Variance estimation when using inverse probability of treatment weighting (IPTW) with survival analysis

Propensity score methods are used to reduce the effects of observed confounding when using observational data to estimate the effects of treatments or exposures. A popular method of using the propensity score is inverse probability of treatment weighting (IPTW). When using this method, a weight is calculated for each subject that is equal to the inverse of the probability of receiving the treatment that was actually received. These weights are then incorporated into the analyses to minimize the effects of observed confounding. Previous research has found that these methods result in unbiased estimation when estimating the effect of treatment on survival outcomes. However, conventional methods of variance estimation were shown to result in biased estimates of standard error. In this study, we conducted an extensive set of Monte Carlo simulations to examine different methods of variance estimation when using a weighted Cox proportional hazards model to estimate the effect of treatment. We considered three variance estimation methods: (i) a naïve model‐based variance estimator; (ii) a robust sandwich‐type variance estimator; and (iii) a bootstrap variance estimator. We considered estimation of both the average treatment effect and the average treatment effect in the treated. We found that the use of a bootstrap estimator resulted in approximately correct estimates of standard errors and confidence intervals with the correct coverage rates. The other estimators resulted in biased estimates of standard errors and confidence intervals with incorrect coverage rates. Our simulations were informed by a case study examining the effect of statin prescribing on mortality. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Induced smoothing for rank‐based regression with recurrent gap time data

Various semiparametric regression models have recently been proposed for the analysis of gap times between consecutive recurrent events. Among them, the semiparametric accelerated failure time (AFT) model is especially appealing owing to its direct interpretation of covariate effects on the gap times. In general, estimation of the semiparametric AFT model is challenging because the rank‐based estimating function is a nonsmooth step function. As a result, solutions to the estimating equations do not necessarily exist. Moreover, the popular resampling‐based variance estimation for the AFT model requires solving rank‐based estimating equations repeatedly and hence can be computationally cumbersome and unstable. In this paper, we extend the induced smoothing approach to the AFT model for recurrent gap time data. Our proposed smooth estimating function permits the application of standard numerical methods for both the regression coefficients estimation and the standard error estimation. Large‐sample properties and an asymptotic variance estimator are provided for the proposed method. Simulation studies show that the proposed method outperforms the existing nonsmooth rank‐based estimating function methods in both point estimation and variance estimation. The proposed method is applied to the data analysis of repeated hospitalizations for patients in the Danish Psychiatric Center Register.     

Variance reduction in randomised trials by inverse probability weighting using the propensity score

In individually randomised controlled trials, adjustment for baseline characteristics is often undertaken to increase precision of the treatment effect estimate. This is usually performed using covariate adjustment in outcome regression models. An alternative method of adjustment is to use inverse probability‐of‐treatment weighting (IPTW), on the basis of estimated propensity scores. We calculate the large‐sample marginal variance of IPTW estimators of the mean difference for continuous outcomes, and risk difference, risk ratio or odds ratio for binary outcomes. We show that IPTW adjustment always increases the precision of the treatment effect estimate. For continuous outcomes, we demonstrate that the IPTW estimator has the same large‐sample marginal variance as the standard analysis of covariance estimator. However, ignoring the estimation of the propensity score in the calculation of the variance leads to the erroneous conclusion that the IPTW treatment effect estimator has the same variance as an unadjusted estimator; thus, it is important to use a variance estimator that correctly takes into account the estimation of the propensity score. The IPTW approach has particular advantages when estimating risk differences or risk ratios. In this case, non‐convergence of covariate‐adjusted outcome regression models frequently occurs. Such problems can be circumvented by using the IPTW adjustment approach. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Variance estimation for clustered recurrent event data with a small number of clusters

