Left-censored Values Research Articles

Geostatistical interpolation of streambed hydrologic attributes with addition of left censored data and anisotropy

Spatial geostatistical interpolation of point measurements of streambed attributes in the hyporheic zone may be constrained by the streambed anisotropy, and data density and spatial distribution may significantly impact the results. Spatial clustering and low spatial data density can be caused by bedrock outcropping at the streambed limiting installation of in-stream piezometers. This study examines parameter error variability of the geostatistical interpolation using anisotropic interpolation methods and increasing the data density by adding left censored values (i.e., data below measurement limit) to locations where measurements were limited by exposed bedrock lining the streambed. The reduction in relative standard error of the interpolation was determined for the spatial distributions of streambed attributes including hydraulic conductivity, seepage flux, and mercury solute flux measured in two different years along a study reach in East Fork Poplar Creek, Tennessee, USA. Two methods to impute the left censored values were compared including the conventional half the detection limit substitution method, and the Stochastic Approximation of Expectation-Maximization (SAEM) algorithm, which both had comparable results. Imputing left censored data increased the data density to recommended ranges, reduced data clustering, increased the spatial dependence for some attributes, and reduced the standard error for each of the three attributes. For the reach considered herein, addition of the left censored values resulted in a larger error reduction than the consideration of anisotropy within the interpolation, which confirms the benefit of data addition to increase data density within data-limited river corridors.

A Bayesian hierarchical approach to account for left-censored and missing radiation doses prone to classical measurement error when analyzing lung cancer mortality due to γ-ray exposure in the French cohort of uranium miners.

Epidemiological data on cohorts of occupationally exposed uranium miners are currently used to assess health risks associated with chronic exposure to low doses of ionizing radiation. Nevertheless, exposure uncertainty is ubiquitous and questions the validity of statistical inference in these cohorts. This paper highlights the flexibility and relevance of the Bayesian hierarchical approach to account for both missing and left-censored (i.e. only known to be lower than a fixed detection limit) radiation doses that are prone to measurement error, when estimating radiation-related risks. Up to the authors' knowledge, this is the first time these three sources of uncertainty are dealt with simultaneously in radiation epidemiology. To illustrate the issue, this paper focuses on the specific problem of accounting for these three sources of uncertainty when estimating the association between occupational exposure to low levels of γ-radiation and lung cancer mortality in the post-55 sub-cohort of French uranium miners. The impact of these three sources of dose uncertainty is of marginal importance when estimating the risk of death by lung cancer among French uranium miners. The corrected excess hazard ratio (EHR) is 0.81 per 100mSv (95% credible interval: [0.28; 1.75]). Interestingly, even if the 95% credible interval of the corrected EHR is wider than the uncorrected one, a statistically significant positive association remains between γ-ray exposure and the risk of death by lung cancer, after accounting for dose uncertainty. Sensitivity analyses show that the results obtained are robust to different assumptions. Because of its flexible and modular nature, the Bayesian hierarchical models proposed in this work could be easily extended to account for high proportions of missing and left-censored dose values or exposure data, prone to more complex patterns of measurement error.

Modelling biogenic amines in fish meat in Central Europe using censored distributions

The aim of the study is to model concentrations of selected biogenic amines in various fish species (Atlantic salmon, Atlantic cod, striped catfish) bought in retail stores in Central Europe. Since the data contains non-detectable values, statistical methods for left-censored values from the exponential and Weibull distributions are applied and used to evaluate and compare the amount of biogenic amines in fish samples. Moreover, a risk of exceeding certain limits of biogenic amine concentrations to protect human health is determined. There are relatively high concentrations of putrescine, cadaverine and histamine in almost all fish species. Moreover, there was a significant difference in mean concentrations (distributions of concentrations, respectively) of histamine, tyramine and spermidine among the species.

