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.     

Ordinal logistic regression models: application in quality of life studies

Quality of life has been increasingly emphasized in public health research in recent years. Typically, the results of quality of life are measured by means of ordinal scales. In these situations, specific statistical methods are necessary because procedures such as either dichotomization or misinformation on the distribution of the outcome variable may complicate the inferential process. Ordinal logistic regression models are appropriate in many of these situations. This article presents a review of the proportional odds model, partial proportional odds model, continuation ratio model, and stereotype model. The fit, statistical inference, and comparisons between models are illustrated with data from a study on quality of life in 273 patients with schizophrenia. All tested models showed good fit, but the proportional odds or partial proportional odds models proved to be the best choice due to the nature of the data and ease of interpretation of the results. Ordinal logistic models perform differently depending on categorization of outcome, adequacy in relation to assumptions, goodness-of-fit, and parsimony.     

Ordinal regression models for epidemiologic data

Health status is often measured in epidemiologic studies on an ordinal scale, but data of this type are generally reduced for analysis to a single dichotomy. Several statistical models have been developed to make full use of information in ordinal response data, but have not been much used in analyzing epidemiologic studies. The authors discuss two of these statistical models--the cumulative odds model and the continuation ratio model. They may be interpreted in terms of odds ratios, can account for confounding variables, have clear and testable assumptions, and have parameters that may be estimated and hypotheses that may be tested using available statistical packages. However, calculations of asymptotic relative efficiency and results of simulations showed that simple logistic regression applied to dichotomized responses can in some realistic situations have more than 75% of the efficiency of ordinal regression models, but only if the ordinal scale is collapsed into a dichotomy close to the optimal point. The application of the proposed models to data from a study of chest x-rays of workers exposed to mineral fibers confirmed that they are easy to use and interpret, but gave results quite similar to those obtained using simple logistic regression after dichotomizing outcome in the conventional way.     

Controlled pattern imputation for sensitivity analysis of longitudinal binary and ordinal outcomes with nonignorable dropout

The controlled imputation method refers to a class of pattern mixture models that have been commonly used as sensitivity analyses of longitudinal clinical trials with nonignorable dropout in recent years. These pattern mixture models assume that participants in the experimental arm after dropout have similar response profiles to the control participants or have worse outcomes than otherwise similar participants who remain on the experimental treatment. In spite of its popularity, the controlled imputation has not been formally developed for longitudinal binary and ordinal outcomes partially due to the lack of a natural multivariate distribution for such endpoints. In this paper, we propose 2 approaches for implementing the controlled imputation for binary and ordinal data based respectively on the sequential logistic regression and the multivariate probit model. Efficient Markov chain Monte Carlo algorithms are developed for missing data imputation by using the monotone data augmentation technique for the sequential logistic regression and a parameter‐expanded monotone data augmentation scheme for the multivariate probit model. We assess the performance of the proposed procedures by simulation and the analysis of a schizophrenia clinical trial and compare them with the fully conditional specification, last observation carried forward, and baseline observation carried forward imputation methods.     

Alternative parameterization of polychotomous models: theory and application to matched case-control studies

A method is proposed for transforming a class of models having an outcome variable with more than two levels into an equivalent binary model. The polychotomous logistic model is used to demonstrate the method. The equivalency to a simple logistic regression model after some data transformation (augmentation) is shown. The method is applied to the data from two case-control studies each with two control groups, and further applications are indicated.     

Using logistic regression in perinatal epidemiology: an introduction for clinical researchers. Part 1: basic concepts

Logistic regression is a statistical modelling technique which may be applied to estimate the simultaneous effect of a set of predictors (e.g. gestational age, birthweight) on the risk of a certain outcome variable (e.g. neonatal death) which can take either one of two possible values (yes/no, alive/dead) or in the situation where one wants to estimate the effect of a particular risk factor (e.g. sex) while adjusting (correcting) for the effect of other risk factors (e.g. gestational age). Since this situation often occurs both in medical or epidemiological research and in daily practice it is important to have a flexible and readily interpretable technique to predict risk of mortality and morbidity. Since the logistic regression technique is a powerful and widely applicable tool which is appearing more and more often in the epidemiological literature, a basic understanding of this technique becomes necessary for the clinical researcher. In this paper we explain logistic regression to medical researchers who do not have any particular statistical background. Part 1 covers the basic concepts. Part 2 will describe the actual representation of the basic concepts in a logistic framework.     

