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.     

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.     

A goodness‐of‐fit test for the ordered stereotype model

This paper presents a new goodness‐of‐fit test for an ordered stereotype model used for an ordinal response variable. The proposed test is based on the well‐known Hosmer–Lemeshow test and its version for the proportional odds regression model. The latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The proposed test takes the use of this new adjusted spacing to partition data. A simulation study shows good performance of the proposed test under a variety of scenarios. Finally, the results of the application in two examples are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

The trend odds model for ordinal data

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

Numeric score‐based conditional and overall change‐in‐status indices for ordered categorical data

Planned interventions and/or natural conditions often effect change on an ordinal categorical outcome (e.g., symptom severity). In such scenarios, it is sometimes desirable to assign a priori scores to observed changes in status, typically giving higher weight to changes of greater magnitude. We define change indices for such data based upon a multinomial model for each row of a c × c table, where the rows represent the baseline status categories. We distinguish an index designed to assess conditional changes within each baseline category from two others designed to capture overall change. One of these overall indices measures expected change across a target population. The other is scaled to capture the proportion of total possible change in the direction indicated by the data, so that it ranges from ?1 (when all subjects finish in the least favorable category) to +1 (when all finish in the most favorable category). The conditional assessment of change can be informative regardless of how subjects are sampled into the baseline categories. In contrast, the overall indices become relevant when subjects are randomly sampled at baseline from the target population of interest, or when the investigator is able to make certain assumptions about the baseline status distribution in that population. We use a Dirichlet‐multinomial model to obtain Bayesian credible intervals for the conditional change index that exhibit favorable small‐sample frequentist properties. Simulation studies illustrate the methods, and we apply them to examples involving changes in ordinal responses for studies of sleep deprivation and activities of daily living. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel modeling framework for ordinal data defined by collapsed counts

Adolescent alcohol use is a serious public health concern. Despite advances in the theoretical conceptualization of pathways to alcohol use, researchers are limited by the statistical techniques currently available. Researchers often fit linear models and restrictive categorical models (e.g., proportional odds models) to ordinal data with many response categories defined by collapsed count data (0 drinking days, 1–2days, 3–6days, etc.). Consequently, existing models ignore the underlying count process, resulting in disjoint between the construct of interest and the models being fitted. Our proposed ordinal modeling approach overcomes this limitation by explicitly linking ordinal responses to a suitable underlying count distribution. In doing so, researchers can use maximum likelihood estimation to fit count models to ordinal data as if they had directly observed the underlying discrete counts. The usefulness of our ordinal negative binomial and ordinal zero‐inflated negative binomial models is verified by simulation studies. We also demonstrate our approach using real empirical data from the 2010 National Survey of Drug Use and Health. Results show the benefit of the proposed ordinal modeling framework compared with existing methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized multifactor dimensionality reduction approaches to identification of genetic interactions underlying ordinal traits

The manifestation of complex traits is influenced by gene–gene and gene–environment interactions, and the identification of multifactor interactions is an important but challenging undertaking for genetic studies. Many complex phenotypes such as disease severity are measured on an ordinal scale with more than two categories. A proportional odds model can improve statistical power for these outcomes, when compared to a logit model either collapsing the categories into two mutually exclusive groups or limiting the analysis to pairs of categories. In this study, we propose a proportional odds model-based generalized multifactor dimensionality reduction (GMDR) method for detection of interactions underlying polytomous ordinal phenotypes. Computer simulations demonstrated that this new GMDR method has a higher power and more accurate predictive ability than the GMDR methods based on a logit model and a multinomial logit model. We applied this new method to the genetic analysis of low-density lipoprotein (LDL) cholesterol, a causal risk factor for coronary artery disease, in the Multi-Ethnic Study of Atherosclerosis, and identified a significant joint action of the CELSR2, SERPINA12, HPGD, and APOB genes. This finding provides new information to advance the limited knowledge about genetic regulation and gene interactions in metabolic pathways of LDL cholesterol. In conclusion, the proportional odds model-based GMDR is a useful tool that can boost statistical power and prediction accuracy in studying multifactor interactions underlying ordinal traits.     

