Empirical Likelihood Inference for Longitudinal Data with Missing Response Variables and Error-Prone Covariates

We consider statistical inference for longitudinal partially linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. The block empirical likelihood procedure is used to estimate the regression coefficients and residual adjusted block empirical likelihood is employed for the baseline function. This leads us to prove a nonparametric version of Wilk's theorem. Compared with methods based on normal approximations, our proposed method does not require a consistent estimators for the asymptotic variance and bias. An application to a longitudinal study is used to illustrate the procedure developed here. A simulation study is also reported.     

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.     

Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

Generalized growth curve models for longitudinal data in application to a randomized controlled trial

Growth curve analysis is beneficial in longitudinal studies, where the pattern of response variables measured repeatedly over time is of interest, yet unknown. In this article, we propose generalized growth curve models under a polynomial regression framework and offer a complete process that identifies the parsimonious growth curves for different groups of interest, as well as compares the curves. A higher order of a polynomial degree generally provides more flexible regression, yet it may suffer from the complicated and overfitted model in practice. Therefore, we employ the model selection procedure that chooses the optimal degree of a polynomial consistently. Consideration of a quadratic inference function (Qu et al., 2000) for estimation on regression parameters is addressed and estimation efficiency is improved by incorporating the within-subject correlation commonly existing in longitudinal data. In biomedical studies, it is of particular interest to compare multiple treatments and provide an effective one. We further conduct the hypothesis test that assesses the equality of the growth curves through an asymptotic chi-square test statistic. The proposed methodology is employed on a randomized controlled longitudinal dataset on depression. The effectiveness of our procedure is also confirmed with simulation studies.     

Using structural equation modelling to detect measurement bias and response shift in longitudinal data

We propose a three step procedure to investigate measurement bias and response shift, a special case of measurement bias in longitudinal data. Structural equation modelling is used in each of the three steps, which can be described as (1) establishing a measurement model using confirmatory factor analysis, (2) detecting measurement bias by testing the equivalence of model parameters across measurement occasions, (3) detecting measurement bias with respect to additional exogenous variables by testing their direct effects on the indicator variables. The resulting model can be used to investigate true change in the attributes of interest, by testing changes in common factor means. Solutions for the issue of constraint interaction and for chance capitalisation in model specification searches are discussed as part of the procedure. The procedure is illustrated by applying it to longitudinal health-related quality-of-life data of HIV/AIDS patients, collected at four semi-annual measurement occasions.     

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.     

Joint analysis of longitudinal measurements and survival times with a cure fraction based on partly linear mixed and semiparametric cure models

In a joint analysis of longitudinal quality of life (QoL) scores and relapse-free survival (RFS) times from a clinical trial on early breast cancer conducted by the Canadian Cancer Trials Group, we observed a complicated trajectory of QoL scores and existence of long-term survivors. Motivated by this observation, we proposed in this paper a flexible joint model for the longitudinal measurements and survival times. A partly linear mixed effect model is used to capture the complicated but smooth trajectory of longitudinal measurements and approximated by B-splines and a semiparametric mixture cure model with the B-spline baseline hazard to model survival times with a cure fraction. These two models are linked by shared random effects to explore the dependence between longitudinal measurements and survival times. A semiparametric inference procedure with an EM algorithm is proposed to estimate the parameters in the joint model. The performance of proposed procedures are evaluated by simulation studies and through the application to the analysis of data from the clinical trial which motivated this research.     

A pseudo likelihood approach to analyze rate difference of binary response data in longitudinal factorial studies

Although Fan showed that the mixed-effects model for repeated measures (MMRM) is appropriate to analyze complete longitudinal binary data in terms of the rate difference, they focused on using the generalized estimating equations (GEE) to make statistical inference. The current article emphasizes validity of the MMRM when the normal-distribution-based pseudo likelihood approach is used to make inference for complete longitudinal binary data. For incomplete longitudinal binary data with missing at random missing mechanism, however, the MMRM, using either the GEE or the normal-distribution-based pseudo likelihood inferential procedure, gives biased results in general and should not be used for analysis.     

A new orthogonality-based estimation for varying-coefficient partially linear models

Varying coefficient partially linear models are usually used for longitudinal data analysis, and an interest is mainly to improve efficiency of regression coefficients. By the orthogonality estimation technology and the quadratic inference function method, we propose a new orthogonality-based estimation method to estimate parameter and nonparametric components in varying coefficient partially linear models with longitudinal data. The proposed procedure can separately estimate the parametric and nonparametric components, and the resulting estimators do not affect each other. Under some mild conditions, we establish some asymptotic properties of the resulting estimators. Furthermore, the finite sample performance of the proposed procedure is assessed by some simulation experiments.     

