Interpreting epidemiological evidence in the presence of multiple endpoints: an alternative analytic approach using the 9-year follow-up of the Seychelles child development study

Purpose  The potential for ill-informed causal inference is a major concern in published longitudinal studies evaluating impaired neurological function in children prenatally exposed to background levels of methyl mercury (MeHg). These studies evaluate a large number of developmental tests. We propose an alternative analysis strategy that reduces the number of comparisons tested in these studies. Methods  Using data from the 9-year follow-up of 643 children in the Seychelles child development study, we grouped 18 individual endpoints into one overall ordinal outcome variable as well as by developmental domains. Subsequently, ordinal logistic regression analyses were performed. Results  We did not find an association between prenatal MeHg exposure and developmental outcomes at 9 years of age. Conclusion  Our proposed framework is more likely to result in a balanced interpretation of a posteriori associations. In addition, this new strategy should facilitate the use of complex epidemiological data in quantitative risk assessment.     

Robust analysis in joint models: an application to a study on muscular dystrophy

Generalized partial ordinal models occur frequently in biomedical investigations where, along with ordinal longitudinal outcomes, there are time‐dependent covariates that act nonparametrically. In these studies, an association between such outcomes and time to an event is of considerable interest to medical practitioners. The primary objective in the present article is to study the robustness of estimators of the parameters of interest in a joint generalized partial ordinal models and a time‐to‐event model, because in many situations, the estimators in such joint models are sensitive to outliers. A Monte Carlo Metropolis–Hastings Newton Raphson algorithm is proposed for robust estimation. A detailed simulation study was performed to justify the behavior of the proposed estimators. By way of motivation, we consider a data set concerning longitudinal outcomes of children involved in a study on muscular dystrophy. Our analysis revealed some interesting findings that may be useful to medical practitioners. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Reconsidering the use of rankings in the valuation of health states: a model for estimating cardinal values from ordinal data

Background  

In survey studies on health-state valuations, ordinal ranking exercises often are used as precursors to other elicitation methods such as the time trade-off (TTO) or standard gamble, but the ranking data have not been used in deriving cardinal valuations. This study reconsiders the role of ordinal ranks in valuing health and introduces a new approach to estimate interval-scaled valuations based on aggregate ranking data.     

Modeling continuous response variables using ordinal regression

We study the application of a widely used ordinal regression model, the cumulative probability model (CPM), for continuous outcomes. Such models are attractive for the analysis of continuous response variables because they are invariant to any monotonic transformation of the outcome and because they directly model the cumulative distribution function from which summaries such as expectations and quantiles can easily be derived. Such models can also readily handle mixed type distributions. We describe the motivation, estimation, inference, model assumptions, and diagnostics. We demonstrate that CPMs applied to continuous outcomes are semiparametric transformation models. Extensive simulations are performed to investigate the finite sample performance of these models. We find that properly specified CPMs generally have good finite sample performance with moderate sample sizes, but that bias may occur when the sample size is small. Cumulative probability models are fairly robust to minor or moderate link function misspecification in our simulations. For certain purposes, the CPMs are more efficient than other models. We illustrate their application, with model diagnostics, in a study of the treatment of HIV. CD4 cell count and viral load 6 months after the initiation of antiretroviral therapy are modeled using CPMs; both variables typically require transformations, and viral load has a large proportion of measurements below a detection limit.     

Joint two‐part Tobit models for longitudinal and time‐to‐event data

In this article, we show how Tobit models can address problems of identifying characteristics of subjects having left‐censored outcomes in the context of developing a method for jointly analyzing time‐to‐event and longitudinal data. There are some methods for handling these types of data separately, but they may not be appropriate when time to event is dependent on the longitudinal outcome, and a substantial portion of values are reported to be below the limits of detection. An alternative approach is to develop a joint model for the time‐to‐event outcome and a two‐part longitudinal outcome, linking them through random effects. This proposed approach is implemented to assess the association between the risk of decline of CD4/CD8 ratio and rates of change in viral load, along with discriminating between patients who are potentially progressors to AIDS from patients who do not. We develop a fully Bayesian approach for fitting joint two‐part Tobit models and illustrate the proposed methods on simulated and real data from an AIDS clinical study.     

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.     

Joint modeling of two longitudinal outcomes and competing risk data

Aortic gradient and aortic regurgitation are echocardiographic markers of aortic valve function. Both are biomarkers repeatedly measured in patients with valve abnormalities, and thus, it is expected that they are biologically interrelated. Loss of follow‐up could be caused by multiple reasons, including valve progression related, such as an intervention or even the death of the patient. In that case, it would be of interest and appropriate to analyze these outcomes jointly. Joint models have recently received much attention because they cover a wide range of clinical applications and have promising results. We propose a joint model consisting of two longitudinal outcomes, one continuous (aortic gradient) and one ordinal (aortic regurgitation), and two time‐to‐events (death and reoperation). Moreover, we allow for more flexibility for the average evolution and the subject‐specific profiles of the continuous repeated outcome by using B‐splines. A disadvantage, however, is that when adopting a non‐linear structure for the model, we may have difficulties when interpreting the results. To overcome this problem, we propose a graphical approach. In this paper, we apply the proposed joint models under the Bayesian framework, using a data set including serial echocardiographic measurements of aortic gradient and aortic regurgitation and measurements of the occurrence of death and reoperation in patients who received a human tissue valve in the aortic position. The interpretation of the results will be discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