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.     

Three‐part joint modeling methods for complex functional data mixed with zero‐and‐one–inflated proportions and zero‐inflated continuous outcomes with skewness

We take a functional data approach to longitudinal studies with complex bivariate outcomes. This work is motivated by data from a physical activity study that measured 2 responses over time in 5‐minute intervals. One response is the proportion of time active in each interval, a continuous proportions with excess zeros and ones. The other response, energy expenditure rate in the interval, is a continuous variable with excess zeros and skewness. This outcome is complex because there are 3 possible activity patterns in each interval (inactive, partially active, and completely active), and those patterns, which are observed, induce both nonrandom and random associations between the responses. More specifically, the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate; a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate, and the completely active pattern means that the proportion of activity is exactly one, and the energy expenditure rate is greater than zero and likely to be higher. To address these challenges, we propose a 3‐part functional data joint modeling approach. The first part is a continuation‐ratio model to reorder the ordinal valued 3 activity patterns. The second part models the proportions when they are in interval (0,1). The last component specifies the skewed continuous energy expenditure rate with Box‐Cox transformations when they are greater than zero. In this 3‐part model, the regression structures are specified as smooth curves measured at various time points with random effects that have a correlation structure. The smoothed random curves for each variable are summarized using a few important principal components, and the association of the 3 longitudinal components is modeled through the association of the principal component scores. The difficulties in handling the ordinal and proportional variables are addressed using a quasi‐likelihood type approximation. We develop an efficient algorithm to fit the model that also involves the selection of the number of principal components. The method is applied to physical activity data and is evaluated empirically by a simulation study.     

Using DCE and ranking data to estimate cardinal values for health states for deriving a preference‐based single index from the sexual quality of life questionnaire

There is an increasing interest in using data derived from ordinal methods, particularly data derived from discrete choice experiments (DCEs), to estimate the cardinal values for health states to calculate quality adjusted life years (QALYs). Ordinal measurement strategies such as DCE may have considerable practical advantages over more conventional cardinal measurement techniques, e.g. time trade‐off (TTO), because they may not require such a high degree of abstract reasoning. However, there are a number of challenges to deriving the cardinal values for health states using ordinal data, including anchoring the values on the full health–dead scale used to calculate QALYs. This paper reports on a study that deals with these problems in the context of using two ordinal techniques, DCE and ranking, to derive the cardinal values for health states derived from a condition‐specific sexual health measure. The results were compared with values generated using a commonly used cardinal valuation technique, the TTO. This study raises some important issues about the use of ordinal data to produce cardinal health state valuations. Copyright © 2009 John Wiley & Sons, Ltd.     

Sample size and power estimation for studies with health related quality of life outcomes: a comparison of four methods using the SF-36

We describe and compare four different methods for estimating sample size and power, when the primary outcome of the study is a Health Related Quality of Life (HRQoL) measure. These methods are: 1. assuming a Normal distribution and comparing two means; 2. using a non-parametric method; 3. Whitehead's method based on the proportional odds model; 4. the bootstrap. We illustrate the various methods, using data from the SF-36. For simplicity this paper deals with studies designed to compare the effectiveness (or superiority) of a new treatment compared to a standard treatment at a single point in time. The results show that if the HRQoL outcome has a limited number of discrete values (< 7) and/or the expected proportion of cases at the boundaries is high (scoring 0 or 100), then we would recommend using Whitehead's method (Method 3). Alternatively, if the HRQoL outcome has a large number of distinct values and the proportion at the boundaries is low, then we would recommend using Method 1. If a pilot or historical dataset is readily available (to estimate the shape of the distribution) then bootstrap simulation (Method 4) based on this data will provide a more accurate and reliable sample size estimate than conventional methods (Methods 1, 2, or 3). In the absence of a reliable pilot set, bootstrapping is not appropriate and conventional methods of sample size estimation or simulation will need to be used. Fortunately, with the increasing use of HRQoL outcomes in research, historical datasets are becoming more readily available. Strictly speaking, our results and conclusions only apply to the SF-36 outcome measure. Further empirical work is required to see whether these results hold true for other HRQoL outcomes. However, the SF-36 has many features in common with other HRQoL outcomes: multi-dimensional, ordinal or discrete response categories with upper and lower bounds, and skewed distributions, so therefore, we believe these results and conclusions using the SF-36 will be appropriate for other HRQoL measures.     

