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When estimating the average causal effect in observational studies, researchers have to tackle both self-selection of treatment and outcome modeling. This is difficult because the parametric form of the outcome model is often unknown and there exists a large number of covariates. In this work, we present a semiparametric strategy for estimating the average causal effect by regressing on the propensity score. Furthermore, we show that regression error terms usually depend on the propensity score as well, which could cause heteroscedastic variances, and thus construct a refined estimator to improve the estimation efficiency. Both estimators are shown to be consistent and asymptotically normally distributed, with the latter one having a smaller asymptotic variance. The simulation studies indicate that our methods compare favorably with many competing estimators. Our methods are easy to implement and avoid hazardous impact due to extreme weights as often seen in weighting estimators. They can also be extended to handle subgroup effects with known structure. We apply the proposed methods to data from the Ohio Medicaid Assessment Survey 2012, estimating the effect of having health insurance on self-reported health status for a population with subsidized insurance plan choices under the Affordable Care Act.

Matching is a routinely used technique to balance covariates and thereby alleviate confounding bias in causal inference with observational data. Most of the matching literatures involve the estimating of propensity score with parametric model, which heavily depends on the model specification. In this paper, we employ machine learning and matching techniques to learn the average causal effect. By comparing a variety of machine learning methods in terms of propensity score under extensive scenarios, we find that the ensemble methods, especially generalized random forests, perform favorably with others. We apply all the methods to the data of tropical storms that occurred on the mainland of China since 1949.

Partly interval censored data frequently occur in many areas including clinical trials, epidemiology research, and medical follow-up studies. When data come from observational studies, we need to carefully adjust for the confounding bias in order to estimate the true treatment effect. Pair matching designs are popular for removing confounding bias without parametric assumptions. With time-to-event outcomes, there are some literature for hypothesis testing with paired right censored data, but not for interval censored data. O’Brien and Fleming extended the Prentice Wilcoxon test to right censored paired data by making use of the Prentice-Wilcoxon scores. Akritas proposed the Akritas test and established its asymptotic properties. We extend Akritas test to partly interval censored data. We estimate the survival distribution function by nonparametric maximum likelihood estimation (NPMLE), and prove the asymptotic validity of the new test. To improve our test under small sample size or extreme distributions, we also propose a modified version using the rank of the score difference. Simulation results indicate that our proposed methods have very good performance.

In this paper, we are interested in exploring the dynamic causal relationships among two sets of three variables in different quarters. One set is futures sugar closing price in Zhengzhou futures exchange market (ZC), spot sugar price in Zhengzhou (ZS) and futures sugar closing price in New York futures exchange market(NC) and the other includes futures sugar opening price in Zhengzhou (ZO), ZS and NC. For each quarter, we first use Bayesian model selection to obtain the optimal causal graph with the highest BD scores and then use Bayesian model averaging approach to explore the causal relationship between every two variables. From the real data analysis, the two conclusions almost coincide, which shows that the two methods are practical.

This paper studies semiparametric regression analysis of panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. To explore the nonlinear interactions between covariates, we propose a class of partially linear models with possibly varying coefficients for the mean function of the counting processes with panel count data. The functional coefficients are estimated by B-spline function approximations. The estimation procedures are based on maximum pseudo-likelihood and likelihood approaches and they are easy to implement. The asymptotic properties of the resulting estimators are established, and their finite-sample performance is assessed by Monte Carlo simulation studies. We also demonstrate the value of the proposed method by the analysis of a cancer data set, where the new modeling approach provides more comprehensive information than the usual proportional mean model.

This paper presents a method of constructing a mixed graph which can be used to analyze the causality for multivariate time series. We construct a partial correlation graph at first which is an undirected graph. For every undirected edge in the partial correlation graph, the measures of linear feedback between two time series can help us decide its direction, then we obtain the mixed graph. Using this method, we construct a mixed graph for futures sugar prices in Zhengzhou (ZF), spot sugar prices in Zhengzhou (ZS) and futures sugar prices in New York (NF). The result shows that there is a bi-directional causality between ZF and ZS, an unidirectional causality from NF to ZF, but no causality between NF and ZS.