Multivariate Statistical Analysis of Trace Elements in Pyrite: Prediction, Bias and Artefacts in Defining Mineral Signatures

Pyrite is the most common sulphide in a wide range of ore deposits and well known to host numerous trace elements, with implications for recovery of valuable metals and for generation of clean concentrates. Trace element signatures of pyrite are also widely used to understand ore-forming processes. Pyrite is an important component of the Olympic Dam Cu–U–Au–Ag orebody, South Australia. Using a multivariate statistical approach applied to a large trace element dataset derived from analysis of random pyrite grains, trace element signatures in Olympic Dam pyrite are assessed. Pyrite is characterised by: (i) a Ag–Bi–Pb signature predicting inclusions of tellurides (as PC1); and (ii) highly variable Co–Ni ratios likely representing an oscillatory zonation pattern in pyrite (as PC2). Pyrite is a major host for As, Co and probably also Ni. These three elements do not correlate well at the grain-scale, indicating high variability in zonation patterns. Arsenic is not, however, a good predictor for invisible Au at Olympic Dam. Most pyrites contain only negligible Au, suggesting that invisible gold in pyrite is not commonplace within the deposit. A minority of pyrite grains analysed do, however, contain Au which correlates with Ag, Bi and Te. The results are interpreted to reflect not only primary patterns but also the effects of multi-stage overprinting, including cycles of partial replacement and recrystallisation. The latter may have caused element release from the pyrite lattice and entrapment as mineral inclusions, as widely observed for other ore and gangue minerals within the deposit. Results also show the critical impact on predictive interpretations made from statistical analysis of large datasets containing a large percentage of left-censored values (i.e., those falling below the minimum limits of detection). The treatment of such values in large datasets is critical as the number of these values impacts on the cluster results. Trimming of datasets to eliminate artefacts introduced by left-censored data should be performed with caution lest bias be unintentionally introduced. The practice may, however, reveal meaningful correlations that might be diluted using the complete dataset.

A generalized Bayesian nonlinear mixed-effects regression model for zero-inflated longitudinal count data in tuberculosis trials.

In this paper, we investigate Bayesian generalized nonlinear mixed-effects (NLME) regression models for zero-inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis(TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero-inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log-transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti-TB drugs. We therefore recommend that zero-inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.

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.

New approaches for censored longitudinal data in joint modelling of longitudinal and survival data, with application to HIV vaccine studies.

In HIV vaccine studies, longitudinal immune response biomarker data are often left-censored due to lower limits of quantification of the employed immunological assays. The censoring information is important for predicting HIV infection, the failure event of interest. We propose two approaches to addressing left censoring in longitudinal data: one that makes no distributional assumptions for the censored data-treating left censored values as a "point mass" subgroup-and the other makes a distributional assumption for a subset of the censored data but not for the remaining subset. We develop these two approaches to handling censoring for joint modelling of longitudinal and survival data via a Cox proportional hazards model fit by h-likelihood. We evaluate the new methods via simulation and analyze an HIV vaccine trial data set, finding that longitudinal characteristics of the immune response biomarkers are highly associated with the risk of HIV infection.

GSimp: A Gibbs sampler based left-censored missing value imputation approach for metabolomics studies.

Left-censored missing values commonly exist in targeted metabolomics datasets and can be considered as missing not at random (MNAR). Improper data processing procedures for missing values will cause adverse impacts on subsequent statistical analyses. However, few imputation methods have been developed and applied to the situation of MNAR in the field of metabolomics. Thus, a practical left-censored missing value imputation method is urgently needed. We developed an iterative Gibbs sampler based left-censored missing value imputation approach (GSimp). We compared GSimp with other three imputation methods on two real-world targeted metabolomics datasets and one simulation dataset using our imputation evaluation pipeline. The results show that GSimp outperforms other imputation methods in terms of imputation accuracy, observation distribution, univariate and multivariate analyses, and statistical sensitivity. Additionally, a parallel version of GSimp was developed for dealing with large scale metabolomics datasets. The R code for GSimp, evaluation pipeline, tutorial, real-world and simulated targeted metabolomics datasets are available at: https://github.com/WandeRum/GSimp.

Dutch Tariff for the Five-Level Version of EQ-5D

BackgroundIn 2009, a new version of the EuroQol five-dimensional questionnaire (EQ-5D) was introduced with five rather than three answer levels per dimension. This instrument is known as the EQ-5D-5L. To make the EQ-5D-5L suitable for use in economic evaluations, societal values need to be attached to all 3125 health states. ObjectivesTo derive a Dutch tariff for the EQ-5D-5L. MethodsHealth state values were elicited during face-to-face interviews in a general population sample stratified for age, sex, and education, using composite time trade-off (cTTO) and a discrete choice experiment (DCE). Data were modeled using ordinary least squares and tobit regression (for cTTO) and a multinomial conditional logit model (for DCE). Model performance was evaluated on the basis of internal consistency, parsimony, goodness of fit, handling of left-censored values, and theoretical considerations. ResultsA representative sample (N = 1003) of the Dutch population participated in the valuation study. Data of 979 and 992 respondents were included in the analysis of the cTTO and the DCE, respectively. The cTTO data were left-censored at −1. The tobit model was considered the preferred model for the tariff on the basis of its handling of the censored nature of the data, which was confirmed through comparison with the DCE data. The predicted values for the EQ-5D-5L ranged from −0.446 to 1. ConclusionsThis study established a Dutch tariff for the EQ-5D-5L on the basis of cTTO. The values represent the preferences of the Dutch population. The tariff can be used to estimate the impact of health care interventions on quality of life, for example, in context of economic evaluations.

