Statistical problems in environmental research

Growing concern about the health effects of exposure to pollutants and other chemicals in the environment has stimulated new research to detect and quantify environmental hazards. This research has generated many interesting and challenging methodological problems for statisticians. One type of statistical research develops new methods for the design and analysis of individual studies. Because current research of this type is too diverse to summarize in a single article, we discuss current work in two areas of application: the carcinogen bioassay in small rodents and epidemiologic studies of air pollution. To assess the risk of a potentially harmful agent, one must frequently combine evidence from different and often quite dissimilar studies. Hence, this paper also discusses the central role of data synthesis in risk assessment, reviews some of the relevant statistical literature, and considers the role of statisticians in evaluating and combining evidence from diverse sources.     

A statistical framework for ecological and aggregate studies

Inference from studies that make use of data at the level of the area, rather than at the level of the individual, is more difficult for a variety of reasons. Some of these difficulties arise because frequently exposures (including confounders) vary within areas. In the most basic form of ecological study the outcome measure is regressed against a simple area level summary of exposure. In the aggregate data approach a survey of exposures and confounders is taken within each area. An alternative approach is to assume a parametric form for the within-area exposure distribution. We provide a framework within which ecological and aggregate data studies may be viewed, and we review some approaches to inference in such studies, clarifying the assumptions on which they are based. General strategies for analysis are provided including an estimator based on Monte Carlo integration that allows inference in the case of a general risk–exposure model. We also consider the implications of the introduction of random effects, and the existence of confounding and errors in variables.     

Properties of preliminary test estimators and shrinkage estimators for evaluating multiple exposures - Application to questionnaire data from the SONIC study

Epidemiology studies increasingly examine multiple exposures in relation to disease by selecting the exposures of interest in a thematic manner. For example, sun exposure, sunburn, and sun protection behavior could be themes for an investigation of sun-related exposures. Several studies now use pre-defined linear combinations of the exposures pertaining to the themes to estimate the effects of the individual exposures. Such analyses may improve the precision of the exposure effects, but they can lead to inflated bias and type I errors when the linear combinations are inaccurate. We investigate preliminary test estimators and empirical Bayes type shrinkage estimators as alternative approaches when it is desirable to exploit the thematic choice of exposures, but the accuracy of the pre-defined linear combinations is unknown. We show that the two types of estimator are intimately related under certain assumptions. The shrinkage estimator derived under the assumption of an exchangeable prior distribution gives precise estimates and is robust to misspecifications of the user-defined linear combinations. The precision gains and robustness of the shrinkage estimation approach are illustrated using data from the SONIC study, where the exposures are the individual questionnaire items and the outcome is (log) total back nevus count.     

Accounting for Response Misclassification and Covariate Measurement Error Using a Random Effects Logit Model

Often in longitudinal data arising out of epidemiologic studies, measurement error in covariates and/or classification errors in binary responses may be present. The goal of the present work is to develop a random effects logistic regression model that corrects for the classification errors in binary responses and/or measurement error in covariates. The analysis is carried out under a Bayesian set up. Simulation study reveals the effect of ignoring measurement error and/or classification errors on the estimates of the regression coefficients.     

Ecological regression analysis of environmental benzene exposure and childhood leukaemia: sensitivity to data inaccuracies, geographical scale and ecological bias

Benzene is classified as a group 1 human carcinogen by the International Agency for Research on Cancer, and it is now accepted that occupational exposure is associated with an increased risk of various leukaemias. However, occupational exposure accounts for less than 1% of all benzene exposures, the major sources being cigarette smoking and vehicle exhaust emissions. Whether such low level exposures to environmental benzene are also associated with the risk of leukaemia is currently not known. In this study, we investigate the relationship between benzene emissions arising from outdoor sources (predominantly road traffic and petrol stations) and the incidence of childhood leukaemia in Greater London. An ecological design was used because of the rarity of the disease, the difficulty of obtaining individual level measurements of benzene exposure and the availability of data. However, some methodological difficulties were encountered, including problems of case registration errors, the choice of geographical areas for analysis, exposure measurement errors and ecological bias. We use a Bayesian hierarchical modelling framework to address these issues, and we investigate the sensitivity of our inference to various modelling assumptions.     

