Unified Inference for Sparse and Dense Longitudinal Data in Time‐varying Coefficient Models

Time‐varying coefficient models are widely used in longitudinal data analysis. These models allow the effects of predictors on response to vary over time. In this article, we consider a mixed‐effects time‐varying coefficient model to account for the within subject correlation for longitudinal data. We show that when kernel smoothing is used to estimate the smooth functions in time‐varying coefficient models for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases might lead to erroneous conclusions for statistical inference. In order to solve this problem, we establish a unified self‐normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and an analysis of Baltimore MACS data.     

M-estimation and model identification based on double SCAD penalization

M-estimation is a widely used method for robust statistical inference. In this article, using a B-spline series approximation with a double smoothly clipped absolute deviation penalization, we solve the problem of simultaneous variable selection and parametric component identification in a non parametric additive model. The theoretical properties of the double non concave penalized M-estimation are established. The proposed approach is resistant to heavy-tailed errors or outliers in the responses. Simulation studies for finite-sample cases are conducted and a real dataset is also analyzed for illustration of this new approach.     

On additive-multiplicative hazards model

In survival analysis and reliability theory, a fundamental problem is the study of lifetime properties of a live organism or system. In this regard, there have been considered and studied several models based on different concepts of ageing such as hazard rate and mean residual life. In this paper, we consider an additive-multiplicative hazard model (AMHM) and study some reliability and ageing properties of the proposed model. We then specify the bivariate models whose conditionals satisfy AMHM. Several properties of the proposed bivariate model are investigated and adequacy of the model is evaluated based on a real data set.     

A bivariate regression model for matched paired survival data: local influence and residual analysis

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we consider a location scale model for bivariate survival times based on the proposal of a copula to model the dependence of bivariate survival data. For the proposed model, we consider inferential procedures based on maximum likelihood. Gains in efficiency from bivariate models are also examined in the censored data setting. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the bivariate regression model for matched paired survival data. Sensitivity analysis methods such as local and total influence are presented and derived under three perturbation schemes. The martingale marginal and the deviance marginal residual measures are used to check the adequacy of the model. Furthermore, we propose a new measure which we call modified deviance component residual. The methodology in the paper is illustrated on a lifetime data set for kidney patients.     

Variable selection for partially varying coefficient single-index model

In this paper, we consider the problem of variable selection for partially varying coefficient single-index model, and present a regularized variable selection procedure by combining basis function approximations with smoothly clipped absolute deviation penalty. The proposed procedure simultaneously selects significant variables in the single-index parametric components and the nonparametric coefficient function components. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. Finite sample performance of the proposed method is illustrated by a simulation study and real data analysis.     

Bayesian analysis of paired survival data using a bivariate exponential distribution

We consider a Bayesian analysis method of paired survival data using a bivariate exponential model proposed by Moran (1967, Biometrika 54:385–394). Important features of Moran’s model include that the marginal distributions are exponential and the range of the correlation coefficient is between 0 and 1. These contrast with the popular exponential model with gamma frailty. Despite these nice properties, statistical analysis with Moran’s model has been hampered by lack of a closed form likelihood function. In this paper, we introduce a latent variable to circumvent the difficulty in the Bayesian computation. We also consider a model checking procedure using the predictive Bayesian P-value.     

Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

M-Estimation for partially functional linear regression model based on splines

ABSTRACT

M-estimation is a widely used technique for robust statistical inference. In this paper, we study robust partially functional linear regression model in which a scale response variable is explained by a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, we use polynomial splines to approximate the slop parameter. The estimation procedure is easy to implement, and it is resistant to heavy-tailederrors or outliers in the response. The asymptotic properties of the proposed estimators are established. Finally, we assess the finite sample performance of the proposed method by Monte Carlo simulation studies.     

Structural identification and variable selection in high-dimensional varying-coefficient models

Varying-coefficient models have been widely used to investigate the possible time-dependent effects of covariates when the response variable comes from normal distribution. Much progress has been made for inference and variable selection in the framework of such models. However, the identification of model structure, that is how to identify which covariates have time-varying effects and which have fixed effects, remains a challenging and unsolved problem especially when the dimension of covariates is much larger than the sample size. In this article, we consider the structural identification and variable selection problems in varying-coefficient models for high-dimensional data. Using a modified basis expansion approach and group variable selection methods, we propose a unified procedure to simultaneously identify the model structure, select important variables and estimate the coefficient curves. The unique feature of the proposed approach is that we do not have to specify the model structure in advance, therefore, it is more realistic and appropriate for real data analysis. Asymptotic properties of the proposed estimators have been derived under regular conditions. Furthermore, we evaluate the finite sample performance of the proposed methods with Monte Carlo simulation studies and a real data analysis.     

On multivariate nonlinear regression models with stationary correlated errors

In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics are polar motion (which is now widely known as “Chandlers Wobble”), where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data were first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 h and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life, etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data, and we use the nonlinear regression methodology developed here to achieve that goal.     

