The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi‐parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI‐GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI‐GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood‐based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI‐GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE‐based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM. Copyright © 2009 John Wiley & Sons, Ltd.     

A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.     

A joint marginalized multilevel model for longitudinal outcomes

The shared-parameter model and its so-called hierarchical or random-effects extension are widely used joint modeling approaches for a combination of longitudinal continuous, binary, count, missing, and survival outcomes that naturally occurs in many clinical and other studies. A random effect is introduced and shared or allowed to differ between two or more repeated measures or longitudinal outcomes, thereby acting as a vehicle to capture association between the outcomes in these joint models. It is generally known that parameter estimates in a linear mixed model (LMM) for continuous repeated measures or longitudinal outcomes allow for a marginal interpretation, even though a hierarchical formulation is employed. This is not the case for the generalized linear mixed model (GLMM), that is, for non-Gaussian outcomes. The aforementioned joint models formulated for continuous and binary or two longitudinal binomial outcomes, using the LMM and GLMM, will naturally have marginal interpretation for parameters associated with the continuous outcome but a subject-specific interpretation for the fixed effects parameters relating covariates to binary outcomes. To derive marginally meaningful parameters for the binary models in a joint model, we adopt the marginal multilevel model (MMM) due to Heagerty [13] and Heagerty and Zeger [14] and formulate a joint MMM for two longitudinal responses. This enables to (1) capture association between the two responses and (2) obtain parameter estimates that have a population-averaged interpretation for both outcomes. The model is applied to two sets of data. The results are compared with those obtained from the existing approaches such as generalized estimating equations, GLMM, and the model of Heagerty [13]. Estimates were found to be very close to those from single analysis per outcome but the joint model yields higher precision and allows for quantifying the association between outcomes. Parameters were estimated using maximum likelihood. The model is easy to fit using available tools such as the SAS NLMIXED procedure.     

Consistency of information criteria for model selection with missing data

ABSTRACT

In this paper, we investigate the consistency of the Expectation Maximization (EM) algorithm-based information criteria for model selection with missing data. The criteria correspond to a penalization of the conditional expectation of the complete data log-likelihood given the observed data and with respect to the missing data conditional density. We present asymptotic properties related to maximum likelihood estimation in the presence of incomplete data and we provide sufficient conditions for the consistency of model selection by minimizing the information criteria. Their finite sample performance is illustrated through simulation and real data studies.     

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19 D. Hedeker, Generalized linear mixed models, in Encyclopedia of Statistics in Behavioral Science, B. Everitt and D. Howell, eds., John Wiley, London, 2005, pp. 729–738. [Google Scholar]] and a simulation.     

Modeling longitudinal count data with dropouts

This paper explores the utility of different approaches for modeling longitudinal count data with dropouts arising from a clinical study for the treatment of actinic keratosis lesions on the face and balding scalp. A feature of these data is that as the disease for subjects on the active arm improves their data show larger dispersion compared with those on the vehicle, exhibiting an over‐dispersion relative to the Poisson distribution. After fitting the marginal (or population averaged) model using the generalized estimating equation (GEE), we note that inferences from such a model might be biased as dropouts are treatment related. Then, we consider using a weighted GEE (WGEE) where each subject's contribution to the analysis is weighted inversely by the subject's probability of dropout. Based on the model findings, we argue that the WGEE might not address the concerns about the impact of dropouts on the efficacy findings when dropouts are treatment related. As an alternative, we consider likelihood‐based inference where random effects are added to the model to allow for heterogeneity across subjects. Finally, we consider a transition model where, unlike the previous approaches that model the log‐link function of the mean response, we model the subject's actual lesion counts. This model is an extension of the Poisson autoregressive model of order 1, where the autoregressive parameter is taken to be a function of treatment as well as other covariates to induce different dispersions and correlations for the two treatment arms. We conclude with a discussion about model selection. Published in 2009 by John Wiley & Sons, Ltd.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

Bayesian inference for nonlinear stochastic SIR epidemic model

ABSTRACT

Inference for epidemic parameters can be challenging, in part due to data that are intrinsically stochastic and tend to be observed by means of discrete-time sampling, which are limited in their completeness. The problem is particularly acute when the likelihood of the data is computationally intractable. Consequently, standard statistical techniques can become too complicated to implement effectively. In this work, we develop a powerful method for Bayesian paradigm for susceptible–infected–removed stochastic epidemic models via data-augmented Markov Chain Monte Carlo. This technique samples all missing values as well as the model parameters, where the missing values and parameters are treated as random variables. These routines are based on the approximation of the discrete-time epidemic by diffusion process. We illustrate our techniques using simulated epidemics and finally we apply them to the real data of Eyam plague.     

