Models For Ordinal Outcomes Research Articles

Comparative Analysis of Frequentist and Bayesian Approaches in Fitting Stereotype Models for Ordinal Outcomes

Introduction: Stereotype regression models provide a parsimonious solution for analyzing ordinal response variables. When the proportional odds assumption is violated, these models offer a viable alternative to more commonly used cumulative logit models. However, their adoption in research remains limited due to a lack of standardization. Our study compares frequentist and Bayesian approaches for fitting stereotype models for ordinal outcomes, elucidating the benefits of each method to encourage broader utilization. Methods: We simulated ordinal data to contrast a Bayesian approach for an ordered stereotype model with two frequentist methods in R: Reduced-Rank Vector Generalized Linear Models (RRVGLM) for unordered scores and Ordered Stereotype Model (OSM) for ordered scores. Metrics included mean squared error (MSE) and bias across multiple simulation scenarios with various sample sizes and the introduction of multicollinear predictors. Lastly, a real dataset was utilized to demonstrate the application of these approaches. Results: Both frequentist methods exhibited errors in simulations and real data when the sample size was small and when multicollinearity was present. In simulation scenarios with small sample size (N=50, 70), frequentist methods often failed to converge or produced large standard errors, while the Bayesian approach always converged and yielded lower MSEs. In scenarios with large sample sizes (N=300, 500), all methods produced comparable MSEs. However, frequentist methods produced slightly less biased estimates. Conclusion: RRVGLM offers fast, accurate results but may encounter errors or produce unordered scores complicating interpretation. In these cases, OSM may provide better results. Bayesian models excel with small sample sizes and complex data with issues such as multicollinearity but require more computation time.

Embolization therapy for type 2 endoleaks after endovascular aortic aneurysm repair: imaging-based predictive factors and clinical outcomes on long-term follow-up

To evaluate the technical, radiological, and clinical outcomes after type 2 endoleak (T2EL) embolization in patients with a growing aneurysm sac after endovascular aortic aneurysm repair (EVAR). Additionally, to determine clinical and imaging-based factors for outcome prediction after embolization of a T2EL. A single-institution, retrospective analysis was performed of 60 patients who underwent a T2EL embolization procedure between September 2005 and August 2016 to treat a growing aneurysm sac diameter following EVAR. The patients' electronic medical records and all available pre- and post-embolization imaging were reviewed. Statistical analysis methods included logistic regression models for binary outcomes, proportional odds models for ordinal outcomes, and linear regression models for continuous outcomes. The Kaplan-Meier method was used to estimate the overall survival probability. Technical, radiological, and clinical success rates after T2EL embolization were 95% (n = 57), 26.7% (n = 16), and 76.7% (n = 46), respectively. Persistent aneurysm sac expansion was found in 31 patients (51.7%). Unsharp or blurred T2EL delineation on pre-interventional computed tomography (CT) was a predictive factor for a post-embolization persistent visible endoleak and persistent growth of the aneurysm sac (P = 0.025). Median survival after T2EL embolization was 5.35 years, with no difference observed between patients with persistent sac expansion compared with patients with stable or decreased sac diameter. Progression of the aneurysm sac diameter was observed in half the study patients, despite technically successful T2EL embolization. Unsharp or blurred T2EL delineation on pre-interventional CT seemed to be an imaging-based predictor for a persistent T2EL and progressive aneurysm sac growth after embolization.

AutoScore-Ordinal: an interpretable machine learning framework for generating scoring models for ordinal outcomes

BackgroundRisk prediction models are useful tools in clinical decision-making which help with risk stratification and resource allocations and may lead to a better health care for patients. AutoScore is a machine learning–based automatic clinical score generator for binary outcomes. This study aims to expand the AutoScore framework to provide a tool for interpretable risk prediction for ordinal outcomes.MethodsThe AutoScore-Ordinal framework is generated using the same 6 modules of the original AutoScore algorithm including variable ranking, variable transformation, score derivation (from proportional odds models), model selection, score fine-tuning, and model evaluation. To illustrate the AutoScore-Ordinal performance, the method was conducted on electronic health records data from the emergency department at Singapore General Hospital over 2008 to 2017. The model was trained on 70% of the data, validated on 10% and tested on the remaining 20%.ResultsThis study included 445,989 inpatient cases, where the distribution of the ordinal outcome was 80.7% alive without 30-day readmission, 12.5% alive with 30-day readmission, and 6.8% died inpatient or by day 30 post discharge. Two point-based risk prediction models were developed using two sets of 8 predictor variables identified by the flexible variable selection procedure. The two models indicated reasonably good performance measured by mean area under the receiver operating characteristic curve (0.758 and 0.793) and generalized c-index (0.737 and 0.760), which were comparable to alternative models.ConclusionAutoScore-Ordinal provides an automated and easy-to-use framework for development and validation of risk prediction models for ordinal outcomes, which can systematically identify potential predictors from high-dimensional data.