This paper studies a monotone constrained varying-coefficient partially linear model with errors in variables using a monotone B-spline approach based on adjusted least squares. The proposed estimator is consistent, asymptotically normal, and performs well in numerical studies.

We consider an Error-in-Variable partially linear model where the covariates of linear part are measured with error which follows a normal distribution with a known covariance matrix. We propose a corrected-loss estimation of the covariate effect. The proposed estimator is asymptotically normal. Simulation studies are presented to show that the proposed method performs well with finite samples, and the proposed method is applied to a real data set.

Linear transformation models provide a class of flexible models for regression analysis of failure time data and several methods have been proposed for fitting them to right-censored failure time data (Chen, 2002; Cheng et al., 1995; Yin and Zeng, 2006). This article considers the fitting of these models to current status data, a special type of interval-censored failure time data. We derive the maximum likelihood estimates and their information bound and establish the efficiency of the estimates. Simulation studies show that the approach is appropriate for practical use and an application is given to illustrate the approach.

Panel count data usually occur in longitudinal follow-up studies that concern occurrence rates of certain recurrent events and their analysis involves two processes. One is the underlying recurrent event process of interest and the other is the observation process that controls observation times. In some situations, the two processes may be correlated and, for this, several estimation procedures have recently been developed (He et?al., 2009; Huang et?al., 2006; Sun et?al., 2007b; Zhao and Tong, 2011). These methods, however, rely on some restrictive models or assumptions such as the Poisson assumption. In this work, a more general and robust estimation approach is proposed for regression analysis of panel count data with related observation times. The asymptotic properties of the resulting estimates are established and the numerical studies conducted indicate that the approach works well for practical situations.

In this article, we consider a linear model in which the covariates are measured with errors. We propose a t-type corrected-loss estimation of the covariate effect, when the measurement error follows the Laplace distribution. The proposed estimator is asymptotically normal. In practical studies, some outliers that diminish the robustness of the estimation occur. Simulation studies show that the estimators are resistant to vertical outliers and an application of 6-minute walk test is presented to show that the proposed method performs well.

In this paper, we are interested in investigating the causal relationships among futures sugar prices in the Zhengzhou futures exchange market (ZF), the spot sugar prices in Zhengzhou (ZS) and the futures sugar prices in New York futures exchange market (NF). A useful tool called Bayesian network is introduced to analyze the problem. Since there are only three variables in our Bayesian network, the algorithm is straightforward: we display all the 25 possible network structures and adopt certain scoring metrics to evaluate them. We applied five different scoring metrics in total. Firstly, for each metric, we obtained 24 scores, each calculated from one of the 24 possible structures i.e. a Directed Acyclic Graph (DAG). Then we eliminated the network structure which represents the independence of the three variables according to our prior knowledge concerning the futures sugar market. After that, the optimal network structure which implies the causal relationships was selected according to the corresponding scoring metric. Finally, after comparing the results from different scoring metrics, we obtained the relatively affirmative conclusion that ZS causes ZF from both the Bayesian Dirichlet (BD) metric, Bayesian Dirichlet-Akaike Information Criterion (BD-AIC) metric, Bayesian Dirichlet-Bayesian Information Criterion (BD-BIC) metric and Bayesian Information Criterion (BIC) metric. The conclusions that NF causes ZF and ZF causes ZS from the Akaike Information Criterion (AIC) metric and ZF causes ZS from the BIC metric were useful and significant to our investigation.

This paper discusses regression analysis of panel count data that arise naturally when recurrent events are considered. For the analysis of panel count data, most of the existing methods have assumed that observation times are completely independent of recurrent events or given covariates, which may not be true in practice. We propose a joint modeling approach that uses an unobserved random variable and a completely unspecified link function to characterize the correlations between the response variable and the observation times. For inference about regression parameters, estimating equation approaches are developed without involving any estimation for latent variables, and the asymptotic properties of the resulting estimators are established. In addition, a technique is provided for assessing the adequacy of the model. The performance of the proposed estimation procedures are evaluated by means of Monte Carlo simulations, and a data set from a bladder tumor study is analyzed as an illustrative example.

