Maximum likelihood estimation in generalized linear models with multiple covariates subject to detection limits

Making Sense of Censored Covariates: Statistical Methods for Studies of Huntington's Disease

The landscape of survival analysis is constantly being revolutionized to answer biomedical challenges, most recently the statistical challenge of censored covariates rather than outcomes. There are many promising strategies to tackle censored covariates, including weighting, imputation, maximum likelihood, and Bayesian methods. Still, this is a relatively fresh area of research, different from the areas of censored outcomes (i.e., survival analysis) or missing covariates. In this review, we discuss the unique statistical challenges encountered when handling censored covariates and provide an in-depth review of existing methods designed to address those challenges. We emphasize each method's relative strengths and weaknesses, providing recommendations to help investigators pinpoint the best approach to handling censored covariates in their data.

Joint modeling approaches for censored predictors due to detection limits with applications to metabolites data

Measures of substance concentration in urine, serum or other biological matrices often have an assay limit of detection. When concentration levels fall below the limit, exact measures cannot be obtained, and thus are left censored. The problem becomes more challenging when the censored data come from heterogeneous populations consisting of exposed and non-exposed subjects. If the censored data come from non-exposed subjects, their measures are always zero and hence censored, forming a latent class governed by a distinct censoring mechanism compared with the exposed subjects. The exposed group's censored measurements are always greater than zero, but less than the detection limit. It is very often that the exposed and non-exposed subjects may have different disease traits or different relationships with outcomes of interest, so we need to disentangle the two different populations for valid inference. In this article, we aim to fill the methodological gaps in the literature by developing a novel joint modeling approach to not only address the censoring issue in predictors, but also untangle different relationships of exposed and non-exposed subjects with the outcome. Simulation studies are performed to assess the numerical performance of our proposed approach when the sample size is small to moderate. The joint modeling approach is also applied to examine associations between plasma metabolites and blood pressure in Bogalusa Heart Study, and identify new metabolites that are highly associated with blood pressure.

Semiparametric analysis of a generalized linear model with multiple covariates subject to detection limits

Studies on the health effects of environmental mixtures face the challenge of limit of detection (LOD) in multiple correlated exposure measurements. Conventional approaches to deal with covariates subject to LOD, including complete-case analysis, substitution methods, and parametric modeling of covariate distribution, are feasible but may result in efficiency loss or bias. With a single covariate subject to LOD, a flexible semiparametric accelerated failure time (AFT) model to accommodate censored measurements has been proposed. We generalize this approach by considering a multivariate AFT model for the multiple correlated covariates subject to LOD and a generalized linear model for the outcome. A two-stage procedure based on semiparametric pseudo-likelihood is proposed for estimating the effects of these covariates on health outcome. Consistency and asymptotic normality of the estimators are derived for an arbitrary fixed dimension of covariates. Simulations studies demonstrate good large sample performance of the proposed methods vs conventional methods in realistic scenarios. We illustrate the practical utility of the proposed method with the LIFECODES birth cohort data, where we compare our approach to existing approaches in an analysis of multiple urinary trace metals in association with oxidative stress in pregnant women.

Urine tungsten and chronic kidney disease in rural Colorado

BACKGROUND

Chronic kidney disease (CKD) is a cause of global morbidity and mortality in agricultural communities. The San Luis Valley (SLV) is a rural agricultural community in southern Colorado with geographic and sociodemographic risk factors for CKD, including a water supply contaminated by heavy metals.

METHODS

We obtained pre-existing sociodemographic, clinical, and urine trace metal data for 1659 subjects from the San Luis Valley Diabetes Study, a prospective cohort study. We assessed prospective associations between urine tungsten (W) and time-to-CKD using accelerated failure time models (n = 1659). Additionally, logistic models were used to assess relationships between urine W and renal injury markers (NGAL, KIM1) using Tobit regression (n = 816), as well as epidemiologically-defined CKD of unknown origin (CKDu) using multiple logistic regression (n = 620).

RESULTS

Elevated urine W was strongly associated with decreased time-to-CKD, even after controlling for hypertension and diabetes. Depending on how CKD was defined, a doubling of urine W was associated with a 27% (95% CI 11%, 46%) to 31% (14%, 51%) higher odds of developing CKD within 5 years. The relationship between urine W and select renal injury markers was not significant, although urine NGAL was modified by diabetes status. Elevated (>95%ile) urinary W was significantly associated with CKDu (OR 5.93, 1.83, 19.21) while adjusting for known CKD risk factors.

