Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

A Semiparametric Bayesian Approach to Multivariate Longitudinal Data

We extend the standard multivariate mixed model by incorporating a smooth time effect and relaxing distributional assumptions. We propose a semiparametric Bayesian approach to multivariate longitudinal data using a mixture of Polya trees prior distribution. Usually, the distribution of random effects in a longitudinal data model is assumed to be Gaussian. However, the normality assumption may be suspect, particularly if the estimated longitudinal trajectory parameters exhibit multimodality and skewness. In this paper we propose a mixture of Polya trees prior density to address the limitations of the parametric random effects distribution. We illustrate the methodology by analyzing data from a recent HIV-AIDS study.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Semiparametric analysis of longitudinal data with informative observation times and censoring times

We focus on regression analysis of irregularly observed longitudinal data which often occur in medical follow-up studies and observational investigations. The model for such data involves two processes: a longitudinal response process of interest and an observation process controlling observation times. Restrictive models and questionable assumptions, such as Poisson assumption and independent censoring time assumption, were posed in previous works for analysing longitudinal data. In this paper, we propose a more general model together with a robust estimation approach for longitudinal data with informative observation times and censoring times, and the asymptotic normalities of the proposed estimators are established. Both simulation studies and real data application indicate that the proposed method is promising.     

The analysis of incontinence episodes and other count data in patients with overactive bladder by Poisson and negative binomial regression

Clinical studies in overactive bladder have traditionally used analysis of covariance or nonparametric methods to analyse the number of incontinence episodes and other count data. It is known that if the underlying distributional assumptions of a particular parametric method do not hold, an alternative parametric method may be more efficient than a nonparametric one, which makes no assumptions regarding the underlying distribution of the data. Therefore, there are advantages in using methods based on the Poisson distribution or extensions of that method, which incorporate specific features that provide a modelling framework for count data. One challenge with count data is overdispersion, but methods are available that can account for this through the introduction of random effect terms in the modelling, and it is this modelling framework that leads to the negative binomial distribution. These models can also provide clinicians with a clearer and more appropriate interpretation of treatment effects in terms of rate ratios. In this paper, the previously used parametric and non‐parametric approaches are contrasted with those based on Poisson regression and various extensions in trials evaluating solifenacin and mirabegron in patients with overactive bladder. In these applications, negative binomial models are seen to fit the data well. Copyright © 2014 John Wiley & Sons, Ltd.     

Sensitivity analyses for partially observed recurrent event data

Recurrent events involve the occurrences of the same type of event repeatedly over time and are commonly encountered in longitudinal studies. Examples include seizures in epileptic studies or occurrence of cancer tumors. In such studies, interest lies in the number of events that occur over a fixed period of time. One considerable challenge in analyzing such data arises when a large proportion of patients discontinues before the end of the study, for example, because of adverse events, leading to partially observed data. In this situation, data are often modeled using a negative binomial distribution with time‐in‐study as offset. Such an analysis assumes that data are missing at random (MAR). As we cannot test the adequacy of MAR, sensitivity analyses that assess the robustness of conclusions across a range of different assumptions need to be performed. Sophisticated sensitivity analyses for continuous data are being frequently performed. However, this is less the case for recurrent event or count data. We will present a flexible approach to perform clinically interpretable sensitivity analyses for recurrent event data. Our approach fits into the framework of reference‐based imputations, where information from reference arms can be borrowed to impute post‐discontinuation data. Different assumptions about the future behavior of dropouts dependent on reasons for dropout and received treatment can be made. The imputation model is based on a flexible model that allows for time‐varying baseline intensities. We assess the performance in a simulation study and provide an illustration with a clinical trial in patients who suffer from bladder cancer. Copyright © 2015 John Wiley & Sons, Ltd.     

Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

A Bayesian Shrinkage Model for Incomplete Longitudinal Binary Data with Application to the Breast Cancer Prevention Trial

We consider inference in randomized longitudinal studies with missing data that is generated by skipped clinic visits and loss to follow-up. In this setting, it is well known that full data estimands are not identified unless unverified assumptions are imposed. We assume a non-future dependence model for the drop-out mechanism and partial ignorability for the intermittent missingness. We posit an exponential tilt model that links non-identifiable distributions and distributions identified under partial ignorability. This exponential tilt model is indexed by non-identified parameters, which are assumed to have an informative prior distribution, elicited from subject-matter experts. Under this model, full data estimands are shown to be expressed as functionals of the distribution of the observed data. To avoid the curse of dimensionality, we model the distribution of the observed data using a Bayesian shrinkage model. In a simulation study, we compare our approach to a fully parametric and a fully saturated model for the distribution of the observed data. Our methodology is motivated by, and applied to, data from the Breast Cancer Prevention Trial.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

Semiparametric models of longitudinal and time-to-event data with applications to HIV viral dynamics and CD4 counts

We propose a semiparametric approach based on proportional hazards and copula method to jointly model longitudinal outcomes and the time-to-event. The dependence between the longitudinal outcomes on the covariates is modeled by a copula-based times series, which allows non-Gaussian random effects and overcomes the limitation of the parametric assumptions in existing linear and nonlinear random effects models. A modified partial likelihood method using estimated covariates at failure times is employed to draw statistical inference. The proposed model and method are applied to analyze a set of progression to AIDS data in a study of the association between the human immunodeficiency virus viral dynamics and the time trend in the CD4/CD8 ratio with measurement errors. Simulations are also reported to evaluate the proposed model and method.     

