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.     

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.     

A general score‐independent test for order‐restricted inference

In the analysis of ordered categorical data, the categories are often assigned a set of subjectively chosen order‐restricted scores. To overcome the arbitrariness involved in the assignment of the scores, several score‐independent tests have been proposed. However, these methods are limited to 2 × K contingency tables, where K is the number of ordered categories. We present an efficiency robust score‐independent test that is applicable to more general situations. The test is embedded into a flexible framework for conditional inference and provides a natural generalization of many familiar tests involving ordered categorical data, such as the generalized Cochran‐Mantel‐Haenszel test for singly or doubly ordered contingency tables, the Page test for randomized block designs and the Tarone‐Ware trend test for survival data. The proposed method is illustrated by several numerical examples.     

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.     

Ordinal response prediction using bootstrap aggregation,with application to a high‐throughput methylation data set

Many investigators conducting translational research are performing high‐throughput genomic experiments and then developing multigenic classifiers using the resulting high‐dimensional data set. In a large number of applications, the class to be predicted may be inherently ordinal. Examples of ordinal outcomes include tumor‐node‐metastasis (TNM) stage (I, II, III, IV); drug toxicity evaluated as none, mild, moderate, or severe; and response to treatment classified as complete response, partial response, stable disease, or progressive disease. While one can apply nominal response classification methods to ordinal response data, in doing so some information is lost that may improve the predictive performance of the classifier. This study examined the effectiveness of alternative ordinal splitting functions combined with bootstrap aggregation for classifying an ordinal response. We demonstrate that the ordinal impurity and ordered twoing methods have desirable properties for classifying ordinal response data and both perform well in comparison to other previously described methods. Developing a multigenic classifier is a common goal for microarray studies, and therefore application of the ordinal ensemble methods is demonstrated on a high‐throughput methylation data set. Copyright © 2009 John Wiley & Sons, Ltd.     

A Cochran–Armitage‐type and a score‐free global test for multivariate ordinal data

We propose a Cochran–Armitage‐type and a score‐free global test that can be used to assess the presence of an association between a set of ordinally scaled covariates and an outcome variable within the range of generalized linear models. Both tests are developed within the framework of the well‐established ‘global test’ methodology and as such are feasible in high‐dimensional data situations under any correlation and enable adjustment for covariates. The Cochran–Armitage‐type test, for which an intimate connection with the traditional score‐based Cochran–Armitage test is shown, rests upon explicit assumptions on the distances between the covariates' ordered categories. The score‐free test, in contrast, parametrizes these distances and thus keeps them flexible, rendering it ideally suited for covariates measured on an ordinal scale. As confirmed by means of simulations, the Cochran–Armitage‐type test focuses its power on set‐outcome relationships where the distances between the covariates' categories are equal or close to those assumed, whereas the score‐free test spreads its power over a wide range of possible set‐outcome relationships, putting more emphasis on monotonic than on non‐monotonic ones. Based on the tests' power properties, it is discussed when to favour one or the other, and the practical merits of both of them are illustrated by an application in the field of rehabilitation medicine. Our proposed tests are implemented in the R package globaltest . Copyright © 2016 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.     

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.     

Measures of agreement between many raters for ordinal classifications

Screening and diagnostic procedures often require a physician's subjective interpretation of a patient's test result using an ordered categorical scale to define the patient's disease severity. Because of wide variability observed between physicians' ratings, many large‐scale studies have been conducted to quantify agreement between multiple experts' ordinal classifications in common diagnostic procedures such as mammography. However, very few statistical approaches are available to assess agreement in these large‐scale settings. Many existing summary measures of agreement rely on extensions of Cohen's kappa. These are prone to prevalence and marginal distribution issues, become increasingly complex for more than three experts, or are not easily implemented. Here we propose a model‐based approach to assess agreement in large‐scale studies based upon a framework of ordinal generalized linear mixed models. A summary measure of agreement is proposed for multiple experts assessing the same sample of patients' test results according to an ordered categorical scale. This measure avoids some of the key flaws associated with Cohen's kappa and its extensions. Simulation studies are conducted to demonstrate the validity of the approach with comparison with commonly used agreement measures. The proposed methods are easily implemented using the software package R and are applied to two large‐scale cancer agreement studies. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

Random‐effects meta‐analysis of the clinical utility of tests and prediction models