CONCLUSIONS

Our data suggest that increased exposure to W is associated with decreased time-to-CKD and may be associated with CKDu. Given persistence of associations after controlling for diabetes and hypertension, W may exert a primary effect on the kidney, although this needs to be evaluated further in future studies.

Estimating Outcome-Exposure Associations when Exposure Biomarker Detection Limits vary Across Batches

Limit of detection (LOD) issues are ubiquitous in exposure assessment. Although there is an extensive literature on modeling exposure data under such imperfect measurement processes, including likelihood-based methods and multiple imputation, the standard practice continues to be naïve single imputation by a constant (e.g., (Equation is included in full-text article.)). In this article, we consider the situation where, due to the practical logistics of data accrual, sampling, and resource constraints, exposure data are analyzed in multiple batches where the LOD and the proportion of censored observations differ across batches. Compounding this problem is the potential for nonrandom assignment of samples to each batch, often driven by enrollment patterns and biosample storage. This issue is particularly important for binary outcome data where batches may have different levels of outcome enrichment. We first consider variants of existing methods to address varying LODs across multiple batches. We then propose a likelihood-based multiple imputation strategy to impute observations that are below the LOD while simultaneously accounting for differential batch assignment. Our simulation study shows that our proposed method has superior estimation properties (i.e., bias, coverage, statistical efficiency) compared to standard alternatives, provided that distributional assumptions are satisfied. Additionally, in most batch assignment configurations, complete-case analysis can be made unbiased by including batch indicator terms in the analysis model, although this strategy is less efficient relative to the proposed method. We illustrate our method by analyzing data from a cohort study in Puerto Rico that is investigating the relation between endocrine disruptor exposures and preterm birth.

Higher soluble CD14 levels are associated with lower visuospatial memory performance in youth with HIV

OBJECTIVE

HIV-associated neurocognitive disorders persist despite early antiretroviral therapy (ART) and optimal viral suppression. We examined the relationship between immunopathogenesis driven by various pathways of immune activation and discrete neurocognitive performance domains in youth with HIV (YWH).

DESIGN

Observational cross-sectional study.

METHODS

YWH, ages 20-28 years, enrolled in Adolescent Medicine Trials Network 071/101 were assessed for biomarkers of macrophage, lymphocyte activation, and vascular inflammation using ELISA/multiplex assays. Standardized neurocognitive tests were performed, and demographically adjusted z-scores were combined to form indices of attention, motor, executive function, verbal, and visuospatial memory. Cross-sectional analysis of the relationship between 18 plasma inflammatory biomarkers and each neurocognitive domain was performed. Linear regression models were fit for each combination of log-transformed biomarker value and neurocognitive domain score, and were adjusted for demographics, socioeconomic status, substance use, depression, CD4 T-cell count, HIV viral load, and ART status.

RESULTS

Study included 128 YWH [mean age 23.8 (SD 1.7) years, 86% men, 68% African American]. Verbal and visuospatial memory domains were most significantly impaired in the cohort (z = -1.59 and -1.0, respectively). Higher sCD14 was associated with impaired visuospatial memory, which remained robust after adjusting for other biomarkers, demographics, and HIV-associated covariates. Among biomarkers of vascular inflammation, sICAM-1 was negatively associated with verbal memory and attention, whereas sVCAM-1 was positively associated with executive function and visuospatial memory. Specific neurocognitive domains were not associated with sCD163, LPS, or CCL2 levels.

CONCLUSION

Impaired visuospatial memory in YWH is associated with immune activation, as reflected by higher sCD14.

