Marginal hazards regression for retrospective studies within cohort with possibly correlated failure time data

Summary .  A retrospective dental study was conducted to evaluate the degree to which pulpal involvement affects tooth survival. Due to the clustering of teeth, the survival times within each subject could be correlated and thus the conventional method for the case–control studies cannot be directly applied. In this article, we propose a marginal model approach for this type of correlated case–control within cohort data. Weighted estimating equations are proposed for the estimation of the regression parameters. Different types of weights are also considered for improving the efficiency. Asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulations studies. The proposed method is applied to the aforementioned dental study.     

Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression

Summary .  In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.     

Semiparametric estimation of the covariate-specific ROC curve in presence of ignorable verification bias

Covariate-specific receiver operating characteristic (ROC) curves are often used to evaluate the classification accuracy of a medical diagnostic test or a biomarker, when the accuracy of the test is associated with certain covariates. In many large-scale screening tests, the gold standard is subject to missingness due to high cost or harmfulness to the patient. In this article, we propose a semiparametric estimation of the covariate-specific ROC curves with a partial missing gold standard. A location-scale model is constructed for the test result to model the covariates' effect, but the residual distributions are left unspecified. Thus the baseline and link functions of the ROC curve both have flexible shapes. With the gold standard missing at random (MAR) assumption, we consider weighted estimating equations for the location-scale parameters, and weighted kernel estimating equations for the residual distributions. Three ROC curve estimators are proposed and compared, namely, imputation-based, inverse probability weighted, and doubly robust estimators. We derive the asymptotic normality of the estimated ROC curve, as well as the analytical form of the standard error estimator. The proposed method is motivated and applied to the data in an Alzheimer's disease research.     

Robust Estimation of Area Under ROC Curve Using Auxiliary Variables in the Presence of Missing Biomarker Values

Summary In medical research, the receiver operating characteristic (ROC) curves can be used to evaluate the performance of biomarkers for diagnosing diseases or predicting the risk of developing a disease in the future. The area under the ROC curve (ROC AUC), as a summary measure of ROC curves, is widely utilized, especially when comparing multiple ROC curves. In observational studies, the estimation of the AUC is often complicated by the presence of missing biomarker values, which means that the existing estimators of the AUC are potentially biased. In this article, we develop robust statistical methods for estimating the ROC AUC and the proposed methods use information from auxiliary variables that are potentially predictive of the missingness of the biomarkers or the missing biomarker values. We are particularly interested in auxiliary variables that are predictive of the missing biomarker values. In the case of missing at random (MAR), that is, missingness of biomarker values only depends on the observed data, our estimators have the attractive feature of being consistent if one correctly specifies, conditional on auxiliary variables and disease status, either the model for the probabilities of being missing or the model for the biomarker values. In the case of missing not at random (MNAR), that is, missingness may depend on the unobserved biomarker values, we propose a sensitivity analysis to assess the impact of MNAR on the estimation of the ROC AUC. The asymptotic properties of the proposed estimators are studied and their finite‐sample behaviors are evaluated in simulation studies. The methods are further illustrated using data from a study of maternal depression during pregnancy.     

Incorporating Correlation for Multivariate Failure Time Data When Cluster Size Is Large

Summary We propose a new estimation method for multivariate failure time data using the quadratic inference function (QIF) approach. The proposed method efficiently incorporates within‐cluster correlations. Therefore, it is more efficient than those that ignore within‐cluster correlation. Furthermore, the proposed method is easy to implement. Unlike the weighted estimating equations in Cai and Prentice (1995, Biometrika 82 , 151–164), it is not necessary to explicitly estimate the correlation parameters. This simplification is particularly useful in analyzing data with large cluster size where it is difficult to estimate intracluster correlation. Under certain regularity conditions, we show the consistency and asymptotic normality of the proposed QIF estimators. A chi‐squared test is also developed for hypothesis testing. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed methods. We also illustrate the proposed methods by analyzing primary biliary cirrhosis (PBC) data.     