Patient Engagement by the Tobacco Quitline After Electronic Referrals.

Referrals through the electronic health record (EHR) system provide an efficient evidence-based method to connect patients to the Tobacco Quitline. However, patients frequently do not respond to Quitline phone calls or accept services. The goal of this study was to characterize factors associated with successful engagement with Quitline following e-referrals by physicians in Maryland. This is a cross-sectional study with hierarchical data modeling. Data for 1790 patients e-referred in 2018-2019 by the University of Maryland Medical System (UMMS) were analyzed. Patients' engagement was assessed using a generalized estimating equation multivariable regression model for ordinal outcomes at two levels: Picking up a phone call from Quitline (1-800-QUIT-NOW) and enrollment in tobacco cessation programs. Older age, female gender, black race, low socioeconomic status, and provider's skills were significantly associated with successful outcomes of Quitline referral. The engagement with Quitline was higher in black non-Hispanic patients compared to other racial/ethnic groups (phone call response odds ratio [OR] = 1.99, 95% confidence interval [CI] = 1.35% to 2.93% and service acceptance OR = 1.89, 95% CI = 1.28% to 2.79%). Patients residing in socioeconomically deprived areas were more likely to respond to Quitline phone calls compared to those from affluent neighborhoods (OR = 1.52, 95% CI = 1.03% to 2.25%). Patients referred by faculty or attending physicians were more likely to respond compared to those referred by residents (OR = 1.23, 95% CI 1.04, 1.44, p = .0141). Multiple factors impact successful engagement with Quitline. Additional means to improve Quitline engagement success may include focused messaging on tobacco cessation benefits to patients, and skillful counseling by the provider. Implementation of the clinical decision support (CDS) tool for electronic referrals to the Tobacco Quitline at the UMMS was successful in providing evidence-based free service to elderly patients and socioeconomically disadvantaged racial and ethnic minorities. The CDS also served to engage physicians in conversation about tobacco use and cessation with every tobacco-using patient. Curricular content for physicians in training should be enriched to expand tobacco use and treatment.

Deep and interpretable regression models for ordinal outcomes

Outcomes with a natural order commonly occur in prediction problems and often the available input data are a mixture of complex data like images and tabular predictors. Deep Learning (DL) models are state-of-the-art for image classification tasks but frequently treat ordinal outcomes as unordered and lack interpretability. In contrast, classical ordinal regression models consider the outcome’s order and yield interpretable predictor effects but are limited to tabular data. We present ordinal neural network transformation models (ontrams), which unite DL with classical ordinal regression approaches. ontrams are a special case of transformation models and trade off flexibility and interpretability by additively decomposing the transformation function into terms for image and tabular data using jointly trained neural networks. The performance of the most flexible ontram is by definition equivalent to a standard multi-class DL model trained with cross-entropy while being faster in training when facing ordinal outcomes. Lastly, we discuss how to interpret model components for both tabular and image data on two publicly available datasets.

Model-based random forests for ordinal regression.

We study and compare several variants of random forests tailored to prognostic models for ordinal outcomes. Models of the conditional odds function are employed to understand the various random forest flavours. Existing random forest variants for ordinal outcomes, such as Ordinal Forests and Conditional Inference Forests, are evaluated in the presence of a non-proportional odds impact of prognostic variables. We propose two novel random forest variants in the model-based transformation forest family, only one of which explicitly assumes proportional odds. These two novel transformation forests differ in the specification of the split procedures for the underlying ordinal trees. One of these split criteria is able to detect changes in non-proportional odds situations and the other one focuses on finding proportional-odds signals. We empirically evaluate the performance of the existing and proposed methods using a simulation study and illustrate the practical aspects of the procedures by a re-analysis of the respiratory sub-item in functional rating scales of patients suffering from Amyotrophic Lateral Sclerosis (ALS).

