Empirical Likelihood Based Longitudinal Data Analysis

In longitudinal data analysis, our primary interest is in the estimation of regression parameters for the marginal expectations of the longitudinal responses, and the longitudinal correlation parameters are of secondary interest. The joint likelihood function for longitudinal data is challenging, particularly due to correlated responses. Marginal models, such as generalized estimating equations (GEEs), have received much attention based on the assumption of the first two moments of the data and a working correlation structure. The confidence regions and hypothesis tests are constructed based on the asymptotic normality. This approach is sensitive to the misspecification of the variance function and the working correlation structure which may yield inefficient and inconsistent estimates leading to wrong conclusions. To overcome this problem, we propose an empirical likelihood (EL) procedure based on a set of estimating equations for the parameter of interest and discuss its characteristics and asymptotic properties. We also provide an algorithm based on EL principles for the estimation of the regression parameters and the construction of its confidence region. We have applied the proposed method in two case examples.

Transition Logic Regression Method to Identify Interactions in Binary Longitudinal Data

Logic regression is an adaptive regression method which searches for Boolean (logic) combinations of binary variables that best explain the variability in the outcome, and thus, it reveals interaction effects which are associated with the response. In this study, we extended logic regression to longitudinal data with binary response and proposed “Transition Logic Regression Method” to find interactions related to response. In this method, interaction effects over time were found by Annealing Algorithm with AIC (Akaike Information Criterion) as the score function of the model. Also, first and second orders Markov dependence were allowed to capture the correlation among successive observations of the same individual in longitudinal binary response. Performance of the method was evaluated with simulation study in various conditions. Proposed method was used to find interactions of SNPs and other risk factors related to low HDL over time in data of 329 participants of longitudinal TLGS study.

Partial Linear Model Averaging Prediction for Longitudinal Data

Prediction plays an important role in data analysis.Model averaging method generally provides better prediction than using any of its components.Even though model averaging has been extensively investigated under independent errors,few authors have considered model averaging for semiparametric models with correlated errors.In this paper,the authors offer an optimal model averaging method to improve the prediction in partially linear model for longitudinal data.The model averaging weights are obtained by minimizing criterion,which is an unbiased estimator of the expected in-sample squared error loss plus a constant.Asymptotic properties,including asymptotic optimality and consistency of averaging weights,are established under two scenarios:(i)All candidate models are misspecified;(ii)Correct models are available in the candidate set.Simulation studies and an empirical example show that the promise of the proposed procedure over other competitive methods.

Composite Quantile Estimation for Kink Model with Longitudinal Data

Kink model is developed to analyze the data where the regression function is two-stage piecewise linear with respect to the threshold covariate but continuous at an unknown kink point.In quantile regression for longitudinal data,kink point where the kink effect happens is often assumed to be heterogeneous across different quantiles.However,the kink point tends to be the same across different quantiles,especially in a region of neighboring quantile levels.Incorporating such homogeneity information could increase the estimation efficiency of the common kink point.In this paper,we propose a composite quantile estimation approach for the common kink point by combining information from multiple neighboring quantiles.Asymptotic normality of the proposed estimator is studied.In addition,we also develop a sup-likelihood-ratio test to check the existence of the kink effect at a given quantile level.A test-inversion confidence interval for the common kink point is also developed based on the quantile rank score test.The simulation studies show that the proposed composite kink estimator is more efficient than the single quantile regression estimator.We also illustrate the proposed method via an application to a longitudinal data set on blood pressure and body mass index.

Empirical Likelihood for Generalized Linear Models with Longitudinal Data

Generalized linear models are usually adopted to model the discrete or nonnegative responses.In this paper,empirical likelihood inference for fixed design generalized linear models with longitudinal data is investigated.Under some mild conditions,the consistency and asymptotic normality of the maximum empirical likelihood estimator are established,and the asymptotic χ^(2) distribution of the empirical log-likelihood ratio is also obtained.Compared with the existing results,the new conditions are more weak and easy to verify.Some simulations are presented to illustrate these asymptotic properties.