A multivariate two-factor skew model

It is well known that many data, such as the financial or demographic data, exhibit asymmetric distributions. In recent years, researchers have concentrated their efforts to model this asymmetry. Skew normal model is one of such models that are skew and yet possess many properties of the normal model. In this paper, a new multivariate skew model is proposed, along with its statistical properties. It includes the multivariate normal distribution and multivariate skew normal distribution as special cases. The quadratic form of this random vector follows a χ² distribution. The roles of the parameters in the model are investigated using contour plots of bivariate densities.     

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.     

Skew-normal semiparametric varying coefficient model and score test

In this paper, we consider inference aspects of skew-normal semiparametric varying coefficient models which provide a useful extension of the normal regression models. The maximum likelihood estimation based on B-spline is proposed. Further, we discuss the score test for homogeneity of the variance in skew-normal semiparametric varying coefficient models. Their asymptotical properties are investigated. Some simulated examples are used to examine our proposed methods.     

Efficient estimation for the proportional hazards model with bivariate current status data

We consider efficient estimation of regression and association parameters jointly for bivariate current status data with the marginal proportional hazards model. Current status data occur in many fields including demographical studies and tumorigenicity experiments and several approaches have been proposed for regression analysis of univariate current status data. We discuss bivariate current status data and propose an efficient score estimation approach for the problem. In the approach, the copula model is used for joint survival function with the survival times assumed to follow the proportional hazards model marginally. Simulation studies are performed to evaluate the proposed estimates and suggest that the approach works well in practical situations. A real life data application is provided for illustration.     

A bivariate regression model with cure fraction

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we introduce a location-scale model for bivariate survival times based on the copula to model the dependence of bivariate survival data with cure fraction. We create the correlation structure between the failure times using the Clayton family of copulas, which is assumed to have any distribution. It turns out that the model becomes very flexible with respect to the choice of the marginal distributions. For the proposed model, we consider inferential procedures based on constrained parameters under maximum likelihood. We derive the appropriate matrices for assessing local influence under different perturbation schemes and present some ways to perform global influence analysis. The relevance of the approach is illustrated using a real data set and a diagnostic analysis is performed to select an appropriate model.     

Bivariate Birnbaum-Saunders accelerated lifetime model: estimation and diagnostic analysis

In this paper, we discuss the bivariate Birnbaum-Saunders accelerated lifetime model, in which we have modeled the dependence structure of bivariate survival data through the use of frailty models. Specifically, we propose the bivariate model Birnbaum-Saunders with the following frailty distributions: gamma, positive stable and logarithmic series. We present a study of inference and diagnostic analysis for the proposed model, more concisely, are proposed a diagnostic analysis based in local influence and residual analysis to assess the fit model, as well as, to detect influential observations. In this regard, we derived the normal curvatures of local influence under different perturbation schemes and we performed some simulation studies for assessing the potential of residuals to detect misspecification in the systematic component, the presence in the stochastic component of the model and to detect outliers. Finally, we apply the methodology studied to real data set from recurrence in times of infections of 38 kidney patients using a portable dialysis machine, we analyzed these data considering independence within the pairs and using the bivariate Birnbaum-Saunders accelerated lifetime model, so that we could make a comparison and verify the importance of modeling dependence within the times of infection associated with the same patient.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Robust variable selection and parametric component identification in varying coefficient models

ABSTRACT

In this paper, we study a novelly robust variable selection and parametric component identification simultaneously in varying coefficient models. The proposed estimator is based on spline approximation and two smoothly clipped absolute deviation (SCAD) penalties through rank regression, which is robust with respect to heavy-tailed errors or outliers in the response. Furthermore, when the tuning parameter is chosen by modified BIC criterion, we show that the proposed procedure is consistent both in variable selection and the separation of varying and constant coefficients. In addition, the estimators of varying coefficients possess the optimal convergence rate under some assumptions, and the estimators of constant coefficients have the same asymptotic distribution as their counterparts obtained when the true model is known. Simulation studies and a real data example are undertaken to assess the finite sample performance of the proposed variable selection procedure.     

Cure rate model with bivariate interval censored data

A mixture model is proposed to analyze a bivariate interval censored data with cure rates. There exist two types of association related with bivariate failure times and bivariate cure rates, respectively. A correlation coefficient is adopted for the association of bivariate cure rates and a copula function is applied for bivariate survival times. The conditional expectation of unknown quantities attributable to interval censored data and cure rates are calculated in the E-step in ES (Expectation-Solving algorithm) and the marginal estimates and the association measures are estimated in the S-step through a two-stage procedure. A simulation study is performed to evaluate the suggested method and a real data from HIV patients is analyzed as a real data example.     

Cumulative Incidence Association Models for Bivariate Competing Risks Data

Association models, like frailty and copula models, are frequently used to analyze clustered survival data and evaluate within-cluster associations. The assumption of noninformative censoring is commonly applied to these models, though it may not be true in many situations. In this paper, we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function (CIF), a nonparametrically identifiable quantity. Copula models are proposed which relate the bivariate CIF to its corresponding univariate CIFs, similarly to independently right censored data, and accommodate frailty models for the bivariate CIF. Two estimating equations are developed to estimate the association parameter, permitting the univariate CIFs to be estimated either parametrically or nonparametrically. Goodness-of-fit tests are presented for formally evaluating the parametric models. Both estimators perform well with moderate sample sizes in simulation studies. The practical use of the methodology is illustrated in an analysis of dementia associations.