Mediation analysis in a case-control study when the mediator is a censored variable

Mediation analysis is an approach for assessing the direct and indirect effects of an initial variable on an outcome through a mediator. In practice, mediation models can involve a censored mediator (eg, a woman's age at menopause). The current research for mediation analysis with a censored mediator focuses on scenarios where outcomes are continuous. However, the outcomes can be binary (eg, type 2 diabetes). Another challenge when analyzing such a mediation model is to use data from a case-control study, which results in biased estimations for the initial variable-mediator association if a standard approach is directly applied. In this study, we propose an approach (denoted as MAC-CC) to analyze the mediation model with a censored mediator given data from a case-control study, based on the semiparametric accelerated failure time model along with a pseudo-likelihood function. We adapted the measures for assessing the indirect and direct effects using counterfactual definitions. We conducted simulation studies to investigate the performance of MAC-CC and compared it to those of the naïve approach and the complete-case approach. MAC-CC accurately estimates the coefficients of different paths, the indirect effects, and the proportions of the total effects mediated. We applied the proposed and existing approaches to the mediation study of genetic variants, a woman's age at menopause, and type 2 diabetes based on a case-control study of type 2 diabetes. Our results indicate that there is no mediating effect from the age at menopause on the association between the genetic variants and type 2 diabetes.

Estimation in generalized linear models under censored covariates with an application to MIREC data

In many biological experiments, certain values of a biomarker are often nondetectable due to low concentrations of an analyte or the limitations of a chemical analysis device, resulting in left-censored values. There is an increasing demand for the analysis of data subject to detection limits in clinical and environmental studies. In this paper, we develop a novel statistical method for the maximum likelihood estimation in generalized linear models with covariates subject to detection limits. Simulations are carried out to study the relative performance of the proposed estimators, as compared to other existing estimators. The proposed method is also applied to a real dataset from the Maternal-Infant Research on Environmental Chemicals cohort study, where we investigate how different chemical mixtures affect the health outcomes of infants and pregnant women.

Linear regression with left-censored covariates and outcome using a pseudolikelihood approach

Environmental toxicology studies often involve sample values that fall below a laboratory procedure's limit of quantification. Such left-censored data give rise to several problems for regression analyses. First, both covariates and outcome may be left censored. Second, the transformed toxicant levels may not be normal but mixtures of normals because of differences in personal characteristics, e.g. exposure history and demographic factors. Third, the outcome and covariates may be linear functions of left-censored variates, such as averages and differences. Fourth, some toxicant levels may be functions of other toxicant levels resulting in a recursive system. In this paper marginal and pseudo-likelihood based methods are proposed for estimation of the means and covariance matrix of variates found in these four settings. Next, linear regression methods are developed allowing outcomes and covariates to be linear combinations of left-censored measures. This is extended to a recursive system of modeling equations. Bootstrap standard errors and confidence intervals are used. Simulation studies demonstrate the proposed methods are accurate for a wide range of study designs and left-censoring probabilities. The proposed methods are illustrated through the analysis of an on-going community-based study of polychlorinated biphenyls, which motivated the proposed methodology.

Maximum Likelihood Estimation in a Semicontinuous Survival Model with Covariates Subject to Detection Limits

Semicontinuous data are common in biological studies, occurring when a variable is continuous over a region but has a point mass at one or more points. In the motivating Genetic and Inflammatory Markers of Sepsis (GenIMS) study, it was of interest to determine how several biomarkers subject to detection limits were related to survival for patients entering the hospital with community acquired pneumonia. While survival times were recorded for all individuals in the study, the primary endpoint of interest was the binary event of 90-day survival, and no patients were lost to follow-up prior to 90 days. In order to use all of the available survival information, we propose a two-part regression model where the probability of surviving to 90 days is modeled using logistic regression and the survival distribution for those experiencing the event prior to this time is modeled with a truncated accelerated failure time model. We assume a series of mixture of normal regression models to model the joint distribution of the censored biomarkers. To estimate the parameters in this model, we suggest a Monte Carlo EM algorithm where multiple imputations are generated for the censored covariates in order to estimate the expectation in the E-step and then weighted maximization is applied to the observed and imputed data in the M-step. We conduct simulations to assess the proposed model and maximization method, and we analyze the GenIMS data set.

Threshold regression to accommodate a censored covariate

In several common study designs, regression modeling is complicated by the presence of censored covariates. Examples of such covariates include maternal age of onset of dementia that may be right censored in an Alzheimer's amyloid imaging study of healthy subjects, metabolite measurements that are subject to limit of detection censoring in a case-control study of cardiovascular disease, and progressive biomarkers whose baseline values are of interest, but are measured post-baseline in longitudinal neuropsychological studies of Alzheimer's disease. We propose threshold regression approaches for linear regression models with a covariate that is subject to random censoring. Threshold regression methods allow for immediate testing of the significance of the effect of a censored covariate. In addition, they provide for unbiased estimation of the regression coefficient of the censored covariate. We derive the asymptotic properties of the resulting estimators under mild regularity conditions. Simulations demonstrate that the proposed estimators have good finite-sample performance, and often offer improved efficiency over existing methods. We also derive a principled method for selection of the threshold. We illustrate the approach in application to an Alzheimer's disease study that investigated brain amyloid levels in older individuals, as measured through positron emission tomography scans, as a function of maternal age of dementia onset, with adjustment for other covariates. We have developed an R package, censCov, for implementation of our method, available at CRAN.