Unified Variable Selection for Varying Coefficient Models with Longitudinal Data

Variable selection for varying coefficient models includes the separation of varying and constant effects,and the selection of variables with nonzero varying effects and those with nonzero constant effects.This paper proposes a unified variable selection approach called the double-penalized quadratic inference functions method for varying coefficient models of longitudinal data.The proposed method can not only separate varying coefficients and constant coefficients,but also estimate and select the nonzero varying coefficients and nonzero constant coefficients.It is suitable for variable selection of linear models,varying coefficient models,and partial linear varying coefficient models.Under regularity conditions,the proposed method is consistent in both separation and selection of varying coefficients and constant coefficients.The obtained estimators of varying coefficients possess the optimal convergence rate of non-parametric function estimation,and the estimators of nonzero constant coefficients are consistent and asymptotically normal.Finally,the authors investigate the finite sample performance of the proposed method through simulation studies and a real data analysis.The results show that the proposed method performs better than the existing competitor.

Average Estimation of Semiparametric Models for High-Dimensional Longitudinal Data

Model average receives much attention in recent years.This paper considers the semiparametric model averaging for high-dimensional longitudinal data.To minimize the prediction error,the authors estimate the model weights using a leave-subject-out cross-validation procedure.Asymptotic optimality of the proposed method is proved in the sense that leave-subject-out cross-validation achieves the lowest possible prediction loss asymptotically.Simulation studies show that the performance of the proposed model average method is much better than that of some commonly used model selection and averaging methods.

Empirical Likelihood for Partially Linear Errors-in-variables Models with Longitudinal Data

Empirical likelihood inference for partially linear errors-in-variables models with longitudinal data is investigated.Under regularity conditions,it is shown that the empirical log-likelihood ratio at the true parameters converges to the standard Chi-squared distribution.Furthermore,we consider some estimates of the unknown parameter and the resulting estimators are shown to be asymptotically normal.Some simulations and a real data analysis are given to illustrate the performance of the proposed method.

Sleep quality and mental health of the elderly in China: evidence from longitudinal data

Against a macro backdrop of accelerating population aging,the mental health and sleep quality of China’s elderly have become subjects of interest because these fac-tors are both closely linked to seniors’quality of life.Based on multi-period track-ing data from the Chinese Longitudinal Healthy Longevity Survey(CLHLS),this study uses a fixed-effects model to examine the correlation between the sleep quality and mental health of China’s elderly and found that:(1)the elderly in China gener-ally had good sleep quality and mental health;(2)better sleep quality has a posi-tive effect on the mental health of the elderly,while better mental health also posi-tively boosts sleep quality of the elderly;and(3)there may be a causal relationship between the sleep quality and mental health of the elderly.

Efficient Estimation of Longitudinal Data Additive Varying Coefficient Regression Models

We consider a longitudinal data additive varying coefficient regression model, in which the coefficients of some factors(covariates) are additive functions of other factors, so that the interactions between different factors can be taken into account effectively. By considering within-subject correlation among repeated measurements over time and additive structure, we propose a feasible weighted two-stage local quasi-likelihood estimation. In the first stage, we construct initial estimators of the additive component functions by B-spline series approximation. With the initial estimators, we transform the additive varying coefficients regression model into a varying coefficients regression model and further apply the local weighted quasi-likelihood method to estimate the varying coefficient functions in the second stage. The resulting second stage estimators are computationally expedient and intuitively appealing. They also have the advantages of higher asymptotic efficiency than those neglecting the correlation structure, and an oracle property in the sense that the asymptotic property of each additive component is the same as if the other components were known with certainty. Simulation studies are conducted to demonstrate finite sample behaviors of the proposed estimators, and a real data example is given to illustrate the usefulness of the proposed methodology.