In response: some species sensitivity distribution statistics revisited--a governmental perspective.

In response: some species sensitivity distribution statistics revisited--a governmental perspective.

Covariance matrix estimation for left-censored data

Multivariate methods often rely on a sample covariance matrix. The conventional estimators of a covariance matrix require complete data vectors on all subjects—an assumption that can frequently not be met. For example, in many fields of life sciences that are utilizing modern measuring technology, such as mass spectrometry, left-censored values caused by denoising the data are a commonplace phenomena. Left-censored values are low-level concentrations that are considered too imprecise to be reported as a single number but known to exist somewhere between zero and the laboratory’s lower limit of detection. Maximum likelihood-based covariance matrix estimators that allow the presence of the left-censored values without substituting them with a constant or ignoring them completely are considered. The presented estimators efficiently use all the information available and thus, based on simulation studies, produce the least biased estimates compared to often used competing estimators. As the genuine maximum likelihood estimate can be solved fast only in low dimensions, it is suggested to estimate the covariance matrix element-wise and then adjust the resulting covariance matrix to achieve positive semi-definiteness. It is shown that the new approach succeeds in decreasing the computation times substantially and still produces accurate estimates. Finally, as an example, a left-censored data set of toxic chemicals is explored.

Differential network analysis with multiply imputed lipidomic data.

The importance of lipids for cell function and health has been widely recognized, e.g., a disorder in the lipid composition of cells has been related to atherosclerosis caused cardiovascular disease (CVD). Lipidomics analyses are characterized by large yet not a huge number of mutually correlated variables measured and their associations to outcomes are potentially of a complex nature. Differential network analysis provides a formal statistical method capable of inferential analysis to examine differences in network structures of the lipids under two biological conditions. It also guides us to identify potential relationships requiring further biological investigation. We provide a recipe to conduct permutation test on association scores resulted from partial least square regression with multiple imputed lipidomic data from the LUdwigshafen RIsk and Cardiovascular Health (LURIC) study, particularly paying attention to the left-censored missing values typical for a wide range of data sets in life sciences. Left-censored missing values are low-level concentrations that are known to exist somewhere between zero and a lower limit of quantification. To make full use of the LURIC data with the missing values, we utilize state of the art multiple imputation techniques and propose solutions to the challenges that incomplete data sets bring to differential network analysis. The customized network analysis helps us to understand the complexities of the underlying biological processes by identifying lipids and lipid classes that interact with each other, and by recognizing the most important differentially expressed lipids between two subgroups of coronary artery disease (CAD) patients, the patients that had a fatal CVD event and the ones who remained stable during two year follow-up.

Pseudo-likelihood Estimation of Multivariate Normal Parameters in the Presence of Left-Censored Data

Environmental data often include left-censored values reported to be less than some limit of detection (LOD). While simple imputation of a specific value such as LOD/2 is common, maximum likelihood methods accounting for censoring provide alternate ways of analyzing such data. Concentration levels of contaminants in water, for example, are typically modeled with normal or lognormal distributions. Corresponding maximum likelihood estimates (MLEs) of means and variances in univariate analyses can be obtained from standard software packages; however, multivariate analyses may be more appropriate when multiple measurements come from the same entity. For example, measures of several dissolved trace metals may be derived from freshwater stream samples. In less-polluted areas, one or more of these measures fall below the LOD. An index of overall contamination may be formed as a linear function of these measures. The desire to estimate this index led to the need to estimate the parameters in the presence of nondetects, which led to our proposed method. We propose a pseudo-likelihood method utilizing pairs of variables that provides MLEs of mean and unstructured covariance parameters corresponding to a multivariate normal or lognormal distribution in the presence of left-censored data. In conducting hypothesis tests and estimating functions of MLEs with standard errors, we apply this method to trace metal concentration data collected from freshwater streams across Virginia.

