Bayesian hierarchical duration model for repeated events: an application to behavioral observations

This article presents a continuous-time Bayesian model for analyzing durations of behavior displays in social interactions. Duration data of social interactions are often complex because of repeated behaviors (events) at individual or group (e.g. dyad) level, multiple behaviors (multistates), and several choices of exit from a current event (competing risks). A multilevel, multistate model is proposed to adequately characterize the behavioral processes. The model incorporates dyad-specific and transition-specific random effects to account for heterogeneity among dyads and interdependence among competing risks. The proposed method is applied to child–parent observational data derived from the School Transitions Project to assess the relation of emotional expression in child–parent interaction to risk for early and persisting child conduct problems.     

MULTILEVEL DISCRETE‐TIME EVENT HISTORY MODELS WITH APPLICATIONS TO THE ANALYSIS OF RECURRENT EMPLOYMENT TRANSITIONS

Data on the timing of events such as births, residential moves and changes in employment status are collected in many longitudinal surveys. These data often have a highly complex structure, with events of several types occurring repeatedly over time to an individual and interdependences between different event processes (e.g. births and employment transitions). The aim of this paper is to review a general class of multilevel discrete‐time event history models for handling recurrent events and transitions between multiple states. It is also shown how standard methods can be extended to allow for time‐varying covariates that are outcomes of an event process that is jointly determined with the process of interest. The considerable potential of these methods for studying transitions through the life course is illustrated in analyses of the effect of the presence and age of children on women's employment transitions, using data from the British Household Panel Survey.     

A comparison of univariate and multivariate multilevel models for repeated measures of use of antenatal care in Uttar Pradesh

Summary.  We compare two different multilevel modelling approaches to the analysis of repeated measures data to assess the effect of mother level characteristics on women's use of prenatal care services in Uttar Pradesh, India. We apply univariate multilevel models to our data and find that the model assumptions are severely violated and the parameter estimates are not stable, particularly for the mother level random effect. To overcome this we apply a multivariate multilevel model. The correlation structure shows that, once the decision has been made regarding use of antenatal care by the mother for her first observed birth in the data, she does not tend to change this decision for higher order births.     

Multilevel multivariate modelling of childhood growth, numbers of growth measurements and adult characteristics

Summary.  A general latent normal model for multilevel data with mixtures of response types is extended in the case of ordered responses to deal with variates having a large number of categories and including count data. An example is analysed by using repeated measures data on child growth and adult measures of body mass index and glucose. Applications are described that are concerned with the flexible prediction of adult measurements from collections of growth measurements and for studying the relationship between the number of measurement occasions and growth trajectories.     

An example of a two-part latent growth curve model for semicontinuous outcomes in the health sciences

A new method of modeling coronary artery calcium (CAC) is needed in order to properly understand the probability of onset and growth of CAC. CAC remains a controversial indicator of cardiovascular disease (CVD) risk, but this may be due to ill-equipped methods of specifying CAC during the analysis phase of studies reporting an analysis where CAC is the primary outcome. The modern method of two-part latent growth modeling may represent a strong alternative to the myriad of existing methods for modeling CAC. We provide a brief overview of existing methods of analysis used for CAC before introducing the general latent growth curve model, how it extends into a two-part (semicontinuous) growth model, and how the ubiquitous problem of missing data can be effectively handled. We then present an example of how to model CAC using this framework. We demonstrate that utilizing this type of modeling strategy can result in traditional predictors of CAC (e.g. age, gender, and high-density lipoprotein cholesterol), exerting a different impact on the two different, yet simultaneous, operationalizations of CAC. This method of analyzing CAC could inform future analyses of CAC and inform subsequent discussions about the nature of its potential to inform long-term CVD risk and heart events.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).     

Analysis of growth curve data by using cubic smoothing splines

Longitudinal data frequently arises in various fields of applied sciences where individuals are measured according to some ordered variable, e.g. time. A common approach used to model such data is based on the mixed models for repeated measures. This model provides an eminently flexible approach to modeling of a wide range of mean and covariance structures. However, such models are forced into a rigidly defined class of mathematical formulas which may not be well supported by the data within the whole sequence of observations. A possible non-parametric alternative is a cubic smoothing spline, which is highly flexible and has useful smoothing properties. It can be shown that under normality assumption, the solution of the penalized log-likelihood equation is the cubic smoothing spline, and this solution can be further expressed as a solution of the linear mixed model. It is shown here how cubic smoothing splines can be easily used in the analysis of complete and balanced data. Analysis can be greatly simplified by using the unweighted estimator studied in the paper. It is shown that if the covariance structure of random errors belong to certain class of matrices, the unweighted estimator is the solution to the penalized log-likelihood function. This result is new in smoothing spline context and it is not only confined to growth curve settings. The connection to mixed models is used in developing a rough testing of group profiles. Numerical examples are presented to illustrate the techniques proposed.     