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. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of artificial neural network models in occupational safety and health utilizing ordinal variables

Safety professionals and practitioners are always searching for methods to accurately assess the association between exposures and possible occupational disorders or diseases and predict the outcome of any variable. Statistical analysis and logistic regression (LR) in particular are among the most popular tools being used today. Artificial neural network (ANN) models are another method of predicting outcomes, which are gradually finding their way into the safety field. Limited studies have shown that they are capable of predicting outcomes more accurately than LR, but they have been tested either on continuous or on dichotomous variables or combinations of them. The objective of this research was to demonstrate that ANN models can perform better than LR models with data sets comprised of all ordinal variables, which has not been done so far. The data set used in this research was collected from construction workers using the Work Compatibility questionnaire. The data set contained only ordinal variables both as input (exposure) and as output (outcome) variables. LR models and ANN models were constructed using the same data set and the performance of all models was compared by using the log-likelihood ratio. The result of this study showed that ANN models performed significantly better than LR models with a data set of all ordinal variables as well as other types of variables such as dichotomous and continuous.     

Cumulative logit models for ordinal data: a case study involving allergic rhinitis severity scores

Ordered categorical data arise in numerous settings, a common example being pain scores in analgesic trials. The modelling of such data is intrinsically more difficult than the modelling of continuous data due to the constraints on the underlying probabilities and the reduced amount of information that discrete outcomes contain. In this paper we discuss the class of cumulative logit models, which provide a natural framework for ordinal data analysis. We show how viewing the categorical outcome as the discretization of an underlying continuous response allows a natural interpretation of model parameters. We also show how covariates are incorporated into the model and how various types of correlation among repeated measures on the same individual may be accounted for. The models are illustrated using longitudinal allergy data consisting of sneezing scores measured on a four-point scale. Our approach throughout is Bayesian and we present a range of simple diagnostics to aid model building.     

Impact of prenatal COVID-19 vaccination on delivery and neonatal outcomes: Results from a New York City cohort

Research suggest prenatal vaccination against coronavirus disease-19 (COVID-19) is safe. However, previous studies utilized retrospectively collected data or examined late pregnancy vaccinations. We investigated the associations of COVID-19 vaccination throughout pregnancy with delivery and neonatal outcomes. We included 1,794 mother-neonate dyads enrolled in the Generation C Study with known prenatal COVID-19 vaccination status and complete covariate and outcome data. We used multivariable quantile regressions to estimate the effect of prenatal COVID-19 vaccination on birthweight, delivery gestational age, and blood loss at delivery; and Poisson generalized linear models for Caesarean delivery (CD) and Neonatal Intensive Care Unit (NICU) admission. Using the above methods, we estimated effects of trimester of vaccine initiation on these outcomes. In our sample, 13.7% (n = 250) received at least one prenatal dose of any COVID-19 vaccine. Vaccination was not associated with birthweight (β = 12.42 g [-90.5, 114.8]), gestational age (β = 0.2 days [-1.1, 1.5]), blood loss (β = -50.6 ml [-107.0, 5.8]), the risks of CD (RR = 0.8; [0.6, 1.1]) or NICU admission (RR = 0.9 [0.5, 1.7]). Trimester of vaccine initiation was also not associated with these outcomes. Our findings suggest that there is no associated risk between prenatal COVID-19 vaccination and adverse delivery and neonatal outcomes in a cohort sample from NYC.     

A latent variable transformation model approach for exploring dysphagia

Multiple outcomes are often collected in applications where the quantity of interest cannot be measured directly or is difficult or expensive to measure. In a head and neck cancer study conducted at Dana‐Farber Cancer Institute, the investigators wanted to determine the effect of clinical and treatment factors on unobservable dysphagia through collected multiple outcomes of mixed types. Latent variable models are commonly adopted in this setting. These models stipulate that multiple collected outcomes are conditionally independent given the latent factor. Mixed types of outcomes (e.g., continuous vs. ordinal) and censored outcomes present statistical challenges, however, as a natural analog of the multivariate normal distribution does not exist for mixed data. Recently, Lin et al. proposed a semiparametric latent variable transformation model for mixed outcome data; however, it may not readily accommodate event time outcomes where censoring is present. In this paper, we extend the work of Lin et al. by proposing both semiparametric and parametric latent variable models that allow for the estimation of the latent factor in the presence of measurable outcomes of mixed types, including censored outcomes. Both approaches allow for a direct estimate of the treatment (or other covariate) effect on the unobserved latent variable, greatly enhancing the interpretability of the models. The semiparametric approach has the added advantage of allowing the relationship between the measurable outcomes and latent variables to be unspecified, rendering more robust inference. The parametric and semiparametric models can also be used together, providing a comprehensive modeling strategy for complicated latent variable problems. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Design and analysis considerations for utilizing a mapping function in a small sample,sequential, multiple assignment,randomized trials with continuous outcomes