Assessing the agreement of biomarker data in the presence of left-censoring.

BackgroundIn many clinical biomarker studies, Lin’s concordance correlation coefficient (CCC) is commonly used to assess the level of agreement of a biomarker measured under two different conditions. However, measurement of a specific biomarker typically cannot provide accurate numerical values below the lower limit of detection (LLD) of the assay, which results in left-censored data. Most researchers discard the data below the LLD or apply simple data imputation methods in the presence of left-censored data, such as replacing values below the LLD with a fixed number less than or equal to the LLD. This is not statistically optimal, because it often leads to biased estimates and overestimates the precision.MethodsWe describe a simple method using a bivariate normal distribution in this situation and apply SAS statistical software to arrive at the maximum likelihood (ML) estimate of the parameters and construct the estimate of the CCC. We conduct a computer simulation study to investigate the statistical properties of the ML method versus the data deletion and simple data imputation method. We also contrast the methods with real data using two urine biomarkers, Interleukin 18 and Cystatin C.ResultsThe computer simulation studies confirm that the ML procedure is superior to the data deletion and simple data imputation procedures. In all of the simulated scenarios, the ML method yields the smallest relative bias and the highest percentage of the 95% confidence intervals that include the true value of the CCC. In the first simulation scenario (sample size of 100 paired data points, 25% left-censoring for both members of the pair, true CCC of 0.238), the relative bias is −1.43% for the ML method, −40.97% for the data deletion method, and it ranges between −12.94% and −21.72% for the simple data imputation methods. Similarly, when the left-censoring for one of the members of the data pairs increases from 25% to 40%, the relative bias displays the same pattern for all methods.ConclusionsWhen estimating the CCC from paired biomarker data in the presence of left-censored values, the ML method works better than data deletion and simple data imputation methods.

A Two-Step Multiple Imputation for Analysis of Repeated Measures With Left-Censored and Missing Data

Left-censored data can occur in many clinical trials in which laboratory measures are obtained with a quantitative analysis assay. Some common examples are longitudinal viral load measures in HIV drug trials and antibody measures in vaccine studies. Conventionally, an analysis of covariance model or constraint longitudinal data analysis model may be applied in the analysis of log-transformed responses, with the left-censored values replaced with the detection limit or half of the limit. When the proportion of left censoring is moderate to large, this single ad hoc imputation method may lead to bias in parameter estimates and inappropriate alpha levels for statistical tests. In this article, we propose a two-step multiple imputation (MI) approach to deal with the censored and missing data. The proposed two-step MI can be implemented using common software such as SAS procedures. Simulations are conducted under various scenarios to evaluate the performance of this method compared with the conventional methods. Applications to real clinical trials are also given to illustrate the potential impact and benefit of the proposed method. Supplementary materials for this article are available online.

Linear Regression With an Independent Variable Subject to a Detection Limit

Linear regression with a left-censored independent variable X due to limit of detection (LOD) was recently considered by 2 groups of researchers: Richardson and Ciampi (Am J Epidemiol. 2003;157:355-363), and Schisterman et al (Am J Epidemiol. 2006;163:374-383). Both groups obtained consistent estimators for the regression slopes by replacing left-censored X with a constant, that is, the expectation of X given X below LOD E(X|X<LOD) in the former group and the sample mean of X given X above LOD in the latter. Schisterman et al argued that their approach would be a better choice because the sample mean of X given X above LOD is available, whereas E(X|X<LOD) is unknown. Other substitution methods, such as replacing the left-censored values with LOD, or LOD/2,have been extensively used in the literature. Simulations were conducted to compare the performance under 2 scenarios in which the independent variable is normally and not normally distributed. Recommendations are given based on theoretical and simulation results. These recommendations are illustrated with one case study.