A semiparametric joint model for cluster size and subunit-specific interval-censored outcomes

Clustered data frequently arise in biomedical studies, where observations, or subunits, measured within a cluster are associated. The cluster size is said to be informative, if the outcome variable is associated with the number of subunits in a cluster. In most existing work, the informative cluster size issue is handled by marginal approaches based on within-cluster resampling, or cluster-weighted generalized estimating equations. Although these approaches yield consistent estimation of the marginal models, they do not allow estimation of within-cluster associations and are generally inefficient. In this paper, we propose a semiparametric joint model for clustered interval-censored event time data with informative cluster size. We use a random effect to account for the association among event times of the same cluster as well as the association between event times and the cluster size. For estimation, we propose a sieve maximum likelihood approach and devise a computationally-efficient expectation-maximization algorithm for implementation. The estimators are shown to be strongly consistent, with the Euclidean components being asymptotically normal and achieving semiparametric efficiency. Extensive simulation studies are conducted to evaluate the finite-sample performance, efficiency and robustness of the proposed method. We also illustrate our method via application to a motivating periodontal disease dataset.     

Efficient resampling methods for nonsmooth estimating functions

We propose a simple and general resampling strategy to estimatevariances for parameter estimators derived from nonsmooth estimatingfunctions. This approach applies to a wide variety of semiparametricand nonparametric problems in biostatistics. It does not requiresolving estimating equations and is thus much faster than theexisting resampling procedures. Its usefulness is illustratedwith heteroscedastic quantile regression and censored data rankregression. Numerical results based on simulated and real dataare provided.     

Extended Mantel-Haenszel estimating procedure for multivariate logistic regression models

A class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case-control studies. It reduces to the well-known Mantel-Haenszel estimator when there is a single binary risk factor. Large-sample properties of the solutions to the proposed estimating equations are established for two distinct situations. Efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered. Application of the proposed method to family data and longitudinal data, where the conditional likelihood approach fails, is discussed. Two examples from case-control studies and one example from a study on familial aggregation are presented.     

The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables

In sample surveys, it is usual to make use of auxiliary information to increase the precision of the estimators. We propose a new chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables and obtain the mean squared error (MSE) equations for the proposed estimators. We find theoretical conditions that make proposed estimators more efficient than the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables.     

The relative efficiency of penetrance estimators for sib pairs

Here we present analytical studies to evaluate the relative efficiency of commonly used penetrance estimators using linkage designs. We investigated three different methods of estimating penetrance using sib pairs: Maximum likehood estimation (MLE) with trait information alone, MLE with both trait and marker information and the MOD score approach. Modeling sib pairs with unknown phase, we evaluated the asymptotic relative efficiency between estimators under either random sampling or single ascertainment for an autosomal dominant or recessive disease. We then provide plots of the asymptotic relative efficiency, enabling researchers to easily determine regions where the MOD score or segregation alone performs with comparable efficiency relative to joint segregation and linkage.     

Effects of variance-function misspecification in analysis of longitudinal data

The approach of generalized estimating equations (GEE) is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within-subject correlations. The variance is often assumed to be a known function of the mean. This article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses. Our numerical studies indicate that (1) correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified; (2) misspecification of the variance function impacts much more on estimators for within-cluster covariates than for cluster-level covariates; and (3) if the variance function is misspecified, correct choice of the correlation structure may not necessarily improve estimation efficiency. We illustrate impacts of different variance functions using a real data set from cow growth.     

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.     

Estimation of recurrence of colorectal adenomas with dependent censoring using weighted logistic regression

In colorectal polyp prevention trials, estimation of the rate of recurrence of adenomas at the end of the trial may be complicated by dependent censoring, that is, time to follow-up colonoscopy and dropout may be dependent on time to recurrence. Assuming that the auxiliary variables capture the dependence between recurrence and censoring times, we propose to fit two working models with the auxiliary variables as covariates to define risk groups and then extend an existing weighted logistic regression method for independent censoring to each risk group to accommodate potential dependent censoring. In a simulation study, we show that the proposed method results in both a gain in efficiency and reduction in bias for estimating the recurrence rate. We illustrate the methodology by analyzing a recurrent adenoma dataset from a colorectal polyp prevention trial.     