Empirical Likelihood Inferences for Semiparametric Varying-Coefficient Partially Linear Models with Longitudinal Data

In this article, empirical likelihood inferences for semiparametric varying-coefficient partially linear models with longitudinal data are investigated. We propose a groupwise empirical likelihood procedure to handle the inter-series dependence of the longitudinal data. By using residual-adjustment, an empirical likelihood ratio function for the nonparametric component is constructed, and a nonparametric version Wilks' phenomenons is proved. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. A simulation study is undertaken to assess the finite sample performance of the proposed confidence regions.     

Longitudinal covariate-adjusted response-adaptive randomized designs

Clinical trials often involve longitudinal data set which has two important characteristics: repeated and correlated measurements and time-varying covariates. In this paper, we propose a general framework of longitudinal covariate-adjusted response-adaptive (LCARA) randomization procedures. We study their properties under widely satisfied conditions. This design skews the allocation probabilities which depend on both patients' first observed covariates and sequentially estimated parameters based on the accrued longitudinal responses and covariates. The asymptotic properties of estimators for the unknown parameters and allocation proportions are established. The special case of binary treatment and continuous responses is studied in detail. Simulation studies and an analysis of the National Cooperative Gallstone Study (NCGS) data are carried out to illustrate the advantages of the proposed LCARA randomization procedure.     

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 new local estimation method for single index models for longitudinal data

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

Robust variable selection in modal varying-coefficient models with longitudinal

In this article we present a robust and efficient variable selection procedure by using modal regression for varying-coefficient models with longitudinal data. The new method is proposed based on basis function approximations and a group version of the adaptive LASSO penalty, which can select significant variables and estimate the non-zero smooth coefficient functions simultaneously. Under suitable conditions, we establish the consistency in variable selection and the oracle property in estimation. A simulation study and two real data examples are undertaken to assess the finite sample performance of the proposed variable selection procedure.     

Variable selection for semiparametric errors-in-variables regression model with longitudinal data

In this paper, we focus on the variable selection for the semiparametric regression model with longitudinal data when some covariates are measured with errors. A new bias-corrected variable selection procedure is proposed based on the combination of the quadratic inference functions and shrinkage estimations. With appropriate selection of the tuning parameters, we establish the consistency and asymptotic normality of the resulting estimators. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure. We further illustrate the proposed procedure with an application.     

ADAPTIVE GROUP SEQUENTIAL PROCEDURES FOR MULTIPLE COMPARISON OF TREATMENTS

ABSTRACT

We present a flexible group sequential procedure for comparing several treatments to a control. Though longitudinal data corresponding to a two stage mixed effects model are considered, ranges of application include any process with independent increments. The procedure allows the experimenter to drop the inferior treatments from the trial as soon as they are detected. It control strongly the familywise error rate. We also discuss a new error spending function (ESF) and study the performance of the procedure using various ESFs and time scales. Finally, the procedure is illustrated on a real example and implementation considerations are discussed.     

Smooth-Threshold GEE Variable Selection in High-Dimensional Partially Linear Models with Longitudinal Data

We consider the problem of variable selection in high-dimensional partially linear models with longitudinal data. A variable selection procedure is proposed based on the smooth-threshold generalized estimating equation (SGEE). The proposed procedure automatically eliminates inactive predictors by setting the corresponding parameters to be zero, and simultaneously estimates the nonzero regression coefficients by solving the SGEE. We establish the asymptotic properties in a high-dimensional framework where the number of covariates p_n increases as the number of clusters n increases. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure.     

Estimation in an empirical bayes model for longitudinal and cross‐sectionally clustered binary data

Some studies generate data that can be grouped into clusters in more than one way. Consider for instance a smoking prevention study in which responses on smoking status are collected over several years in a cohort of students from a number of different schools. This yields longitudinal data, also cross‐sectionaliy clustered in schools. The authors present a model for analyzing binary data of this type, combining generalized estimating equations and estimation of random effects to address the longitudinal and cross‐sectional dependence, respectively. The estimation procedure for this model is discussed, as are the results of a simulation study used to investigate the properties of its estimates. An illustration using data from a smoking prevention trial is given.     

A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction

The joint analysis of longitudinal measurements and survival data is useful in clinical trials and other medical studies. In this paper, we consider a joint model which assumes a linear mixed $tt$ model for longitudinal measurements and a promotion time cure model for survival data and links these two models through a latent variable. A semiparametric inference procedure with an EM algorithm implementation is developed for the parameters in the joint model. The proposed procedure is evaluated in a simulation study and applied to analyze the quality of life and time to recurrence data from a clinical trial on women with early breast cancer. The Canadian Journal of Statistics 40: 207–224; 2012 © 2012 Statistical Society of Canada