Associations between mixtures of urinary phthalate metabolites with gestational age at delivery: a time to event analysis using summative phthalate risk scores

BACKGROUND

Preterm birth is a significant public health concern and exposure to phthalates has been shown to be associated with an increased odds of preterm birth. Even modest reductions in gestational age at delivery could entail morbid consequences for the neonate and analyzing data with this additional information may be useful. In the present analysis, we consider gestational age at delivery as our outcome of interest and examine associations with multiple phthalates.

METHODS

Women were recruited early in pregnancy as part of a prospective, longitudinal birth cohort at the Brigham and Women's Hospital in Boston, Massachusetts. Urine samples were collected at up to four time points during gestation for urinary phthalate metabolite measurement, and birth outcomes were recorded at delivery. From this population, we selected all 130 cases of preterm birth (< 37 weeks of gestation) as well as 352 random controls. We conducted analysis with both geometric average of the exposure concentrations across the first three visits as well as using repeated measures of the exposure. Two different time to event models were used to examine associations between nine urinary phthalate metabolite concentrations and time to delivery. Two different approaches to constructing a summative phthalate risk score were also considered.

RESULTS

The single-pollutant analysis using a Cox proportional hazards model showed the strongest association with a hazard ratio (HR) of 1.21 (95% confidence interval (CI): 1.09, 1.33) per interquartile range (IQR) change in average log-transformed mono-2-ethyl-5-carboxypentyl phthalate (MECPP) concentration. Using the accelerated failure time model, we observed a 1.19% (95% CI: 0.26, 2.11%) decrease in gestational age in association with an IQR change in average log-transformed MECPP. We next examined associations with an environmental risk score (ERS). The fourth quartile of ERS was significantly associated with a HR of 1.44 (95% CI: 1.19, 1.75) and a reduction of 2.55% (95% CI: 0.76, 4.30%) in time to delivery (in days) compared to the first quartile.

CONCLUSIONS

On average, pregnant women with higher urinary metabolite concentrations of individual phthalates have shorter time to delivery. The strength of the observed associations are amplified with the risk scores when compared to individual pollutants.

Association of adverse birth outcomes with prenatal exposure to vanadium: a population-based cohort study

BACKGROUND

Vanadium, an important pollutant produced from anthropogenic activities, has been suggested to be embryotoxic and fetotoxic in animal studies. However, little is known about its effects on humans. We aimed to assess the association of prenatal exposure to vanadium with the risk of adverse birth outcomes in babies born to women in China.

METHODS

For this population-based cohort study, the Healthy Baby Cohort, women were recruited from three cities in Hubei Province, China. Women included in this analysis were recruited from Wuhan Women and Children Medical Care Center, Wuhan. We measured urinary concentrations of vanadium and other metals simultaneously using inductively coupled plasma mass spectrometry. We used multivariable logistic regressions, with adjustment for potential confounders, to estimate the associations of natural logarithm transformed creatinine-corrected urinary vanadium (Ln-vanadium) concentrations as continuous variables and categorised into quartiles (Q; Q1: ≤0·84 μg/g creatinine, Q2: 0·84-1·40 μg/g creatinine, Q3: 1·40-2·96 μg/g creatinine, Q4: >2·96 μg/g creatinine, with the lowest quartile set as reference) with preterm delivery, early-term delivery, low birthweight, and being small for gestational age. We applied restricted cubic spline models to evaluate the dose-response relationships.

