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We propose a new set of test statistics to examine the association between two ordinal categorical variables X and Y after adjusting for continuous and/or categorical covariates Z. Our approach first fits multinomial (e.g., proportional odds) models of X and Y, separately, on Z. For each subject, we then compute the conditional distributions of X and Y given Z. If there is no relationship between X and Y after adjusting for Z, then these conditional distributions will be independent, and the observed value of (X, Y) for a subject is expected to follow the product distribution of these conditional distributions. We consider two simple ways of testing the null of conditional independence, both of which treat X and Y equally, in the sense that they do not require specifying an outcome and a predictor variable. The first approach adds these product distributions across all subjects to obtain the expected distribution of (X, Y) under the null and then contrasts it with the observed unconditional distribution of (X, Y). Our second approach computes \"residuals\" from the two multinomial models and then tests for correlation between these residuals; we define a new individual-level residual for models with ordinal outcomes. We present methods for computing p-values using either the empirical or asymptotic distributions of our test statistics. Through simulations, we demonstrate that our test statistics perform well in terms of power and Type I error rate when compared to proportional odds models which treat X as either a continuous or categorical predictor. We apply our methods to data from a study of visual impairment in children and to a study of cervical abnormalities in human immunodeficiency virus (HIV)-infected women. Supplemental materials for the article are available online.

Background In randomized clinical trials (RCTs) with non-compliance, evaluating the causal effects of interventions would lead to a more precise estimation of treatment effect when the estimand of interest is the effect of treatment amongst compliers. While there is a large body of literature addressing the issue of non-compliance for continuous, binary, and time-to-event outcomes, this issue is seldom discussed for ordinal outcomes. Methods In this paper, we consider one-sided non-compliance. We introduce an extension of the inverse probability weighting (IPW) method for handling non-compliance involving an ordinal outcome by fully utilizing the information of non-compliance and defining it as a categorical variable to describe the extent of non-compliance. This is in contrast to the usual convention where compliance is regarded as a binary variable. We provide the identification and asymptotic distribution of the proposed method. We compare the proposed method (IPW_Dnew) with intention-to-treat (ITT), per protocol (PP), instrumental variable (IV), and IPW method via a simulation study and real-life data from the JOBS II intervention trial and the IMMACULATE trial. Results Simulation results demonstrate that the proposed method performs better than other methods in terms of bias, coverage, mean squared error, power and Type I error under various scenarios, particularly in situations with selection bias and partial compliance. In the empirical study, a substantial estimate of partial compliance by IPW_Dnew implies that there may be a partial compliance effect. Conclusion For ordinal outcome in the presence of non-compliance, we suggest using the proposed method to estimate the causal effect of treatment amongst compliers and partial compliers, especially when there exists selection bias.

Background Risk 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. Methods The 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%. Results This 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. Conclusion AutoScore-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.

Background Randomised controlled trials (RCTs) aim to estimate the causal effect of one or more interventions relative to a control. One type of outcome that can be of interest in an RCT is an ordinal outcome, which is useful to answer clinical questions regarding complex and evolving patient states. The target parameter of interest for an ordinal outcome depends on the research question and the assumptions the analyst is willing to make. This review aimed to provide an overview of how ordinal outcomes have been used and analysed in RCTs. Methods The review included RCTs with an ordinal primary or secondary outcome published between 2017 and 2022 in four highly ranked medical journals (the British Medical Journal , New England Journal of Medicine , The Lancet , and the Journal of the American Medical Association ) identified through PubMed. Details regarding the study setting, design, the target parameter, and statistical methods used to analyse the ordinal outcome were extracted. Results The search identified 309 studies, of which 144 were eligible for inclusion. The most used target parameter was an odds ratio, reported in 78 (54%) studies. The ordinal outcome was dichotomised for analysis in 47 ( 33 % ) studies, and the most common statistical model used to analyse the ordinal outcome on the full ordinal scale was the proportional odds model (64 [ 44 % ] studies). Notably, 86 (60%) studies did not explicitly check or describe the robustness of the assumptions for the statistical method(s) used. Conclusions The results of this review indicate that in RCTs that use an ordinal outcome, there is variation in the target parameter and the analytical approaches used, with many dichotomising the ordinal outcome. Few studies provided assurance regarding the appropriateness of the assumptions and methods used to analyse the ordinal outcome. More guidance is needed to improve the transparent reporting of the analysis of ordinal outcomes in future trials.

