A random effects model for multivariate failure time data from multicenter clinical trials

A random effects model for analyzing multivariate failure time data is proposed. The work is motivated by the need for assessing the mean treatment effect in a multicenter clinical trial study, assuming that the centers are a random sample from an underlying population. An estimating equation for the mean hazard ratio parameter is proposed. The proposed estimator is shown to be consistent and asymptotically normally distributed. A variance estimator, based on large sample theory, is proposed. Simulation results indicate that the proposed estimator performs well in finite samples. The proposed variance estimator effectively corrects the bias of the naive variance estimator, which assumes independence of individuals within a group. The methodology is illustrated with a clinical trial data set from the Studies of Left Ventricular Dysfunction. This shows that the variability of the treatment effect is higher than found by means of simpler models.     

Application of pattern-mixture models to outcomes that are potentially missing not at random using pseudo maximum likelihood estimation

In this work, we fit pattern-mixture models to data sets with responses that are potentially missing not at random (MNAR, Little and Rubin, 1987). In estimating the regression parameters that are identifiable, we use the pseudo maximum likelihood method based on exponential families. This procedure provides consistent estimators when the mean structure is correctly specified for each pattern, with further information on the variance structure giving an efficient estimator. The proposed method can be used to handle a variety of continuous and discrete outcomes. A test built on this approach is also developed for model simplification in order to improve efficiency. Simulations are carried out to compare the proposed estimation procedure with other methods. In combination with sensitivity analysis, our approach can be used to fit parsimonious semi-parametric pattern-mixture models to outcomes that are potentially MNAR. We apply the proposed method to an epidemiologic cohort study to examine cognition decline among elderly.     

Exploiting gene-environment independence for analysis of case-control studies: an empirical Bayes-type shrinkage estimator to trade-off between bias and efficiency

Summary .   Standard prospective logistic regression analysis of case–control data often leads to very imprecise estimates of gene-environment interactions due to small numbers of cases or controls in cells of crossing genotype and exposure. In contrast, under the assumption of gene-environment independence, modern "retrospective" methods, including the "case-only" approach, can estimate the interaction parameters much more precisely, but they can be seriously biased when the underlying assumption of gene-environment independence is violated. In this article, we propose a novel empirical Bayes-type shrinkage estimator to analyze case–control data that can relax the gene-environment independence assumption in a data-adaptive fashion. In the special case, involving a binary gene and a binary exposure, the method leads to an estimator of the interaction log odds ratio parameter in a simple closed form that corresponds to an weighted average of the standard case-only and case–control estimators. We also describe a general approach for deriving the new shrinkage estimator and its variance within the retrospective maximum-likelihood framework developed by Chatterjee and Carroll (2005, Biometrika 92, 399–418). Both simulated and real data examples suggest that the proposed estimator strikes a balance between bias and efficiency depending on the true nature of the gene-environment association and the sample size for a given study.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.     

Nonparametric analysis of recurrent events and death

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.     

Borrowing information across genes and experiments for improved error variance estimation in microarray data analysis

Statistical inference for microarray experiments usually involves the estimation of error variance for each gene. Because the sample size available for each gene is often low, the usual unbiased estimator of the error variance can be unreliable. Shrinkage methods, including empirical Bayes approaches that borrow information across genes to produce more stable estimates, have been developed in recent years. Because the same microarray platform is often used for at least several experiments to study similar biological systems, there is an opportunity to improve variance estimation further by borrowing information not only across genes but also across experiments. We propose a lognormal model for error variances that involves random gene effects and random experiment effects. Based on the model, we develop an empirical Bayes estimator of the error variance for each combination of gene and experiment and call this estimator BAGE because information is Borrowed Across Genes and Experiments. A permutation strategy is used to make inference about the differential expression status of each gene. Simulation studies with data generated from different probability models and real microarray data show that our method outperforms existing approaches.     

Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.     

Nonparametric estimation of the concordance correlation coefficient under univariate censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.     

Corrected estimator of sensitive population proportion using unknown repeated trials in the unrelated question randomized response model

In this paper, we have pointed out a major mistake in the research paper of Singh and Mathur [(2004). Unknown repeated trials in the unrelated question randomized response model, Biometrical Journal, 46:375–378]. We have corrected this mistake and proposed the corresponding corrected estimator of sensitive population proportion. Furthermore, we have obtained the variance of our proposed estimator. Likewise, Singh and Mathur, we have also compared the variance of our proposed estimator with that of the Greenberg et al.’s estimator theoretically as well as numerically.     

A penalized spline approach to functional mixed effects model analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.     