Often in biomedical studies, the event of interest is recurrent and within-subject events cannot usually be assumed independent. In semi-parametric estimation of the proportional rates model, a working independence assumption leads to an estimating equation for the regression parameter vector, with within-subject correlation accounted for through a robust (sandwich) variance estimator; these methods have been extended to the case of clustered subjects. We consider variance estimation in the setting where subjects are clustered and the study consists of a small number of moderate-to-large-sized clusters. We demonstrate through simulation that the robust estimator is quite inaccurate in this setting. We propose a corrected version of the robust variance estimator, as well as jackknife and bootstrap estimators. Simulation studies reveal that the corrected variance is considerably more accurate than the robust estimator, and slightly more accurate than the jackknife and bootstrap variance. The proposed methods are used to compare hospitalization rates between Canada and the U.S. in a multi-centre dialysis study. Copyright (c) 2005 John Wiley & Sons, Ltd.     

Covariance estimators for generalized estimating equations (GEE) in longitudinal analysis with small samples

Generalized estimating equations (GEE) is a general statistical method to fit marginal models for longitudinal data in biomedical studies. The variance–covariance matrix of the regression parameter coefficients is usually estimated by a robust “sandwich” variance estimator, which does not perform satisfactorily when the sample size is small. To reduce the downward bias and improve the efficiency, several modified variance estimators have been proposed for bias‐correction or efficiency improvement. In this paper, we provide a comprehensive review on recent developments of modified variance estimators and compare their small‐sample performance theoretically and numerically through simulation and real data examples. In particular, Wald tests and t‐tests based on different variance estimators are used for hypothesis testing, and the guideline on appropriate sample sizes for each estimator is provided for preserving type I error in general cases based on numerical results. Moreover, we develop a user‐friendly R package “geesmv” incorporating all of these variance estimators for public usage in practice. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of relative risk and prevalence ratio

Relative risks (RRs) and prevalence ratios (PRs) are measures of association that are more intuitively interpretable than odds ratios (ORs). Many health science studies report OR estimates, however, even when their designs permit and study questions target RRs and/or PRs. This is, partially, attributable to the popularity and technical advantage (i.e. no restriction on the parameter space) of logistic regression for estimating ORs. To improve this practice, several biostatistical approaches for estimating RR/PR, adjusting for potential confounders, have been proposed. In this paper, we consider two RR/PR estimating methods: (1) the modification of log‐binomial regression with the COPY method; and (2) an inverse‐probability‐of‐treatment‐weighted (IPTW) log‐binomial regression we newly propose. For the COPY method, we rigorously establish the existence and uniqueness of the maximum‐likelihood estimator, provided certain degeneracies in the data do not occur. Moreover, the global maximum of the COPY‐modified likelihood is shown to occur at an interior point of the restricted parameter space. This result explains why the COPY method avoids convergence problems of log‐binomial models frequently. For the IPTW estimator, we show that its simple procedure results in standardized estimates of RR/PR, and discuss its potential challenges, extensions, and an improvement through propensity‐score‐based grouping of observations. Furthermore, we compare the performances of four RR/PR estimation methods, including the COPY method and IPTW log‐binomial regression, on simulated data. We demonstrate a lack of robustness of the COPY method against misspecification of the true relationship between binary outcome and explanatory variables, and show robustness of the IPTW approach in this regard. Copyright © 2010 John Wiley & Sons, Ltd.     

Inference for the median residual life function in sequential multiple assignment randomized trials

In survival analysis, median residual lifetime is often used as a summary measure to assess treatment effectiveness; it is not clear, however, how such a quantity could be estimated for a given dynamic treatment regimen using data from sequential randomized clinical trials. We propose a method to estimate a dynamic treatment regimen‐specific median residual life (MERL) function from sequential multiple assignment randomized trials. We present the MERL estimator, which is based on inverse probability weighting, as well as, two variance estimates for the MERL estimator. One variance estimate follows from Lunceford, Davidian and Tsiatis' 2002 survival function‐based variance estimate and the other uses the sandwich estimator. The MERL estimator is evaluated, and its two variance estimates are compared through simulation studies, showing that the estimator and both variance estimates produce approximately unbiased results in large samples. To demonstrate our methods, the estimator has been applied to data from a sequentially randomized leukemia clinical trial. Copyright © 2013 John Wiley & Sons, Ltd.     