Real-time monitoring of carbon monoxide using value-at-risk measure and control charting

One of the most important environmental health issues is air pollution, causing the deterioration of the population's quality of life, principally in cities where the urbanization level seems limitless. Among ambient pollutants, carbon monoxide (CO) is well known for its biological toxicity. Many studies report associations between exposure to CO and excess mortality. In this context, the present work provides an advanced modelling scheme for real-time monitoring of pollution data and especially of carbon monoxide pollution in city level. The real-time monitoring is based on an appropriately adjusted multivariate time series model that is used in finance and gives accurate one-step-ahead forecasts. On the output of the time series, we apply an empirical monitoring scheme that is used for the early detection of abnormal increases of CO levels. The proposed methodology is applied in the city of Athens and as the analysis revealed has a valuable performance.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

Regression calibration for logistic regression with multiple surrogates for one exposure

Methods have been developed by several authors to address the problem of bias in regression coefficients due to errors in exposure measurement. These approaches typically assume that there is one surrogate for each exposure. Occupational exposures are quite complex and are often described by characteristics of the workplace and the amount of time that one has worked in a particular area. In this setting, there are several surrogates which are used to define an individual's exposure. To analyze this type of data, regression calibration methodology is extended to adjust the estimates of exposure-response associations for the bias and additional uncertainty due to exposure measurement error from multiple surrogates. The health outcome is assumed to be binary and related to the quantitative measure of exposure by a logistic link function. The model for the conditional mean of the quantitative exposure measurement in relation to job characteristics is assumed to be linear. This approach is applied to a cross-sectional epidemiologic study of lung function in relation to metal working fluid exposure and the corresponding exposure assessment study with quantitative measurements from personal monitors. A simulation study investigates the performance of the proposed estimator for various values of the baseline prevalence of disease, exposure effect and measurement error variance. The efficiency of the proposed estimator relative to the one proposed by Carroll et al. [1995. Measurement Error in Nonlinear Models. Chapman & Hall, New York] is evaluated numerically for the motivating example. User-friendly and fully documented Splus and SAS routines implementing these methods are available (http://www.hsph.harvard.edu/faculty/spiegelman/multsurr.html).     

Autocorrelation in real-time continuous monitoring of microenvironments

Interpretation of continuous measurements in microenvironmental studies and exposure assessments can be complicated by autocorrelation, the implications of which are often not fully addressed. We discuss some statistical issues that arose in the analysis of microenvironmental particulate matter concentration data collected in 1998 by the Harvard School of Public Health. We present a simulation study that suggests that Generalized Estimating Equations, a technique often used to adjust for autocorrelation, may produce inflated Type I errors when applied to microenvironmental studies of small or moderate sample size, and that Linear Mixed Effects models may be more appropriate in small-sample settings. Environmental scientists often appeal to longer averaging times to reduce autocorrelation. We explore the functional relationship between averaging time, autocorrelation, and standard errors of both mean and variance, showing that longer averaging times impair statistical inferences about main effects. We conclude that, given widely available techniques that adjust for autocorrelation, longer averaging times may be inappropriate in microenvironmental studies.     

