Regression Analysis Of Sparse Longitudinal Data

In medical research and clinical studies,follow-up data have significant practical value and guiding significance.This type of data is a mixture of cross-sectional data and time-series data,falling under the category of longitudinal data.Such data exhibit numerous challenging characteristics when processing.For instance,measurements of patients’ physiological indicators may be discontinuous,sparse,and irregular.Moreover,measurements of different indicators may occur synchronously or asynchronously.Measurements of the same physiological indicator are often not independent but rather exhibit a certain level of correlation.Existing methodologies are not directly applicable,hence garnering substantial attention from statisticians.A commonly employed ad hoc method for processing longitudinal data is last observation carried forward,where the last observation is used to impute unobserved values.However,this method is too rough,ignoring the change of data and potentially introducing significant bias.Another commonly used approach involves using linear mixed-effects models to characterize longitudinal observation processes,and obtaining estimates of target parameters through joint survival models.As a likelihood-based method,linear mixed-effects models assume that random effects and error terms follow a normal distribution.However,in practice,such assumptions are too strong and difficult to validate.When these assumptions do not hold,this method can introduce significant errors.Additionally,joint modeling requires iterative computations,resulting in large computational burdens.This paper considers theoretically and computationally simpler kernel smoothing methods,which need weaker model assumptions,to address issues such as sparsity,asynchrony,and correlation in longitudinal data.The specific content consists of the following three parts:1.Regression analysis of additive hazards model with sparse longitudinal covariates.As a complement to the classical proportional hazards model,additive hazards model provides an alternative way to analyze survival data.However,existing inference procedures only consider cases where covariates are time-independent.The ad hoc last observation carried forward method is deemed too crude,leading to biased estimations.We propose a kernel smoothing method to handle the estimation with sparse longitudinal covariates.Three different weighting approaches are provided to analyze longitudinal data with different characteristics.Among them,the weighting strategy based on last observation carried forward requires the least computational resources.In contrast,the weighting method based on the observations before failure event uses as much information as possible.If the covariates are still observed after failure events,we propose a weighting strategy that involves all observations.The third weighting approach has the fastest convergence rate and is suitable for cases where the failure event is recurring.In this paper,the convergence rate of proposed methods is discussed,each method’s consistency and asymptotic normality are proved.Through numerical studies,we compare the performance of the proposed methods with ad hoc method,validating our theoretical findings.To illustrate the practical utility of the proposed approach,we apply the proposed method to real data from an Alzheimer study.2.Regression analysis of multiplicative hazards model with time-dependent coefficient for sparse longitudinal covariates.The traditional proportional hazards model and additive hazards model assume that the impact of covariates on the failure event is time-independent.To complement these traditional models,we consider a multiplicative hazards model,in which the effect of covariates is time-varying.In this paper,we propose a kernel-smoothing-based estimating equation approach to estimate time-varying coefficients with sparse longitudinal covariates.To capture the trajectory of time-varying coefficients,we construct simultaneous confidence bands using multiplier bootstrap method.We establish the consistency and asymptotic normality of the proposed estimator and validate the effectiveness of the constructed simultaneous confidence bands.Numerical studies demonstrate that the proposed estimation method is unbiased with a moderate sample size.Furthermore,the coverage of the proposed simultaneous confidence band outperforms the pointwise confidence band,ensuring an appropriate coverage rate.Finally,the analysis of the cerebral infarction dataset illustrates the application of the proposed method.3.Regression analysis of sparse longitudinal data with mixed synchronous and asynchronous longitudinal covariates.In medical and epidemiological studies,the presence of omitted variables and asynchronous observations are common challenges.In this paper,partial linear model method and centralization method are proposed to deal with omitted variable bias.Furthermore,for scenarios involving mixed synchronous and asynchronous observations,we consider omitted variables based two-step method and one-step method.The two-step method exhibits a faster convergence rate,allowing for the estimation of synchronous coefficients to achieve √n convergence,albeit with more stringent model assumptions.On the other hand,the one-step estimation method converges more slowly but requires weaker model assumptions.The above two methods complement each other.A large number of simulation studies provide numerical support for the theoretical findings.Finally,we apply the proposed method to the Alzheimer’s Disease Neuroimaging Initiative Study,illustrating the practical application of the proposed method.