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.     

The proportional odds model for multivariate interval-censored failure time data

The proportional odds model is one of the most commonly used regression models in failure time data analysis and has been discussed by many authors (Appl. Stat. 1983; 32:165-171; J. Am. Stat. Assoc. 1999; 94:125-136; J. Am. Stat. Assoc. 1997; 92:960-967; Biometrics 2000; 56:511-518; J. Am. Stat. Assoc. 2001; 96:1446-1457). It specifies that covariates have multiplicative effects on the odds function and is often used when, for example, the covariate effect diminishes over time. Most of the existing methods for the model are for univariate failure time data. In this paper, we discuss how to fit the proportional odds model to multivariate interval-censored failure time data. For inference, the maximum likelihood approach is developed and evaluated by simulation studies, which suggest that the method works well for practical situations. The method is applied to a set of bivariate interval-censored data arising from an AIDS clinical trial.     

Variable selection for proportional odds model

In this paper we study the problem of variable selection for the proportional odds model, which is a useful alternative to the proportional hazards model and might be appropriate when the proportional hazards assumption is not satisfied. We propose to fit the proportional odds model by maximizing the marginal likelihood subject to a shrinkage-type penalty, which encourages sparse solutions and hence facilitates the process of variable selection. Two types of shrinkage penalties are considered: the LASSO and the adaptive-LASSO (ALASSO) penalty. In the ALASSO penalty, different weights are imposed on different coefficients such that important variables are more protectively retained in the final model while unimportant ones are more likely to be shrunk to zeros. We further provide an efficient computation algorithm to implement the proposed methods, and demonstrate their performance through simulation studies and an application to real data. Numerical results indicate that both methods can produce accurate and interpretable models, and the ALASSO tends to work better than the usual LASSO.     

Bayesian joint ordinal and survival modeling for breast cancer risk assessment

We propose a joint model to analyze the structure and intensity of the association between longitudinal measurements of an ordinal marker and time to a relevant event. The longitudinal process is defined in terms of a proportional‐odds cumulative logit model. Time‐to‐event is modeled through a left‐truncated proportional‐hazards model, which incorporates information of the longitudinal marker as well as baseline covariates. Both longitudinal and survival processes are connected by means of a common vector of random effects. General inferences are discussed under the Bayesian approach and include the posterior distribution of the probabilities associated to each longitudinal category and the assessment of the impact of the baseline covariates and the longitudinal marker on the hazard function. The flexibility provided by the joint model makes possible to dynamically estimate individual event‐free probabilities and predict future longitudinal marker values. The model is applied to the assessment of breast cancer risk in women attending a population‐based screening program. The longitudinal ordinal marker is mammographic breast density measured with the Breast Imaging Reporting and Data System (BI‐RADS) scale in biennial screening exams. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

糖代谢异常影响因素累积比数Logistic回归分析

目的 分析中老年人糖代谢异常影响因素,为2型糖尿病(T₂DM)防治工作提供科学依据.方法 采用整群抽样方法抽取河北省保定市>40岁常住居民3 212人进行问卷调查、体检和实验室检查,采用累积比数Logistic回归法分析糖代谢异常的影响因素.结果 本次调查获得有效问卷3 102份.其中T₂DM患者682例,糖尿病前期(IGR)患者284例,患病率分别为21.99%和9.15%,按2000年全国人口普查数据进行标化的患病率分别为19.03%和8.68%;糖尿病家族史(OR=2.74)、腰围增加(OR=1.47)、甘油三酯异常(OR=1.44)、腰臀比异常(OR=1.41)、从事专业技术类工作(OR=1.40)、收缩压高(OR=1.27)、年龄增大(OR=1.17)、体质指数异常(OR=1.16)、脾气暴躁或易激动(OR=1.14)及女性有重大胎儿生育史(OR=1.41)为保定市中老年人糖代谢异常发生的危险因素,女性(OR=0.80)和经常食用水果(OR=0.83)为其保护因素.结论 糖代谢异常的发生受多种因素影响;通过调整饮食结构、加强体育锻炼降低体重、控制血压和血脂、保持良好心态以及戒烟、限酒可有效预防T₂DM.     

Parametric robust inferences for correlated ordinal data

The aim of this article is to provide asymptotically valid likelihood inferences about regression parameters for correlated ordinal response variables. The legitimacy of this novel approach requires no knowledge of the underlying joint distributions so long as their second moments exist. The efficacy of the proposed parametric approach is demonstrated via simulations and the analyses of two real data sets.     