Small-sample adjustments in using the sandwich variance estimator in generalized estimating equations

The generalized estimating equation (GEE) approach is widely used in regression analyses with correlated response data. Under mild conditions, the resulting regression coefficient estimator is consistent and asymptotically normal with its variance being consistently estimated by the so-called sandwich estimator. Statistical inference is thus accomplished by using the asymptotic Wald chi-squared test. However, it has been noted in the literature that for small samples the sandwich estimator may not perform well and may lead to much inflated type I errors for the Wald chi-squared test. Here we propose using an approximate t- or F-test that takes account of the variability of the sandwich estimator. The level of type I error of the proposed t- or F-test is guaranteed to be no larger than that of the Wald chi-squared test. The satisfactory performance of the proposed new tests is confirmed in a simulation study. Our proposal also has some advantages when compared with other new approaches based on direct modifications of the sandwich estimator, including the one that corrects the downward bias of the sandwich estimator. In addition to hypothesis testing, our result has a clear implication on constructing Wald-type confidence intervals or regions.     

基于秩次R类稳健回归

目的从参数估计、稳健性质、回归诊断应用等方面介绍基于广义秩次的一类稳健回归分析方法—R和GR估计。方法在SAS的IML模块下模拟其对非正态误差分布表现、正态误差下的估计效率并进行实例分析。结果误差为cauchy分布时,R估计优于LS估计,X空间存在离群值时,GR估计优于R和LS估计,误差为正态分布时,R与CR估计效率达95%。结论R和GR估计为是一种估计效率较高的稳健回归方法,其中GR估计可同时避免X和Y空间离群点。     

Application of the bootstrap procedure provides an alternative to standard statistical procedures in the estimation of the vitamin B-6 requirement.

The bootstrap procedure is a versatile statistical tool for the estimation of standard errors and confidence intervals. It is useful when standard statistical methods are not available or are poorly behaved, e.g., for nonlinear functions or when assumptions of a statistical model have been violated. Inverse regression estimation is an example of a statistical tool with a wide application in human nutrition. In a recent study, inverse regression was used to estimate the vitamin B-6 requirement of young women. In the present statistical application, both standard statistical methods and the bootstrap technique were used to estimate the mean vitamin B-6 requirement, standard errors and 95% confidence intervals for the mean. The bootstrap procedure produced standard error estimates and confidence intervals that were similar to those calculated by using standard statistical estimators. In a Monte Carlo simulation exploring the behavior of the inverse regression estimators, bootstrap standard errors were found to be nearly unbiased, even when the basic assumptions of the regression model were violated. On the other hand, the standard asymptotic estimator was found to behave well when the assumptions of the regression model were met, but behaved poorly when the assumptions were violated. In human metabolic studies, which are often restricted to small sample sizes, or when statistical methods are not available or are poorly behaved, bootstrap estimates for calculating standard errors and confidence intervals may be preferred. Investigators in human nutrition may find that the bootstrap procedure is superior to standard statistical procedures in cases similar to the examples presented in this paper.     

The use of bootstrapping when using propensity‐score matching without replacement: a simulation study

Propensity‐score matching is frequently used to estimate the effect of treatments, exposures, and interventions when using observational data. An important issue when using propensity‐score matching is how to estimate the standard error of the estimated treatment effect. Accurate variance estimation permits construction of confidence intervals that have the advertised coverage rates and tests of statistical significance that have the correct type I error rates. There is disagreement in the literature as to how standard errors should be estimated. The bootstrap is a commonly used resampling method that permits estimation of the sampling variability of estimated parameters. Bootstrap methods are rarely used in conjunction with propensity‐score matching. We propose two different bootstrap methods for use when using propensity‐score matching without replacementand examined their performance with a series of Monte Carlo simulations. The first method involved drawing bootstrap samples from the matched pairs in the propensity‐score‐matched sample. The second method involved drawing bootstrap samples from the original sample and estimating the propensity score separately in each bootstrap sample and creating a matched sample within each of these bootstrap samples. The former approach was found to result in estimates of the standard error that were closer to the empirical standard deviation of the sampling distribution of estimated effects. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons, Ltd.     

