Weighted causal inference methods with mismeasured covariates and misclassified outcomes

Inverse probability weighting (IPW) estimation has been widely used in causal inference. Its validity relies on the important condition that the variables are precisely measured. This condition, however, is often violated, which distorts the IPW method and thus yields biased results. In this paper, we study the IPW estimation of average treatment effects for settings with mismeasured covariates and misclassified outcomes. We develop estimation methods to correct for measurement error and misclassification effects simultaneously. Our discussion covers a broad scope of treatment models, including typically assumed logistic regression models and general treatment assignment mechanisms. Satisfactory performance of the proposed methods is demonstrated by extensive numerical studies.     

Regression calibration to correct correlated errors in outcome and exposure

Measurement error arises through a variety of mechanisms. A rich literature exists on the bias introduced by covariate measurement error and on methods of analysis to address this bias. By comparison, less attention has been given to errors in outcome assessment and nonclassical covariate measurement error. We consider an extension of the regression calibration method to settings with errors in a continuous outcome, where the errors may be correlated with prognostic covariates or with covariate measurement error. This method adjusts for the measurement error in the data and can be applied with either a validation subset, on which the true data are also observed (eg, a study audit), or a reliability subset, where a second observation of error prone measurements are available. For each case, we provide conditions under which the proposed method is identifiable and leads to consistent estimates of the regression parameter. When the second measurement on the reliability subset has no error or classical unbiased measurement error, the proposed method is consistent even when the primary outcome and exposures of interest are subject to both systematic and random error. We examine the performance of the method with simulations for a variety of measurement error scenarios and sizes of the reliability subset. We illustrate the method's application using data from the Women's Health Initiative Dietary Modification Trial.     

Regression calibration for models with two predictor variables measured with error and their interaction,using instrumental variables and longitudinal data

Regression calibration provides a way to obtain unbiased estimators of fixed effects in regression models when one or more predictors are measured with error. Recent development of measurement error methods has focused on models that include interaction terms between measured‐with‐error predictors, and separately, methods for estimation in models that account for correlated data. In this work, we derive explicit and novel forms of regression calibration estimators and associated asymptotic variances for longitudinal models that include interaction terms, when data from instrumental and unbiased surrogate variables are available but not the actual predictors of interest. The longitudinal data are fit using linear mixed models that contain random intercepts and account for serial correlation and unequally spaced observations. The motivating application involves a longitudinal study of exposure to two pollutants (predictors) – outdoor fine particulate matter and cigarette smoke – and their association in interactive form with levels of a biomarker of inflammation, leukotriene E4 (LTE ₄, outcome) in asthmatic children. Because the exposure concentrations could not be directly observed, we used measurements from a fixed outdoor monitor and urinary cotinine concentrations as instrumental variables, and we used concentrations of fine ambient particulate matter and cigarette smoke measured with error by personal monitors as unbiased surrogate variables. We applied the derived regression calibration methods to estimate coefficients of the unobserved predictors and their interaction, allowing for direct comparison of toxicity of the different pollutants. We used simulations to verify accuracy of inferential methods based on asymptotic theory. Copyright © 2013 John Wiley & Sons, Ltd.     

The estimation of calibration equations for variables with heteroscedastic measurement errors

In clinical chemistry and medical research, there is often a need to calibrate the values obtained from an old or discontinued laboratory procedure to the values obtained from a new or currently used laboratory method. The objective of the calibration study is to identify a transformation that can be used to convert the test values of one laboratory measurement procedure into the values that would be obtained using another measurement procedure. However, in the presence of heteroscedastic measurement error, there is no good statistical method available for estimating the transformation. In this paper, we propose a set of statistical methods for a calibration study when the magnitude of the measurement error is proportional to the underlying true level. The corresponding sample size estimation method for conducting a calibration study is discussed as well. The proposed new method is theoretically justified and evaluated for its finite sample properties via an extensive numerical study. Two examples based on real data are used to illustrate the procedure. Copyright © 2014 John Wiley & Sons, Ltd.     

A disattenuated correlation estimate when variables are measured with error: illustration estimating cross-platform correlations

Previous cross-platform reproducibility studies have compared consistency of intensities as well as consistency of fold changes across different platforms using Pearson's correlation coefficient. In this study, we propose the use of measurement error models for estimating gene-specific correlations. Additionally, gene-specific reliability estimates are shown to be useful in prioritizing clones for sequence verification rather than selecting clones using a simple random sample. The proposed 'disattenuated' correlation may prove useful in a wide variety of studies when both X and Y are measured with error, such as in confirmation studies of microarray gene expression values, wherein more reliable laboratory assays such as real-time polymerase chain reaction are used.     

Regression calibration in studies with correlated variables measured with error.