Combining multiple imputation and inverse-probability weighting

Two approaches commonly used to deal with missing data are multiple imputation (MI) and inverse-probability weighting (IPW). IPW is also used to adjust for unequal sampling fractions. MI is generally more efficient than IPW but more complex. Whereas IPW requires only a model for the probability that an individual has complete data (a univariate outcome), MI needs a model for the joint distribution of the missing data (a multivariate outcome) given the observed data. Inadequacies in either model may lead to important bias if large amounts of data are missing. A third approach combines MI and IPW to give a doubly robust estimator. A fourth approach (IPW/MI) combines MI and IPW but, unlike doubly robust methods, imputes only isolated missing values and uses weights to account for remaining larger blocks of unimputed missing data, such as would arise, e.g., in a cohort study subject to sample attrition, and/or unequal sampling fractions. In this article, we examine the performance, in terms of bias and efficiency, of IPW/MI relative to MI and IPW alone and investigate whether the Rubin's rules variance estimator is valid for IPW/MI. We prove that the Rubin's rules variance estimator is valid for IPW/MI for linear regression with an imputed outcome, we present simulations supporting the use of this variance estimator in more general settings, and we demonstrate that IPW/MI can have advantages over alternatives. IPW/MI is applied to data from the National Child Development Study.     

A novel penalized inverse-variance weighted estimator for Mendelian randomization with applications to COVID-19 outcomes

Mendelian randomization utilizes genetic variants as instrumental variables (IVs) to estimate the causal effect of an exposure variable on an outcome of interest even in the presence of unmeasured confounders. However, the popular inverse-variance weighted (IVW) estimator could be biased in the presence of weak IVs, a common challenge in MR studies. In this article, we develop a novel penalized inverse-variance weighted (pIVW) estimator, which adjusts the original IVW estimator to account for the weak IV issue by using a penalization approach to prevent the denominator of the pIVW estimator from being close to zero. Moreover, we adjust the variance estimation of the pIVW estimator to account for the presence of balanced horizontal pleiotropy. We show that the recently proposed debiased IVW (dIVW) estimator is a special case of our proposed pIVW estimator. We further prove that the pIVW estimator has smaller bias and variance than the dIVW estimator under some regularity conditions. We also conduct extensive simulation studies to demonstrate the performance of the proposed pIVW estimator. Furthermore, we apply the pIVW estimator to estimate the causal effects of five obesity-related exposures on three coronavirus disease 2019 (COVID-19) outcomes. Notably, we find that hypertensive disease is associated with an increased risk of hospitalized COVID-19; and peripheral vascular disease and higher body mass index are associated with increased risks of COVID-19 infection, hospitalized COVID-19, and critically ill COVID-19.     

A Monte Carlo evaluation of five interval estimators for the relative risk in sparse data

The relative risk (RR) is one of the most frequently used indices to measure the strength of association between a disease and a risk factor in etiological studies or the efficacy of an experimental treatment in clinical trials. In this paper, we concentrate attention on interval estimation of RR for sparse data, in which we have only a few patients per stratum, but a moderate or large number of strata. We consider five asymptotic interval estimators for RR, including a weighted least-squares (WLS) interval estimator with an ad hoc adjustment procedure for sparse data, an interval estimator proposed elsewhere for rare events, an interval estimator based on the Mantel-Haenszel (MH) estimator with a logarithmic transformation, an interval estimator calculated from a quadratic equation, and an interval estimator derived from the ratio estimator with a logarithmic transformation. On the basis of Monte Carlo simulations, we evaluate and compare the performance of these five interval estimators in a variety of situations. We note that, except for the cases in which the underlying common RR across strata is around 1, using the WLS interval estimator with the adjustment procedure for sparse data can be misleading. We note further that using the interval estimator suggested elsewhere for rare events tends to be conservative and hence leads to loss of efficiency. We find that the other three interval estimators can consistently perform well even when the mean number of patients for a given treatment is approximately 3 patients per stratum and the number of strata is as small as 20. Finally, we use a mortality data set comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the estimators discussed in this paper.     

On an Efficient Regression Type Estimator

The regression type estimator proposed by KAUR (1985) is considered. Another expression for the approximated mean square error (AMSE), to a first degree of approximation, is obtained. This AMSE is also minimized with respect to a parameter α. Three numerical examples are included. These numerical examples show that this estimator is not significantly more efficient than regression estimator and with respect to ratio and sample mean estimators, it does not always exhibit a high efficiency, as was contended by KAUR (1985). Moreover, an upper bound for the relative precision of the proposed estimator with respect to linear regression estimator is derived.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.     

A Stochastic Regression Model for General Trend Analysis of Longitudinal Continuous Data

A predictive continuous time model is developed for continuous panel data to assess the effect of time‐varying covariates on the general direction of the movement of a continuous response that fluctuates over time. This is accomplished by reparameterizing the infinitesimal mean of an Ornstein–Uhlenbeck processes in terms of its equilibrium mean and a drift parameter, which assesses the rate that the process reverts to its equilibrium mean. The equilibrium mean is modeled as a linear predictor of covariates. This model can be viewed as a continuous time first‐order autoregressive regression model with time‐varying lag effects of covariates and the response, which is more appropriate for unequally spaced panel data than its discrete time analog. Both maximum likelihood and quasi‐likelihood approaches are considered for estimating the model parameters and their performances are compared through simulation studies. The simpler quasi‐likelihood approach is suggested because it yields an estimator that is of high efficiency relative to the maximum likelihood estimator and it yields a variance estimator that is robust to the diffusion assumption of the model. To illustrate the proposed model, an application to diastolic blood pressure data from a follow‐up study on cardiovascular diseases is presented. Missing observations are handled naturally with this model.