A monte carlo comparison of the smoothing,scoring and em algorithms for dispersion matrix estimation with incomplete growth curve data

Incomplete growth curve data often result from missing or mistimed observations in a repeated measures design. Virtually all methods of analysis rely on the dispersion matrix estimates. A Monte Carlo simulation was used to compare three methods of estimation of dispersion matrices for incomplete growth curve data. The three methods were: 1) maximum likelihood estimation with a smoothing algorithm, which finds the closest positive semidefinite estimate of the pairwise estimated dispersion matrix; 2) a mixed effects model using the EM (estimation maximization) algorithm; and 3) a mixed effects model with the scoring algorithm. The simulation included 5 dispersion structures, 20 or 40 subjects with 4 or 8 observations per subject and 10 or 30% missing data. In all the simulations, the smoothing algorithm was the poorest estimator of the dispersion matrix. In most cases, there were no significant differences between the scoring and EM algorithms. The EM algorithm tended to be better than the scoring algorithm when the variances of the random effects were close to zero, especially for the simulations with 4 observations per subject and two random effects.     

Multilevel models for repeated binary outcomes: attitudes and voting over the electoral cycle

Models for fitting longitudinal binary responses are explored by using a panel study of voting intentions. A standard multilevel repeated measures logistic model is shown to be inadequate owing to a substantial proportion of respondents who maintain a constant response over time. A multivariate binary response model is shown to be a better fit to the data.     

Multilevel models for longitudinal variables prognostic for survival

The issue of modelling the joint distribution of survival time and of prognostic variables measured periodically has recently become of interest in the AIDS literature but is of relevance in other applications. The focus of this paper is on clinical trials where follow-up measurements of potentially prognostic variables are often collected but not routinely used. These measurements can be used to study the biological evolution of the disease of interest; in particular the effect of an active treatment can be examined by comparing the time profiles of patients in the active and placebo group. It is proposed to use multilevel regression analysis to model the individual repeated observations as function of time and, possibly, treatment. To address the problem of informative drop-out—which may arise if deaths (or any other censoring events) are related to the unobserved values of the prognostic variables—we analyse sequentially overlapping portions of the follow-up information. An example arising from a randomized clinical trial for the treatment of primary biliary cirrhosis is examined in detail.     

Simulated maximum likelihood estimation in joint models for multiple longitudinal markers and recurrent events of multiple types,in the presence of a terminal event

In medical studies we are often confronted with complex longitudinal data. During the follow-up period, which can be ended prematurely by a terminal event (e.g. death), a subject can experience recurrent events of multiple types. In addition, we collect repeated measurements from multiple markers. An adverse health status, represented by ‘bad’ marker values and an abnormal number of recurrent events, is often associated with the risk of experiencing the terminal event. In this situation, the missingness of the data is not at random and, to avoid bias, it is necessary to model all data simultaneously using a joint model. The correlations between the repeated observations of a marker or an event type within an individual are captured by normally distributed random effects. Because the joint likelihood contains an analytically intractable integral, Bayesian approaches or quadrature approximation techniques are necessary to evaluate the likelihood. However, when the number of recurrent event types and markers is large, the dimensionality of the integral is high and these methods are too computationally expensive. As an alternative, we propose a simulated maximum-likelihood approach based on quasi-Monte Carlo integration to evaluate the likelihood of joint models with multiple recurrent event types and markers.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

A Discrete-Time Multilevel Mixture Model for Event History Data with Long-Term Survivors,with an Application to an Analysis of Contraceptive Sterilization in Bangladesh

Event history models typically assume that the entire population is at risk of experiencing the event of interest throughout the observation period. However, there will often be individuals, referred to as long-term survivors, who may be considered a priori to have a zero hazard throughout the study period. In this paper, a discrete-time mixture model is proposed in which the probability of long-term survivorship and the timing of event occurrence are modelled jointly. Another feature of event history data that often needs to be considered is that they may come from a population with a hierarchical structure. For example, individuals may be nested within geographical regions and individuals in the same region may have similar risks of experiencing the event of interest due to unobserved regional characteristics. Thus, the discrete-time mixture model is extended to allow for clustering in the likelihood and timing of an event within regions. The model is further extended to allow for unobserved individual heterogeneity in the hazard of event occurrence. The proposed model is applied in an analysis of contraceptive sterilization in Bangladesh. The results show that a woman's religion and education level affect her probability of choosing sterilization, but not when she gets sterilized. There is also evidence of community-level variation in sterilization timing, but not in the probability of sterilization.     