Regression calibration is a technique that corrects biases in regression results in situations where exposure variables are measured with error. The existence of a calibration substudy, where accurate and crude measurement methods are related by a second regression analysis, is assumed. The cost of measurement error in multivariate analyses is loss of statistical power. In this paper, calibration data from California Seventh-day Adventists are used to simulate study populations and new calibration studies. Applying regression calibration logistic analyses, the authors estimate power for pairs of nutritional variables. The results demonstrate substantial loss of power if variables measured with error are strongly correlated. Biases in estimated effects in cases where regression calibration is not performed can be large and are corrected by regression calibration. When the true coefficient has zero value, the corresponding coefficient in a crude analysis will usually have a nonzero expected value. Then type I error probabilities are not nominal, and the erroneous appearance of statistical significance can readily occur, particularly in large studies. Major determinants of power with use of regression calibration are collinearity between the variables measured with error and the size of correlations between crude and corresponding true variables. Where there is important collinearity, useful gains in power accrue with calibration study size up to 1,000 subjects.     

Regression analysis with covariates that have heteroscedastic measurement error

We consider the estimation of the regression of an outcome Y on a covariate X, where X is unobserved, but a variable W that measures X with error is observed. A calibration sample that measures pairs of values of X and W is also available; we consider calibration samples where Y is measured (internal calibration) and not measured (external calibration). One common approach for measurement error correction is Regression Calibration (RC), which substitutes the unknown values of X by predictions from the regression of X on W estimated from the calibration sample. An alternative approach is to multiply impute the missing values of X given Y and W based on an imputation model, and then use multiple imputation (MI) combining rules for inferences. Most of current work assumes that the measurement error of W has a constant variance, whereas in many situations, the variance varies as a function of X. We consider extensions of the RC and MI methods that allow for heteroscedastic measurement error, and compare them by simulation. The MI method is shown to provide better inferences in this setting. We also illustrate the proposed methods using a data set from the BioCycle study.     

Incorporating validation subsets into discrete proportional hazards models for mismeasured outcomes

Standard proportional hazards methods are inappropriate for mismeasured outcomes. Previous work has shown that outcome mismeasurement can bias estimation of hazard ratios for covariates. We previously developed an adjusted proportional hazards method that can produce accurate hazard ratio estimates when outcome measurement is either non-sensitive or non-specific. That method requires that mismeasurement rates (the sensitivity and specificity of the diagnostic test) are known. Here, we develop an approach to handle unknown mismeasurement rates. We consider the case where the true failure status is known for a subset of subjects (the validation set) until the time of observed failure or censoring. Five methods of handling these mismeasured outcomes are described and compared. The first method uses only subjects on whom complete data are available (validation subset), whereas the second method uses only mismeasured outcomes (naive method). Three other methods include available data from both validated and non-validated subjects. Through simulation, we show that inclusion of the non-validated subjects can improve efficiency relative to use of the complete case data only and that inclusion of some true outcomes (the validation subset) can reduce bias relative to use of mismeasured outcomes only. We also compare the performance of the validation methods proposed using an example data set.     

A mechanistic nonlinear model for censored and mismeasured covariates in longitudinal models,with application in AIDS studies

When modeling longitudinal data, the true values of time‐varying covariates may be unknown because of detection‐limit censoring or measurement error. A common approach in the literature is to empirically model the covariate process based on observed data and then predict the censored values or mismeasured values based on this empirical model. Such an empirical model can be misleading, especially for censored values since the (unobserved) censored values may behave very differently than observed values due to the underlying data‐generation mechanisms or disease status. In this paper, we propose a mechanistic nonlinear covariate model based on the underlying data‐generation mechanisms to address censored values and mismeasured values. Such a mechanistic model is based on solid scientific or biological arguments, so the predicted censored or mismeasured values are more reasonable. We use a Monte Carlo EM algorithm for likelihood inference and apply the methods to an AIDS dataset, where viral load is censored by a lower detection limit. Simulation results confirm that the proposed models and methods offer substantial advantages over existing empirical covariate models for censored and mismeasured covariates.     