Penalized estimation for proportional hazards models with current status data

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B‐splines and use a hybrid approach based on the Fisher‐scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite‐sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness‐of‐fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications.     

Semi‐parametric estimation of random effects in a logistic regression model using conditional inference

This paper describes a new approach to the estimation in a logistic regression model with two crossed random effects where special interest is in estimating the variance of one of the effects while not making distributional assumptions about the other effect. A composite likelihood is studied. For each term in the composite likelihood, a conditional likelihood is used that eliminates the influence of the random effects, which results in a composite conditional likelihood consisting of only one‐dimensional integrals that may be solved numerically. Good properties of the resulting estimator are described in a small simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

Modified robust variance estimator for generalized estimating equations with improved small-sample performance

Generalized estimating equations (GEE (Biometrika 1986; 73(1):13-22) is a general statistical method to fit marginal models for correlated or clustered responses, and it uses a robust sandwich estimator to estimate the variance-covariance matrix of the regression coefficient estimates. While this sandwich estimator is robust to the misspecification of the correlation structure of the responses, its finite sample performance deteriorates as the number of clusters or observations per cluster decreases. To address this limitation, Pan (Biometrika 2001; 88(3):901-906) and Mancl and DeRouen (Biometrics 2001; 57(1):126-134) investigated two modifications to the original sandwich variance estimator. Motivated by the ideas underlying these two modifications, we propose a novel robust variance estimator that combines the strengths of these estimators. Our theoretical and numerical results show that the proposed estimator attains better efficiency and achieves better finite sample performance compared with existing estimators. In particular, when the sample size or cluster size is small, our proposed estimator exhibits lower bias and the resulting confidence intervals for GEE estimates achieve better coverage rates performance. We illustrate the proposed method using data from a dental study.     

Methods to adjust for misclassification in the quantiles for the generalized linear model with measurement error in continuous exposures

In many biomedical studies, covariates of interest may be measured with errors. However, frequently in a regression analysis, the quantiles of the exposure variable are often used as the covariates in the regression analysis. Because of measurement errors in the continuous exposure variable, there could be misclassification in the quantiles for the exposure variable. Misclassification in the quantiles could lead to bias estimation in the association between the exposure variable and the outcome variable. Adjustment for misclassification will be challenging when the gold standard variables are not available. In this paper, we develop two regression calibration estimators to reduce bias in effect estimation. The first estimator is normal likelihood‐based. The second estimator is linearization‐based, and it provides a simple and practical correction. Finite sample performance is examined via a simulation study. We apply the methods to a four‐arm randomized clinical trial that tested exercise and weight loss interventions in women aged 50–75years. Copyright © 2015 John Wiley & Sons, Ltd.     

Blinded sample size re‐estimation in three‐arm trials with ‘gold standard’ design

In this article, we study blinded sample size re‐estimation in the ‘gold standard’ design with internal pilot study for normally distributed outcomes. The ‘gold standard’ design is a three‐arm clinical trial design that includes an active and a placebo control in addition to an experimental treatment. We focus on the absolute margin approach to hypothesis testing in three‐arm trials at which the non‐inferiority of the experimental treatment and the assay sensitivity are assessed by pairwise comparisons. We compare several blinded sample size re‐estimation procedures in a simulation study assessing operating characteristics including power and type I error. We find that sample size re‐estimation based on the popular one‐sample variance estimator results in overpowered trials. Moreover, sample size re‐estimation based on unbiased variance estimators such as the Xing–Ganju variance estimator results in underpowered trials, as it is expected because an overestimation of the variance and thus the sample size is in general required for the re‐estimation procedure to eventually meet the target power. To overcome this problem, we propose an inflation factor for the sample size re‐estimation with the Xing–Ganju variance estimator and show that this approach results in adequately powered trials. Because of favorable features of the Xing–Ganju variance estimator such as unbiasedness and a distribution independent of the group means, the inflation factor does not depend on the nuisance parameter and, therefore, can be calculated prior to a trial. Moreover, we prove that the sample size re‐estimation based on the Xing–Ganju variance estimator does not bias the effect estimate. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimating and testing direct genetic effects in directed acyclic graphs using estimating equations