A parametric model for long-term follow-up data from phase III breast cancer clinical trials

We propose a parametric version of a univariate gamma frailty model. The proposed model is shown to be flexible enough to model long-term follow-up survival data from breast cancer clinical trials when the treatment effect diminishes as time progresses, a case for which neither the proportional hazards nor proportional odds assumptions are satisfied. The observed information matrix is computed to evaluate the variances of parameter estimates. A simple parametric test statistic to test proportional odds assumption is also constructed. The model is applied to a data set from a phase III clinical trial on breast cancer.     

A Bayesian ordinal model for heterogeneity in a multi-centre myocardial infarction clinical trial

A Bayesian methodology is developed to investigate the homogeneity of the treatment effects in a multi-centre clinical trial with an ordinal response. A hierarchical model is formulated for the ordinal response, and the marginal posterior distributions of the covariates, overall treatment and the centre effects are calculated using the Gibbs sampler. The methodology is applied to data arising from a multi-centre clinical trial of therapies for acute myocardial infarction. In this trial, the overall results show that the treatment is effective. However, there appears to be substantial differences in both the baseline risk and treatment effect across centres. Thus, the observed treatment effects may not be generalized to a broader patient population, and exploratory analyses to ascertain reasons for the treatment-by-centre interaction and its possible effect on the study conclusions would be useful.     

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.     

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.     

Pairwise comparisons with ordered categorical data

Clinical trials frequently involve pairwise comparisons of different treatments to evaluate their relative efficacy. In this study, we examine methods for conducting pairwise tests of treatments with ordered categorical responses. A modified version of the Wilcoxon–Mann–Whitney test based on a logistic regression model assuming proportional odds is a popular choice for comparing two treatments. This paper discusses the extension of this test to pairwise comparisons involving more than two treatments. However, when the proportional odds assumption is not valid, the Wilcoxon–Mann–Whitney‐type test procedure cannot control the overall type I error rate at the prespecified level of significance. We therefore propose a better strategy in which a latent normal model is employed. We presented a simulated comparative study of power and the overall type I error rate to illustrate the superiority of the latent normal model. Examples are also given for illustrative purposes. Copyright © 2013 John Wiley & Sons, Ltd.     

Goodness-of-fit tests for ordinal response regression models

It is well documented that the commonly used Pearson chi-square and deviance statistics are not adequate for assessing goodness-of-fit in logistic regression models when continuous covariates are modelled. In recent years, several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently. However, these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case. We present the modified Pearson chi-square and deviance tests that are appropriate for assessing goodness-of-fit in ordinal response models when both categorical and continuous covariates are present. The methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate. These tests also provide immediate information as to where a model may not fit well. In addition, the methods are simple to understand and implement, and are non-specific. That is, they do not require prespecification of a type of lack-of-fit to detect.     

Beta‐binomial model for meta‐analysis of odds ratios

In meta‐analysis of odds ratios (ORs), heterogeneity between the studies is usually modelled via the additive random effects model (REM). An alternative, multiplicative REM for ORs uses overdispersion. The multiplicative factor in this overdispersion model (ODM) can be interpreted as an intra‐class correlation (ICC) parameter. This model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta‐distributed, resulting in beta‐binomial distributions. We propose two new estimators of the ICC for meta‐analysis in this setting. One is based on the inverted Breslow‐Day test, and the other on the improved gamma approximation by Kulinskaya and Dollinger (2015, p. 26) to the distribution of Cochran's Q. The performance of these and several other estimators of ICC on bias and coverage is studied by simulation. Additionally, the Mantel‐Haenszel approach to estimation of ORs is extended to the beta‐binomial model, and we study performance of various ICC estimators when used in the Mantel‐Haenszel or the inverse‐variance method to combine ORs in meta‐analysis. The results of the simulations show that the improved gamma‐based estimator of ICC is superior for small sample sizes, and the Breslow‐Day‐based estimator is the best for . The Mantel‐Haenszel‐based estimator of OR is very biased and is not recommended. The inverse‐variance approach is also somewhat biased for ORs≠1, but this bias is not very large in practical settings. Developed methods and R programs, provided in the Web Appendix, make the beta‐binomial model a feasible alternative to the standard REM for meta‐analysis of ORs. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Comparing odds ratios measuring improvement or deterioration across repeated exposure or treatment sessions

Unbiased estimating equations are used to estimate and compare the odds ratios representing the odds of heightened anxiety following acute exposures relative to baseline in a cross-over study examining indoor air contaminants. Although estimating equations are used in the creation of the estimates and standard errors, inference is conditional on subject. The proposed method produces estimates that are less biased and computationally simpler than pseudo-likelihood. In addition, specification of the form of the random-effects distributions is not required.     