Estimating equations with nonignorably missing response data

Troxel, Lipsitz, and Brennan (1997, Biometrics 53, 857-869) considered parameter estimation from survey data with nonignorable nonresponse and proposed weighted estimating equations to remove the biases in the complete-case analysis that ignores missing observations. This paper suggests two alternative modifications for unbiased estimation of regression parameters when a binary outcome is potentially observed at successive time points. The weighting approach of Robins, Rotnitzky, and Zhao (1995, Journal of the American Statistical Association 90, 106-121) is also modified to obtain unbiased estimating functions. The suggested estimating functions are unbiased only when the missingness probability is correctly specified, and misspecification of the missingness model will result in biases in the estimates. Simulation studies are carried out to assess the performance of different methods when the covariate is binary or normal. For the simulation models used, the relative efficiency of the two new methods to the weighting methods is about 3.0 for the slope parameter and about 2.0 for the intercept parameter when the covariate is continuous and the missingness probability is correctly specified. All methods produce substantial biases in the estimates when the missingness model is misspecified or underspecified. Analysis of data from a medical survey illustrates the use and possible differences of these estimating functions.     

Order-restricted inference for clustered ROC data with application to fingerprint matching accuracy

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.     

Median regression with censored cost data

Because of the skewness of the distribution of medical costs, we consider modeling the median as well as other quantiles when establishing regression relationships to covariates. In many applications, the medical cost data are also right censored. In this article, we propose semiparametric procedures for estimating the parameters in median regression models based on weighted estimating equations when censoring is present. Numerical studies are conducted to show that our estimators perform well with small samples and the resulting inference is reliable in circumstances of practical importance. The methods are applied to a dataset for medical costs of patients with colorectal cancer.     

On the estimation of average treatment effects with right-censored time to event outcome and competing risks

We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, censoring, and treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that these estimators are regular asymptotically linear estimators and estimate their variances based on estimates of their influence functions. In empirical studies, we assess the robustness of the estimators and the coverage of confidence intervals. The methods are further illustrated using data from a Danish registry study.     

Testing for bias in weighted estimating equations

It is very common in regression analysis to encounter incompletely observed covariate information. A recent approach to analyse such data is weighted estimating equations (Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994), JASA, 89, 846-866, and Zhao, L. P., Lipsitz, S. R. and Lew, D. (1996), Biometrics, 52, 1165-1182). With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse of the probability of being observed. We propose a test statistic to assess if the weighted estimating equations produce biased estimates. Our test statistic is similar to the test statistic proposed by DuMouchel and Duncan (1983) for weighted least squares estimates for sample survey data. The method is illustrated using data from a randomized clinical trial on chemotherapy for multiple myeloma.     

Class of Estimators of Mean and Variance Using Auxiliary Information when Correlation Coefficient is also Known

For estimating the mean of a finite population using information on an auxiliary variable, a class of estimators which also uses the value of the correlation coefficient between the two variables which is assumed known, is defined. Expression for its asymptotic mean squared error and its minimum value is obtained. An expression by which the minimum mean squared error of this class is smaller than those which use only the sample mean and the sample variance of the auxiliary variable is obtained. A similar class of estimators is considered for the estimation of the population variance. The gain in efficiency is illustrated for six populations considered in literature.     

Improved Horvitz–Thompson Estimation of Model Parameters from Two-phase Stratified Samples: Applications in Epidemiology

The case-cohort study involves two-phase samplings: simple random sampling from an infinite superpopulation at phase one and stratified random sampling from a finite cohort at phase two. Standard analyses of case-cohort data involve solution of inverse probability weighted (IPW) estimating equations, with weights determined by the known phase two sampling fractions. The variance of parameter estimates in (semi)parametric models, including the Cox model, is the sum of two terms: (i) the model-based variance of the usual estimates that would be calculated if full data were available for the entire cohort; and (ii) the design-based variance from IPW estimation of the unknown cohort total of the efficient influence function (IF) contributions. This second variance component may be reduced by adjusting the sampling weights, either by calibration to known cohort totals of auxiliary variables correlated with the IF contributions or by their estimation using these same auxiliary variables. Both adjustment methods are implemented in the R survey package. We derive the limit laws of coefficients estimated using adjusted weights. The asymptotic results suggest practical methods for construction of auxiliary variables that are evaluated by simulation of case-cohort samples from the National Wilms Tumor Study and by log-linear modeling of case-cohort data from the Atherosclerosis Risk in Communities Study. Although not semiparametric efficient, estimators based on adjusted weights may come close to achieving full efficiency within the class of augmented IPW estimators.