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

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. © 1996 Wiley-Liss, Inc.     

Three methods for analysing correlated ROC curves: a comparison in real data sets from multi-reader,multi-case studies with a factorial design

This paper compares three published methods for analysing multiple correlated ROC curves: a method using generalized estimating equations with marginal non-proportional ordinal regression models; a method using jackknifed pseudovalues of summary statistics; a method using a corrected F-test from analysis of variance of summary statistics. Use of these methods is illustrated through six real data examples from studies with the common factorial design, that is, multiple readers interpreting images obtained with each test modality on each study subject. The issue of the difference between typical summary statistics and summary statistics from typical ROC curves is explored. The examples also address similarities and differences among the analytical methods. In particular, while point estimates of differences between test modalities are similar, the standard errors of these differences do not agree for all three methods. A simulation study supports the standard errors provided by the generalized estimating equations with marginal non-proportional ordinal regression models.     

Likelihood-based modelling of age-related normal ranges for ordinal measurements: changes in visual acuity through early childhood

This paper investigates the construction of age-related standards for ordinal outcome data. Asymmetric logistic models are used to describe the age-related changes in the cumulative probabilities for each ordinal level. Maximum likelihood estimation of model parameters allows the use of likelihood ratio tests to ascertain the appropriate model complexity. In contrast to methodologies for constructing age-related standards where the outcome is continuous, we show how the methodology leads directly to centile estimation for individuals. The method is illustrated using visual acuity measurements collected from 2968 children between 2 and 9 years of age made on a 30-point ordinal scale. We show how, in this instance, smoothing of parameters across ordinal categories leads to reduced validity of the centiles, justifying the need for specialized methodology for non-continuous outcomes.     

OPTIMAL DISCRIMINANT ANALYSIS FOR ORDINAL RESPONSES

Optimal classification formulation is adapted to the context of discrimination when the response is ordinal. The resulting method, optimal discriminant analysis for ordinal responses (ODAO), is presented and compared with two reference discrimination techniques used in this context (proportional-odds ordinal logistic regression and normal discrimination) using a study of prognosis following burn injuries and simulated data. The ODAO method clearly outperforms both reference methods in terms of classification accuracy (in training and validation samples), robustness to outliers, simplicity of use and applicability in clinical settings. ODAO is a promising method for improving classification performance in discrimination with ordinal responses and merits further investigation. © 1997 by John Wiley & Sons, Ltd.     

A mixed effects model for the analysis of ordinal longitudinal pain data subject to informative drop-out

We extend the model of Pulkstenis et al. that models binary longitudinal data, subject to informative drop-out through remedication, to the ordinal response case. We present a selection model shared-parameter approach that specifies mixed models for both ordinal response and discrete survival time to remedication. In this fashion, the random parameter present in both models completely characterizes the relationship between response and time to remedication inducing their conditional independence. With a log-log link function for both response and study 'survival', as well as specification of a log-gamma distribution for the random effect, we obtain a closed-form expression for the marginal log-likelihood of response and time to remedication that does not require approximation or numerical integration techniques. A data analysis is performed and simulation results presented which support the consistency of parameter and standard error estimates.     

The trend odds model for ordinal data

Ordinal data appear in a wide variety of scientific fields. These data are often analyzed using ordinal logistic regression models that assume proportional odds. When this assumption is not met, it may be possible to capture the lack of proportionality using a constrained structural relationship between the odds and the cut‐points of the ordinal values. We consider a trend odds version of this constrained model, wherein the odds parameter increases or decreases in a monotonic manner across the cut‐points. We demonstrate algebraically and graphically how this model is related to latent logistic, normal, and exponential distributions. In particular, we find that scale changes in these potential latent distributions are consistent with the trend odds assumption, with the logistic and exponential distributions having odds that increase in a linear or nearly linear fashion. We show how to fit this model using SAS Proc NLMIXED and perform simulations under proportional odds and trend odds processes. We find that the added complexity of the trend odds model gives improved power over the proportional odds model when there are moderate to severe departures from proportionality. A hypothetical data set is used to illustrate the interpretation of the trend odds model, and we apply this model to a swine influenza example wherein the proportional odds assumption appears to be violated. Copyright © 2012 John Wiley & Sons, Ltd.     

POPULATION-AVERAGED AND CLUSTER-SPECIFIC MODELS FOR CLUSTERED ORDINAL RESPONSE DATA

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.     

Ordered multinomial regression for genetic association analysis of ordinal phenotypes at Biobank scale

Logistic regression is the primary analysis tool for binary traits in genome-wide association studies (GWAS). Multinomial regression extends logistic regression to multiple categories. However, many phenotypes more naturally take ordered, discrete values. Examples include (a) subtypes defined from multiple sources of clinical information and (b) derived phenotypes generated by specific phenotyping algorithms for electronic health records (EHR). GWAS of ordinal traits have been problematic. Dichotomizing can lead to a range of arbitrary cutoff values, generating inconsistent, hard to interpret results. Using multinomial regression ignores trait value hierarchy and potentially loses power. Treating ordinal data as quantitative can lead to misleading inference. To address these issues, we analyze ordinal traits with an ordered, multinomial model. This approach increases power and leads to more interpretable results. We derive efficient algorithms for computing test statistics, making ordinal trait GWAS computationally practical for Biobank scale data. Our method is available as a Julia package OrdinalGWAS.jl. Application to a COPDGene study confirms previously found signals based on binary case–control status, but with more significance. Additionally, we demonstrate the capability of our package to run on UK Biobank data by analyzing hypertension as an ordinal trait.     