Regression Estimation for Longitudinal Data with Nonignorable Intermittent Nonresponse and Dropout

We mainly focus on regression estimation in a longitudinal study with nonignorable intermittent nonresponse and dropout.To handle the identifiability issue,we take a time-independent covariate as nonresponse instrument which is independent of nonresponse propensity conditioned on other covariates and responses to ensure the identifiability of nonresponse propensity.The nonresponse propensity is assumed to be a parametric model,and the corresponding parameters are estimated by using the generalized method of moments approach.Then the marginal response means are estimated by inverse probability weighting method.Furthermore,to improve the robustness of estimators,we derive an augmented inverse probability weighting estimator which is shown to be consistent and asymptotically normally distributed.Simulation studies and a real-data analysis show that the proposed approach yields highly efficient estimators.

Parsimonious Mean-Covariance Modeling for Longitudinal Data with ARMA Errors

Based on the generalized estimating equation approach,the authors propose a parsimonious mean-covariance model for longitudinal data with autoregressive and moving average error process,which not only unites the existing autoregressive Cholesky factor model and moving average Cholesky factor model but also provides a wide variety of structures of covariance matrix.The resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed under mild conditions.The authors demonstrate the effectiveness,parsimoniousness and desirable performance of the proposed approach by analyzing the CD4-I-cell counts data set and conducting extensive simulations.

Statistical inference for multivariate longitudinal data with irregular auto-correlated error process

Multivariate longitudinal data arise frequently in a variety of applications,where multiple outcomes are measured repeatedly from the same subject.In this paper,we first propose a two-stage weighted least square estimation procedure for the regression coefficients when the random error follows an irregular autoregressive(AR)process,and establish asymptotic normality properties for the resulting estimators.We then apply the smoothly clipped absolute deviation(SCAD)variable selection approach to determine the order of the AR error process.We further propose a test statistic to check whether multiple responses are correlated at the same observation time,and derive the asymptotic distribution of the proposed test statistic.Several simulated examples and real data analysis are presented to illustrate the finite-sample performance of the proposed method.

Bayesian Joint Semiparametric Mean–Covariance Modeling for Longitudinal Data

Joint parsimonious modeling the mean and covariance is important for analyzing longitudinal data,because it accounts for the efficiency of parameter estimation and easy interpretation of variability.The main potential risk is that it may lead to inefficient or biased estimators of parameters while misspecification occurs.A good alternative is the semiparametric model.In this paper,a Bayesian approach is proposed for modeling the mean and covariance simultaneously by using semiparametric models and the modified Cholesky decomposition.We use a generalized prior to avoid the knots selection while using B-spline to approximate the nonlinear part and propose a Markov Chain Monte Carlo scheme based on Metropolis–Hastings algorithm for computations.Simulation studies and real data analysis show that the proposed approach yields highly efficient estimators for the parameters and nonparametric parts in the mean,meanwhile providing parsimonious estimation for the covariance structure.

Personalized treatment selection via the covariate-specific treatment effect curve for longitudinal data

Treatment selection based on patient characteristics has been widely recognised in modern medicine.In this paper,we propose a generalised partially linear single-index mixed-effects modelling strategy for treatment selection and heterogeneous treatment effect estimation in longitudinal clinical and observational studies.We model the treatment effect as an unknown functional curve of a weighted linear combination of time-dependent covariates.This method enables us to investigate covariate-specific treatment effects and make personalised treatment selection in a flexible fashion.We develop a method that combines local linear regression and penalised quasi-likelihood to estimate the weight for each covariate,the unknown treatment effect curve and the parameters for mixed-effects.Based on pointwise confidence intervals for the treatment effect curve,we can make individualised treatment decisions from the information of patient characteristics.A simulation study is conducted to evaluate finite sample performance of the proposed method.We also illustrate the method via analysis of a real data example.

D-optimal population designs in linear mixed effects models for multiple longitudinal data

The main purpose of this paper is to investigate D-optimal population designs in multi-response linear mixed models for longitudinal data.Observations of each response variable within subjects are assumed to have a first-order autoregressive structure,possibly with observation error.The equivalence theorems are provided to characterise theD-optimal population designs for the estimation of fixed effects in the model.The semi-Bayesian D-optimal design which is robust against the serial correlation coefficient is also considered.Simulation studies show that the correlation between multi-response variables has tiny effects on the optimal design,while the experimental costs are important factors in the optimal designs.