First-order marginalised transition random effects models with probit link function

Marginalised models, also known as marginally specified models, have recently become a popular tool for analysis of discrete longitudinal data. Despite being a novel statistical methodology, these models introduce complex constraint equations and model fitting algorithms. On the other hand, there is a lack of publicly available software to fit these models. In this paper, we propose a three-level marginalised model for analysis of multivariate longitudinal binary outcome. The implicit function theorem is introduced to approximately solve the marginal constraint equations explicitly. probit link enables direct solutions to the convolution equations. Parameters are estimated by maximum likelihood via a Fisher–Scoring algorithm. A simulation study is conducted to examine the finite-sample properties of the estimator. We illustrate the model with an application to the data set from the Iowa Youth and Families Project. The R package pnmtrem is prepared to fit the model.     

Determination of the Selection Statistics and Best Significance Level in Backward Stepwise Logistic Regression

The weighted generalized estimating equation (WGEE), an extension of the generalized estimating equation (GEE) method, is a method for analyzing incomplete longitudinal data. An inappropriate specification of the working correlation structure results in the loss of efficiency of the GEE estimation. In this study, we evaluated the efficiency of WGEE estimation for incomplete longitudinal data when the working correlation structure was misspecified. As a result, we found that the efficiency of the WGEE estimation was lower when an improper working correlation structure was selected, similar to the case of the GEE method. Furthermore, we modified the criterion proposed by Gosho et al. (2011 Gosho, M., Hamada, C. and Yoshimura, I. 2011. Criterion for the selection of a working correlation structure in the generalized estimating equation approach for longitudinal balanced data. Communications in Statistics -Theory and Methods, 40: 3839–3856. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for selecting a working correlation structure, such that the GEE and WGEE methods can be applied to incomplete longitudinal data, and we investigated the performance of the modified criterion. The results revealed that when the modified criterion was adopted, the proportion that the true correlation structure was selected was likely higher than that in the case of adopting other competing approaches.     

A Bayesian model for longitudinal count data with non-ignorable dropout

Asthma is an important chronic disease of childhood. An intervention programme for managing asthma was designed on principles of self-regulation and was evaluated by a randomized longitudinal study.The study focused on several outcomes, and, typically, missing data remained a pervasive problem. We develop a pattern-mixture model to evaluate the outcome of intervention on the number of hospitalizations with non-ignorable dropouts. Pattern-mixture models are not generally identifiable as no data may be available to estimate a number of model parameters. Sensitivity analyses are performed by imposing structures on the unidentified parameters.We propose a parameterization which permits sensitivity analyses on clustered longitudinal count data that have missing values due to non-ignorable missing data mechanisms. This parameterization is expressed as ratios between event rates across missing data patterns and the observed data pattern and thus measures departures from an ignorable missing data mechanism. Sensitivity analyses are performed within a Bayesian framework by averaging over different prior distributions on the event ratios. This model has the advantage of providing an intuitive and flexible framework for incorporating the uncertainty of the missing data mechanism in the final analysis.     

A stochastic variant of the EM algorithm to fit mixed (discrete and continuous) longitudinal data with nonignorable missingness

Abstract

In longitudinal studies data are collected on the same set of units for more than one occasion. In medical studies it is very common to have mixed Poisson and continuous longitudinal data. In such studies, for different reasons, some intended measurements might not be available resulting in a missing data setting. When the probability of missingness is related to the missing values, the missingness mechanism is termed nonrandom. The stochastic expectation-maximization (SEM) algorithm and the parametric fractional imputation (PFI) method are developed to handle nonrandom missingness in mixed discrete and continuous longitudinal data assuming different covariance structures for the continuous outcome. The proposed techniques are evaluated using simulation studies. Also, the proposed techniques are applied to the interstitial cystitis data base (ICDB) data.     