A Flexible Skewed Link Model for Ordinal Outcomes: An Application to Infertility

BACKGROUND: An important issue in modeling categorical response data is the choice of the links. The commonly used complementary log-log link is inclined to link misspecification due to its positive and fixed skewness parameter.
 AIM: The objective of this paper is to introduce a flexible skewed link function for modeling ordinal data with some covariates.
 METHODS: We introduce a flexible skewed link model for the cumulative ordinal regression model based on Chen model.
 RESULTS: The main advantage suggested by the proposed links is the skewed link provide much more identifiable than the existing skewed links. The propriety of posterior distributions under proper and improper priors is explored in detail. An efficient Markov chain Monte Carlo algorithm is developed for sampling from the posterior distribution.
 CONCLUSION: The proposed methodology is motivated and illustrated by ovary hyperstimulation syndrome data.

Sexual function recovery after robot-assisted radical prostatectomy: Outcomes from an Italian referral centre and predicting nomogram.

Aims of this study were to assess sexual recovery after robotic-assisted radical prostatectomy (RARP) and to build a nomogram predicting 1-year sexual function. From May 2015 to July 2016, all patients eligible for RARP at our institution were invited to enter the study. The Expanded Prostate cancer Index Composite (EPIC) questionnaire was administered pre-operatively, then at 45days, and at 3, 6, 9, and 12months post-operatively. According to sexual function scores, patients were divided into four classes. Multivariate analysis was used to investigate the influence of patient- and disease-related features on sexual recovery. A total of 643 patients were included. Age was associated with baseline potency (p<.0001). Bioptic Gleason score (GS; p=.0002), American Society of Anesthesiologists (ASA) score (=.002ASA Physical Status Classification System ) and Charlson Comorbidity Index (CCI; p=.02) were negatively associated with potency. Baseline sexual function was associated with potency recovery. A nomogram resulted from fitting a proportional odds logistic model for ordinal outcomes, with 1-year sexual function as a dependent variable and baseline sexual potency, age, body mass index (BMI), clinical stage, biopsy GS, initial prostate-specific antigen (iPSA), ASA score, and CCI as predictors. After further validation, this nomogram could be a useful tool for the pre-operative counselling.

Heterogeneity Matters: Predicting Self-Esteem in Online Interventions Based on Ecological Momentary Assessment Data.

Self-esteem is a crucial factor for an individual's well-being and mental health. Low self-esteem is associated with depression and anxiety. Data about self-esteem is oftentimes collected in Internet-based interventions through Ecological Momentary Assessments and is usually provided on an ordinal scale. We applied models for ordinal outcomes in order to predict the self-esteem of 130 patients based on diary data of an online depression treatment and thereby illustrated a path of how to analyze EMA data in Internet-based interventions. Specifically, we analyzed the relationship between mood, worries, sleep, enjoyed activities, social contact, and the self-esteem of patients. We explored several ordinal models with varying degrees of heterogeneity and estimated them using Bayesian statistics. Thereby, we demonstrated how accounting for patient-heterogeneity influences the prediction performance of self-esteem. Our results show that models that allow for more heterogeneity performed better regarding various performance measures. We also found that higher mood levels and enjoyed activities are associated with higher self-esteem. Sleep, social contact, and worries were significant predictors for only some individuals. Patient-individual parameters enable us to better understand the relationships between the variables on a patient-individual level. The analysis of relationships between self-esteem and other psychological factors on an individual level can therefore lead to valuable information for therapists and practitioners.