FINDINGS

Data from 7297 women recruited between Sept 22, 2012, and Oct 22, 2014, were included in this study. Urinary Ln-vanadium concentrations showed non-linear dose-response relationships with risk of preterm delivery (S-shaped, p<0·0001) and low birthweight (J-shaped, p=0·0001); the adjusted odds ratios (ORs) for increasing quartiles of urinary vanadium were 1·76 (95% CI 1·05-2·95) for Q2, 3·17 (1·96-5·14) for Q3, and 8·86 (5·66-13·86) for Q4 for preterm delivery, and 2·29 (95% CI 1·08-4·84) for Q2, 3·22 (1·58-6·58) for Q3, and 3·56 (1·79-7·10) for Q4 for low birthweight. Ln-vanadium concentrations were linearly associated with the risk of early-term delivery (linear, p<0·0001) and being small for gestational age (linear, p=0·0027), with adjusted ORs of 1·15 (95% CI 1·10-1·21) for early-term delivery and 1·12 (1·04-1·21) for being small for gestational age per unit increase in Ln-vanadium concentrations.

INTERPRETATION

Our findings reveal a relationship between prenatal exposure to higher levels of vanadium and increased risk of adverse birth outcomes, suggesting that vanadium might be a potential toxic metal for human beings. Further studies are needed to replicate the observed associations and investigate the interaction effects of prenatal exposure to different metals on adverse birth outcomes.

FUNDING

National Key R&D Plan of China, National Natural Science Foundation of China, and Fundamental Research Funds for the Central Universities, Key Laboratory of Environment and Health.

Improved conditional imputation for linear regression with a randomly censored predictor

This article describes a nonparametric conditional imputation analytic method for randomly censored covariates in linear regression. While some existing methods make assumptions about the distribution of covariates or underestimate standard error due to lack of imputation error, the proposed approach is distribution-free and utilizes resampling to correct for variance underestimation. The performance of the novel method is assessed using simulations, and results are contrasted with methods currently used for a limit of detection censored design, including the complete case approach and other nonparametric approaches. Theoretical justifications for the proposed method are provided, and its application is demonstrated through a study of association between lipoprotein cholesterol in offspring and parental history of cardiovascular disease.

Estimation of exposure distribution adjusting for association between exposure level and detection limit

In environmental exposure studies, it is common to observe a portion of exposure measurements to fall below experimentally determined detection limits (DLs). The reverse Kaplan-Meier estimator, which mimics the well-known Kaplan-Meier estimator for right-censored survival data with the scale reversed, has been recommended for estimating the exposure distribution for the data subject to DLs because it does not require any distributional assumption. However, the reverse Kaplan-Meier estimator requires the independence assumption between the exposure level and DL and can lead to biased results when this assumption is violated. We propose a kernel-smoothed nonparametric estimator for the exposure distribution without imposing any independence assumption between the exposure level and DL. We show that the proposed estimator is consistent and asymptotically normal. Simulation studies demonstrate that the proposed estimator performs well in practical situations. A colon cancer study is provided for illustration. Copyright © 2017 John Wiley & Sons, Ltd.

Estimation of indirect effect when the mediator is a censored variable

A mediation model explores the direct and indirect effects of an initial variable ( X) on an outcome variable ( Y) by including a mediator ( M). In many realistic scenarios, investigators observe censored data instead of the complete data. Current research in mediation analysis for censored data focuses mainly on censored outcomes, but not censored mediators. In this study, we proposed a strategy based on the accelerated failure time model and a multiple imputation approach. We adapted a measure of the indirect effect for the mediation model with a censored mediator, which can assess the indirect effect at both the group and individual levels. Based on simulation, we established the bias in the estimations of different paths (i.e. the effects of X on M [ a], of M on Y [ b] and of X on Y given mediator M [ c']) and indirect effects when analyzing the data using the existing approaches, including a naïve approach implemented in software such as Mplus, complete-case analysis, and the Tobit mediation model. We conducted simulation studies to investigate the performance of the proposed strategy compared to that of the existing approaches. The proposed strategy accurately estimates the coefficients of different paths, indirect effects and percentages of the total effects mediated. We applied these mediation approaches to the study of SNPs, age at menopause and fasting glucose levels. Our results indicate that there is no indirect effect of association between SNPs and fasting glucose level that is mediated through the age at menopause.