非胰岛素依赖型糖尿病病人影响因素的非条件Logistic回归分析

在苏南农村现况调查资料的基础上，对筛选出的８８例非胰岛素依赖型糖尿病（ＮＩＤＤＭ）患者进行病例对照研究，应用非条件Ｌｏｇｉｓｔｉｃ单元及多元回归分析，结果显示ＮＩＤＤＭ病人具高血压史，心血管病史，营养过丰和肥胖等的比例显著高于对照组（Ｐ〈０．０１）。其中高血压ＯＲ值达４．５０，去除其他因素后仍达２．９４，提示高血压可能是独立的预测因素之一，多元拟合模型还显示ＮＩＤＤＭ对心血管疾病的发生有直接影响。     

妊娠晚期贫血影响因素的Logistic回归分析

目的 探讨妊娠晚期贫血的影响因素.方法 数据来自104例孕妇的资料,以她们妊娠中期的血清铁蛋白(SF)、游离红细胞原卟啉(FEP)、红细胞容积分布宽度(RDW)和平均红细胞容积(MCV)四项指标作为自变量,以妊娠晚期游离红细胞原卟啉(FEP)作为因变量,作Logistic回归分析.结果 只有妊娠中期游离红细胞原卟啉(FEP)对妊娠晚期贫血的影响有显着意义(P<0.001),其偏回归系数为12.048,妊娠晚期贫血Logistic回归分析的理论分类与实际分类的总符合率为894%,结果较好,而妊娠中期血清铁蛋白(SF)、红细胞容积分布宽度(RDW)和平均红细胞容积(MCV)对妊娠晚期贫血的影响均无显着性意义(P>0.05).结论 妊娠中期游离红细胞原卟啉对妊娠晚期贫血有重要作用.     

A comparison of regression calibration, moment reconstruction and imputation for adjusting for covariate measurement error in regression

Regression calibration (RC) is a popular method for estimating regression coefficients when one or more continuous explanatory variables, X, are measured with an error. In this method, the mismeasured covariate, W, is substituted by the expectation E(X|W), based on the assumption that the error in the measurement of X is non-differential. Using simulations, we compare three versions of RC with two other 'substitution' methods, moment reconstruction (MR) and imputation (IM), neither of which rely on the non-differential error assumption. We investigate studies that have an internal calibration sub-study. For RC, we consider (i) the usual version of RC, (ii) RC applied only to the 'marker' information in the calibration study, and (iii) an 'efficient' version (ERC) in which the estimators (i) and (ii) are combined. Our results show that ERC is preferable when there is non-differential measurement error. Under this condition, there are cases where ERC is less efficient than MR or IM, but they rarely occur in epidemiology. We show that the efficiency gain of usual RC and ERC over the other methods can sometimes be dramatic. The usual version of RC carries similar efficiency gains to ERC over MR and IM, but becomes unstable as measurement error becomes large, leading to bias and poor precision. When differential measurement error does pertain, then MR and IM have considerably less bias than RC, but can have much larger variance. We demonstrate our findings with an analysis of dietary fat intake and mortality in a large cohort study.     

解释变量非正交时logistic回归系数的估计

目的；研究解释变量非正交时，ｌｏｇｉｓｔｉｃ回归模型参数的估计。方法：引用线性回归系数主成分估计的思想，提出主成分改进的ｌｏｇｉｓｔｉｃ回归系数的加权最小二乘估计。结果：改进的方法能克服多元共线性引起的一般回归系数加权最小二乘估计方差扩大现象，估计值优于一般加权最小二乘估计。结论：用主成分改进的加权最小二乘估计为基础来筛选变量，从而避免变量间共线关系对变量选择的影响。     

Sensitivity of regression calibration to non‐perfect validation data with application to the Norwegian Women and Cancer Study

Measurement error occurs when we observe error‐prone surrogates, rather than true values. It is common in observational studies and especially so in epidemiology, in nutritional epidemiology in particular. Correcting for measurement error has become common, and regression calibration is the most popular way to account for measurement error in continuous covariates. We consider its use in the context where there are validation data, which are used to calibrate the true values given the observed covariates. We allow for the case that the true value itself may not be observed in the validation data, but instead, a so‐called reference measure is observed. The regression calibration method relies on certain assumptions.This paper examines possible biases in regression calibration estimators when some of these assumptions are violated. More specifically, we allow for the fact that (i) the reference measure may not necessarily be an ‘alloyed gold standard’ (i.e., unbiased) for the true value; (ii) there may be correlated random subject effects contributing to the surrogate and reference measures in the validation data; and (iii) the calibration model itself may not be the same in the validation study as in the main study; that is, it is not transportable. We expand on previous work to provide a general result, which characterizes potential bias in the regression calibration estimators as a result of any combination of the violations aforementioned. We then illustrate some of the general results with data from the Norwegian Women and Cancer Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Regression modeling of combined data from multiple sample surveys

Combined data from multiple sample surveys are often used in population‐based epidemiologic studies. Combining data can be beneficial in that sampling errors are reduced and coverage biases are corrected. Also, it is often necessary in order to use information lacking in one survey but available in another. We propose an estimation equations method for generalized linear models from the combined data. The estimation procedures for logistic regression models and Poisson regression models are developed. An example of estimating the relative risk of death by smoking status is used as an illustration and a simulation study is performed to examine the performance of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

