Covariate Adjusted Correlation Analysis via Varying Coefficient Models

Abstract.  We propose covariate adjusted correlation (Cadcor) analysis to target the correlation between two hidden variables that are observed after being multiplied by an unknown function of a common observable confounding variable. The distorting effects of this confounding may alter the correlation relation between the hidden variables. Covariate adjusted correlation analysis enables consistent estimation of this correlation, by targeting the definition of correlation through the slopes of the regressions of the hidden variables on each other and by establishing a connection to varying-coefficient regression. The asymptotic distribution of the resulting adjusted correlation estimate is established. These distribution results, when combined with proposed consistent estimates of the asymptotic variance, lead to the construction of approximate confidence intervals and inference for adjusted correlations. We illustrate our approach through an application to the Boston house price data. Finite sample properties of the proposed procedures are investigated through a simulation study.     

To adjust or not to adjust for baseline when analyzing repeated binary responses? The case of complete data when treatment comparison at study end is of interest

The benefits of adjusting for baseline covariates are not as straightforward with repeated binary responses as with continuous response variables. Therefore, in this study, we compared different methods for analyzing repeated binary data through simulations when the outcome at the study endpoint is of interest. Methods compared included chi‐square, Fisher's exact test, covariate adjusted/unadjusted logistic regression (Adj.logit/Unadj.logit), covariate adjusted/unadjusted generalized estimating equations (Adj.GEE/Unadj.GEE), covariate adjusted/unadjusted generalized linear mixed model (Adj.GLMM/Unadj.GLMM). All these methods preserved the type I error close to the nominal level. Covariate adjusted methods improved power compared with the unadjusted methods because of the increased treatment effect estimates, especially when the correlation between the baseline and outcome was strong, even though there was an apparent increase in standard errors. Results of the Chi‐squared test were identical to those for the unadjusted logistic regression. Fisher's exact test was the most conservative test regarding the type I error rate and also with the lowest power. Without missing data, there was no gain in using a repeated measures approach over a simple logistic regression at the final time point. Analysis of results from five phase III diabetes trials of the same compound was consistent with the simulation findings. Therefore, covariate adjusted analysis is recommended for repeated binary data when the study endpoint is of interest. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear mode regression with covariate measurement error

We consider estimating the mode of a response given an error‐prone covariate. It is shown that ignoring measurement error typically leads to inconsistent inference for the conditional mode of the response given the true covariate, as well as misleading inference for regression coefficients in the conditional mode model. To account for measurement error, we first employ the Monte Carlo corrected score method (Novick & Stefanski, 2002) to obtain an unbiased score function based on which the regression coefficients can be estimated consistently. To relax the normality assumption on measurement error this method requires, we propose another method where deconvoluting kernels are used to construct an objective function that is maximized to obtain consistent estimators of the regression coefficients. Besides rigorous investigation on asymptotic properties of the new estimators, we study their finite sample performance via extensive simulation experiments, and find that the proposed methods substantially outperform a naive inference method that ignores measurement error. The Canadian Journal of Statistics 47: 262–280; 2019 © 2019 Statistical Society of Canada     

A semiparametric additive rates model for recurrent event data

Recurrent event data often arise in biomedical studies, with examples including hospitalizations, infections, and treatment failures. In observational studies, it is often of interest to estimate the effects of covariates on the marginal recurrent event rate. The majority of existing rate regression methods assume multiplicative covariate effects. We propose a semiparametric model for the marginal recurrent event rate, wherein the covariates are assumed to add to the unspecified baseline rate. Covariate effects are summarized by rate differences, meaning that the absolute effect on the rate function can be determined from the regression coefficient alone. We describe modifications of the proposed method to accommodate a terminating event (e.g., death). Proposed estimators of the regression parameters and baseline rate are shown to be consistent and asymptotically Gaussian. Simulation studies demonstrate that the asymptotic approximations are accurate in finite samples. The proposed methods are applied to a state-wide kidney transplant data set.     