Estimation of Nested and Zero-Inflated Ordered Probit Models

We introduce three new STATA commands, nop, ziop2 and ziop3, for the estimation of a three-part nested ordered probit model, the two-part zero-inflated ordered probit models of Harris and Zhao (2007, Journal of Econometrics 141: 1073-1099) and Brooks, Harris and Spencer (2012, Economics Letters 117: 683-686), and a three-part zero-inflated ordered probit model for ordinal outcomes, with both exogenous and endogenous switching. The three-part models allow the probabilities of positive, neutral (zero) and negative outcomes to be generated by distinct processes. The zero-inflated models address a preponderance of zeros and allow them to emerge in different latent regimes. We provide postestimation commands to compute probabilistic predictions and various measures of their accuracy, to access the goodness of fit, and to perform model comparison using the Vuong test (Vuong 1989, Econometrica 57: 307-333) with the corrections based on the Akaike and Schwarz information criteria. We investigate the finite-sample performance of the maximum likelihood estimators by Monte Carlo simulations, discuss the relations among the models, and illustrate the new commands with an empirical application to the U.S. federal funds rate target.

Dose finding with longitudinal data: simpler models, richer outcomes.

Phase I oncology clinical trials are designed to identify the optimal dose that will be recommended for phase II trials. This dose is typically defined as the dose associated with a certain probability of severe toxicity at cycle 1, although toxicity is repeatedly measured over cycles on an ordinal scale. Recently, a proportional odds mixed-effect model for ordinal outcomes has been proposed to (i) identify the optimal dose accounting for repeated events and (ii) to provide some framework to explore time trend. We compare this approach to a method based on repeated binary variables and to a method based on an under-parameterized model of the dose-time toxicity relationship. We show that repeated binary and ordinal outcomes both improve the accuracy of dose-finding trials in the same proportion; ordinal outcomes are, however, superior to detect time trend even in the presence of nonproportional odds models. Moreover, less parameterized models led to the best operating characteristics. These approaches are illustrated on two dose-finding phase I trials. Integration of repeated measurements is appealing in phase I dose-finding trials.

A random pattern mixture model for ordinal outcomes with informative dropouts.

We extend a random pattern mixture joint model for longitudinal ordinal outcomes and informative dropouts. The patients are generalized to 'pattern' groups based on known covariates that are potentially surrogated for the severity of the underlying condition. The random pattern effects are defined as the latent effects linking the dropout process and the ordinal longitudinal outcome. Conditional on the random pattern effects, the longitudinal outcome and the dropout times are assumed independent. Estimates are obtained via the Expectation-maximization algorithm. We applied the model to the end-stage renal disease data. Anemia was found to be significantly affected by the baseline iron treatment when the dropout information was adjusted via the study model; as opposed to an independent or shared parameter model. Simulations were performed to evaluate the performance of the random pattern mixture model under various assumptions.

Methods for Multilevel Ordinal Data in Prevention Research.

This paper discusses statistical models for multilevel ordinal data that may be more appropriate for prevention outcomes than models that assume continuous measurement and normality. Prevention outcomes often have distributions that make them inappropriate for many popular statistical models that assume normality and are more appropriately considered ordinal outcomes. Despite this, the modeling of ordinal outcomes is often not well understood. This article discusses ways to analyze multilevel ordinal outcomes that are clustered or longitudinal, including the proportional odds regression model for ordinal outcomes, which assumes that the covariate effects are the same across the levels of the ordinal outcome. The article will cover how to test this assumption and what to do if it is violated. It will also discuss application of these models using computer software programs.

Discrimination ability of prediction models for ordinal outcomes: Relationships between existing measures and a new measure

In this paper, we focus on measures to evaluate discrimination of prediction models for ordinal outcomes. We review existing extensions of the dichotomous c-index-which is equivalent to the area under the receiver operating characteristic (ROC) curve--suggest a new measure, and study their relationships. The volume under the ROC surface (VUS) scores sets of cases including one case from each outcome category. VUS considers sets as either correctly or incorrectly ordered by the model. All other existing measures assess pairs of cases. We propose an ordinal c-index (ORC) that is set-based but, contrary to VUS, scores sets more gradually by indicating the closeness of the model-based ordering to the perfect ordering. As a result, the ORC does not decrease rapidly as the number of outcome categories increases. It turns out that the ORC can be rewritten as the average of pairwise c-indexes. Hence, the ORC has both a set- and pair-based interpretation. There are several relationships between the existing measures, leading to only two types of existing measures: a prevalence-weighted average of pairwise c-indexes and the VUS. Our suggested measure ORC positions itself in between as it is set-based but turns out to equal an unweighted average of pairwise c-indexes. The measures are demonstrated through a case study on the prediction of six-month outcome after traumatic brain injury. In conclusion, the set-based nature and graded scoring system make the ORC an attractive measure with a simple interpretation, together with its prevalence-independence that is a natural property of a discrimination measure.