Background There is a growing interest in assessment of the quality of hospital care, based on outcome measures. Many quality of care comparisons rely on binary outcomes, for example mortality rates. Due to low numbers, the observed differences in outcome are partly subject to chance. We aimed to quantify the gain in efficiency by ordinal instead of binary outcome analyses for hospital comparisons. We analyzed patients with traumatic brain injury (TBI) and stroke as examples. Methods We sampled patients from two trials. We simulated ordinal and dichotomous outcomes based on the modified Rankin Scale (stroke) and Glasgow Outcome Scale (TBI) in scenarios with and without true differences between hospitals in outcome. The potential efficiency gain of ordinal outcomes, analyzed with ordinal logistic regression, compared to dichotomous outcomes, analyzed with binary logistic regression was expressed as the possible reduction in sample size while keeping the same statistical power to detect outliers. Results In the IMPACT study (9578 patients in 265 hospitals, mean number of patients per hospital = 36), the analysis of the ordinal scale rather than the dichotomized scale (‘unfavorable outcome’), allowed for up to 32% less patients in the analysis without a loss of power. In the PRACTISE trial (1657 patients in 12 hospitals, mean number of patients per hospital = 138), ordinal analysis allowed for 13% less patients. Compared to mortality, ordinal outcome analyses allowed for up to 37 to 63% less patients. Conclusions Ordinal analyses provide the statistical power of substantially larger studies which have been analyzed with dichotomization of endpoints. We advise to exploit ordinal outcome measures for hospital comparisons, in order to increase efficiency in quality of care measurements. Trial registration We do not report the results of a health care intervention.

Assessing the clinical importance of an exposure effect on a quality of life (QoL) score often requires quantifying the effect in terms of a difference in scores. Using the linear regression model (LRM) for this purpose assumes the ordinal score is a proxy for an underlying continuous variable, but the analysis offers no assessment for the validity of the assumption. We propose an approach that assesses the proxy assumption and estimates the exposure effect by using the cumulative link model (CLM). CLM is a well-established regression model that assumes an ordinal score is an ordered category generated from applying thresholds to a latent continuous variable. Our approach assesses the proxy assumption by testing whether these thresholds are equidistant. We compared the performance of CLM and LRM using simulated ordinal data and illustrated their application to the effect of time since diagnosis on five subscales of fatigue among breast cancer survivors measured using the Multidimensional Fatigue Inventory. CLM had good performance in estimating the difference in means with simulated ordinal data satisfying the proxy assumption, even when the outcome had only a few categories. When the proxy assumption was inadequate, both the CLM and LRM had biased estimates with poor coverage. The proxy assumption was appropriate for four of the five subscales in our real data application to fatigue scores, which highlighted the importance of assessing the proxy assumption to avoid reporting invalid estimates in terms of the difference in scores. The proxy assumption is critical to the interpretation of the exposure effect on the difference in mean QoL scores. CLM offers a valid test for the presence of an association, a method for assessing the proxy assumption, and when the assumption is adequate, an assessment for clinical significance using the difference in means.

We propose a novel multivariate model for analyzing hybrid traits and identifying genetic factors for comorbid conditions. Comorbidity is a common phenomenon in mental health in which an individual suffers from multiple disorders simultaneously. For example, in the Study of Addiction: Genetics and Environment (SAGE), alcohol and nicotine addiction were recorded through multiple assessments that we refer to as hybrid traits. Statistical inference for studying the genetic basis of hybrid traits has not been well developed. Recent rank-based methods have been utilized for conducting association analyses of hybrid traits but do not inform the strength or direction of effects. To overcome this limitation, a parametric modeling framework is imperative. Although such parametric frameworks have been proposed in theory, they are neither well developed nor extensively used in practice due to their reliance on complicated likelihood functions that have high computational complexity. Many existing parametric frameworks tend to instead use pseudo-likelihoods to reduce computational burdens. Here, we develop a model fitting algorithm for the full likelihood. Our extensive simulation studies demonstrate that inference based on the full likelihood can control the type-I error rate, and gains power and improves the effect size estimation when compared with several existing methods for hybrid models. These advantages remain even if the distribution of the latent variables is misspecified. After analyzing the SAGE data, we identify three genetic variants (rs7672861, rs958331, rs879330) that are significantly associated with the comorbidity of alcohol and nicotine addiction at the chromosome-wide level. Moreover, our approach has greater power in this analysis than several existing methods for hybrid traits.Although the analysis of the SAGE data motivated us to develop the model, it can be broadly applied to analyze any hybrid responses.