Sensitivity analysis of longitudinal data with intermittent missing values

Several models for longitudinal data with nonrandom missingness are available. The selection model of Diggle and Kenward is one of these models. It has been mentioned by many authors that this model depends on untested modelling assumptions, such as the response distribution, from the observed data. So, a sensitivity analysis of the study’s conclusions for such assumptions is needed. The stochastic EM algorithm is proposed and developed to handle continuous longitudinal data with nonrandom intermittent missing values when the responses have non-normal distribution. This is a step in investigating the sensitivity of the parameter estimates to the change of the response distribution. The proposed technique is applied to real data from the International Breast Cancer Study Group.     

Non parametric analysis of gap times for bivariate recurrent event data with cure fraction

Recurrent event data often arise in longitudinal studies. In many applications, subjects may experience two different types of events alternatively over time or a pair of subjects may experience recurrent events of the same type. Medical advances have made it possible for some patients to be cured such that the disease of interest does not recur. In this article, we consider non parametric analysis of bivariate recurrent event data with cure fraction. Using the inverse-probability weighted (IPW) approach, we propose non parametric estimators for the proportion of cured patients and for the joint distribution functions of bivariate recurrence times of the uncured ones. The asymptotic properties of the proposed estimators are established. Simulation study indicates that the proposed estimators perform well in finite samples.     

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.     

Optimal dynamic treatment regimes

Summary. A dynamic treatment regime is a list of decision rules, one per time interval, for how the level of treatment will be tailored through time to an individual's changing status. The goal of this paper is to use experimental or observational data to estimate decision regimes that result in a maximal mean response. To explicate our objective and to state the assumptions, we use the potential outcomes model. The method proposed makes smooth parametric assumptions only on quantities that are directly relevant to the goal of estimating the optimal rules. We illustrate the methodology proposed via a small simulation.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

A Bayesian nonparametric causal model

Typically, in the practice of causal inference from observational studies, a parametric model is assumed for the joint population density of potential outcomes and treatment assignments, and possibly this is accompanied by the assumption of no hidden bias. However, both assumptions are questionable for real data, the accuracy of causal inference is compromised when the data violates either assumption, and the parametric assumption precludes capturing a more general range of density shapes (e.g., heavier tail behavior and possible multi-modalities). We introduce a flexible, Bayesian nonparametric causal model to provide more accurate causal inferences. The model makes use of a stick-breaking prior, which has the flexibility to capture any multi-modalities, skewness and heavier tail behavior in this joint population density, while accounting for hidden bias. We prove the asymptotic consistency of the posterior distribution of the model, and illustrate our causal model through the analysis of small and large observational data sets.     

Mixed Correlated Bivariate Ordinal and Negative Binomial Longitudinal Responses with Nonignorable Missing Values

Regression models with random effects are proposed for joint analysis of negative binomial and ordinal longitudinal data with nonignorable missing values under fully parametric framework. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the ordinal response of osteoporosis of the spine and negative binomial response is the number of joint damage. A sensitivity of the results to the assumptions is also investigated. The effect of some covariates on all responses are investigated simultaneously.     

Copula‐based semiparametric analysis for time series data with detection limits

The analysis of time series data with detection limits is challenging due to the high‐dimensional integral involved in the likelihood. Existing methods are either computationally demanding or rely on restrictive parametric distributional assumptions. We propose a semiparametric approach, where the temporal dependence is captured by parametric copula, while the marginal distribution is estimated non‐parametrically. Utilizing the properties of copulas, we develop a new copula‐based sequential sampling algorithm, which provides a convenient way to calculate the censored likelihood. Even without full parametric distributional assumptions, the proposed method still allows us to efficiently compute the conditional quantiles of the censored response at a future time point, and thus construct both point and interval predictions. We establish the asymptotic properties of the proposed pseudo maximum likelihood estimator, and demonstrate through simulation and the analysis of a water quality data that the proposed method is more flexible and leads to more accurate predictions than Gaussian‐based methods for non‐normal data. The Canadian Journal of Statistics 47: 438–454; 2019 © 2019 Statistical Society of Canada     

Nonparametric modeling of clustered customer survival data

We incorporate a random clustering effect into the nonparametric version of Cox Proportional Hazards model to characterize clustered survival data. The simulation studies provide evidence that clustered survival data can be better characterized through a nonparametric model. Predictive accuracy of the nonparametric model is affected by number of clusters and distribution of the random component accounting for clustering effect. As the functional form of the covariate departs from linearity, the nonparametric model is becoming more advantageous over the parametric counterpart. Finally, nonparametric is better than parametric model when data are highly heterogenous and/or there is misspecification error.