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.     

Estimation for varying coefficient partially nonlinear models with distorted measurement errors

In this paper, we propose a new varying coefficient partially nonlinear model where both the response and predictors are not directly observed, but are observed by unknown distorting functions of a commonly observable covariate. Because of the complexity of the model, existing estimation methods cannot be directly employed. For this, we propose using an efficient nonparametric regression to estimate the unknown distortion functions concerning the covariates and response on the distorting variable, and further, we obtain the profile nonlinear least squares estimators for the parameters and the coefficient functions using the calibrated variables. Furthermore, we establish the asymptotic properties of the resulting estimators. To illustrate our proposed methodology, we carry out some simulated and real examples.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada     

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.     

Quantile regression estimation for distortion measurement error data

We study the quantile estimation methods for the distortion measurement error data when variables are unobserved and distorted with additive errors by some unknown functions of an observable confounding variable. After calibrating the error-prone variables, we propose the quantile regression estimation procedure and composite quantile estimation procedure. Asymptotic properties of the proposed estimators are established, and we also investigate the asymptotic relative efficiency compared with the least-squares estimator. Simulation studies are conducted to evaluate the performance of the proposed methods, and a real dataset is analyzed as an illustration.     

Marginal Regression Model with Time-Varying Coefficients for Panel Data

In this article, we formulate a class of semiparametric marginal means models with a mixture of time-varying and time-independent parameters for analyzing panel data. For inference about the regression parameters, an estimation procedure is developed and asymptotic properties of the proposed estimators are established. In addition, some tests are presented for investigating whether or not covariate effects vary with time. The finite-sample behavior of the proposed methods is examined in simulation studies, and the data from an AIDS clinical trial study are used to illustrate the methodology.     

Regression Analysis of Longitudinal Data with Time‐Dependent Covariates and Informative Observation Times

Abstract. Longitudinal data frequently occur in many studies, and longitudinal responses may be correlated with observation times. In this paper, we propose a new joint modelling for the analysis of longitudinal data with time‐dependent covariates and possibly informative observation times via two latent variables. For inference about regression parameters, estimating equation approaches are developed and asymptotic properties of the proposed estimators are established. In addition, a lack‐of‐fit test is presented for assessing the adequacy of the model. The proposed method performs well in finite‐sample simulation studies, and an application to a bladder tumour study is provided.     

Regression analysis of case-cohort studies in the presence of dependent interval censoring

The case-cohort design is widely used as a means of reducing the cost in large cohort studies, especially when the disease rate is low and covariate measurements may be expensive, and has been discussed by many authors. In this paper, we discuss regression analysis of case-cohort studies that produce interval-censored failure time with dependent censoring, a situation for which there does not seem to exist an established approach. For inference, a sieve inverse probability weighting estimation procedure is developed with the use of Bernstein polynomials to approximate the unknown baseline cumulative hazard functions. The proposed estimators are shown to be consistent and the asymptotic normality of the resulting regression parameter estimators is established. A simulation study is conducted to assess the finite sample properties of the proposed approach and indicates that it works well in practical situations. The proposed method is applied to an HIV/AIDS case-cohort study that motivated this investigation.     

Simultaneous estimation and inference for multiple response variables

Multivariate data arise frequently in biomedical and health studies where multiple response variables are collected across subjects. Unlike a univariate procedure fitting each response separately, a multivariate regression model provides a unique opportunity in studying the joint evolution of various response variables. In this paper, we propose two estimation procedures that improve estimation efficiency for the regression parameter by accommodating correlations among the response variables. The proposed procedures do not require knowledge of the true correlation structure nor does it estimate the parameters associated with the correlation. Theoretical and simulation results confirm that the proposed estimators are more efficient than the one obtained from the univariate approach. We further propose simple and powerful inference procedures for a goodness-of-fit test that possess the chi-squared asymptotic properties. Extensive simulation studies suggest that the proposed tests are more powerful than the Wald test based on the univariate procedure. The proposed methods are also illustrated through the mother’s stress and children’s morbidity study.     