Multiply robust inference for statistical interactions

A primary focus of an increasing number of scientific studies is to determine whether two exposures interact in the effect that they produce on an outcome of interest. Interaction is commonly assessed by fitting regression models in which the linear predictor includes the product between those exposures. When the main interest lies in the interaction, this approach is not entirely satisfactory because it is prone to (possibly severe) bias when the main exposure effects or the association between outcome and extraneous factors are misspecified. In this article, we therefore consider conditional mean models with identity or log link which postulate the statistical interaction in terms of a finite-dimensional parameter, but which are otherwise unspecified. We show that estimation of the interaction parameter is often not feasible in this model because it would require nonparametric estimation of auxiliary conditional expectations given high-dimensional variables. We thus consider 'multiply robust estimation' under a union model that assumes at least one of several working submodels holds. Our approach is novel in that it makes use of information on the joint distribution of the exposures conditional on the extraneous factors in making inferences about the interaction parameter of interest. In the special case of a randomized trial or a family-based genetic study in which the joint exposure distribution is known by design or by Mendelian inheritance, the resulting multiply robust procedure leads to asymptotically distribution-free tests of the null hypothesis of no interaction on an additive scale. We illustrate the methods via simulation and the analysis of a randomized follow-up study.     

The Collection and Management of Occupational Exposure Data

This presentation addresses a number of issues pertinent to the collection and management of occupational exposure data and offers suggestions to those persons who are developing systems to handle large volumes of occupational exposure data. A perspective is taken that aims to meet the traditional objectives of industrial hygienists while accommodating epidemiologic needs for linking occupational exposure and health outcome data. The suggestions are based on experience gained through the retrospective use of industrial hygiene data in a large number of epidemiologic studies.     

Use of Cohort Information in the Design and Analysis of Case-Control Studies

Abstract.  Four case studies are presented to illustrate how information available on cohort members can be used to inform the control selection in epidemiologic case-control studies. The basic framework is the nested case-control paradigm and accompanying analysis methods. Emphasis is on development of intuition for choosing study design candidates, the form of the estimators, and extensions of the basic theory to solve design and analysis problems.     

Modelling daily multivariate pollutant data at multiple sites

Summary. This paper considers the spatiotemporal modelling of four pollutants measured daily at eight monitoring sites in London over a 4-year period. Such multiple-pollutant data sets measured over time at multiple sites within a region of interest are typical. Here, the modelling was carried out to provide the exposure for a study investigating the health effects of air pollution. Alternative objectives include the design problem of the positioning of a new monitoring site, or for regulatory purposes to determine whether environmental standards are being met. In general, analyses are hampered by missing data due, for example, to a particular pollutant not being measured at a site, a monitor being inactive by design (e.g. a 6-day monitoring schedule) or because of an unreliable or faulty monitor. Data of this type are modelled here within a dynamic linear modelling framework, in which the dependences across time, space and pollutants are exploited. Throughout the approach is Bayesian, with implementation via Markov chain Monte Carlo sampling.     

Multistage sampling for latent variable models

I consider the design of multistage sampling schemes for epidemiologic studies involving latent variable models, with surrogate measurements of the latent variables on a subset of subjects. Such models arise in various situations: when detailed exposure measurements are combined with variables that can be used to assign exposures to unmeasured subjects; when biomarkers are obtained to assess an unobserved pathophysiologic process; or when additional information is to be obtained on confounding or modifying variables. In such situations, it may be possible to stratify the subsample on data available for all subjects in the main study, such as outcomes, exposure predictors, or geographic locations. Three circumstances where analytic calculations of the optimal design are possible are considered: (i) when all variables are binary; (ii) when all are normally distributed; and (iii) when the latent variable and its measurement are normally distributed, but the outcome is binary. In each of these cases, it is often possible to considerably improve the cost efficiency of the design by appropriate selection of the sampling fractions. More complex situations arise when the data are spatially distributed: the spatial correlation can be exploited to improve exposure assignment for unmeasured locations using available measurements on neighboring locations; some approaches for informative selection of the measurement sample using location and/or exposure predictor data are considered.     

Logistic regression analysis of epidemiologic data: theory and practice

The use of logistic regression analysis is widely applicable to epidemiologic studies concerned with quantifying an association between a study factor (i.e., an exposure variable) and a health outcome (i.e., disease status). This paper reviews the general characteristics of the logistic model and illustrates its use in epidemiologic inquiry. Particular emphasis is given to the control of extraneous variables in the context of follow-up and case-control studies. Techniques for both unconditional and conditional maximum likelihood estimation of the parameters in the logistic model are described and illustrated. A general analysis strategy is also presented which incorporates the assessment of both interaction and confounding in quantifying an exposure-disease association of interest.     