Bivariate Left-Censored Bayesian Model for Predicting Exposure: Preliminary Analysis of Worker Exposure during the Deepwater Horizon Oil Spill

In April 2010, the Deepwater Horizon oil rig caught fire and exploded, releasing almost 5 million barrels of oil into the Gulf of Mexico over the ensuing 3 months. Thousands of oil spill workers participated in the spill response and clean-up efforts. The GuLF STUDY being conducted by the National Institute of Environmental Health Sciences is an epidemiological study to investigate potential adverse health effects among these oil spill clean-up workers. Many volatile chemicals were released from the oil into the air, including total hydrocarbons (THC), which is a composite of the volatile components of oil including benzene, toluene, ethylbenzene, xylene, and hexane (BTEXH). Our goal is to estimate exposure levels to these toxic chemicals for groups of oil spill workers in the study (hereafter called exposure groups, EGs) with likely comparable exposure distributions. A large number of air measurements were collected, but many EGs are characterized by datasets with a large percentage of censored measurements (below the analytic methods' limits of detection) and/or a limited number of measurements. We use THC for which there was less censoring to develop predictive linear models for specific BTEXH air exposures with higher degrees of censoring. We present a novel Bayesian hierarchical linear model that allows us to predict, for different EGs simultaneously, exposure levels of a second chemical while accounting for censoring in both THC and the chemical of interest. We illustrate the methodology by estimating exposure levels for several EGs on the Development Driller III, a rig vessel charged with drilling one of the relief wells. The model provided credible estimates in this example for geometric means, arithmetic means, variances, correlations, and regression coefficients for each group. This approach should be considered when estimating exposures in situations when multiple chemicals are correlated and have varying degrees of censoring.

Maternal and fetal exposure to cadmium, lead, manganese and mercury: The MIREC study

Given the susceptibility of the fetus to toxicants, it is important to estimate their exposure. Approximately 2000 pregnant women were recruited in 2008-2011 from 10 cities across Canada. Cd, Pb, Mn and total Hg were measured in maternal blood from the 1st and 3rd trimesters, umbilical cord blood, and infant meconium. Nutrient intakes of vitamin D, iron, and calcium (Ca) were assessed using a food frequency questionnaire and a dietary supplement questionnaire. Median concentrations in 1st trimester maternal blood (n = 1938) were 0.20, 8.79 and 0.70 μg/L for Cd, Mn and Hg, respectively, and 0.60 μg/dL for Pb. While the median difference between the paired 1st and 3rd trimester concentrations of Cd was 0, there was a significant decrease in Pb (0.04 μg/dL) and Hg (0.12 μg/L) and an increase in Mn (3.30 μg/L) concentrations over the course of the pregnancy. While Cd was rarely detected in cord blood (19%) or meconium (3%), median Pb (0.77 μg/dL), Mn (31.87 μg/L) and Hg (0.80 μg/L) concentrations in cord blood were significantly higher than in maternal blood. Significant negative associations were observed between estimated Ca intake and maternal Cd, Pb, Mn and Hg, as well as cord blood Pb. Vitamin D intake was associated with lower maternal Cd, Pb, and Mn as well as Pb in cord blood. Even at current metal exposure levels, increasing dietary Ca and vitamin D intake during pregnancy may be associated with lower maternal blood Pb and Cd concentrations and lower Pb in cord blood.

Linear Regression with a Randomly Censored Covariate: Application to an Alzheimer's Study

The association between maternal age of onset of dementia and amyloid deposition (measured by in vivo positron emission tomography (PET) imaging) in cognitively normal older offspring is of interest. In a regression model for amyloid, special methods are required due to the random right censoring of the covariate of maternal age of onset of dementia. Prior literature has proposed methods to address the problem of censoring due to assay limit of detection, but not random censoring. We propose imputation methods and a survival regression method that do not require parametric assumptions about the distribution of the censored covariate. Existing imputation methods address missing covariates, but not right censored covariates. In simulation studies, we compare these methods to the simple, but inefficient complete case analysis, and to thresholding approaches. We apply the methods to the Alzheimer's study.

Bayesian inference for generalized linear mixed models with predictors subject to detection limits: an approach that leverages information from auxiliary variables

This paper is motivated from a retrospective study of the impact of vitamin D deficiency on the clinical outcomes for critically ill patients in multi-center critical care units. The primary predictors of interest, vitamin D2 and D3 levels, are censored at a known detection limit. Within the context of generalized linear mixed models, we investigate statistical methods to handle multiple censored predictors in the presence of auxiliary variables. A Bayesian joint modeling approach is proposed to fit the complex heterogeneous multi-center data, in which the data information is fully used to estimate parameters of interest. Efficient Monte Carlo Markov chain algorithms are specifically developed depending on the nature of the response. Simulation studies demonstrate the outperformance of the proposed Bayesian approach over other existing methods. An application to the data set from the vitamin D deficiency study is presented. Possible extensions of the method regarding the absence of auxiliary variables, semiparametric models, as well as the type of censoring are also discussed.

