AN APPROACH FOR JOINTLY MODELING MULTIVARIATE LONGITUDINAL MEASUREMENTS AND DISCRETE TIME-TO-EVENT DATA

In many medical studies, patients are followed longitudinally and interest is on assessing the relationship between longitudinal measurements and time to an event. Recently, various authors have proposed joint modeling approaches for longitudinal and time-to-event data for a single longitudinal variable. These joint modeling approaches become intractable with even a few longitudinal variables. In this paper we propose a regression calibration approach for jointly modeling multiple longitudinal measurements and discrete time-to-event data. Ideally, a two-stage modeling approach could be applied in which the multiple longitudinal measurements are modeled in the first stage and the longitudinal model is related to the time-to-event data in the second stage. Biased parameter estimation due to informative dropout makes this direct two-stage modeling approach problematic. We propose a regression calibration approach which appropriately accounts for informative dropout. We approximate the conditional distribution of the multiple longitudinal measurements given the event time by modeling all pairwise combinations of the longitudinal measurements using a bivariate linear mixed model which conditions on the event time. Complete data are then simulated based on estimates from these pairwise conditional models, and regression calibration is used to estimate the relationship between longitudinal data and time-to-event data using the complete data. We show that this approach performs well in estimating the relationship between multivariate longitudinal measurements and the time-to-event data and in estimating the parameters of the multiple longitudinal process subject to informative dropout. We illustrate this methodology with simulations and with an analysis of primary biliary cirrhosis (PBC) data.     

A Frailty Model in Crossover Studies with Time-to-Event Response Variable

In this article, we develop a model to study treatment, period, carryover, and other applicable effects in a crossover design with a time-to-event response variable. Because time-to-event outcomes on different treatment regimens within the crossover design are correlated for an individual, we adopt a proportional hazards frailty model. If the frailty is assumed to have a gamma distribution, and the hazard rates are piecewise constant, then the likelihood function can be determined via closed-form expressions. We illustrate the methodology via an application to a data set from an asthma clinical trial and run simulations that investigate sensitivity of the model to data generated from different distributions.     

Mixture of latent trait analyzers for model-based clustering of categorical data

Model-based clustering methods for continuous data are well established and commonly used in a wide range of applications. However, model-based clustering methods for categorical data are less standard. Latent class analysis is a commonly used method for model-based clustering of binary data and/or categorical data, but due to an assumed local independence structure there may not be a correspondence between the estimated latent classes and groups in the population of interest. The mixture of latent trait analyzers model extends latent class analysis by assuming a model for the categorical response variables that depends on both a categorical latent class and a continuous latent trait variable; the discrete latent class accommodates group structure and the continuous latent trait accommodates dependence within these groups. Fitting the mixture of latent trait analyzers model is potentially difficult because the likelihood function involves an integral that cannot be evaluated analytically. We develop a variational approach for fitting the mixture of latent trait models and this provides an efficient model fitting strategy. The mixture of latent trait analyzers model is demonstrated on the analysis of data from the National Long Term Care Survey (NLTCS) and voting in the U.S. Congress. The model is shown to yield intuitive clustering results and it gives a much better fit than either latent class analysis or latent trait analysis alone.     

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.     

Joint model-based clustering of nonlinear longitudinal trajectories and associated time-to-event data analysis,linked by latent class membership: with application to AIDS clinical studies

Longitudinal and time-to-event data are often observed together. Finite mixture models are currently used to analyze nonlinear heterogeneous longitudinal data, which, by releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, can cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, and be associated with clinically important time-to-event data. This article develops a joint modeling approach to a finite mixture of NLME models for longitudinal data and proportional hazard Cox model for time-to-event data, linked by individual latent class indicators, under a Bayesian framework. The proposed joint models and method are applied to a real AIDS clinical trial data set, followed by simulation studies to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and Cox model are fitted separately.     

Recurrent time-to-event models with ordinal outcomes

A model to accommodate time-to-event ordinal outcomes was proposed by Berridge and Whitehead. Very few studies have adopted this approach, despite its appeal in incorporating several ordered categories of event outcome. More recently, there has been increased interest in utilizing recurrent events to analyze practical endpoints in the study of disease history and to help quantify the changing pattern of disease over time. For example, in studies of heart failure, the analysis of a single fatal event no longer provides sufficient clinical information to manage the disease. Similarly, the grade/frequency/severity of adverse events may be more important than simply prolonged survival in studies of toxic therapies in oncology. We propose an extension of the ordinal time-to-event model to allow for multiple/recurrent events in the case of marginal models (where all subjects are at risk for each recurrence, irrespective of whether they have experienced previous recurrences) and conditional models (subjects are at risk of a recurrence only if they have experienced a previous recurrence). These models rely on marginal and conditional estimates of the instantaneous baseline hazard and provide estimates of the probabilities of an event of each severity for each recurrence over time. We outline how confidence intervals for these probabilities can be constructed and illustrate how to fit these models and provide examples of the methods, together with an interpretation of the results.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

Spatial and non-spatial model-based protection procedures for the release of business microdata