In many longitudinal studies, observation times as well as censoring times may be correlated with longitudinal responses. This paper considers a multiplicative random effects model for the longitudinal response where these correlations may exist and a joint modeling approach is proposed via a shared latent variable. For inference about regression parameters, estimating equation approaches are developed and asymptotic properties of the proposed estimators are established. The finite sample behavior of the methods is examined through simulation studies and an application to a data set from a bladder cancer study is provided for illustration.

A maximum likelihood method with spline smoothing is proposed for linear transformation models with varying coefficients. The estimation and inference procedures are computationally easy. Under some regularity conditions, the estimators are proved to be consistent and asymptotically normal. A simulation study using the Stanford transplant data is presented to show that the proposed method performs well with a finite sample and is easy to use in practice. Copyright 2010, Oxford University Press.

Recurrent event data occur in many clinical and observational studies (Cook and Lawless, Analysis of recurrent event data, 2007) and in these situations, there may exist a terminal event such as death that is related to the recurrent event of interest (Ghosh and Lin, Biometrics 56:554-562, 2000; Wang et al., J Am Stat Assoc 96:1057-1065, 2001; Huang and Wang, J Am Stat Assoc 99:1153-1165, 2004; Ye et al., Biometrics 63:78-87, 2007). In addition, sometimes there may exist more than one type of recurrent events, that is, one faces multivariate recurrent event data with some dependent terminal event (Chen and Cook, Biostatistics 5:129-143, 2004). It is apparent that for the analysis of such data, one has to take into account the dependence both among different types of recurrent events and between the recurrent and terminal events. In this paper, we propose a joint modeling approach for regression analysis of the data and both finite and asymptotic properties of the resulting estimates of unknown parameters are established. The methodology is applied to a set of bivariate recurrent event data arising from a study of leukemia patients.

This article considers a semiparametric varying-coefficient partially linear regression model with current status data. The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable. A Sieve maximum likelihood estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. The convergence rate of the estimator for the unknown smooth function is obtained and the estimator for the unknown parameter is shown to be asymptotically efficient and normally distributed. Simulation studies are conducted to examine the small-sample properties of the proposed estimates and a real dataset is used to illustrate our approach.

For partial linear model Y = Xτβ0 + g0(T) + ∈ with unknown β0 ∈ ȑd and an unknown smooth function g0, this paper considers the Huber-Dutter estimators of β0, scale σ for the errors and the function g0 approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β0 and σ are shown to be asymptotically normal with the rate of convergence n−1/2 and the B-spline Huber-Dutter estimator of g0 achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator.

With an increasing number of causal genes discovered for complex human disorders, it is crucial to assess the genetic risk of disease onset for individuals who are carriers of these causal mutations and compare the distribution of age-at-onset with that in non-carriers. In many genetic epidemiological studies aiming at estimating causal gene effect on disease, the age-at-onset of disease is subject to censoring. In addition, some individuals' mutation carrier or non-carrier status can be unknown due to the high cost of in-person ascertainment to collect DNA samples or death in older individuals. Instead, the probability of these individuals' mutation status can be obtained from various sources. When mutation status is missing, the available data take the form of censored mixture data. Recently, various methods have been proposed for risk estimation from such data, but none is efficient for estimating a nonparametric distribution. We propose a fully efficient sieve maximum likelihood estimation method, in which we estimate the logarithm of the hazard ratio between genetic mutation groups using B-splines, while applying nonparametric maximum likelihood estimation for the reference baseline hazard function. Our estimator can be calculated via an expectation-maximization algorithm which is much faster than existing methods. We show that our estimator is consistent and semiparametrically efficient and establish its asymptotic distribution. Simulation studies demonstrate superior performance of the proposed method, which is applied to the estimation of the distribution of the age-at-onset of Parkinson's disease for carriers of mutations in the leucine-rich repeat kinase 2 gene.

