Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial

The 'Clinical antipsychotic trials in intervention effectiveness' study, was designed to evaluate whether there were significant differences between several antipsychotic medications in effectiveness, tolerability, cost and quality of life of subjects with schizophrenia. Overall, 74 % of patients discontinued the study medication for various reasons before the end of 18 months in phase I of the study. When such a large percentage of study participants fail to complete the study schedule, it is not clear whether the apparent profile in effectiveness reflects genuine changes over time or is influenced by selection bias, with participants with worse (or better) outcome values being more likely to drop out or to discontinue. To assess the effect of dropouts for different reasons on inferences, we construct a joint model for the longitudinal outcome and cause-specific dropouts that allows for interval-censored dropout times. Incorporating the information regarding the cause of dropout improves inferences and provides better understanding of the association between cause-specific dropout and the outcome process. We use simulations to demonstrate the advantages of the joint modelling approach in terms of bias and efficiency.     

含非随机缺失数据的面板数据参数估计方法

抽样调查领域常采用对多个受访者进行跟踪调查得到面板数据,进而对总体特性进行统计推断,在面板数据中常含缺失数据,大多数处理面板缺失数据的软件都是直接删去含缺失值的受访者以得到完全数据集,当数据缺失机制为非随机缺失时会导致总体参数估计结果有偏。本文针对数据缺失机制为非随机缺失情形下,如何对面板数据进行统计分析进行了阐述,主要采用的是基于模型的似然推断法,对目标变量、缺失指示变量和随机效应向量的联合分布建模,在已有选择模型和模式混合模型的基础上,引入随机效应,研究目标变量期望的计算方法,并研究随机效应杂合模型下参数的估计方法,在变量分布相对简单的情形下给出了用极大似然法推断总体参数的估计步骤,最后通过模拟分析比较方法的优劣。     

Regression Modeling with Recurrent Events and Time-Dependent Interval-Censored Marker Data

In life history studies involving patients with chronic diseases it is often of interest to study the relationship between a marker process and a more clinically relevant response process. This interest may arise from a desire to gain a better understanding of the underlying pathophysiology, a need to evaluate the utility of the marker as a potential surrogate outcome, or a plan to conduct inferences based on joint models. We consider data from a trial of breast cancer patients with bone metastases. In this setting, the marker process is a point process which records the onset times and cumulative number of bone lesions which reflects the extent of metastatic bone involvement. The response is also a point process, which records the times patients experience skeletal complications resulting from these bone lesions. Interest lies in assessing how the development of new bone lesions affects the incidence of skeletal complications. By considering the marker as an internal time-dependent covariate in the point process model for skeletal complications we develop and apply methods which allow one to express the association via regression. A complicating feature of this study is that new bone lesions are only detected upon periodic radiographic examination, which makes the marker processes subject to interval-censoring. A modified EM algorithm is used to deal with this incomplete data problem.     

Application of sensitivity analysis to incomplete longitudinal CD4 count data

In this paper, we investigate the effect of tuberculosis pericarditis (TBP) treatment on CD4 count changes over time and draw inferences in the presence of missing data. We accounted for missing data and conducted sensitivity analyses to assess whether inferences under missing at random (MAR) assumption are sensitive to not missing at random (NMAR) assumptions using the selection model (SeM) framework. We conducted sensitivity analysis using the local influence approach and stress-testing analysis. Our analyses showed that the inferences from the MAR are robust to the NMAR assumption and influential subjects do not overturn the study conclusions about treatment effects and the dropout mechanism. Therefore, the missing CD4 count measurements are likely to be MAR. The results also revealed that TBP treatment does not interact with HIV/AIDS treatment and that TBP treatment has no significant effect on CD4 count changes over time. Although the methods considered were applied to data in the IMPI trial setting, the methods can also be applied to clinical trials with similar settings.     

Parametric bootstrap inferences for panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients and the variance components of panel data regression models with complete panels. The PB pivot variables are proposed based on sufficient statistics of the parameters. On the other hand, we also derive generalized inferences and improved generalized inferences for variance components in this article. Some simulation results are presented to compare the performance of the PB approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly the same as that of generalized inferences with respect to the expected lengths and powers. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the incomplete panels. Finally, the proposed approaches are illustrated by using a real data example.     