In this paper we discuss methodology for the safe release of business microdata. In particular we extend the model-based protection procedure of Franconi and Stander (2002, The Statistician 51: 1–11) by allowing the model to take account of the spatial structure underlying the geographical information in the microdata. We discuss the use of the Gibbs sampler for performing the computations required by this spatial approach. We provide an empirical comparison of these non-spatial and spatial disclosure limitation methods based on the Italian sample from the Community Innovation Survey. We quantify the level of protection achieved for the released microdata and the error induced when various inferences are performed. We find that although the spatial method often induces higher inferential errors, it almost always provides more protection. Moreover the aggregated areas from the spatial procedure can be somewhat more spatially smooth, and hence possibly more meaningful, than those from the non-spatial approach. We discuss possible applications of these model-based protection procedures to more spatially extensive data sets.     

A semiparametric type II Tobit model for time-to-event data with application to a merger and acquisition study

Merger and acquisition is an important corporate strategy. We collect recent merger and acquisition data for companies on the China A-share stock market to explore the relationship between corporate ownership structure and speed of merger success. When studying merger success, selection bias occurs if only completed mergers are analyzed. There is also a censoring problem when duration time is used to measure the speed. In this article, for time-to-event outcomes, we propose a semiparametric version of the type II Tobit model that can simultaneously handle selection bias and right censoring. The proposed model can also easily incorporate time-dependent covariates. A nonparametric maximum likelihood estimator is proposed. The resulting estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Some Monte Carlo studies are carried out to assess the finite-sample performance of the proposed approach. Using the proposed model, we find that higher power balance of a company is associated with faster merger success.     

The value of the aggregate data approach in meta-analysis with time-to-event outcomes

Collecting individual patient data has been described as the 'gold standard' for undertaking meta-analysis. If studies involve time-to-event outcomes, conducting a meta-analysis based on aggregate data can be problematical. Two meta-analyses of randomized controlled trials with time-to-event outcomes are used to illustrate the practicality and value of several proposed methods to obtain summary statistic estimates. In the first example the results suggest that further effort should be made to find unpublished trials. In the second example the use of aggregate data for trials where no individual patient data have been supplied allows the totality of evidence to be assessed and indicates previously unrecognized heterogeneity.     

Model-based hierarchical clustering with Bregman divergences and Fishers mixture model: application to depth image analysis

Model-based clustering is a method that clusters data with an assumption of a statistical model structure. In this paper, we propose a novel model-based hierarchical clustering method for a finite statistical mixture model based on the Fisher distribution. The main foci of the proposed method are: (a) provide efficient solution to estimate the parameters of a Fisher mixture model (FMM); (b) generate a hierarchy of FMMs and (c) select the optimal model. To this aim, we develop a Bregman soft clustering method for FMM. Our model estimation strategy exploits Bregman divergence and hierarchical agglomerative clustering. Whereas, our model selection strategy comprises a parsimony-based approach and an evaluation graph-based approach. We empirically validate our proposed method by applying it on simulated data. Next, we apply the method on real data to perform depth image analysis. We demonstrate that the proposed clustering method can be used as a potential tool for unsupervised depth image analysis.     

The competing risks Cox model with auxiliary case covariates under weaker missing-at-random cause of failure

In the analysis of time-to-event data with multiple causes using a competing risks Cox model, often the cause of failure is unknown for some of the cases. The probability of a missing cause is typically assumed to be independent of the cause given the time of the event and covariates measured before the event occurred. In practice, however, the underlying missing-at-random assumption does not necessarily hold. Motivated by colorectal cancer molecular pathological epidemiology analysis, we develop a method to conduct valid analysis when additional auxiliary variables are available for cases only. We consider a weaker missing-at-random assumption, with missing pattern depending on the observed quantities, which include the auxiliary covariates. We use an informative likelihood approach that will yield consistent estimates even when the underlying model for missing cause of failure is misspecified. The superiority of our method over naive methods in finite samples is demonstrated by simulation study results. We illustrate the use of our method in an analysis of colorectal cancer data from the Nurses’ Health Study cohort, where, apparently, the traditional missing-at-random assumption fails to hold.     

Identification of local clusters for count data: a model-based Moran's I test

We set out I_DR as a loglinear-model-based Moran's I test for Poisson count data that resembles the Moran's I residual test for Gaussian data. We evaluate its type I and type II error probabilities via simulations, and demonstrate its utility via a case study. When population sizes are heterogeneous, I_DR is effective in detecting local clusters by local association terms with an acceptable type I error probability. When used in conjunction with local spatial association terms in loglinear models, I_DR can also indicate the existence of first-order global cluster that can hardly be removed by local spatial association terms. In this situation, I_DR should not be directly applied for local cluster detection. In the case study of St. Louis homicides, we bridge loglinear model methods for parameter estimation to exploratory data analysis, so that a uniform association term can be defined with spatially varied contributions among spatial neighbors. The method makes use of exploratory tools such as Moran's I scatter plots and residual plots to evaluate the magnitude of deviance residuals, and it is effective to model the shape, the elevation and the magnitude of a local cluster in the model-based test.     