Estimating Population Distributions When Some Data Are Below a Limit of Detection by Using a Reverse Kaplan-Meier Estimator

Data with some values below a limit of detection (LOD) can be analyzed using methods of survival analysis for left-censored data. The reverse Kaplan-Meier (KM) estimator provides an effective method for estimating the distribution function and thus population percentiles for such data. Although developed in the 1970s and strongly advocated since then, it remains rarely used, partly due to limited software availability. In this paper, the reverse KM estimator is described and is illustrated using serum dioxin data from the University of Michigan Dioxin Exposure Study (UMDES) and the National Health and Nutrition Examination Survey (NHANES). Percentile estimates for left-censored data using the reverse KM estimator are compared with replacing values below the LOD with the LOD/2 or LOD/ radical2. When some LODs are in the upper range of the complete values, and/or the percent censored is high, the different methods can yield quite different percentile estimates. The reverse KM estimator, which is the nonparametric maximum likelihood estimator, is the preferred method. Software options are discussed: The reverse KM can be calculated using software for the KM estimator. The JMP and SAS (SAS Institute, Cary, NC) and Minitab (Minitab, Inc, State College, PA), software packages calculate the reverse KM directly using their Turnbull estimator routines. The reverse KM estimator is recommended for estimation of the distribution function and population percentiles in preference to commonly used methods such as substituting LOD/2 or LOD/ radical2 for values below the LOD, assuming a known parametric distribution, or using imputation to replace the left-censored values.

Nonparametric predictive pairwise comparison for real-valued data with terminated tails

This paper presents a statistical method for comparison of two groups of real-valued data, based on nonparametric predictive inference (NPI), with the tails of the data possibly terminated, leading to small values being left-censored and large values being right-censored. Such tails termination can occur due to several reasons, including limits of detection, consideration of outliers, and specific designs of experiments. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. We present NPI lower and upper probabilities for the event that the value of a future observation from one group is less than the value of a future observation from the other group, and we discuss several special cases that relate to well-known statistical problems.

A Bayesian approach estimating treatment effects on biomarkers containing zeros with detection limits

Often in randomized clinical trials and observational studies in occupational and environmental health, a non-negative continuously distributed response variable denoting some metabolites of environmental toxicants is measured in treatment and control groups. When observations occur in both unexposed and exposed subjects, the biomarker measurement can be bimodally distributed with an extra spike at zero reflecting those unexposed. In the presence of left censoring due to values falling below biomarker assay detection limits, those unexposed with true zeros are indistinguishable from those exposed with left-censored values. Since interventions usually do not enhance or eliminate exposure, they do not have any impact on those unexposed. Thus, only the subset of individuals who are exposed should be used to make comparisons to estimate the effect of interventions. In this article, we present Bayesian approaches using non-standard mixture distributions to account for true zeros. The performance of the proposed Bayesian methods is compared with the maximum likelihood methods presented in Chu et al. (Stat. Med. 2005; 24:2053-2067) through simulation studies and a randomized chemoprevention trial conducted in Qidong, People's Republic of China.

Estimation of dynamical model parameters taking into account undetectable marker values

BackgroundMathematical models are widely used for studying the dynamic of infectious agents such as hepatitis C virus (HCV). Most often, model parameters are estimated using standard least-square procedures for each individual. Hierarchical models have been proposed in such applications. However, another issue is the left-censoring (undetectable values) of plasma viral load due to the lack of sensitivity of assays used for quantification. A method is proposed to take into account left-censored values for estimating parameters of non linear mixed models and its impact is demonstrated through a simulation study and an actual clinical trial of anti-HCV drugs.MethodsThe method consists in a full likelihood approach distinguishing the contribution of observed and left-censored measurements assuming a lognormal distribution of the outcome. Parameters of analytical solution of system of differential equations taking into account left-censoring are estimated using standard software.ResultsA simulation study with only 14% of measurements being left-censored showed that model parameters were largely biased (from -55% to +133% according to the parameter) with the exception of the estimate of initial outcome value when left-censored viral load values are replaced by the value of the threshold. When left-censoring was taken into account, the relative bias on fixed effects was equal or less than 2%. Then, parameters were estimated using the 100 measurements of HCV RNA available (with 12% of left-censored values) during the first 4 weeks following treatment initiation in the 17 patients included in the trial. Differences between estimates according to the method used were clinically significant, particularly on the death rate of infected cells. With the crude approach the estimate was 0.13 day-1 (95% confidence interval [CI]: 0.11; 0.17) compared to 0.19 day-1 (CI: 0.14; 0.26) when taking into account left-censoring. The relative differences between estimates of individual treatment efficacy according to the method used varied from 0.001% to 37%.ConclusionWe proposed a method that gives unbiased estimates if the assumed distribution is correct (e.g. lognormal) and that is easy to use with standard software.