The bioaccessibility of soil-based mercury as determined by physiological based extraction tests and human biomonitoring in children

Environmental contaminants associated with soil particles are generally less bioavailable than contaminants associated with other exposure media where chemicals are often found in more soluble forms. In vitro methods, such as Physiological Based Extraction Tests (PBET), can provide estimates of bioaccessibility for soil-based contaminants. The results of these tests can be used to predict exposure to contaminants from soil ingestion pathways within human health risk assessment (HHRA). In the current investigation, an HHRA was conducted to examine the risks associated with elevated concentrations of mercury in soils in the northern Canadian smelter community of Flin Flon, Manitoba. A PBET was completed for residential soils and indicated mean bioaccessibilities of 1.2% and 3.0% for total mercury using gastric phase and gastric+intestinal phase methodologies, respectively. However, as many regulators only allow for the consideration of in vitro results for lead and arsenic in the HHRA process, in vitro bioaccessibility results for mercury were not utilized in the current HHRA. Based on the need to assume 100% bioaccessibility for inorganic mercury in soil, results from the HHRA indicated the need for further assessment of exposure and risk. A biomonitoring study was undertaken for children between 2 and 15 years of age in the community to examine urinary inorganic mercury concentrations. Overall, 375 children provided valid urine samples for analysis. Approximately 50% of urine samples had concentrations of urinary inorganic mercury below the limit of detection (0.1 μg/L), with an average creatinine adjusted concentration of 0.11 μg/g. Despite high variability in mercury soil concentrations within sub-communities, soil concentrations did not appear to influence urinary mercury concentrations. The results of the current investigation indicate that mercury bioaccessibility in residential soils in the Flin Flon area was likely limited and that HHRA estimates would have been better approximated through inclusion of the in vitro study results.

Frailty models for pneumonia to death with a left-censored covariate

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson's numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin-10 was subject to LOD.

Female exposure to phenols and phthalates and time to pregnancy: the Maternal-Infant Research on Environmental Chemicals (MIREC) Study

OBJECTIVE

To assess the potential effect of bisphenol A (BPA), triclosan (TCS), and phthalates on women's fecundity, as measured by time to pregnancy (TTP).

DESIGN

Pregnancy-based retrospective TTP study.

SETTING

Not applicable.

PATIENT(S)

A total of 2,001 women during the first trimester of pregnancy recruited between 2008 and 2011 (the Maternal-Infant Research on Environmental Chemicals (MIREC) Study), with 1,742 women included in the BPA analysis, 1,699 in the TCS analysis, and 1,597 in the phthalates analysis.

INTERVENTION(S)

None.

MAIN OUTCOME MEASURE(S)

Fecundability odds ratios (FORs) estimated using the Cox model modified for discrete time data.

RESULT(S)

The BPA concentrations were not statistically significantly associated with diminished fecundity either in crude or adjusted models. Women in the highest quartile of TCS (>72 ng/mL) had evidence of decreased fecundity (FOR 0.84; 95% confidence interval, 0.72-0.97) compared with the three lower quartiles as the reference group. Exposure to phthalates was suggestive of a shorter TTP, as indicated by FORs greater than 1, although the 95% confidence interval always included 1.

CONCLUSION(S)

Elevated TCS exposure may be associated with diminished fecundity. BPA and phthalates showed no negative impact; on the contrary, some phthalates might be associated with a shorter time to pregnancy. A major limitation of the study was that only one measurement of exposure was available for each woman after conception. Further research is necessary to test these findings.