Stochastic EM algorithm of a finite mixture model from hurdle Poisson distribution with missing responses

ABSTRACT

In this article, a finite mixture model of hurdle Poisson distribution with missing outcomes is proposed, and a stochastic EM algorithm is developed for obtaining the maximum likelihood estimates of model parameters and mixing proportions. Specifically, missing data is assumed to be missing not at random (MNAR)/non ignorable missing (NINR) and the corresponding missingness mechanism is modeled through probit regression. To improve the algorithm efficiency, a stochastic step is incorporated into the E-step based on data augmentation, whereas the M-step is solved by the method of conditional maximization. A variation on Bayesian information criterion (BIC) is also proposed to compare models with different number of components with missing values. The considered model is a general model framework and it captures the important characteristics of count data analysis such as zero inflation/deflation, heterogeneity as well as missingness, providing us with more insight into the data feature and allowing for dispersion to be investigated more fully and correctly. Since the stochastic step only involves simulating samples from some standard distributions, the computational burden is alleviated. Once missing responses and latent variables are imputed to replace the conditional expectation, our approach works as part of a multiple imputation procedure. A simulation study and a real example illustrate the usefulness and effectiveness of our methodology.     

Skew-mixed effects model for multivariate longitudinal data with categorical outcomes and missingness

A longitudinal study commonly follows a set of variables, measured for each individual repeatedly over time, and usually suffers from incomplete data problem. A common approach for dealing with longitudinal categorical responses is to use the Generalized Linear Mixed Model (GLMM). This model induces the potential relation between response variables over time via a vector of random effects, assumed to be shared parameters in the non-ignorable missing mechanism. Most GLMMs assume that the random-effects parameters follow a normal or symmetric distribution and this leads to serious problems in real applications. In this paper, we propose GLMMs for the analysis of incomplete multivariate longitudinal categorical responses with a non-ignorable missing mechanism based on a shared parameter framework with the less restrictive assumption of skew-normality for the random effects. These models may contain incomplete data with monotone and non-monotone missing patterns. The performance of the model is evaluated using simulation studies and a well-known longitudinal data set extracted from a fluvoxamine trial is analyzed to determine the profile of fluvoxamine in ambulatory clinical psychiatric practice.     

Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.     

Nonparametric product partition models for multiple change-points analysis

ABSTRACT

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data.     

Likelihood analysis of joint marginal and conditional models for longitudinal categorical data

The authors develop a Markov model for the analysis of longitudinal categorical data which facilitates modelling both marginal and conditional structures. A likelihood formulation is employed for inference, so the resulting estimators enjoy the optimal properties such as efficiency and consistency, and remain consistent when data are missing at random. Simulation studies demonstrate that the proposed method performs well under a variety of situations. Application to data from a smoking prevention study illustrates the utility of the model and interpretation of covariate effects. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Estimating additive models with missing responses

ABSTRACT

For multivariate regressors, the Nadaraya–Watson regression estimator suffers from the well-known curse of dimensionality. Additive models overcome this drawback. To estimate the additive components, it is usually assumed that we observe all the data. However, in many applied statistical analysis missing data occur. In this paper, we study the effect of missing responses on the additive components estimation. The estimators are based on marginal integration adapted to the missing situation. The proposed estimators turn out to be consistent under mild assumptions. A simulation study allows to compare the behavior of our procedures, under different scenarios.     

A generalized linear mixed model for longitudinal binary data with a marginal logit link function

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall's τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.     

Longitudinal data analysis in the presence of informative sampling: weighted distribution or joint modelling

ABSTRACT

Weighted distributions, as an example of informative sampling, work appropriately under the missing at random mechanism since they neglect missing values and only completely observed subjects are used in the study plan. However, length-biased distributions, as a special case of weighted distributions, remove the subjects with short length deliberately, which surely meet the missing not at random mechanism. Accordingly, applying length-biased distributions jeopardizes the results by producing biased estimates. Hence, an alternate method has to be used such that the results are improved by means of valid inferences. We propose methods that are based on weighted distributions and joint modelling procedure and compare them in analysing longitudinal data. After introducing three methods in use, a set of simulation studies and analysis of two real longitudinal datasets affirm our claim.