Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

A model selection approach for the identification of quantitative trait loci in experimental crosses

Summary. We consider the problem of identifying the genetic loci (called quantitative trait loci (QTLs)) contributing to variation in a quantitative trait, with data on an experimental cross. A large number of different statistical approaches to this problem have been described; most make use of multiple tests of hypotheses, and many consider models allowing only a single QTL. We feel that the problem is best viewed as one of model selection. We discuss the use of model selection ideas to identify QTLs in experimental crosses. We focus on a back-cross experiment, with strictly additive QTLs, and concentrate on identifying QTLs, considering the estimation of their effects and precise locations of secondary importance. We present the results of a simulation study to compare the performances of the more prominent methods.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

Understanding and Addressing the Unbounded “Likelihood” Problem

The joint probability density function, evaluated at the observed data, is commonly used as the likelihood function to compute maximum likelihood estimates. For some models, however, there exist paths in the parameter space along which this density-approximation likelihood goes to infinity and maximum likelihood estimation breaks down. In all applications, however, observed data are really discrete due to the round-off or grouping error of measurements. The “correct likelihood” based on interval censoring can eliminate the problem of an unbounded likelihood. This article categorizes the models leading to unbounded likelihoods into three groups and illustrates the density-approximation breakdown with specific examples. Although it is usually possible to infer how given data were rounded, when this is not possible, one must choose the width for interval censoring, so we study the effect of the round-off on estimation. We also give sufficient conditions for the joint density to provide the same maximum likelihood estimate as the correct likelihood, as the round-off error goes to zero.     

Mixture cure model with an application to interval mapping of quantitative trait loci

When censored time-to-event data are used to map quantitative trait loci (QTL), the existence of nonsusceptible subjects entails extra challenges. If the heterogeneous susceptibility is ignored or inappropriately handled, we may either fail to detect the responsible genetic factors or find spuriously significant locations. In this article, an interval mapping method based on parametric mixture cure models is proposed, which takes into consideration of nonsusceptible subjects. The proposed model can be used to detect the QTL that are responsible for differential susceptibility and/or time-to-event trait distribution. In particular, we propose a likelihood-based testing procedure with genome-wide significance levels calculated using a resampling method. The performance of the proposed method and the importance of considering the heterogeneous susceptibility are demonstrated by simulation studies and an application to survival data from an experiment on mice infected with Listeria monocytogenes.     

Non-concave penalization in linear mixed-effect models and regularized selection of fixed effects

Mixed-effect models are very popular for analyzing data with a hierarchical structure. In medical applications, typical examples include repeated observations within subjects in a longitudinal design, patients nested within centers in a multicenter design. However, recently, due to the medical advances, the number of fixed-effect covariates collected from each patient can be quite large, e.g., data on gene expressions of each patient, and all of these variables are not necessarily important for the outcome. So, it is very important to choose the relevant covariates correctly for obtaining the optimal inference for the overall study. On the other hand, the relevant random effects will often be low-dimensional and pre-specified. In this paper, we consider regularized selection of important fixed-effect variables in linear mixed-effect models along with maximum penalized likelihood estimation of both fixed and random-effect parameters based on general non-concave penalties. Asymptotic and variable selection consistency with oracle properties are proved for low-dimensional cases as well as for high dimensionality of non-polynomial order of sample size (number of parameters is much larger than sample size). We also provide a suitable computationally efficient algorithm for implementation. Additionally, all the theoretical results are proved for a general non-convex optimization problem that applies to several important situations well beyond the mixed model setup (like finite mixture of regressions) illustrating the huge range of applicability of our proposal.     

Bayesian hierarchical regression models for detecting QTLs in plant experiments

Quantitative trait loci (QTL) mapping is a growing field in statistical genetics. In plants, QTL detection experiments often feature replicates or clones within a specific genetic line. In this work, a Bayesian hierarchical regression model is applied to simulated QTL data and to a dataset from the Arabidopsis thaliana plants for locating the QTL mapping associated with cotyledon opening. A conditional model search strategy based on Bayesian model averaging is utilized to reduce the computational burden.     

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.     

A Distribution-free Approach in Statistical Modelling with Repeated Measurements and Missing Values

Mixed effect models, which contain both fixed effects and random effects, are frequently used in dealing with correlated data arising from repeated measurements (made on the same statistical units). In mixed effect models, the distributions of the random effects need to be specified and they are often assumed to be normal. The analysis of correlated data from repeated measurements can also be done with GEE by assuming any type of correlation as initial input. Both mixed effect models and GEE are approaches requiring distribution specifications (likelihood, score function). In this article, we consider a distribution-free least square approach under a general setting with missing value allowed. This approach does not require the specifications of the distributions and initial correlation input. Consistency and asymptotic normality of the estimation are discussed.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

Efficient estimation in partially linear single‐index models for longitudinal data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single‐index models for longitudinal data that may be unbalanced. In particular, a new three‐stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within‐subject correlation. We show that the parameter estimators are asymptotically semiparametrically efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite‐sample performance of the proposed method is demonstrated by simulation studies. In addition, two real‐data examples are analyzed to illustrate the methodology.     