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.     

Ordinal and cardinal measures of health inequality: an empirical comparison

When measuring health inequality using ordinal data, analysts typically must choose between indices specifically based upon ordinal data and more standard indices using ordinal data, which has been transformed into cardinal data. This paper compares inequality rankings across a number of different approaches and finds considerable sensitivity to the choice between ordinal‐ and cardinal‐based indices. There is relatively little sensitivity to the ethical choices made by the analyst in terms of the weight attached to different parts of the distribution. Copyright © 2009 John Wiley & Sons, Ltd.     

Proportional‐odds models for repeated composite and long ordinal outcome scales

In many medical studies, researchers widely use composite or long ordinal scores, that is, scores that have a large number of categories and a natural ordering often resulting from the sum of a number of short ordinal scores, to assess function or quality of life. Typically, we analyse these using unjustified assumptions of normality for the outcome measure, which are unlikely to be even approximately true. Scores of this type are better analysed using methods reserved for more conventional (short) ordinal scores, such as the proportional‐odds model. We can avoid the need for a large number of cut‐point parameters that define the divisions between the score categories for long ordinal scores in the proportional‐odds model by the inclusion of orthogonal polynomial contrasts. We introduce the repeated measures proportional‐odds logistic regression model and describe for long ordinal outcomes modifications to the generalized estimating equation methodology used for parameter estimation. We introduce data from a trial assessing two surgical interventions, briefly describe and re‐analyse these using the new model and compare inferences from the new analysis with previously published results for the primary outcome measure (hip function at 12 months postoperatively). We use a simulation study to illustrate how this model also has more general application for conventional short ordinal scores, to select amongst competing models of varying complexity for the cut‐point parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

A spatial scan statistic for ordinal data

Spatial scan statistics are widely used for count data to detect geographical disease clusters of high or low incidence, mortality or prevalence and to evaluate their statistical significance. Some data are ordinal or continuous in nature, however, so that it is necessary to dichotomize the data to use a traditional scan statistic for count data. There is then a loss of information and the choice of cut-off point is often arbitrary. In this paper, we propose a spatial scan statistic for ordinal data, which allows us to analyse such data incorporating the ordinal structure without making any further assumptions. The test statistic is based on a likelihood ratio test and evaluated using Monte Carlo hypothesis testing. The proposed method is illustrated using prostate cancer grade and stage data from the Maryland Cancer Registry. The statistical power, sensitivity and positive predicted value of the test are examined through a simulation study.     

Regression with an ordered categorical response

A survey on Mseleni joint disease in South Africa involved the scoring of pelvic X-rays of women to measure osteoporosis. The scores were ordinal by construction and ranged from 0 to 12. It is standard practice to use ordinary regression techniques with an ordinal response that has that many categories. We give evidence for these data that the constraints on the response result in a misleading regression analysis. McCullagh's proportional-odds model is designed specifically for the regression analysis of ordinal data. We demonstrate the technique on these data, and show how it fills the gap between ordinary regression and logistic regression (for discrete data with two categories). In addition, we demonstrate non-parametric versions of these models that do not make any linearity assumptions about the regression function.     

Zero inflation in ordinal data: incorporating susceptibility to response through the use of a mixture model.

The aim of this paper is to produce a methodology that will allow users of ordinal scale data to more accurately model the distribution of ordinal outcomes in which some subjects are susceptible to exhibiting the response and some are not (i.e. the dependent variable exhibits zero inflation). This situation occurs with ordinal scales in which there is an anchor that represents the absence of the symptom or activity, such as 'none', 'never' or 'normal,' and is particularly common when measuring abnormal behavior, symptoms, and side effects. Due to the unusually large number of zeros, traditional statistical tests of association can be non-informative. We propose a mixture model for ordinal data with a built-in probability of non-response, which allows modeling of the range (e.g. severity) of the scale, while simultaneously modeling the presence/absence of the symptom. Simulations show that the model is well behaved and a likelihood ratio test can be used to choose between the zero-inflated and the traditional proportional odds model. The model, however, does have minor restrictions on the nature of the covariates that must be satisfied in order for the model to be identifiable. The method is particularly relevant for public health research such as large epidemiological surveys where more careful documentation of the reasons for response may be difficult.     