Modeling and inference for an ordinal effect size measure

An ordinal measure of effect size is a simple and useful way to describe the difference between two ordered categorical distributions. This measure summarizes the probability that an outcome from one distribution falls above an outcome from the other, adjusted for ties. We develop and compare confidence interval methods for the measure. Simulation studies show that with independent multinomial samples, confidence intervals based on inverting the score test and a pseudo-score-type test perform well. This score method also seems to work well with fully-ranked data, but for dependent samples a simple Wald interval on the logit scale can be better with small samples. We also explore how the ordinal effect size measure relates to an effect measure commonly used for normal distributions, and we consider a logit model for describing how it depends on explanatory variables. The methods are illustrated for a study comparing treatments for shoulder-tip pain.     

Bayesian inference for the stereotype regression model: Application to a case–control study of prostate cancer

The stereotype regression model for categorical outcomes, proposed by Anderson (J. Roy. Statist. Soc. B. 1984; 46 :1–30) is nested between the baseline‐category logits and adjacent category logits model with proportional odds structure. The stereotype model is more parsimonious than the ordinary baseline‐category (or multinomial logistic) model due to a product representation of the log‐odds‐ratios in terms of a common parameter corresponding to each predictor and category‐specific scores. The model could be used for both ordered and unordered outcomes. For ordered outcomes, the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable, but are inherently multi‐dimensional in nature. As pointed out by Greenland (Statist. Med. 1994; 13 :1665–1677), an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome‐stratified sampling as in case–control studies. In addition, for matched case–control studies, the stereotype model is amenable to classical conditional likelihood principle, whereas there is no reduction due to sufficiency under the proportional odds model. In spite of these attractive features, the model has been applied less, as there are issues with maximum likelihood estimation and likelihood‐based testing approaches due to non‐linearity and lack of identifiability of the parameters. We present comprehensive Bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach. We illustrate our methodology by analyzing data from The Flint Men's Health Study, a case–control study of prostate cancer in African‐American men aged 40–79 years. We use clinical staging of prostate cancer in terms of Tumors, Nodes and Metastasis as the categorical response of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

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

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

The use of bootstrap methods for estimating sample size and analysing health-related quality of life outcomes

Health-related quality of life (HRQoL) measures are increasingly used in trials as primary outcome measures. Investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes. HRQoL outcomes, like the SF-36, are usual measured on an ordinal scale, although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes (the ordered categories) reflect contiguous intervals along this continuum. The ordinal scaling of HRQoL measures means they tend to generate data that have discrete, bounded and skewed distributions. Thus, standard methods of analysis that assume Normality and constant variance may not be appropriate. For this reason, conventional statistical advice would suggest non-parametric methods be used to analyse HRQoL data. The bootstrap is one such computer intensive non-parametric method for estimating sample sizes and analysing data.We describe three methods of estimating sample sizes for two-group cross-sectional comparisons of HRQoL outcomes. We then compared the power of the three methods for a two-group cross-sectional study design using bootstrap simulation. The results showed that under the location shift alternative hypothesis, conventional methods of sample size estimation performed well, particularly Whitehead's method. Whitehead's method is recommended if the HRQoL outcome has a limited number of discrete values (<7) and/or the expected proportion of cases at either of the bounds is high. If a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate, than conventional methods.Finally, we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters, in an example data set. We then compared and contrasted the bootstrap with standard methods of analysing HRQoL outcomes. In the data set studied, with the SF-36 outcome, the use of the bootstrap for estimating sample sizes and analysing HRQoL data produces results similar to conventional statistical methods. These results suggest that bootstrap methods are not more appropriate for analysing HRQoL outcome data than standard methods.     

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.     