Multilevel Mixed Linear Models for Survival Data

For the analysis of correlated survival data mixed linear models are useful alternatives to frailty models. By their use the survival times can be directly modelled, so that the interpretation of the fixed and random effects is straightforward. However, because of intractable integration involved with the use of marginal likelihood the class of models in use has been severely restricted. Such a difficulty can be avoided by using hierarchical-likelihood, which provides a statistically efficient and fast fitting algorithm for multilevel models. The proposed method is illustrated using the chronic granulomatous disease data. A simulation study is carried out to evaluate the performance.     

A Bayesian Approach to Model Interdependent Event Histories by Graphical Models

In event history analysis, the problem of modeling two interdependent processes is still not completely solved. In a frequentist framework, there are two most general approaches: the causal approach and the system approach. The recent growing interest in Bayesian statistics suggests some interesting works on survival models and event history analysis in a Bayesian perspective. In this work we present a possible solution for the analysis of dynamic interdependence by a Bayesian perspective in a graphical duration model framework, using marked point processes. Main results from the Bayesian approach and the comparison with the frequentist one are illustrated on a real example: the analysis of the dynamic relationship between fertility and female employment.     

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.     

The complexity of school and neighbourhood effects and movements of pupils on school differences in models of educational achievement

Summary.  Traditional studies of school differences in educational achievement use multilevel modelling techniques to take into account the nesting of pupils within schools. However, educational data are known to have more complex non-hierarchical structures. The potential importance of such structures is apparent when considering the effect of pupil mobility during secondary schooling on educational achievement. Movements of pupils between schools suggest that we should model pupils as belonging to the series of schools that are attended and not just their final school. Since these school moves are strongly linked to residential moves, it is important to explore additionally whether achievement is also affected by the history of neighbourhoods that are lived in. Using the national pupil database, this paper combines multiple membership and cross-classified multilevel models to explore simultaneously the relationships between secondary school, primary school, neighbourhood and educational achievement. The results show a negative relationship between pupil mobility and achievement, the strength of which depends greatly on the nature and timing of these moves. Accounting for pupil mobility also reveals that schools and neighbourhoods are more important than shown by previous analysis. A strong primary school effect appears to last long after a child has left that phase of schooling. The additional effect of neighbourhoods, in contrast, is small. Crucially, the rank order of school effects across all types of pupil is sensitive to whether we account for the complexity of the multilevel data structure.     

Reml estimation for repeated measures analysis

Models for repeated measures or growth curves consist of a mean response plus error and the errors are usually correlated. Both maximum likelihood and residual maximum likelihood (REML) estimators of a regression model with dependent errors are derived for cases in which the variance matrix of the error model admits a convenient Cholesky factorisation. This factorisation may be linked to methods for producing recursive estimates of the regression parameters and recursive residuals to provide a convenient computational method. The method is used to develop a general approach to repeated measures analysis.     

Marginal Regression of Multivariate Event Times Based on Linear Transformation Models

Multivariate event time data are common in medical studies and have received much attention recently. In such data, each study subject may potentially experience several types of events or recurrences of the same type of event, or event times may be clustered. Marginal distributions are specified for the multivariate event times in multiple events and clustered events data, and for the gap times in recurrent events data, using the semiparametric linear transformation models while leaving the dependence structures for related events unspecified. We propose several estimating equations for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to the well-known bladder cancer tumor recurrences data is also given to illustrate the methodology.     

A Simulation Comparison of Approximate Tests for Fixed Effects in Random Coefficients Growth Curve Models

Often, the response variables on sampling units are observed repeatedly over time. The sampling units may come from different populations, such as treatment groups. This setting is routinely modeled by a random coefficients growth curve model, and the techniques of general linear mixed models are applied to address the primary research aim. An alternative approach is to reduce each subject’s data to summary measures, such as within-subject averages or regression coefficients. One may then test for equality of means of the summary measures (or functions of them) among treatment groups. Here, we compare by simulation the performance characteristics of three approximate tests based on summary measures and one based on the full data, focusing mainly on accuracy of p-values. We find that performances of these procedures can be quite different for small samples in several different configurations of parameter values. The summary-measures approach performed at least as well as the full-data mixed models approach.