Effective simultaneous confidence bands for repeated measurements in linear mixed-effect models

Linear mixed-effect (LME) models have been extensively accepted to analyse repeated measurements due to their flexibility and ability to handle subject-specific matters. The inclusion of random effects has resulted in much benefit with respect to estimation, but it is complicated to measure their impact on hypothesis testing. While the same complication is present in the construction of simultaneous confidence bands (SCBs), degrees of freedom (df) for SCBs have rarely been discussed unlike those for test statistics. This motivates us to propose the adoption of approximate df to construct SCBs in LME models. Simulation studies were performed to compare the performances of different calculations for the df. The results of simulations demonstrate the efficacy of the use of approximate df. In addition, our proposal allows line-segment SCBs developed under covariance models to function with LME models. Applications with real longitudinal datasets present consistent results with the simulation study.     

Application of Quasi-Least Squares to Analyse Replicated Autoregressive Time Series Regression Models

Time series regression models have been widely studied in the literature by several authors. However, statistical analysis of replicated time series regression models has received little attention. In this paper, we study the application of the quasi-least squares method to estimate the parameters in a replicated time series model with errors that follow an autoregressive process of order p. We also discuss two other established methods for estimating the parameters: maximum likelihood assuming normality and the Yule-Walker method. When the number of repeated measurements is bounded and the number of replications n goes to infinity, the regression and the autocorrelation parameters are consistent and asymptotically normal for all three methods of estimation. Basically, the three methods estimate the regression parameter efficiently and differ in how they estimate the autocorrelation. When p=2, for normal data we use simulations to show that the quasi-least squares estimate of the autocorrelation is undoubtedly better than the Yule-Walker estimate. And the former estimate is as good as the maximum likelihood estimate almost over the entire parameter space.     

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.     

A perspective of composite sampling

Composite samples are formed by physically mixing samples. Usually, composite samples are used to reduce the overall cost associated with analytical procedures that must be performed on each sample, but they can also be used to protect the privacy of individuals.

Composite sampling can reduce the cost of identifying individual cases that have a certain trait, such as those with a rare disease or those exceeding pollution-level standards. Not much is lost by applying this method as long as the trait is relatively rare.

Composite sampling can reduce the cost of estimating the mean of some process. When samples are composited, the ability to estimate the variance is lost. In spite of this, the potential savings are so great that composite samples have been used.

Much of this paper deasl with the variance of estimators based on composite sampling when the porportions of hte original samples comprising the composite sample are actually random. Taking repeated samples and measurements on several composite samples complicates the prodcedure, but allows the estimation of between and within variation as well as measurement error.     

COVARIATE-ADJUSTED REGRESSION FOR LONGITUDINAL DATA INCORPORATING CORRELATION BETWEEN REPEATED MEASUREMENTS

We propose an estimation method that incorporates the correlation/covariance structure between repeated measurements in covariate-adjusted regression models for distorted longitudinal data. In this distorted data setting, neither the longitudinal response nor (possibly time-varying) predictors are directly observable. The unobserved response and predictors are assumed to be distorted/contaminated by unknown functions of a common observable confounder. The proposed estimation methodology adjusts for the distortion effects both in estimation of the covariance structure and in the regression parameters using generalized least squares. The finite-sample performance of the proposed estimators is studied numerically by means of simulations. The consistency and convergence rates of the proposed estimators are also established. The proposed method is illustrated with an application to data from a longitudinal study of cognitive and social development in children.     

Computationally feasible estimation of the covariance structure in generalized linear mixed models

In this paper, we discuss how a regression model, with a non-continuous response variable, which allows for dependency between observations, should be estimated when observations are clustered and measurements on the subjects are repeated. The cluster sizes are assumed to be large. We find that the conventional estimation technique suggested by the literature on generalized linear mixed models (GLMM) is slow and sometimes fails due to non-convergence and lack of memory on standard PCs. We suggest to estimate the random effects as fixed effects by generalized linear model and to derive the covariance matrix from these estimates. A simulation study shows that our proposal is feasible in terms of mean-square error and computation time. We recommend that our proposal be implemented in the software of GLMM techniques so that the estimation procedure can switch between the conventional technique and our proposal, depending on the size of the clusters.     

Joint Modeling of Event Time and Nonignorable Missing Longitudinal Data

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinaly measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.     

Estimation of a residual distribution with small numbers of repeated measurements

The authors consider the estimation of a residual distribution for different measurement problems with a common measurement error process. The problem is motivated by issues arising in the analysis of gene expression data but should have application in other similar settings. It is implicitly assumed throughout that there are large numbers of measurements but small numbers of repeated measurements. As a consequence, the distribution of the estimated residuals is a biased estimate of the residual distribution. The authors present two methods for the estimation of the residual distribution with some restriction on the form of the distribution. They give an upper bound for the rate of convergence for an estimator based on the characteristic function and compare its performance with that of another estimator with simulations.     

Nonparametric Estimation of Average Growth Curve with General Nonstationary Error Process

The nonparametric estimation of the growth curve has been extensively studied in both stationary and some nonstationary particular situations. In this work, we consider the statistical problem of estimating the average growth curve for a fixed design model with nonstationary error process. The nonstationarity considered here is of a general form, and this article may be considered as an extension of previous results. The optimal bandwidth is shown to depend on the singularity of the autocovariance function of the error process along the diagonal. A Monte Carlo study is conducted in order to assess the influence of the number of subjects and the number of observations per subject on the estimation.