Two-part models for repeatedly measured ordinal data with “don't know” category

Ordinal data (eg, “low,” “medium,” “high”; graded response on a Likert scale) with an additional “don't know” category are frequently encountered in the medical, social, and behavioral science literature. The handling of a “don't know” option presents unique challenges as it often “destroys” the ordinal nature of the data. Commonly, nominal models are employed which ignore the partial ordering and have a complicated interpretation, especially in situations with repeatedly measured outcomes. We propose two-part models that easily accommodate longitudinal partially ordered (semiordinal) data. The most easily interpretable formulation consists of a random effect logistic submodel for “don't know” vs all the other categories combined, and a random effect ordinal submodel for the ordered categories. Correlated random effects account for statistical dependence within individual. An extension allowing for nonproportionality of odds for the predictor effects in the ordinal submodel is also considered. Maximum likelihood estimation is performed using adaptive Gaussian quadrature in SAS PROC NLMIXED. A simulation study is performed to evaluate the performance of the estimation algorithm in terms of bias and efficiency, and to compare the results of joint and separate models of the two parts, and of proportional and nonproportional model formulations. The methods are motivated and illustrated on a dataset from a study of adolescents' perceptions of nicotine strength of JUUL e-cigarettes. Using the proposed approach we show that adolescents perceive 5% nicotine content as relatively low, a misconception more pronounced among past month nonusers than among past month users of JUUL e-cigarettes.     

Simultaneous confidence intervals using ordinal effect measures for ordered categorical outcomes

When outcomes are ordered categorical, a model using an ordinal effect size measure is a good alternative of the cumulative logit model to compare several independent group differences. We present a method of constructing simultaneous confidence intervals for the ordinal effect size measures, using the studentized range distribution with the score test statistic. A simulation study shows that the proposed method performs well in terms of coverage probability, and it seems better than the method using a Bonferroni correction for Wald‐type statistics and methods that account for the dependencies among pairwise ordinal effect size measures using the multivariate normal distribution (or the multivariate t‐distribution for small samples). Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐varying effect models for ordinal responses with applications in substance abuse research

Ordinal responses are very common in longitudinal data collected from substance abuse research or other behavioral research. This study develops a new statistical model with free SAS macros that can be applied to characterize time‐varying effects on ordinal responses. Our simulation study shows that the ordinal‐scale time‐varying effects model has very low estimation bias and sometimes offers considerably better performance when fitting data with ordinal responses than a model that treats the response as continuous. Contrary to a common assumption that an ordinal scale with several levels can be treated as continuous, our results indicate that it is not so much the number of levels on the ordinal scale but rather the skewness of the distribution that makes a difference on relative performance of linear versus ordinal models. We use longitudinal data from a well‐known study on youth at high risk for substance abuse as a motivating example to demonstrate that the proposed model can characterize the time‐varying effect of negative peer influences on alcohol use in a way that is more consistent with the developmental theory and existing literature, in comparison with the linear time‐varying effect model. Copyright © 2014 John Wiley & Sons, Ltd.     

Proportional odds model for dose-finding clinical trial designs with ordinal toxicity grading

Currently many dose-finding clinical trial designs, including the continual reassessment method (CRM) and the standard ' 3 + 3' design, dichotomize toxicity outcomes based on the pre-specified dose-limiting toxicity (DLT) criteria. This loss of information is particularly inefficient due to the small sample sizes in phase I trials. Common Toxicity Criteria (CTCAEv3.0) classify adverse events into grades 1-5, which range from 1 as a mild adverse event to 5 as death related to an adverse event. In this paper, we extend the CRM to include ordinal toxicity outcomes as specified by CTCAEv3.0 using the proportional odds model (POM) and compare results with the dichotomous CRM. A sensitivity analysis of the new design compares various target DLT rates, sample sizes, and cohort sizes. This design is also assessed under various dose-toxicity relationship models including POMs as well as those that violate the proportional odds assumption. A simulation study shows that the proportional odds CRM performs as well as the dichotomous CRM on all criteria compared (including safety criteria such as percentage of patients treated at highly toxic or suboptimal dose levels) and with improved estimation of the maximum tolerated dose when the PO assumption is not violated. These findings suggest that it is beneficial to incorporate ordinal toxicity endpoints into phase I trial designs.