Longitudinal network models and permutation-uniform Markov chains

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.     

Zero-Inflated Poisson Regression for Longitudinal Data

Medical and public health research often involve the analysis of repeated or longitudinal count data that exhibit excess zeros such as the number of yearly doctor visits by a group of individuals over a number of years. Zero-inflated Poisson (ZIP) regression models can be used to account for excess zeros in count data. We propose an extension of the ZIP model that is appropriate for longitudinal data. Our extension includes a non stationary, observation-driven time series model based correlation structure. We discuss estimation of the model parameters and the inefficiency of the estimators when the correlation structure is mis-specified. The model's application to the analysis of health care utilization data is also discussed.     

Longitudinal Poisson modeling: an application for CD4 counting in HIV-infected patients

In this paper, we present different “frailty” models to analyze longitudinal data in the presence of covariates. These models incorporate the extra-Poisson variability and the possible correlation among the repeated counting data for each individual. Assuming a CD4 counting data set in HIV-infected patients, we develop a hierarchical Bayesian analysis considering the different proposed models and using Markov Chain Monte Carlo methods. We also discuss some Bayesian discrimination aspects for the choice of the best model.     

Using a Box–Cox transformation in the analysis of longitudinal data with incomplete responses

We analyse longitudinal data on CD4 cell counts from patients who participated in clinical trials that compared two therapeutic treatments: zidovudine and didanosine. The investigators were interested in modelling the CD4 cell count as a function of treatment, age at base-line and disease stage at base-line. Serious concerns can be raised about the normality assumption of CD4 cell counts that is implicit in many methods and therefore an analysis may have to start with a transformation. Instead of assuming that we know the transformation (e.g. logarithmic) that makes the outcome normal and linearly related to the covariates, we estimate the transformation, by using maximum likelihood, within the Box–Cox family. There has been considerable work on the Box–Cox transformation for univariate regression models. Here, we discuss the Box–Cox transformation for longitudinal regression models when the outcome can be missing over time, and we also implement a maximization method for the likelihood, assumming that the missing data are missing at random.     

SOME MODELS FOR OVERDISPERSED BINOMIAL DATA

Various models are currently used to model overdispersed binomial data. It is not always clear which model is appropriate for a given situation. Here we examine the assumptions and discuss the problems and pitfalls of some of these models. We focus on clustered data with one level of nesting, briefly touching on more complex strata and longitudinal data. The estimation procedures are illustrated and some critical comments are made about the various models. We indicate which models are restrictive and how and which can be extended to model more complex situations. In addition some inadequacies in testing procedures are noted. Recommendations as to which models should be used, and when, are made.     

Sequential D-optimal designs for generalized linear mixed models

We discuss the construction of D-optimal sequential designs for the analysis of longitudinal data or repeated measurements using generalized linear mixed models (GLMMs). We investigate the performance of the design through a simulation study, which indicates that the proposed design can be very successful in improving the efficiency of the ML estimators in GLMMs relative to some common competitors. Our simulations also suggest that the usual normal-theory inference procedures remain valid under the sequential sampling schemes. We also present an example using real data obtained from a clinical 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.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

A reanalysis of a longitudinal scleroderma clinical trial using non-ignorable missingness models

When analyzing incomplete longitudinal clinical trial data, it is often inappropriate to assume that the occurrence of missingness is at random, especially in cases where visits are entirely missed. We present a framework that simultaneously models multivariate incomplete longitudinal data and a non-ignorable missingness mechanism using a Bayesian approach. A criterion measure is presented for comparing models. We demonstrate the feasibility of the methodology through reanalysis of two of the longitudinal measures from a clinical trial of penicillamine treatment for scleroderma patients. We compare the results for univariate and bivariate, ignorable and non-ignorable missingness models.     

Two-step and likelihood methods for joint models of longitudinal and survival data

We compare the commonly used two-step methods and joint likelihood method for joint models of longitudinal and survival data via extensive simulations. The longitudinal models include LME, GLMM, and NLME models, and the survival models include Cox models and AFT models. We find that the full likelihood method outperforms the two-step methods for various joint models, but it can be computationally challenging when the dimension of the random effects in the longitudinal model is not small. We thus propose an approximate joint likelihood method which is computationally efficient. We find that the proposed approximation method performs well in the joint model context, and it performs better for more “continuous” longitudinal data. Finally, a real AIDS data example shows that patients with higher initial viral load or lower initial CD4 are more likely to drop out earlier during an anti-HIV treatment.     