The analysis of extreme pollution levels: A case study

Our case study focuses on Milan. Italian law specifies strict guidelines for the permissibility of high levels of a variety of air pollutants in cities. In Milan, a highly sophisticated network of recording stations has been constructed to monitor pollutant levels. The aim of this paper is to obtain a summary of the temporal behaviour of the pollutant series, with particular reference to extreme levels. Simple exploratory analysis reveals a number of sources of stochastic variation and possible dependence on covariate effects, which are subsequently modelled, exploiting recent developments in the modelling and inference for temporal extremes. Using this methodology, we examine the issues of data trends, non-stationarity, meteorological effects and temporal dependence, all of which have substantive implications in the design of pollution control regulations. Moreover, the asymptotic basis of these extreme value models justifies the interpretation of our results, even at levels that are exceptionally high.     

Weighted empirical adaptive variance estimators for correlated data regression

Estimating equations based on marginal generalized linear models are useful for regression modelling of correlated data, but inference and testing require reliable estimates of standard errors. We introduce a class of variance estimators based on the weighted empirical variance of the estimating functions and show that an adaptive choice of weights allows reliable estimation both asymptotically and by simulation in finite samples. Connections with previous bootstrap and jackknife methods are explored. The effect of reliable variance estimation is illustrated in data on health effects of air pollution in King County, Washington.     

Poisson Regression and the Case-Crossover Design: Similarities and Differences

Poisson regression and case-crossover are frequently used methods to estimate transient risks of environmental exposures such as particulate air pollution on acute events such as mortality. Roughly speaking, a case-crossover design results from a Poisson regression by conditioning on the total number of failures. We show that the case-crossover design is somewhat more generally applicable than Poisson regression. Stratification in the case-crossover design is analogous to Poisson regression with dummy variables, or to a marked Poisson regression. Poisson regression makes it possible to express case-crossover likelihood functions as multinomial likelihoods without making reference to cases, controls, or matching. This derivation avoids the counterintuitive notion of basing inferences on exposures that occur post-failure.     

Multiple-bias modelling for analysis of observational data

Summary.  Conventional analytic results do not reflect any source of uncertainty other than random error, and as a result readers must rely on informal judgments regarding the effect of possible biases. When standard errors are small these judgments often fail to capture sources of uncertainty and their interactions adequately. Multiple-bias models provide alternatives that allow one systematically to integrate major sources of uncertainty, and thus to provide better input to research planning and policy analysis. Typically, the bias parameters in the model are not identified by the analysis data and so the results depend completely on priors for those parameters. A Bayesian analysis is then natural, but several alternatives based on sensitivity analysis have appeared in the risk assessment and epidemiologic literature. Under some circumstances these methods approximate a Bayesian analysis and can be modified to do so even better. These points are illustrated with a pooled analysis of case–control studies of residential magnetic field exposure and childhood leukaemia, which highlights the diminishing value of conventional studies conducted after the early 1990s. It is argued that multiple-bias modelling should become part of the core training of anyone who will be entrusted with the analysis of observational data, and should become standard procedure when random error is not the only important source of uncertainty (as in meta-analysis and pooled analysis).     

A unified model for covariate measurement error adjustment in an occupational health study while accounting for non-detectable exposures

Summary.  The paper focuses on an occupational health study where the goal is to associate a worker's true log-normal-scale mean dust exposure over the year with forced expiratory volume. A previous analysis used repeated shift-long dust exposure measurements, taken over a year, as a surrogate to address the issue of the mean exposure being unobservable. However, in this study the associated measurement error is further complicated by the fact that some exposure measurements fall below a detectable limit. We extend the previous analysis via full maximum likelihood, to account appropriately for non-detectable exposures.