In genetic association studies, it is important to distinguish direct and indirect genetic effects in order to build truly functional models. For this purpose, we consider a directed acyclic graph setting with genetic variants, primary and intermediate phenotypes, and confounding factors. In order to make valid statistical inference on direct genetic effects on the primary phenotype, it is necessary to consider all potential effects in the graph, and we propose to use the estimating equations method with robust Huber–White sandwich standard errors. We evaluate the proposed causal inference based on estimating equations (CIEE) method and compare it with traditional multiple regression methods, the structural equation modeling method, and sequential G‐estimation methods through a simulation study for the analysis of (completely observed) quantitative traits and time‐to‐event traits subject to censoring as primary phenotypes. The results show that CIEE provides valid estimators and inference by successfully removing the effect of intermediate phenotypes from the primary phenotype and is robust against measured and unmeasured confounding of the indirect effect through observed factors. All other methods except the sequential G‐estimation method for quantitative traits fail in some scenarios where their test statistics yield inflated type I errors. In the analysis of the Genetic Analysis Workshop 19 dataset, we estimate and test genetic effects on blood pressure accounting for intermediate gene expression phenotypes. The results show that CIEE can identify genetic variants that would be missed by traditional regression analyses. CIEE is computationally fast, widely applicable to different fields, and available as an R package.     

Regression-assisted deconvolution

We present a semi-parametric deconvolution estimator for the density function of a random variable biX that is measured with error, a common challenge in many epidemiological studies. Traditional deconvolution estimators rely only on assumptions about the distribution of X and the error in its measurement, and ignore information available in auxiliary variables. Our method assumes the availability of a covariate vector statistically related to X by a mean-variance function regression model, where regression errors are normally distributed and independent of the measurement errors. Simulations suggest that the estimator achieves a much lower integrated squared error than the observed-data kernel density estimator when models are correctly specified and the assumption of normal regression errors is met. We illustrate the method using anthropometric measurements of newborns to estimate the density function of newborn length.     

Avoiding zero between‐study variance estimates in random‐effects meta‐analysis

Fixed‐effects meta‐analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random‐effects meta‐analysis and meta‐regression are therefore typically used to accommodate explained and unexplained between‐study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between‐study standard deviation, resulting in fixed‐effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between‐study standard deviation. When no prior information is available regarding the magnitude of the between‐study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta‐analysis and meta‐regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between‐study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity. Copyright © 2013 John Wiley & Sons, Ltd.     

Performance of the modified Poisson regression approach for estimating relative risks from clustered prospective data

Modified Poisson regression, which combines a log Poisson regression model with robust variance estimation, is a useful alternative to log binomial regression for estimating relative risks. Previous studies have shown both analytically and by simulation that modified Poisson regression is appropriate for independent prospective data. This method is often applied to clustered prospective data, despite a lack of evidence to support its use in this setting. The purpose of this article is to evaluate the performance of the modified Poisson regression approach for estimating relative risks from clustered prospective data, by using generalized estimating equations to account for clustering. A simulation study is conducted to compare log binomial regression and modified Poisson regression for analyzing clustered data from intervention and observational studies. Both methods generally perform well in terms of bias, type I error, and coverage. Unlike log binomial regression, modified Poisson regression is not prone to convergence problems. The methods are contrasted by using example data sets from 2 large studies. The results presented in this article support the use of modified Poisson regression as an alternative to log binomial regression for analyzing clustered prospective data when clustering is taken into account by using generalized estimating equations.