EMPIRICAL LIKELIHOOD APPROACH FOR LONGITUDINAL DATA WITH MISSING VALUES AND TIME-DEPENDENT COVARIATES

Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this problem,we propose weighted corrected estimating equations under the missing at random mechanism,followed by developing a shrinkage empirical likelihood estimation approach for the parameters of interest when time-dependent covariates are present.Such procedure improves efficiency over generalized estimation equations approach with working independent assumption,via combining the independent estimating equations and the extracted additional information from the estimating equations that are excluded by the independence assumption.The contribution from the remaining estimating equations is weighted according to the likelihood of each equation being a consistent estimating equation and the information it carries.We show that the estimators are asymptotically normally distributed and the empirical likelihood ratio statistic and its profile counterpart follow central chi-square distributions asymptotically when evaluated at the true parameter.The practical performance of our approach is demonstrated through numerical simulations and data analysis.

Bayesian Nonlinear Quantile Regression Approach for Longitudinal Ordinal Data

Longitudinal data with ordinal outcomes commonly arise in clinical and social studies,where the purpose of interest is usually quantile curves rather than a simple reference range.In this paper we consider Bayesian nonlinear quantile regression for longitudinal ordinal data through a latent variable.An efficient Metropolis–Hastings within Gibbs algorithm was developed for model fitting.Simulation studies and a real data example are conducted to assess the performance of the proposed method.Results show that the proposed approach performs well.

Linear mixed-effects model for longitudinal complex data with diversified characteristics

The increasing richness of data encourages a comprehensive understanding of economic and financial activities,where variables of interest may include not only scalar(point-like)indicators,but also functional(curve-like)and compositional(pie-like)ones.In many research topics,the variables are also chronologically collected across individuals,which falls into the paradigm of longitudinal analysis.The complicated nature of data,however,increases the difficulty of modeling these variables under the classic longitudinal frame-work.In this study,we investigate the linear mixed-effects model(LMM)for such complex data.Different types of variables arefirst consistently represented using the corresponding basis expansions so that the classic LMM can then be conducted on them,which gener-alizes the theoretical framework of LMM to complex data analysis.A number of simulation studies indicate the feasibility and effectiveness of the proposed model.We further illustrate its practical utility in a real data study on Chinese stock market and show that the proposed method can enhance the performance and interpretability of the regression for complex data with diversified characteristics.

Factors associated with changes in physical activity and sedentary behaviour during one year among university-based young adults

The purpose of this study was to identify correlates of changes in physical activity(PA)and sedentary behaviour(SB)among university-based young adults in Bangladesh.Data were from a 1-year prospective study with 2 assessment points(baseline n=573,20.7±1.35 years,45%female;retention rate 69%,analytical sample=395).Participants completed a self-administered written survey on PA,SB,health and lifestyle be-haviours,and sociodemographics.Changes in PA were categorised as:negligible(±<60 min/week),>60 min/week decrease,or>60 min/week increase.Changes in SB were categorised as negligible(±<120 min/week),>120 min/week decrease,and>120 min/week increase.Multinomial logistic regression analysis was used to identify the correlates.About quarters(72%)of participants had insufficient PA at both assessment points.Of those who were sufficiently active at Wave 1,5%became insufficiently active at Wave 2.One quarter of par-ticipants(23%)had high SB at Wave 1 and Wave 2.Of those who had low SB at Wave 1,16%had high SB at Wave 2.Being male[OR=2.04(95%CI:1.06–3.93)],baseline phone time of>2 h/day[OR=3.14(95%CI:1.04–7.04)]and not participating in organised sports at baseline[OR=2.56(95%CI:1.24–5.29)]were associated with a decrease in PA by>60 min/week.Participants who frequently experienced stress at baseline had higher odds of increasing SB by>120 min/day[OR=1.83(95%CI:1.04–3.23)].SB is more variable than PA over 1 year in university-based young adults in Bangladesh.Males,those with high phone time,those not engaging with organised sports,and those with frequent stress may change to a more inactive lifestyle.