Testing the proportional odds assumption in multiply imputed ordinal longitudinal data

A popular choice when analyzing ordinal data is to consider the cumulative proportional odds model to relate the marginal probabilities of the ordinal outcome to a set of covariates. However, application of this model relies on the condition of identical cumulative odds ratios across the cut-offs of the ordinal outcome; the well-known proportional odds assumption. This paper focuses on the assessment of this assumption while accounting for repeated and missing data. In this respect, we develop a statistical method built on multiple imputation (MI) based on generalized estimating equations that allows to test the proportionality assumption under the missing at random setting. The performance of the proposed method is evaluated for two MI algorithms for incomplete longitudinal ordinal data. The impact of both MI methods is compared with respect to the type I error rate and the power for situations covering various numbers of categories of the ordinal outcome, sample sizes, rates of missingness, well-balanced and skewed data. The comparison of both MI methods with the complete-case analysis is also provided. We illustrate the use of the proposed methods on a quality of life data from a cancer clinical trial.     

A Bayesian model for multinomial sampling with misclassified data

In this paper the issue of making inferences with misclassified data from a noisy multinomial process is addressed. A Bayesian model for making inferences about the proportions and the noise parameters is developed. The problem is reformulated in a more tractable form by introducing auxiliary or latent random vectors. This allows for an easy-to-implement Gibbs sampling-based algorithm to generate samples from the distributions of interest. An illustrative example related to elections is also presented.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Recovering information in the case of underdetermined problems and incomplete economic data

A framework is developed to recover parameters in the case of incomplete data and underdetermined economic models. Within this context, the maximum entropy formalism is used as the criterion for recovering and making inferences relative to the unknown parameters. Examples are given to suggest the general nature of the problem specification and the corresponding inferences.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

Sensitivity to imputation models and assumptions in receiver operating characteristic analysis with incomplete data

Modern statistical methods using incomplete data have been increasingly applied in a wide variety of substantive problems. Similarly, receiver operating characteristic (ROC) analysis, a method used in evaluating diagnostic tests or biomarkers in medical research, has also been increasingly popular problem in both its development and application. While missing-data methods have been applied in ROC analysis, the impact of model mis-specification and/or assumptions (e.g. missing at random) underlying the missing data has not been thoroughly studied. In this work, we study the performance of multiple imputation (MI) inference in ROC analysis. Particularly, we investigate parametric and non-parametric techniques for MI inference under common missingness mechanisms. Depending on the coherency of the imputation model with the underlying data generation mechanism, our results show that MI generally leads to well-calibrated inferences under ignorable missingness mechanisms.     

Handling missing data in a composite outcome with partially observed components: simulation study based on clustered paediatric routine data

Composite scores are useful in providing insights and trends about complex and multidimensional quality of care processes. However, missing data in subcomponents may hinder the overall reliability of a composite measure. In this study, strategies for handling missing data in Paediatric Admission Quality of Care (PAQC) score, an ordinal composite outcome, were explored through a simulation study. Specifically, the implications of the conventional method employed in addressing missing PAQC score subcomponents, consisting of scoring missing PAQC score components with a zero, and a multiple imputation (MI)-based strategy, were assessed. The latent normal joint modelling MI approach was used for the latter. Across simulation scenarios, MI of missing PAQC score elements at item level produced minimally biased estimates compared to the conventional method. Moreover, regression coefficients were more prone to bias compared to standards errors. Magnitude of bias was dependent on the proportion of missingness and the missing data generating mechanism. Therefore, incomplete composite outcome subcomponents should be handled carefully to alleviate potential for biased estimates and misleading inferences. Further research on other strategies of imputing at the component and composite outcome level and imputing compatibly with the substantive model in this setting, is needed.KEYWORDS: Composite outcome, multiple imputation, paediatrics, PAQC score, pneumonia     

An application of the mixed-effects model and pattern mixture model to treatment groups with differential missingness suspected not-missing-at-random

Likelihood-based, mixed-effects models for repeated measures (MMRMs) are occasionally used in primary analyses for group comparisons of incomplete continuous longitudinal data. Although MMRM analysis is generally valid under missing-at-random assumptions, it is invalid under not-missing-at-random (NMAR) assumptions. We consider the possibility of bias of estimated treatment effect using standard MMRM analysis in a motivational case, and propose simple and easily implementable pattern mixture models within the framework of mixed-effects modeling, to handle the NMAR data with differential missingness between treatment groups. The proposed models are a new form of pattern mixture model that employ a categorical time variable when modeling the outcome and a continuous time variable when modeling the missingness-data patterns. The models can directly provide an overall estimate of the treatment effect of interest using the average of the distribution of the missingness indicator and a categorical time variable in the same manner as MMRM analysis. Our simulation results indicate that the bias of the treatment effect for MMRM analysis was considerably larger than that for the pattern mixture model analysis under NMAR assumptions. In the case study, it would be dangerous to interpret only the results of the MMRM analysis, and the proposed pattern mixture model would be useful as a sensitivity analysis for treatment effect evaluation.     

Evaluation of missing data mechanisms in two and three dimensional incomplete tables

The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies.     