Nonparametric covariate adjustment for receiver operating characteristic curves

The accuracy of a diagnostic test is typically characterized using the receiver operating characteristic (ROC) curve. Summarizing indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. The authors propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and possibly non‐normal errors are discussed and analyzed separately. Nonparametric regression is used for estimating mean and variance functions in both scenarios. In the model that relaxes the assumption of normality, the authors propose a covariate‐adjusted Mann–Whitney estimator for AUC estimation which effectively uses available data to construct working samples at any covariate value of interest and is computationally efficient for implementation. This provides a generalization of the Mann–Whitney approach for comparing two populations by taking covariate effects into account. The authors derive asymptotic properties for the AUC estimators in both settings, including asymptotic normality, optimal strong uniform convergence rates and mean squared error (MSE) consistency. The MSE of the AUC estimators was also assessed in smaller samples by simulation. Data from an agricultural study were used to illustrate the methods of analysis. The Canadian Journal of Statistics 38:27–46; 2010 © 2009 Statistical Society of Canada     

Simulation-based Inference in a Zero-inflated Bernoulli Regression Model

The logistic regression model has become a standard tool to investigate the relationship between a binary outcome and a set of potential predictors. When analyzing binary data, it often arises that the observed proportion of zeros is greater than expected under the postulated logistic model. Zero-inflated binomial (ZIB) models have been developed to fit binary data that contain too many zeros. Maximum likelihood estimators in these models have been proposed and their asymptotic properties established. Several aspects of ZIB models still deserve attention however, such as the estimation of odds-ratios and event probabilities. In this article, we propose estimators of these quantities and we investigate their properties both theoretically and via simulations. Based on these results, we provide recommendations about the range of conditions (minimum sample size, maximum proportion of zeros in excess) under which a reliable statistical inference on the odds-ratios and event probabilities can be obtained in a ZIB regression model. A real-data example illustrates the proposed estimators.     

Maximum Likelihood Estimators in Regression Models for Error‐prone Group Testing Data

Since Dorfman's seminal work on the subject, group testing has been widely adopted in epidemiological studies. In Dorfman's context of detecting syphilis, group testing entails pooling blood samples and testing the pools, as opposed to testing individual samples. A negative pool indicates all individuals in the pool free of syphilis antigen, whereas a positive pool suggests one or more individuals carry the antigen. With covariate information collected, researchers have considered regression models that allow one to estimate covariate‐adjusted disease probability. We study maximum likelihood estimators of covariate effects in these regression models when the group testing response is prone to error. We show that, when compared with inference drawn from individual testing data, inference based on group testing data can be more resilient to response misclassification in terms of bias and efficiency. We provide valuable guidance on designing the group composition to alleviate adverse effects of misclassification on statistical inference.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

Semiparametric mixture of additive regression models

In this article, we propose a semiparametric mixture of additive regression models, in which the regression functions are additive and non parametric while the mixing proportions and variances are constant. Compared with the mixture of linear regression models, the proposed methodology is more flexible in modeling the non linear relationship between the response and covariate. A two-step procedure based on the spline-backfitted kernel method is derived for computation. Moreover, we establish the asymptotic normality of the resultant estimators and examine their good performance through a numerical example.     

Semiparametric log-linear regression for longitudinal measurements subject to outcome-dependent follow-up

A common problem for longitudinal data analyses is that subjects follow-up is irregular, often related to the past outcome or other factors associated with the outcome measure that are not included in the regression model. Analyses unadjusted for outcome-dependent follow-up yield biased estimates. We propose a longitudinal data analysis that can provide consistent estimates in regression models that are subject to outcome-dependent follow-up. We focus on semiparametric marginal log-link regression with arbitrary unspecified baseline function. Based on estimating equations, the proposed class of estimators are root n consistent and asymptotically normal. We present simulation studies that assess the performance of the estimators under finite samples. We illustrate our approach using data from a health services research study.