Local influence diagnostics for incomplete overdispersed longitudinal counts

We develop local influence diagnostics to detect influential subjects when generalized linear mixed models are fitted to incomplete longitudinal overdispersed count data. The focus is on the influence stemming from the dropout model specification. In particular, the effect of small perturbations around an MAR specification are examined. The method is applied to data from a longitudinal clinical trial in epileptic patients. The effect on models allowing for overdispersion is contrasted with that on models that do not.     

Jointly modeling skew longitudinal survival data with missingness and mismeasured covariates

Jointly modeling longitudinal and survival data has been an active research area. Most researches focus on improving the estimating efficiency but ignore many data features frequently encountered in practice. In the current study, we develop the joint models that concurrently accounting for longitudinal and survival data with multiple features. Specifically, the proposed model handles skewness, missingness and measurement errors in covariates which are typically observed in the collection of longitudinal survival data from many studies. We employ a Bayesian inferential method to make inference on the proposed model. We applied the proposed model to an real data study. A few alternative models under different conditions are compared. We conduct extensive simulations in order to evaluate how the method works.     

Modeling longitudinal binomial responses: implications from two dueling paradigms

The generalized estimating equations (GEEs) and generalized linear mixed-effects model (GLMM) are the two most popular paradigms to extend models for cross-sectional data to a longitudinal setting. Although the two approaches yield well-interpreted models for continuous outcomes, it is quite a different story when applied to binomial responses. We discuss major modeling differences between the GEE- and GLMM-derived models by presenting new results regarding the model-driven differences. Our results show that GLMM induces some artifacts in the marginal models at assessment times, making it inappropriate when applied to such responses from real study data. The different interpretations of parameters resulting from the conceptual difference between the two modeling approaches also carry quite significant implications and ramifications with respect to data and power analyses. Although a special case involving a scale difference in parameters between GEE and GLMM has been noted in the literature, its implications in real data analysis has not been thoroughly addressed. Further, this special case has a very limited covariate structure and does not apply to most real studies, especially multi-center clinical trials. The new results presented fill a substantial gap in the literature regarding the model-driven differences between the two dueling paradigms.     

Local estimation for varying-coefficient models with longitudinal data

Varying-coefficient models are very useful for longitudinal data analysis. In this paper, we focus on varying-coefficient models for longitudinal data. We develop a new estimation procedure using Cholesky decomposition and profile least squares techniques. Asymptotic normality for the proposed estimators of varying-coefficient functions has been established. Monte Carlo simulation studies show excellent finite-sample performance. We illustrate our methods with a real data example.     

Bayesian quantile regression for ordinal longitudinal data

Since the pioneering work by Koenker and Bassett [27], quantile regression models and its applications have become increasingly popular and important for research in many areas. In this paper, a random effects ordinal quantile regression model is proposed for analysis of longitudinal data with ordinal outcome of interest. An efficient Gibbs sampling algorithm was derived for fitting the model to the data based on a location-scale mixture representation of the skewed double-exponential distribution. The proposed approach is illustrated using simulated data and a real data example. This is the first work to discuss quantile regression for analysis of longitudinal data with ordinal outcome.     

Recent History Functional Linear Models for Sparse Longitudinal Data

We consider the recent history functional linear models, relating a longitudinal response to a longitudinal predictor where the predictor process only in a sliding window into the recent past has an effect on the response value at the current time. We propose an estimation procedure for recent history functional linear models that is geared towards sparse longitudinal data, where the observation times across subjects are irregular and total number of measurements per subject is small. The proposed estimation procedure builds upon recent developments in literature for estimation of functional linear models with sparse data and utilizes connections between the recent history functional linear models and varying coefficient models. We establish uniform consistency of the proposed estimators, propose prediction of the response trajectories and derive their asymptotic distribution leading to asymptotic point-wise confidence bands. We include a real data application and simulation studies to demonstrate the efficacy of the proposed methodology.     

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.     

State space mixed models for longitudinal observations with binary and binomial responses

We propose a new class of state space models for longitudinal discrete response data where the observation equation is specified in an additive form involving both deterministic and random linear predictors. These models allow us to explicitly address the effects of trend, seasonal or other time-varying covariates while preserving the power of state space models in modeling serial dependence in the data. We develop a Markov chain Monte Carlo algorithm to carry out statistical inference for models with binary and binomial responses, in which we invoke de Jong and Shephard’s (Biometrika 82(2):339–350, 1995) simulation smoother to establish an efficient sampling procedure for the state variables. To quantify and control the sensitivity of posteriors on the priors of variance parameters, we add a signal-to-noise ratio type parameter in the specification of these priors. Finally, we illustrate the applicability of the proposed state space mixed models for longitudinal binomial response data in both simulation studies and data examples.     

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.