The use of data from multiple studies or centers for the validation of a clinical test or a multivariable prediction model allows researchers to investigate the test's/model's performance in multiple settings and populations. Recently, meta‐analytic techniques have been proposed to summarize discrimination and calibration across study populations. Here, we rather consider performance in terms of net benefit, which is a measure of clinical utility that weighs the benefits of true positive classifications against the harms of false positives. We posit that it is important to examine clinical utility across multiple settings of interest. This requires a suitable meta‐analysis method, and we propose a Bayesian trivariate random‐effects meta‐analysis of sensitivity, specificity, and prevalence. Across a range of chosen harm‐to‐benefit ratios, this provides a summary measure of net benefit, a prediction interval, and an estimate of the probability that the test/model is clinically useful in a new setting. In addition, the prediction interval and probability of usefulness can be calculated conditional on the known prevalence in a new setting. The proposed methods are illustrated by 2 case studies: one on the meta‐analysis of published studies on ear thermometry to diagnose fever in children and one on the validation of a multivariable clinical risk prediction model for the diagnosis of ovarian cancer in a multicenter dataset. Crucially, in both case studies the clinical utility of the test/model was heterogeneous across settings, limiting its usefulness in practice. This emphasizes that heterogeneity in clinical utility should be assessed before a test/model is routinely implemented.     

Multivariate non‐parametric methods for Mann–Whitney statistics to analyse cross‐over studies with two treatment sequences

A non‐parametric strategy for the analysis of ordinal data from cross‐over studies with two treatment sequences and d(⩾2) periods is examined through Mann–Whitney rank measures of association. For each period, these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group. Such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment. Methods for U‐statistics are used to produce a consistent estimate of the covariance matrix for the (d+u) Mann–Whitney estimates. The effects of periods and treatments on the respective Mann–Whitney estimates are evaluated through linear (or log‐linear) models. For estimation of the parameters in these models, a modified weighted least squares method is applied through a (2d−1)⩽(d+u) dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the Mann–Whitney estimates. The proposed methods are applicable to response variables with an interval or an ordered categorical scale. Their scope additionally has capabilities for controlling strata in the design of a cross‐over study or concomitant variables for which covariance adjustment is of interest for reduction of variance. Applications of the methods are illustrated through three cross‐over studies with different specifications for the two sequences of two treatments during two to four periods. Copyright © 1999 John Wiley & Sons, Ltd.     

A consistency‐adjusted alpha‐adaptive strategy for sequential testing

In a clinical trial with two clinically important endpoints, each of which can fully characterize a treatment benefit to support an efficacy claim by itself, a minimum degree of consistency in the findings is expected; otherwise interpretation of study findings can be problematic. Clinical trial literature contains examples where lack of consistency in the findings of clinically relevant endpoints led to difficulties in interpreting study results. The aim of this paper is to introduce this consistency concept at the study design stage and investigate the consequences of its implementation in the statistical analysis plan. The proposed methodology allows testing of hierarchically ordered endpoints to proceed as long as a pre‐specified consistency criterion is met. In addition, while an initial allocation of the alpha level is specified for the ordered endpoints at the design stage, the methodology allows the alpha level allocated to the second endpoint to be adaptive to the findings of the first endpoint. In addition, the methodology takes into account the correlation between the endpoints in calculating the significance level and the power of the test for the next endpoint. The proposed Consistency‐Adjusted Alpha‐Adaptive Strategy (CAAAS) is very general. Several of the well‐known multiplicity adjustment approaches arise as special cases of this strategy by appropriate selection of the consistency level and the form of alpha‐adaptation function. We discuss control of the Type I error rate as well as power of the proposed methodology and consider its application to clinical trial data. Published in 2010 by John Wiley & Sons, Ltd.     

A new model for ordinal pain data from a pharmaceutical study

For the pain data analysed previously by Cox and Chuang, this paper proposes a new model that assumes monotone scores for ordered response categories. This proposed model possesses several attractive features and allows a stochastic ordering of the drugs under comparison. Such a model also provides insight regarding the ordinal scale used to classify response. Estimation of the parameters in the model is obtained by use of BMDP3R.     