We present a general method for estimating the effect of a treatment on an ordinal outcome in randomized trials. The method is robust in that it does not rely on the proportional odds assumption. Our estimator leverages information in prognostic baseline variables, and has all of the following properties: (i) it is consistent; (ii) it is locally efficient; (iii) it is guaranteed to have equal or better asymptotic precision than both the inverse probability-weighted and the unadjusted estimators. To the best of our knowledge, this is the first estimator of the causal relation between a treatment and an ordinal outcome to satisfy these properties. We demonstrate the estimator in simulations based on resampling from a completed randomized clinical trial of a new treatment for stroke; we show potential gains of up to 39% in relative efficiency compared to the unadjusted estimator. The proposed estimator could be a useful tool for analyzing randomized trials with ordinal outcomes, since existing methods either rely on model assumptions that are untenable in many practical applications, or lack the efficiency properties of the proposed estimator. We provide R code implementing the estimator.

Background Multiple imputation (MI) is an established technique for handling missing data in observational studies. Joint modelling (JM) and fully conditional specification (FCS) are commonly used methods for imputing multilevel data. However, MI methods for multilevel ordinal outcome variables have not been well studied, especially when cluster size is informative on the outcome. The purpose of this study is to describe and compare different MI strategies for dealing with multilevel ordinal outcomes when informative cluster size (ICS) exists. Methods We conducted comprehensive Monte Carlo simulation studies to compare the performance of five strategies: complete case analysis (CCA), FCS, FCS+CS (including cluster size (CS) in the imputation model), JM, and JM+CS under various scenarios. We evaluated their performance using a proportional odds logistic regression model estimated with cluster weighted generalized estimating equations (CWGEE). Results The simulation results showed that including CS in the imputation model can significantly improve estimation accuracy when ICS exists. FCS provided more accurate and robust estimation than JM, followed by CCA for multilevel ordinal outcomes. We further applied these strategies to a real dental study to assess the association between metabolic syndrome and clinical attachment loss scores. The results based on FCS + CS indicated that the power of the analysis would increase after carrying out the appropriate MI strategy. Conclusions MI is an effective tool to increase the accuracy and power of the downstream statistical analysis for missing ordinal outcomes. FCS slightly outperforms JM when imputing multilevel ordinal outcomes. When there is plausible ICS, we recommend including CS in the imputation phase.

Functional markers and their quantitative features (eg, maximum value, time to maximum, area under the curve [AUC], etc) are increasingly being used in clinical studies to diagnose diseases. It is thus of interest to assess the diagnostic utility of functional markers by assessing alignment between their quantitative features and an ordinal gold standard test that reflects the severity of disease. The concept of broad sense agreement (BSA) has recently been introduced for studying the relationship between continuous and ordinal measurements, and provides a promising tool to address such a question. Our strategy is to adopt a general class of summary fiinctionals (SFs), each of which flexibly captures a different quantitative feature of a functional marker, and study its alignment according to an ordinal outcome via BSA. We further illustrate the proposed framework using three special classes of SFs (AUC-type, magnitude-specific, and time-specific) that are widely used in clinical settings. The proposed BSA estimator is proven to be consistent and asymptotically normal given a consistent estimator for the SF. We further provide an inferential framework for comparing a pair of candidate SFs in terms of their importance on the ordinal outcome. Our simulation results demonstrate satisfactory finite-sample performance of the proposed framework. We demonstrate the application of our methods using a renal study.