For partial linear model Y = X τ β 0 + g 0(T) + e with unknown β 0 ∈¸ R d and an unknown smooth function g 0, this paper considers the Huber-Dutter estimators of β 0, scale σ for the errors and the function g 0 respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β 0 and σ are asymptotically normal with convergence rate n -1/2 and the B-spline Huber-Dutter estimator of g 0 achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β 0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.

In this article, we propose a class of Box-Cox transformation models for recurrent event data, which includes the proportional means models as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the proposed models, we apply a profile pseudo-partial likelihood method to estimate the model parameters via estimating equation approaches and establish large sample properties of the estimators and examine its performance in moderate-sized samples through simulation studies. In addition, some graphical and numerical procedures are presented for model checking. An example of application on a set of multiple-infection data taken from a clinic study on chronic granulomatous disease (CGD) is also illustrated.

Variable selection is an important issue in all regression analysis and in this paper, we discuss this in the context of regression analysis of recurrent event data. Recurrent event data often occur in long-term studies in which individuals may experience the events of interest more than once and their analysis has recently attracted a great deal of attention (Andersen et al., Statistical models based on counting processes, 1993; Cook and Lawless, Biometrics 52:1311-1323, 1996, The analysis of recurrent event data, 2007; Cook et al., Biometrics 52:557-571, 1996; Lawless and Nadeau, Technometrics 37:158-168, 1995; Lin et al., J R Stat Soc B 69:711-730, 2000). However, it seems that there are no established approaches to the variable selection with respect to recurrent event data. For the problem, we adopt the idea behind the nonconcave penalized likelihood approach proposed in Fan and Li (J Am Stat Assoc 96:1348-1360, 2001) and develop a nonconcave penalized estimating function approach. The proposed approach selects variables and estimates regression coefficients simultaneously and an algorithm is presented for this process. We show that the proposed approach performs as well as the oracle procedure in that it yields the estimates as if the correct submodel was known. Simulation studies are conducted for assessing the performance of the proposed approach and suggest that it works well for practical situations. The proposed methodology is illustrated by using the data from a chronic granulomatous disease study.

Panel count data often occur in long-term studies that concern occurrence rate of a recurrent event. Methods have been proposed for regression analysis of panel count data, but most of the existing research focuses on situations where observation times are independent of longitudinal response variables and therefore rely on conditional inference procedures given the observation times. This article considers a different situation where the independence assumption may not hold. That is, the observation times and the response variable may be correlated. For inference, estimating equation approaches are proposed for estimation of regression parameters and both large and finite sample properties of the proposed estimates are established. An illustrative example from a cancer study is provided.

This paper discusses regression analysis of panel count data that often arise in longitudinal studies concerning occurrence rates of certain recurrent events. Panel count data mean that each study subject is observed only at discrete time points rather than under continuous observation. Furthermore, both observation and follow-up times can vary from subject to subject and may be correlated with the recurrent events. For inference, we propose some shared frailty models and estimating equations are developed for estimation of regression parameters. The proposed estimates are consistent and have asymptotically a normal distribution. The finite sample properties of the proposed estimates are investigated through simulation and an illustrative example from a cancer study is provided.

This paper discusses the analysis of interval-censored failure time data, which has recently attracted a great amount of attention (Li and Pu, Lifetime Data Anal 9:57–70, 2003; Sun, The statistical analysis of interval-censored data, 2006; Tian and Cai, Biometrika 93(2):329–342, 2006; Zhang et al., Can J Stat 33:61–70, 2005). Interval-censored data mean that the survival time of interest is observed only to belong to an interval and they occur in many fields including clinical trials, demographical studies, medical follow-up studies, public health studies and tumorgenicity experiments. A major difficulty with the analysis of interval-censored data is that one has to deal with a censoring mechanism that involves two related variables. For the inference, we present a transformation approach that transforms general interval-censored data into current status data, for which one only needs to deal with one censoring variable and the inference is thus much easy. We apply this general idea to regression analysis of interval-censored data using the additive hazards model and numerical studies indicate that the method performs well for practical situations. An illustrative example is provided.

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.