A goodness‐of‐fit test for the proportional odds regression model

We examine goodness‐of‐fit tests for the proportional odds logistic regression model—the most commonly used regression model for an ordinal response variable. We derive a test statistic based on the Hosmer–Lemeshow test for binary logistic regression. Using a simulation study, we investigate the distribution and power properties of this test and compare these with those of three other goodness‐of‐fit tests. The new test has lower power than the existing tests; however, it was able to detect a greater number of the different types of lack of fit considered in this study. Moreover, the test allows for the results to be summarized in a contingency table of observed and estimated frequencies, which is a useful supplementary tool to assess model fit. We illustrate the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents. The test proposed in this paper is similar to a recently developed goodness‐of‐fit test for multinomial logistic regression. A unified approach for testing goodness of fit is now available for binary, multinomial, and ordinal logistic regression models. Copyright © 2012 John Wiley & Sons, Ltd.     

Fitting stratified proportional odds models by amalgamating conditional likelihoods

Classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum-specific nuisance parameters. When the outcome variable has multiple ordered categories, a natural choice for the outcome model is a stratified proportional odds or cumulative logit model. However, classical conditioning techniques do not apply to the general K-category cumulative logit model (K>2) with varying stratum-specific intercepts as there is no reduction due to sufficiency; the nuisance parameters remain in the conditional likelihood. We propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale. The method allows for categorical and continuous covariates in a general regression framework. We provide a robust sandwich estimate of the variance of the proposed estimator. For binary exposures, we show equivalence of our approach to the estimators already proposed in the literature. The proposed recipe can be implemented very easily in standard software. We illustrate the methods via three real data examples related to biomedical research. Simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished.     

多分类 logistic 回归在冠心病危险因素研究中的应用

目的：应用多分类logistic回归找出冠心病危险因素与冠状动脉病变程度间的关系,建立冠状动脉病变程度危险因素的“最优”回归方程。方法采取系统抽样的方法,对来自2005年1月至2009年12月期间山东省潍坊市某几所医院心血管内科初步诊断为冠心病并进行冠脉造影的患者病例,抽取256例作为多分类logistic回归分析的样本。冠心病组根据狭窄病变累及血管范围分为1支病变（单支病变组）、2支病变（双支病变组）和3支及以上病变（多支病变组）。通过多分类logistic回归方法分析冠心病与冠脉病变程度相关关系。结果以冠脉病变程度为因变量,各个因素为自变量。根据单因素分析的结果,从20个研究因素中筛选出8个有统计学意义的影响因素,经过数据相关性分析、共线性诊断及专业解释等,筛选出8个影响因素进入多分类logistic回归中,最后得到与冠脉病变程度有关的有统计学意义的影响因素6个,分别是年龄、合并疾病、心率、血糖、脂蛋白（a）;保护因素1个：X17（载脂蛋白A1）。根据筛选出的6个影响因素建立“最优”回归方程。结论应用多分类logistic回归找出与冠脉病变程度有关的危险因素,并定量分析出各危险因素在冠脉病变不同程度上的概率值。     

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.     

多分类有序反应变量logistic回归应用条件的检验

目的 对多分类有序反应变量logistic回归的应用条件寻求科学合理的检验方法。方法 使用卡方分布的理论，SAS软件及抽样调查方法。结果 设计出多分类有序反应变量logistic回归应用条件的卡方检验方法，推导出反应变量取各水平的概率计算公式及卡方检验中理论值、自由度的计算公式，并在作者主持的国家医师资格临床实践技能考试研究中取得了成功效果。结论 多分类有序反应变量logistic回归得到完善和补充，具有较大的理论和实际意义。     

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.     

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.     

Exploratory causal modeling in epidemiology: are all factors created equal?