A Flexible Approach to Time-varying Coefficients in the Cox Regression Setting

Research on methods for studying time-to-event data (survival analysis) has been extensive in recent years. The basic model in use today represents the hazard function for an individual through a proportional hazards model (Cox, 1972). Typically, it is assumed that a covariate's effect on the hazard function is constant throughout the course of the study. In this paper we propose a method to allow for possible deviations from the standard Cox model, by allowing the effect of a covariate to vary over time. This method is based on a dynamic linear model. We present our method in terms of a Bayesian hierarchical model. We fit the model to the data using Markov chain Monte Carlo methods. Finally, we illustrate the approach with several examples. This revised version was published online in July 2006 with corrections to the Cover Date.     

Diagnostic checks for discrete data regression models using posterior predictive simulations

Model checking with discrete data regressions can be difficult because the usual methods such as residual plots have complicated reference distributions that depend on the parameters in the model. Posterior predictive checks have been proposed as a Bayesian way to average the results of goodness-of-fit tests in the presence of uncertainty in estimation of the parameters. We try this approach using a variety of discrepancy variables for generalized linear models fitted to a historical data set on behavioural learning. We then discuss the general applicability of our findings in the context of a recent applied example on which we have worked. We find that the following discrepancy variables work well, in the sense of being easy to interpret and sensitive to important model failures: structured displays of the entire data set, general discrepancy variables based on plots of binned or smoothed residuals versus predictors and specific discrepancy variables created on the basis of the particular concerns arising in an application. Plots of binned residuals are especially easy to use because their predictive distributions under the model are sufficiently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals defined from an underlying continuous model and quantile–quantile plots of these residuals.     

Cluster detection and clustering with random start forward searches

The forward search is a method of robust data analysis in which outlier free subsets of the data of increasing size are used in model fitting; the data are then ordered by closeness to the model. Here the forward search, with many random starts, is used to cluster multivariate data. These random starts lead to the diagnostic identification of tentative clusters. Application of the forward search to the proposed individual clusters leads to the establishment of cluster membership through the identification of non-cluster members as outlying. The method requires no prior information on the number of clusters and does not seek to classify all observations. These properties are illustrated by the analysis of 200 six-dimensional observations on Swiss banknotes. The importance of linked plots and brushing in elucidating data structures is illustrated. We also provide an automatic method for determining cluster centres and compare the behaviour of our method with model-based clustering. In a simulated example with eight clusters our method provides more stable and accurate solutions than model-based clustering. We consider the computational requirements of both procedures.     

Prediction of transplant-free survival in idiopathic pulmonary fibrosis patients using joint models for event times and mixed multivariate longitudinal data

We implement a joint model for mixed multivariate longitudinal measurements, applied to the prediction of time until lung transplant or death in idiopathic pulmonary fibrosis. Specifically, we formulate a unified Bayesian joint model for the mixed longitudinal responses and time-to-event outcomes. For the longitudinal model of continuous and binary responses, we investigate multivariate generalized linear mixed models using shared random effects. Longitudinal and time-to-event data are assumed to be independent conditional on available covariates and shared parameters. A Markov chain Monte Carlo algorithm, implemented in OpenBUGS, is used for parameter estimation. To illustrate practical considerations in choosing a final model, we fit 37 different candidate models using all possible combinations of random effects and employ a deviance information criterion to select a best-fitting model. We demonstrate the prediction of future event probabilities within a fixed time interval for patients utilizing baseline data, post-baseline longitudinal responses, and the time-to-event outcome. The performance of our joint model is also evaluated in simulation studies.     

A study of R2 measure under the accelerated failure time models

For right-censored data, the accelerated failure time (AFT) model is an alternative to the commonly used proportional hazards regression model. It is a linear model for the (log-transformed) outcome of interest, and is particularly useful for censored outcomes that are not time-to-event, such as laboratory measurements. We provide a general and easily computable definition of the R² measure of explained variation under the AFT model for right-censored data. We study its behavior under different censoring scenarios and under different error distributions; in particular, we also study its robustness when the parametric error distribution is misspecified. Based on Monte Carlo investigation results, we recommend the log-normal distribution as a robust error distribution to be used in practice for the parametric AFT model, when the R² measure is of interest. We apply our methodology to an alcohol consumption during pregnancy data set from Ukraine.     

Approaches for event history analysis based on complex longitudinal survey data

A researcher using complex longitudinal survey data for event history analysis has to make several choices that affect the analysis results. These choices include the following: whether a design-based or a model-based approach for the analysis is taken, which subset of data to use and, if a design-based approach is chosen, which weights to use. We discuss different choices and illustrate their effects using longitudinal register data linked at person-level with the Finnish subset of the European Community Household Panel data. The use of register data enables us to construct an event history data set without nonresponse and attrition. Design-based estimates from these data are used as benchmarks against design-based and model-based estimates from subsets of data usually available for a survey data analyst. Our illustration suggests that the often recommended way to use panel data for longitudinal analyses, data from total respondents and weights from the last wave analysed may not be the best way to go. Instead, using all available data and weights from the first survey wave appears to be a safe choice for longitudinal analyses based on multipurpose survey data.