A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Quality-of-life (QOL) is an important outcome in clinical research, particularly in cancer clinical trials. Typically, data are collected longitudinally from patients during treatment and subsequent follow-up. Missing data are a common problem, and missingness may arise in a non-ignorable fashion. In particular, the probability that a patient misses an assessment may depend on the patient's QOL at the time of the scheduled assessment. We propose a Markov chain model for the analysis of categorical outcomes derived from QOL measures. Our model assumes that transitions between QOL states depend on covariates through generalized logit models or proportional odds models. To account for non-ignorable missingness, we incorporate logistic regression models for the conditional probabilities of observing measurements, given their actual values. The model can accommodate time-dependent covariates. Estimation is by maximum likelihood, summing over all possible values of the missing measurements. We describe options for selecting parsimonious models, and we study the finite-sample properties of the estimators by simulation. We apply the techniques to data from a breast cancer clinical trial in which QOL assessments were made longitudinally, and in which missing data frequently arose.     

Active-control trials with binary data: a comparison of methods for testing superiority or non-inferiority using the odds ratio

This paper considers methods for testing for superiority or non-inferiority in active-control trials with binary data, when the relative treatment effect is expressed as an odds ratio. Three asymptotic tests for the log-odds ratio based on the unconditional binary likelihood are presented, namely the likelihood ratio, Wald and score tests. All three tests can be implemented straightforwardly in standard statistical software packages, as can the corresponding confidence intervals. Simulations indicate that the three alternatives are similar in terms of the Type I error, with values close to the nominal level. However, when the non-inferiority margin becomes large, the score test slightly exceeds the nominal level. In general, the highest power is obtained from the score test, although all three tests are similar and the observed differences in power are not of practical importance.     

Analysis of survival data by the proportional odds model

A model is presented for the analysis of lifetime data in which the rates of mortality for separate groups of patients converge with time. A non-parametric estimate is given for the survivor function. The theoretical basis for the model assumes that prognostic factors have a multiplicative effect on the odds against survival beyond any given time. The model is fitted to data using maximum likelihood estimation, and an example of its use in the analysis of a lung cancer trial is given.     

Improved genetic association tests for an ordinal outcome representing the disease progression process

We are interested in detecting genetic variants that influence transition between discrete stages of a disease progression process, such as the natural history of progression to cervical cancer with the following four stages: (1) normal-human papillomavirus (HPV) exposed, (2) persistent infection with oncogenic HPV, (3) cervical intraepithelial neoplasia grades 2 or 3 (CIN2/3), and (4) cervical cancer. Standard statistical tests derived from the proportional odds model or polytomous regression model can be used to study this type of ordinal outcome. But these methods are either too sensitive to the proportion odds assumption or fail to take advantage of the restriction on the parameter space for the genetic variants. Two alternative tests, the maximum score test (MAX) and the adaptive P-value combination test (Adapt-P), are proposed with the aim of striking a balance between efficiency and robustness. A simulation study demonstrates that MAX and Adapt-P have the most robust performance among all considered tests under various realistic scenarios. As a demonstration, we applied the considered tests to a genetic association study of cervical cancer.     

Non-parametric assessment of non-inferiority with censored data

We suggest non-parametric tests for showing non-inferiority of a new treatment compared to a standard therapy when data are censored. To this end the difference and the odds ratio curves of the entire survivor functions over a certain time period are considered. Two asymptotic approaches for solving these testing problems are investigated, which are based on bootstrap approximations. The performance of the test procedures is investigated in a simulation study, and some guidance on which test to use in specific situations is derived. The proposed methods are applied to a trial in which two thrombolytic agents for the treatment on acute myocardial infarction were compared, and to a study on irradiation therapies for advanced non-small-cell lung cancer. Non-inferiority over a large time period of the study can be shown in both cases.     

Semi‐parametric latent process model for longitudinal ordinal data: Application to cognitive decline

Ordinal and quantitative discrete data are frequent in biomedical and neuropsychological studies. We propose a semi‐parametric model for the analysis of the change over time of such data in longitudinal studies. A threshold model is defined where the outcome value depends on the current value of an underlying Gaussian latent process. The latent process model is a Gaussian linear mixed model with a non‐parametric function of time, f(t), to model the expected change over time. This model includes random‐effects and a stochastic error process to flexibly handle correlation between repeated measures. The function f(t) and all the model parameters are estimated by penalized likelihood using a cubic‐spline approximation for f(t). The smoothing parameter is estimated by an approximate cross‐validation criterion. Confidence bands may be computed for the estimated curves for the latent process and, using a Monte Carlo approach, for the outcome in its natural scale. The method is applied to the Paquid cohort data to compare the time‐course over 14 years of two cognitive scores in a sample of 350 future Alzheimer patients and in a matched sample of healthy subjects. Copyright © 2010 John Wiley & Sons, Ltd.