Power and sample size evaluation for the Cochran-Mantel-Haenszel mean score (Wilcoxon rank sum) test and the Cochran-Armitage test for trend

The power of a chi‐square test, and thus the required sample size, are a function of the noncentrality parameter that can be obtained as the limiting expectation of the test statistic under an alternative hypothesis specification. Herein, we apply this principle to derive simple expressions for two tests that are commonly applied to discrete ordinal data. The Wilcoxon rank sum test for the equality of distributions in two groups is algebraically equivalent to the Mann–Whitney test. The Kruskal–Wallis test applies to multiple groups. These tests are equivalent to a Cochran–Mantel–Haenszel mean score test using rank scores for a set of C‐discrete categories. Although various authors have assessed the power function of the Wilcoxon and Mann–Whitney tests, herein it is shown that the power of these tests with discrete observations, that is, with tied ranks, is readily provided by the power function of the corresponding Cochran–Mantel–Haenszel mean scores test for two and R > 2 groups. These expressions yield results virtually identical to those derived previously for rank scores and also apply to other score functions. The Cochran–Armitage test for trend assesses whether there is an monotonically increasing or decreasing trend in the proportions with a positive outcome or response over the C‐ordered categories of an ordinal independent variable, for example, dose. Herein, it is shown that the power of the test is a function of the slope of the response probabilities over the ordinal scores assigned to the groups that yields simple expressions for the power of the test. Copyright © 2011 John Wiley & Sons, Ltd.     

Generalized estimating equations to estimate the ordered stereotype logit model for panel data

By modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories, and category-specific scalar effects, the ordered stereotype logit model is a flexible regression model for ordinal response variables. In this article, we propose a generalized estimating equations (GEE) approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices, which are not required to be correctly specified. A simulation study compares the performance of GEE estimators based on various working correlation matrices and working covariance matrices using local odds ratios. Estimation of the model is illustrated using a real-world dataset. The results from the simulation study suggest that GEE estimation of this model is feasible in medium-sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix. For low true correlations, the efficiency gains seem to be rather small and if the working covariance structure is too flexible, the corresponding estimator may even be less efficient compared with the GEE estimator assuming independence. Like for GEE estimators more generally, if the true correlations over time are high, then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence.     

Mixture drug‐count response model for the high‐dimensional drug combinatory effect on myopathy

Drug‐drug interactions (DDIs) are a common cause of adverse drug events (ADEs). The electronic medical record (EMR) database and the FDA's adverse event reporting system (FAERS) database are the major data sources for mining and testing the ADE associated DDI signals. Most DDI data mining methods focus on pair‐wise drug interactions, and methods to detect high‐dimensional DDIs in medical databases are lacking. In this paper, we propose 2 novel mixture drug‐count response models for detecting high‐dimensional drug combinations that induce myopathy. The “count” indicates the number of drugs in a combination. One model is called fixed probability mixture drug‐count response model with a maximum risk threshold (FMDRM‐MRT). The other model is called count‐dependent probability mixture drug‐count response model with a maximum risk threshold (CMDRM‐MRT), in which the mixture probability is count dependent. Compared with the previous mixture drug‐count response model (MDRM) developed by our group, these 2 new models show a better likelihood in detecting high‐dimensional drug combinatory effects on myopathy. CMDRM‐MRT identified and validated (54; 374; 637; 442; 131) 2‐way to 6‐way drug interactions, respectively, which induce myopathy in both EMR and FAERS databases. We further demonstrate FAERS data capture much higher maximum myopathy risk than EMR data do. The consistency of 2 mixture models' parameters and local false discovery rate estimates are evaluated through statistical simulation studies.     

Computationally simple analysis of matched,outcome‐based studies of ordinal disease states

Outcome‐based sampling is an efficient study design for rare conditions, such as glioblastoma. It is often used in conjunction with matching, for increased efficiency and to potentially avoid bias due to confounding. A study was conducted at the Massachusetts General Hospital that involved retrospective sampling of glioblastoma patients with respect to multiple‐ordered disease states, as defined by three categories of overall survival time. To analyze such studies, we posit an adjacent categories logit model and exploit its allowance for prospective analysis of a retrospectively sampled study and its advantageous removal of set and level specific nuisance parameters through conditioning on sufficient statistics. This framework allows for any sampling design and is not limited to one level of disease within each set, such as in previous publications. We describe how this ordinal conditional model can be fit using standard conditional logistic regression procedures. We consider an alternative pseudo‐likelihood approach that potentially offers robustness under partial model misspecification at the expense of slight loss of efficiency under correct model specification for small sample sizes. We apply our methods to the Massachusetts General Hospital glioblastoma study. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.