Small sample, sequential, multiple assignment, randomized trials (snSMARTs) are multistage trials with the overall goal of determining the best treatment after a fixed amount of time. In snSMART trials, patients are first randomized to one of three treatments and a binary (e.g. response/nonresponse) outcome is measured at the end of the first stage. Responders to first stage treatment continue their treatment. Nonresponders to first stage treatment are rerandomized to one of the remaining treatments. The same binary outcome is measured at the end of the first and second stages, and data from both stages are pooled together to find the best first stage treatment. However, in many settings the primary endpoint may be continuous, and dichotomizing this continuous variable may reduce statistical efficiency. In this article, we extend the snSMART design and methods to allow for continuous outcomes. Instead of requiring a binary outcome at the first stage for rerandomization, the probability of staying on the same treatment or switching treatment is a function of the first stage outcome. Rerandomization based on a mapping function of a continuous outcome allows for snSMART designs without requiring a binary outcome. We perform simulation studies to compare the proposed design with continuous outcomes to standard snSMART designs with binary outcomes. The proposed design results in more efficient treatment effect estimates and similar outcomes for trial patients.     

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.     

A tutorial on Bayesian bivariate meta‐analysis of mixed binary‐continuous outcomes with missing treatment effects

Bivariate random‐effects meta‐analysis (BVMA) is a method of data synthesis that accounts for treatment effects measured on two outcomes. BVMA gives more precise estimates of the population mean and predicted values than two univariate random‐effects meta‐analyses (UVMAs). BVMA also addresses bias from incomplete reporting of outcomes. A few tutorials have covered technical details of BVMA of categorical or continuous outcomes. Limited guidance is available on how to analyze datasets that include trials with mixed continuous‐binary outcomes where treatment effects on one outcome or the other are not reported. Given the advantages of Bayesian BVMA for handling missing outcomes, we present a tutorial for Bayesian BVMA of incompletely reported treatment effects on mixed bivariate outcomes. This step‐by‐step approach can serve as a model for our intended audience, the methodologist familiar with Bayesian meta‐analysis, looking for practical advice on fitting bivariate models. To facilitate application of the proposed methods, we include our WinBUGS code. As an example, we use aggregate‐level data from published trials to demonstrate the estimation of the effects of vitamin K and bisphosphonates on two correlated bone outcomes, fracture, and bone mineral density. We present datasets where reporting of the pairs of treatment effects on both outcomes was ‘partially’ complete (i.e., pairs completely reported in some trials), and we outline steps for modeling the incompletely reported data. To assess what is gained from the additional work required by BVMA, we compare the resulting estimates to those from separate UVMAs. We discuss methodological findings and make four recommendations. Copyright © 2015 John Wiley & Sons, Ltd.     

Joint modeling of binary response and survival for clustered data in clinical trials

In clinical trials, it is often desirable to evaluate the effect of a prognostic factor such as a marker response on a survival outcome. However, the marker response and survival outcome are usually associated with some potentially unobservable factors. In this case, the conventional statistical methods that model these two outcomes separately may not be appropriate. In this paper, we propose a joint model for marker response and survival outcomes for clustered data, providing efficient statistical inference by considering these two outcomes simultaneously. We focus on a special type of marker response: a binary outcome, which is investigated together with survival data using a cluster-specific multivariate random effect variable. A multivariate penalized likelihood method is developed to make statistical inference for the joint model. However, the standard errors obtained from the penalized likelihood method are usually underestimated. This issue is addressed using a jackknife resampling method to obtain a consistent estimate of standard errors. We conduct extensive simulation studies to assess the finite sample performance of the proposed joint model and inference methods in different scenarios. The simulation studies show that the proposed joint model has excellent finite sample properties compared to the separate models when there exists an underlying association between the marker response and survival data. Finally, we apply the proposed method to a symptom control study conducted by Canadian Cancer Trials Group to explore the prognostic effect of covariates on pain control and overall survival.     

Multilevel moderated mediation model with ordinal outcome

Although increasingly complex models have been proposed in mediation literature, there is no model nor software that incorporates the multiple possible generalizations of the simple mediation model jointly. We propose a flexible moderated mediation model allowing for (1) a hierarchical structure of clustered data, (2) more and possibly correlated mediators, and (3) an ordinal outcome. The motivating data set is obtained from a European study in nursing research. Patients' willingness to recommend their treating hospital was recorded in an ordinal way. The research question is whether such recommendation directly depends on system‐level features in the organization of nursing care, or whether these associations are mediated by 2 measurements of nursing care left undone and possibly moderated by nurse education. We have developed a Bayesian approach and accompanying program that takes all the above generalizations into account.