixture of normal distributions, respectively. Results with three trivariate populations show that for all distributions, logistic discrimination classifies close to the optimal under Neyman-Pearson allocation. These results suggest that logistic discrimination is preferable to other widely-used methods for multiple group classification with non-normal data, and is comparable to classification by multiple linear discrimination with normal data.     

Physical functioning and mortality in older women: an assessment of energy costs and level of difficulty

OBJECTIVE: We tested the hypothesis that the types of activity and the energy equivalent assigned to each activity affect the relationship between self-reported physical functioning and mortality. METHODS: We analyzed the relationship between physical functioning and cardiovascular and noncardiovascular mortality in 1230 women (median age 70 years) observed for 7.5 years. We evaluated five separate scores of physical functioning that differed in the method of scoring the responses. Cox proportional hazard models included baseline age, self-reported physical functioning, medical morbidity, and assessment of health. RESULTS: For cardiac and noncardiovascular mortality, greater self-reported functioning was associated independently with a decreased hazard of death. The effects of physical functioning were sensitive to the form of the score used; that is, a score based on ordinal responses was associated with a greater reduction in hazard of death difference in survival between high and low function score: Ordinal: -15.2% (95% confidence interval [CI] -25.2--4.0); dichotomous: -11.6% (95%CI -18.9--3.9). CONCLUSION: There is a consistent relationship between functional limitation and all causes of mortality. The association is sensitive to the form of the score. Future physical function scores should be based on ordinal responses to individual items used in the scores.     

A novel modeling framework for ordinal data defined by collapsed counts

Adolescent alcohol use is a serious public health concern. Despite advances in the theoretical conceptualization of pathways to alcohol use, researchers are limited by the statistical techniques currently available. Researchers often fit linear models and restrictive categorical models (e.g., proportional odds models) to ordinal data with many response categories defined by collapsed count data (0 drinking days, 1–2days, 3–6days, etc.). Consequently, existing models ignore the underlying count process, resulting in disjoint between the construct of interest and the models being fitted. Our proposed ordinal modeling approach overcomes this limitation by explicitly linking ordinal responses to a suitable underlying count distribution. In doing so, researchers can use maximum likelihood estimation to fit count models to ordinal data as if they had directly observed the underlying discrete counts. The usefulness of our ordinal negative binomial and ordinal zero‐inflated negative binomial models is verified by simulation studies. We also demonstrate our approach using real empirical data from the 2010 National Survey of Drug Use and Health. Results show the benefit of the proposed ordinal modeling framework compared with existing methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Accuracy of three dry-chemistry methods for lipid profiling and risk factor classification

The purpose of this project was to determine the accuracy in lipids measurement and risk factor classification using Reflotron, Cholestech, and Ektachem DT-60 dry-chemistry analyzers. Plasma and capillary venous blood from fasting subjects (n = 47) were analyzed for total cholesterol (TC), high density lipoprotein (HDL-C), and triglycerides (TG) using these analyzers and a CDC certified laboratory. Accuracy was evaluated by comparing the results of each portable analyzer against the CDC reference method. One-way ANOVAs were performed for TC, HDL-C, and TG between all portable analyzers and the reference method. Chi-square was used for risk classification (2001 NIH Guidelines). Compared to the reference method, the Ektachem and Reflotron provided significantly lower values for TC (p < .05). In addition, the Cholestech and Ektachem values for HDL-C were higher than the CDC (p < .05). The Reflotron and Cholestech provided higher values of TG than the CDC (p < .05). Chi-squares analyses for risk classification were not significant (p > .45) between analyzers. According to these results, the Ektachem and Cholestech analyzers met the current NCEP III guidelines for accuracy in measurement of TC, while only Ektachem met guidelines for TG. All 3 analyzers provided a good overall risk classification; however, values of HDL-C should be only used for screening purposes.     

Development of a clinical prediction model for an ordinal outcome: the World Health Organization Multicentre Study of Clinical Signs and Etiological Agents of Pneumonia,Sepsis and Meningitis in Young Infants

This paper describes the methodologies used to develop a prediction model to assist health workers in developing countries in facing one of the most difficult health problems in all parts of the world: the presentation of an acutely ill young infant. Statistical approaches for developing the clinical prediction model faced at least two major difficulties. First, the number of predictor variables, especially clinical signs and symptoms, is very large, necessitating the use of data reduction techniques that are blinded to the outcome. Second, there is no uniquely accepted continuous outcome measure or final binary diagnostic criterion. For example, the diagnosis of neonatal sepsis is ill-defined. Clinical decision makers must identify infants likely to have positive cultures as well as to grade the severity of illness. In the WHO/ARI Young Infant Multicentre Study we have found an ordinal outcome scale made up of a mixture of laboratory and diagnostic markers to have several clinical advantages as well as to increase the power of tests for risk factors. Such a mixed ordinal scale does present statistical challenges because it may violate constant slope assumptions of ordinal regression models. In this paper we develop and validate an ordinal predictive model after choosing a data reduction technique. We show how ordinality of the outcome is checked against each predictor. We describe new but simple techniques for graphically examining residuals from ordinal logistic models to detect problems with variable transformations as well as to detect non-proportional odds and other lack of fit. We examine an alternative type of ordinal logistic model, the continuation ratio model, to determine if it provides a better fit. We find that it does not but that this model is easily modified to allow the regression coefficients to vary with cut-offs of the response variable. Complex terms in this extended model are penalized to allow only as much complexity as the data will support. We approximate the extended continuation ratio model with a model with fewer terms to allow us to draw a nomogram for obtaining various predictions. The model is validated for calibration and discrimination using the bootstrap. We apply much of the modelling strategy described in Harrell, Lee and Mark (Statist. Med. 15 , 361–387 (1998)) for survival analysis, adapting it to ordinal logistic regression and further emphasizing penalized maximum likelihood estimation and data reduction. © 1998 John Wiley & Sons, Ltd.