A joint model for incomplete data in crossover trials

Crossover designs are popular in early phases of clinical trials and in bioavailability and bioequivalence studies. Assessment of carryover effects, in addition to the treatment effects, is a critical issue in crossover trails. The observed data from a crossover trial can be incomplete because of potential dropouts. A joint model for analyzing incomplete data from crossover trials is proposed in this article; the model includes a measurement model and an outcome dependent informative model for the dropout process. The informative-dropout model is compared with the ignorable-dropout model as specific cases of the latter are nested subcases of the proposed joint model. Markov chain sampling methods are used for Bayesian analysis of this model. The joint model is used to analyze depression score data from a clinical trial in women with late luteal phase dysphoric disorder. Interestingly, carryover effect is found to have a strong effect in the informative dropout model, but it is less significant when dropout is considered ignorable.     

A multilevel analysis of longitudinal ordinal data: evaluation of the level of physical performance of women receiving adjuvant therapy for breast cancer

Longitudinal health-related quality-of-life (QOL) data are often collected as part of clinical studies. Here two analyses of QOL data from a prospective study of breast cancer patients evaluate how physical performance is related to factors such as age, menopausal status and type of adjuvant treatment. The first analysis uses summary statistic methods. The same questions are then addressed using a multilevel model. Because of the structure of the physical performance response, regression models for the analysis of ordinal data are used. The analyses of base-line and follow-up QOL data at four time points over two years from 257 women show that reported base-line physical performance was consistently associated with later performance and that women who had received chemotherapy in the month before the QOL assessment had a greater physical performance burden. There is a slight power gain of the multilevel model over the summary statistic analysis. The multilevel model also allows relationships with time-dependent covariates to be included, highlighting treatment-related factors affecting physical performance that could not be considered within the summary statistic analysis. Checking of the multilevel model assumptions is exemplified.     

Bayesian Longitudinal Data Analysis with Mixed Models and Thick-tailed Distributions using MCMC

Linear mixed effects models are frequently used to analyse longitudinal data, due to their flexibility in modelling the covariance structure between and within observations. Further, it is easy to deal with unbalanced data, either with respect to the number of observations per subject or per time period, and with varying time intervals between observations. In most applications of mixed models to biological sciences, a normal distribution is assumed both for the random effects and for the residuals. This, however, makes inferences vulnerable to the presence of outliers. Here, linear mixed models employing thick-tailed distributions for robust inferences in longitudinal data analysis are described. Specific distributions discussed include the Student-t, the slash and the contaminated normal. A Bayesian framework is adopted, and the Gibbs sampler and the Metropolis-Hastings algorithms are used to carry out the posterior analyses. An example with data on orthodontic distance growth in children is discussed to illustrate the methodology. Analyses based on either the Student-t distribution or on the usual Gaussian assumption are contrasted. The thick-tailed distributions provide an appealing robust alternative to the Gaussian process for modelling distributions of the random effects and of residuals in linear mixed models, and the MCMC implementation allows the computations to be performed in a flexible manner.     

Bayesian covariance estimation and inference in latent Gaussian process models

This paper describes inference methods for functional data under the assumption that the functional data of interest are smooth latent functions, characterized by a Gaussian process, which have been observed with noise over a finite set of time points. The methods we propose are completely specified in a Bayesian environment that allows for all inferences to be performed through a simple Gibbs sampler. Our main focus is in estimating and describing uncertainty in the covariance function. However, these models also encompass functional data estimation, functional regression where the predictors are latent functions, and an automatic approach to smoothing parameter selection. Furthermore, these models require minimal assumptions on the data structure as the time points for observations do not need to be equally spaced, the number and placement of observations are allowed to vary among functions, and special treatment is not required when the number of functional observations is less than the dimensionality of those observations. We illustrate the effectiveness of these models in estimating latent functional data, capturing variation in the functional covariance estimate, and in selecting appropriate smoothing parameters in both a simulation study and a regression analysis of medfly fertility data.     

Trend analysis with response incompatible formats and measurement error

The increasing popularity of longitudinal studies, along with the rapid advances in science and technology, has created a potential incompatibility between data formats, which leads to an inference problem when applying conventional statistical methods. This inference problem is further compounded by measurement error, since incompatible data format often arise in the context of measuring latent constructs. Without a systematic study of the impact of scale differences, ad-hoc approaches generally lead to inconsistent estimates and thus, invalid statistical inferences. In this paper, we examine the asymptotic properties and identify conditions that guarantee consistent estimation within the context of a trend analysis with response incompatible formats and measurement error. For model estimation, we introduce two competing methods that use a generalized estimating equation approach to provide inferences for the parameters of interest, and highlight the relative strengths of each method. The approach is illustrated by data obtained from a multi-centre AIDS cohort study (MACS), where a trend analysis of an immunologic marker of HIV infection is of interest.