Quality assessment of ordinal scale reproducibility: log-linear models provided useful information on scale structure

OBJECTIVE: In health research, ordinal scales are extensively used. Reproducibility of ratings using these scales is important to assess their quality. This study aimed to compare two methods analyzing reproducibility: weighted Kappa statistic and log-linear models. STUDY DESIGN AND SETTING: Contributions of each method to the reproducibility assessment of ratings using ordinal scales were compared using intra- and interobserver data chosen in three different fields: Crow's feet scale in dermatology, dysplasia scale in oncology, updated Sydney scale in gastroenterology. RESULTS: Both methods provided an agreement level. In addition, log-linear models allowed evaluation of the structure of agreement. For the Crow's feet scale, both methods gave equivalent high agreement levels. For the dysplasia scale, log-linear models highlighted scale defects and Kappa statistic showed a moderate agreement. For the updated Sydney scale, log-linear models underlined a null distinguishability between two adjacent categories, whereas Kappa statistic gave a high global agreement level. CONCLUSION: Methods that can investigate level and structure of agreement between ordinal ratings are valuable tools, since they may highlight heterogeneities within the scales structure and suggest modifications to improve their reproducibility.     

Quantile estimation following non-parametric phase I clinical trials with ordinal response

A non-parametric multi-dimensional isotonic regression estimator is developed for use in estimating a set of target quantiles from an ordinal toxicity scale. We compare this estimator to the standard parametric maximum likelihood estimator from a proportional odds model for extremely small data sets. A motivating example is from phase I oncology clinical trials, where various non-parametric designs have been proposed that lead to very small data sets, often with ordinal toxicity response data. Our comparison of estimators is performed in conjunction with three of these non-parametric sequential designs for ordinal response data, two from the literature and a new design based on a random walk rule. We also compare with a non-parametric design for binary response trials, by keeping track of ordinal data for estimation purposes, but dichotomizing the data in the design phase. We find that a multidimensional isotonic regression-based estimator far exceeds the others in terms of accuracy and efficiency. A rule by Simon et al. (J. Natl. Cancer Inst. 1997; 89:1138-1147) yields particularly efficient estimators, more so than the random walk rule, but has higher numbers of dose-limiting toxicity. A small data set from a leukemia clinical trial is analysed using our multidimensional isotonic regression-based estimator.     

Inter-rater reliability of pressure ulcer staging: ordinal probit Bayesian hierarchical model that allows for uncertain rater response

This article describes a method for estimating the inter-rater reliability of pressure ulcer (PU) staging (stages I-IV) from raters in National Database of Nursing Quality Indicators (NDNQI) participating hospitals. The method models ordinal spanning data utilizing an ordinal probit Bayesian hierarchical model (BHM) across several hospitals in which raters monitor patient's PUs. An ulcer that cannot be accurately assessed because the base of the wound cannot be seen is defined as unstageable. Our novel approach allows for an unstageable PU rating to be included in the analysis. We compare the ordinal probit BHM to an approximate random-effects (standard approach in the literature) model that assumes that the raw ordinal data are continuous.     

Analysis of failure time data with ordinal categories of response.

When failure times are observed, additional information concerning the type of failure is often recorded. A method which simultaneously models the failure times and additional information in the form of ordinal categories is discussed. An application to clinical trial data, in which the failure times are times of onset of headache, and the headaches are classified into the ordinal categories mild, moderate and severe, illustrates how this method may be used and how the final model can be interpreted. The continuation ratio model, which is used in this method, is described in detail.     

Nonparametric estimation of broad sense agreement between ordinal and censored continuous outcomes

The concept of broad sense agreement (BSA) has recently been proposed for studying the relationship between a continuous measurement and an ordinal measurement. They developed a nonparametric procedure for estimating the BSA index, which is only applicable to completely observed data. In this work, we consider the problem of evaluating BSA index when the continuous measurement is subject to censoring. We propose a nonparametric estimation method built upon a derivation of a new functional representation of the BSA index, which allows for accommodating censoring by plugging in the nonparametric survival function estimators. We establish the consistency and asymptotic normality for the proposed BSA estimator. We also investigate an alternative approach based on the strategy of multiple imputation, which is shown to have better empirical performance with small sample sizes than the plug-in method. Extensive simulation studies are conducted to evaluate our proposals. We illustrate our methods via an application to a Surgical Intensive Care Unit study.     

A semi-parametric approach to impute mixed continuous and categorical data

We propose an extension of the method presented in Helenowski and Demirtas (2013) involving imputing mixed continuous and binary data to data involving categorical variables with three or more levels. In a bivariate case, the medians for the continuous variable will be computed by each level of the categorical variable and the categorical variable will be ranked as an ordinal variable with respect to these medians, so that each ordinal level assigned to a categorical level is determined by the rank order of medians of the continuous variable for that category. In a multivariate case, the categorical variables are ordered with respect to the continuous variable for which the range among the medians is the largest. Here, ‘bivariate’ indicates that the data set includes two variables while ‘multivariate’ indicates that the data set includes three or more variables. The pairwise correlation between the continuous and ordinal variable is then computed. Data will then be transformed to normally distributed values, imputed via joint modeling, and back-transformed to the original scale via the Barton and Schruben (1993) technique for the continuous variable and quantiles based on the original probabilities of the categorical variable. The algorithm is re-iterated until the absolute difference of the pairwise correlations from the original and imputed data is less than some constant c chosen to maximize the coverage rate and minimize standardized bias. Results from simulations applied to artificial data and to real data involving 74 colorectal patients indicate that our technique as promising.