The purpose of this study was to demonstrate the consequences of analyzing sequentially caused relationships with models assuming equally proximate causation. Monte Carlo simulations of data with well defined causations were performed. The logistic modeling approach was strongly misleading if a distant causal factor was treated as a factor being equally distant to the outcome as a proximal causal factor. In contrast, simple pathway analysis was able to correctly identify the true causation. In causal pathways, the relative risk of an intermediate cause with respect to the outcome needs to have a certain magnitude for the effect of the distant variable to be passed on. The results further show that the true relative risk of the distant variable is not dependent on its baseline prevalence. In contrast, the prevalence of the intermediate variable must be small enough to carry the influence of the distant variable through the causal chain. Practical epidemiologic exploration of etiological factors is presently dominated by stepwise multiple regression. This type of exploration is not model free but is often intuitively based on the structural assumption of equal proximity of all potential factors to the outcome. Equal proximity, however, is not likely in many etiologies, especially not if the causal factors under consideration are of different quality, like psychological and biological factors. In cases of causal pathways with some factors more distant and others more proximal to the outcome, the former tend to be dismissed by equal proximity modeling. Upstream exploration of more distant etiological factors is hindered by endemic stepwise multiple regression modeling, treating all variables as being equal in proximity to the outcome.     

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.     

A model for incomplete longitudinal multivariate ordinal data

In studies where multiple outcome items are repeatedly measured over time, missing data often occur. A longitudinal item response theory model is proposed for analysis of multivariate ordinal outcomes that are repeatedly measured. Under the MAR assumption, this model accommodates missing data at any level (missing item at any time point and/or missing time point). It allows for multiple random subject effects and the estimation of item discrimination parameters for the multiple outcome items. The covariates in the model can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is described utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher-scoring solution, which provides standard errors for all model parameters, is used. A data set from a longitudinal prevention study is used to motivate the application of the proposed model. In this study, multiple ordinal items of health behavior are repeatedly measured over time. Because of a planned missing design, subjects answered only two-third of all items at a given point.     

Sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome

In clinical research, investigators are interested in inferring the average causal effect of a treatment. However, the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes. Although some definitions have been proposed, they are limited in that they are not identical to the well‐defined causal risk for a binary outcome, which is the simplest ordinal outcome. In this paper, we propose the use of a causal parameter for an ordinal outcome, defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition. For a binary outcome, this proportion is identical to the causal risk. Unfortunately, the proposed causal parameter cannot be identified, even under randomization. Therefore, we present a numerical method to calculate the sharp nonparametric bounds within a sample, reflecting the impact of confounding. When the assumption of independent potential outcomes is included, the causal parameter can be identified when randomization is in play. Then, we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization‐based approach, which are an extension of the existing methods for a binary outcome. Our methodologies are illustrated using data from an emetic prevention clinical trial.     

Ordinal regression model and the linear regression model were superior to the logistic regression models

OBJECTIVE: Ordinal scales often generate scores with skewed data distributions. The optimal method of analyzing such data is not entirely clear. The objective was to compare four statistical multivariable strategies for analyzing skewed health-related quality of life (HRQOL) outcome data. HRQOL data were collected at 1 year following catheterization using the Seattle Angina Questionnaire (SAQ), a disease-specific quality of life and symptom rating scale. STUDY DESIGN AND SETTING: In this methodological study, four regression models were constructed. The first model used linear regression. The second and third models used logistic regression with two different cutpoints and the fourth model used ordinal regression. To compare the results of these four models, odds ratios, 95% confidence intervals, and 95% confidence interval widths (i.e., ratios of upper to lower confidence interval endpoints) were assessed. RESULTS: Relative to the two logistic regression analysis, the linear regression model and the ordinal regression model produced more stable parameter estimates with smaller confidence interval widths. CONCLUSION: A combination of analysis results from both of these models (adjusted SAQ scores and odds ratios) provides the most comprehensive interpretation of the data.     

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.     

Ranked set sampling for efficient estimation of a population proportion

Ranked set sampling (RSS) is a sampling procedure that can be considerably more efficient than simple random sampling (SRS). It involves preliminary ranking of the variable of interest to aid in sample selection. Although ranking processes for continuous variables that are implemented through either subjective judgement or via the use of a concomitant variable have been studied extensively in the literature, the use of RSS in the case of a binary variable has not been investigated thoroughly. In this paper we propose the use of logistic regression to aid in the ranking of a binary variable of interest. We illustrate the application of RSS to estimation of a population proportion with an example based on the National Health and Nutrition Examination Survey III data set. Our results indicate that this use of logistic regression improves the accuracy of the preliminary ranking in RSS and leads to substantial gains in precision for estimation of a population proportion.