Phthalate and bisphenol A exposure among pregnant women in Canada--results from the MIREC study

Bisphenol A (BPA) and phthalates are endocrine disruptors possibly linked to adverse reproductive and neurodevelopmental outcomes. These chemicals have commonly been measured in urine in population surveys; however, such data are limited for large populations of pregnant women, especially for the critical first trimester of pregnancy. The aim of the study was to measure BPA and phthalate metabolites in first trimester urine samples collected in a large national-scale pregnancy cohort study and to identify major predictors of exposure. Approximately 2000 women were recruited in the first trimester of pregnancy from ten sites across Canada. A questionnaire was administered to obtain demographic and socio-economic data on participants and a spot urine sample was collected and analyzed for total BPA (GC-MS/MS) and 11 phthalate metabolites (LC-MS/MS). The geometric mean (GM) maternal urinary concentration of total BPA, uncorrected for specific gravity, was 0.80 (95% CI 0.76-0.85) μg/L. Almost 88% of the women had detectable urinary concentrations of BPA. An analysis of urinary concentrations of BPA by maternal characteristics with specific gravity as a covariate in the linear model showed that the geometric mean concentrations: (1) decreased with increasing maternal age, (2) were higher in current smokers or women who quit during pregnancy compared to never smokers, and (3) tended to be higher in women who provided a fasting urine sample and who were born in Canada, and had lower incomes and education. Several of the phthalate metabolites analyzed were not prevalent in this population (MCHP, MMP, MiNP, MOP), with percentages detectable at less than 15%. The phthalate metabolites with the highest measured concentrations were MEP (GM: 32.02 μg/L) and MnBP (GM: 11.59 μg/L). MBzP urinary concentrations decreased with maternal age but did not differ by time of urine collection; whereas the DEHP metabolites tended to be higher in older women and when the urine was collected later in the day. This study provides the first biomonitoring results for the largest population of pregnant women sampled in the first trimester of pregnancy. The results indicate that exposure among this population of pregnant women to these chemicals is comparable to or even lower than that observed in a Canadian national population-based survey.

Statistical Methods for Generalized Linear Models with Covariates Subject to Detection Limits

Censored observations are a common occurrence in biomedical data sets. Although a large amount of research has been devoted to estimation and inference for data with censored responses, very little research has focused on proper statistical procedures when predictors are censored. In this paper, we consider statistical methods for dealing with multiple predictors subject to detection limits within the context of generalized linear models. We investigate and adapt several conventional methods and develop a new multiple imputation approach for analyzing data sets with predictors censored due to detection limits. We establish the consistency and asymptotic normality of the proposed multiple imputation estimator and suggest a computationally simple and consistent variance estimator. We also demonstrate that the conditional mean imputation method often leads to inconsistent estimates in generalized linear models, while several other methods are either computationally intensive or lead to parameter estimates that are biased or more variable compared to the proposed multiple imputation estimator. In an extensive simulation study, we assess the bias and variability of different approaches within the context of a logistic regression model and compare variance estimation methods for the proposed multiple imputation estimator. Lastly, we apply several methods to analyze the data set from a recently-conducted GenIMS study.

A Parametric Survival Model When a Covariate is Subject to Left-Censoring

PROBLEM STATEMENT

Modeling survival data with a set of covariates usually assumes that the values of the covariates are fully observed. However, in a variety of applications, some values of a covariate may be left-censored due to inadequate instrument sensitivity to quantify the biospecimen. When data are left-censored, the true values are missing but are known to be smaller than the detection limit. The most commonly used ad-hoc method to deal with nondetect values is to substitute the nondetect values by the detection limit. Such ad-hoc analysis of survival data with an explanatory variable subject to left-censoring may provide biased and inefficient estimators of hazard ratios and survivor functions.

METHOD

We consider a parametric proportional hazards model to analyze time-to-event data. We propose a likelihood method for the estimation and inference of model parameters. In this likelihood approach, instead of replacing the nondetect values by the detection limit, we adopt a numerical integration technique to evaluate the observed data likelihood in the presence of a left-censored covariate. Monte Carlo simulations were used to demonstrate various properties of the proposed regression estimators including the consistency and efficiency.

RESULTS

The simulation study shows that the proposed likelihood approach provides approximately unbiased estimators of the model parameters. The proposed method also provides estimators that are more efficient than those obtained under the ad-hoc method. Also, unlike the ad-hoc estimators, the coverage probabilities of the proposed estimators are at their nominal level. Analysis of a large cohort study, genetic and inflammatory marker of sepsis study, shows discernibly different results based on the proposed method.

CONCLUSION

Naive use of detection limit in a parametric survival model may provide biased and inefficient estimators of hazard ratios and survivor functions. The proposed likelihood approach provides approximately unbiased and efficient estimators of hazard ratios and survivor functions.