大肠癌相关因素的条件logistic回归分析

目的 探讨与大肠癌发病相关的危险因素,为大肠癌的防治提供科学依据.方法 采用1:1配对病例对照研究方法,对大连市新发200例大肠癌患者及相同数量的对照使用统一设计的调查表进行调查.所获资料根据资料的类型采用χ~2检验、秩和检验或单因素条件logistic回归分析进行单因素分析,然后对初步筛选出的与大肠癌发病关系密切的研究因素,纳入多因素条件logistic回归模型进行多因素分析.结果 多因素条件logistic回归分析结果 显示,便秘史、其他癌症家族史、十年前较多食用腌制食品以及平常爱生闷气为大肠癌的相关危险因素,而十年前较多食用新鲜蔬菜为大肠癌的相关保护因素.单因素分析结果 还显示,居住地周围有排污工厂、熬夜史、腹部放射线检查史、大肠癌家族史、痔疮史、阑尾炎及阑尾手术史、十年前较多食用贝类、煎、炸、烤制食品、动物肝脏、经常感到有压力、不愿和别人沟通、处事态度偏于悲观及与同事相处一般等为大肠癌的危险因素.同时,十年前较多摄入鸡肉、豆及其制品、生蒜、体育锻炼及其频次、十年前人均月收入及生活费用高为大肠癌的保护因素.结论 大连市大肠癌的发病是多因素综合作用的结果 ;便秘史、其他癌症家族史、经常爱生闷气、十年前较少摄入新鲜蔬菜及过多摄入腌制食品等因素与大肠癌的发生的关系值得关注.     

伤害监测病例伤害严重程度影响因素的有序分类Logistic回归分析

[目的]了解伤害监测病例伤害严重程度的影响因素,为下一步工作提供依据。[方法]采用有序分类Logistic回归分析方法,对米易县2006-2007年伤害监测哨点医院报告的14993例伤害病例进行分析。[结果]有序分类Logistic回归分析结果,影响严重伤害（中度、重度伤害）发生的危险因素是伤害原因中的中毒（OR=2．448）,伤害发生地点中的其他（OR=11．032）、工业和建筑场所（OR=2．735）、农场／农田（OR=2．701）、家中（OR=2．610）、公路街道（OR=2．083）;保护因素分别有女性（深=0．843）,中间年龄组5～14岁（OR=0．666）、15～24岁（OR=0．552）、25～59岁（OR=0．650）;与神经系统比较发生于其他部位的伤害是较轻的伤害。[结论]男性、老年人、中毒、自残／自杀是严重伤害发生的最主要危险因素。     

The impact of covariate measurement error on risk prediction

In the development of risk prediction models, predictors are often measured with error. In this paper, we investigate the impact of covariate measurement error on risk prediction. We compare the prediction performance using a costly variable measured without error, along with error‐free covariates, to that of a model based on an inexpensive surrogate along with the error‐free covariates. We consider continuous error‐prone covariates with homoscedastic and heteroscedastic errors, and also a discrete misclassified covariate. Prediction performance is evaluated by the area under the receiver operating characteristic curve (AUC), the Brier score (BS), and the ratio of the observed to the expected number of events (calibration). In an extensive numerical study, we show that (i) the prediction model with the error‐prone covariate is very well calibrated, even when it is mis‐specified; (ii) using the error‐prone covariate instead of the true covariate can reduce the AUC and increase the BS dramatically; (iii) adding an auxiliary variable, which is correlated with the error‐prone covariate but conditionally independent of the outcome given all covariates in the true model, can improve the AUC and BS substantially. We conclude that reducing measurement error in covariates will improve the ensuing risk prediction, unless the association between the error‐free and error‐prone covariates is very high. Finally, we demonstrate how a validation study can be used to assess the effect of mismeasured covariates on risk prediction. These concepts are illustrated in a breast cancer risk prediction model developed in the Nurses' Health Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Inflation of the type I error rate when a continuous confounding variable is categorized in logistic regression analyses

This paper demonstrates an inflation of the type I error rate that occurs when testing the statistical significance of a continuous risk factor after adjusting for a correlated continuous confounding variable that has been divided into a categorical variable. We used Monte Carlo simulation methods to assess the inflation of the type I error rate when testing the statistical significance of a risk factor after adjusting for a continuous confounding variable that has been divided into categories. We found that the inflation of the type I error rate increases with increasing sample size, as the correlation between the risk factor and the confounding variable increases, and with a decrease in the number of categories into which the confounder is divided. Even when the confounder is divided in a five-level categorical variable, the inflation of the type I error rate remained high when both the sample size and the correlation between the risk factor and the confounder were high.