Geoadditive regression modeling of stream biological condition

Indices of biotic integrity have become an established tool to quantify the condition of small non-tidal streams and their watersheds. To investigate the effects of watershed characteristics on stream biological condition, we present a new technique for regressing IBIs on watershed-specific explanatory variables. Since IBIs are typically evaluated on an ordinal scale, our method is based on the proportional odds model for ordinal outcomes. To avoid overfitting, we do not use classical maximum likelihood estimation but a component-wise functional gradient boosting approach. Because component-wise gradient boosting has an intrinsic mechanism for variable selection and model choice, determinants of biotic integrity can be identified. In addition, the method offers a relatively simple way to account for spatial correlation in ecological data. An analysis of the Maryland Biological Streams Survey shows that nonlinear effects of predictor variables on stream condition can be quantified while, in addition, accurate predictions of biological condition at unsurveyed locations are obtained.

Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt-PA stroke trial.

Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non-ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non-ignorable missing data caused by the failure times. We apply the method to the NINDS rt-PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales, can be treated in the same way.

A Conceptual Framework for Ordered Logistic Regression Models

Ordinal-level measures are very common in social science research. Researchers often analyze ordinal dependent variables using the proportional odds logistic regression model. However, this ‘‘traditional’’ method is one of many different types of logistic regression models available for the analysis of ordered response variables. In this article, the author identifies 12 distinct models that rely on logistic regression and fit within a framework of three major approaches with variations within each approach based on the application of the proportional odds assumption. This typology provides a degree of conceptual clarity that is missing in the extant literature on logistic regression models for ordinal outcomes. The author illustrates the similarities and differences among the different models with examples from the General Social Survey and the American National Election Study.

Multivariate dynamic model for ordinal outcomes

Individual or stand-level biomass is not easy to measure. The current methods employed, based on cutting down a representative sample of plantations, make it possible to assess the biomasses for various compartments (bark, dead branches, leaves, …). However, this felling makes individual longitudinal follow-up impossible. In this context, we propose a method to evaluate individual biomasses by compartments when these are ordinals. Biomass is measured visually and observations are therefore not destructive. The technique is based on a probit model redefined in terms of latent variables. A generalization of the univariate case to the multivariate case is then natural and takes into account of dependency between compartment biomasses. These models are then extended to the longitudinal case by developing a Dynamic Multivariate Ordinal Probit Model. The performance of the MCMC algorithm used for the estimation is illustrated by means of simulations built from known biomass models. The quality of the estimates and the impact of certain parameters, are then discussed.

Frequency of traumatic cow injuries in relation to housing systems in Swiss organic dairy herds.

A study was performed to estimate the extent of cow injuries as an indicator of animal welfare in organic dairy farms in Switzerland. The study was conducted during the winter feeding period of 1997/98 and was part of a larger project on animal health, nutrition and production in certified organic dairy herds in Switzerland. Potential predictors of injuries related to animal housing were quantified and relations between occurrence of injuries and nutritional status and production were evaluated. The investigation was designed as a cross-sectional study and included a representative sample of 152 farms and 1856 cows. Every farm was visited once and each cow was scored for claw-, skin- and joint-lesions and body condition. Statistical analyses were performed appropriate for a stratified and one-stage cluster sample weighted for the entire population of organic dairy farms in Switzerland. Possible individual and environmental predictors of cow injuries were analysed using multinomial logit models for ordinal outcomes. The overall frequencies of injuries were 10.4% for joint lesions, 12.8% for soft-tissue injuries and 3.6% for claw lesions. Most joint injuries (84.9%) were observed at the hock joint, 9.4% at the carpus and 3.1% at the knee joint. The analysis of the association between frequency of injuries and potential predictors revealed heterogeneous results and the risk profiles were different between the categories of injuries measured in the study. These observations support the assumption that variations in type and severity of injuries are due to specific differences of animal management, housing design and due to different characteristics of individual cows.