Spatial measurement error and correction by spatial SIMEX in linear regression models when using predicted air pollution exposures

Abstract

Spatial modeling of air pollution exposures is widespread in air pollution epidemiology research as a way to improve exposure assessment. However, there are key sources of exposure model uncertainty when air pollution is modeled, including estimation error and model misspecification. We examine the use of predicted air pollution levels in linear health effect models under a measurement error framework. For the prediction of air pollution exposures, we consider a universal Kriging framework, which may include land-use regression terms in the mean function and a spatial covariance structure for the residuals. We derive the bias induced by estimation error and by model misspecification in the exposure model, and we find that a misspecified exposure model can induce asymptotic bias in the effect estimate of air pollution on health. We propose a new spatial simulation extrapolation (SIMEX) procedure, and we demonstrate that the procedure has good performance in correcting this asymptotic bias. We illustrate spatial SIMEX in a study of air pollution and birthweight in Massachusetts.

1. Introduction

There is strong evidence in epidemiological studies that both short-term and long-term exposures to air pollution are related to cardiovascular morbidity and mortality (Brook and others, 2010). Spatial modeling of air pollution levels using Kriging methods is now commonplace in air pollution epidemiology research. Existing pollution monitoring networks are used to collect data on regional air pollution concentrations, and spatial prediction models are then used to estimate location-specific exposures at the home address of each subject in a study. However, the regional heterogeneity of air pollution may be difficult to characterize. For example, ambient levels of have been shown to vary considerably within a given city region, in part due to traffic sources (Brauer and others, 2003; Clougherty and others, 2008).

This measurement error setting where the set of locations in the health analysis does not match the set of locations where exposures are observed is called spatial misalignment (Gryparis and others, 2009). Spatially predicted exposures are often used directly as the individual-specific exposure estimates for the health analysis. This approach treats the predicted exposures as observed, without accounting for the uncertainty in the prediction process. Ignoring measurement error can lead to biased health effect estimates and overstated confidence in the resulting risk assessments (Carroll and others, 2006). The issue of spatial misalignment measurement error in air pollution epidemiology has received considerable attention in a recent literature (Szpiro and others, 2011; Szpiro and Paciorek, 2013; Madsen and others, 2008; Lopiano and others, 2013; Gryparis and others, 2009; Peng and Bell, 2010; Bergen and others, 2013; Chang and others, 2011). However, the role of spatial exposure model misspecification has not formally been investigated.

The work presented here adds to this existing literature in two ways. First, we explicitly evaluate the impact of model misspecification, whereas the majority of previous studies focus instead on the impact of uncertainty associated with estimation of unknown parameters in a known exposure model. In practice, model misspecification will always be an issue given the spatial–temporal complexity of pollution emissions. Second, we propose methods that can be implemented when the exposure prediction algorithm does not yield uncertainties for the exposure model parameter estimates. Examples of such approaches in the environmental science literature include a daily Kriging model implemented in ArcGIS (Liao and others, 2006) and multi-stage models or missing data imputation schemes that make it difficult to propagate the uncertainty in each stage through to the final exposure predictions (Kloog and others, 2012, 2014; Nordio and others, 2013).

This article investigates two factors of exposure estimation that may affect resulting health effect estimates: estimation error and model misspecification. In practice, spatial air pollution models are fit with sparse monitoring data. Hence, we examine the effects of estimation error in the Kriging model parameters under small sample size. In addition, the underlying exposure model that generates air pollution levels in any given region is not known. Thus, we investigate the impact of model misspecification in the spatial model by omitting a spatial covariate. To correct for bias in the health effect estimates, we introduce a spatial version of the simulation extrapolation (SIMEX) method. To our knowledge, our proposed spatial SIMEX procedure is the first treatment of SIMEX allowing for spatially correlated measurement errors.

The remainder of this paper is arranged as follows. In Section 2, we introduce our modeling framework with specific exposure models of interest. In Section 3, we analytically examine the bias for each exposure model and we derive the probability limits of the misspecified parameters. We propose a new spatial SIMEX correction method in Section 4 where correlated classical error is added to the exposure predictions to correct for bias. In Section 5, we present a simulation study to investigate bias and to demonstrate the performance of spatial SIMEX. We then illustrate the spatial SIMEX correction in a study of air pollution and birthweight in Section 6. We end with a concluding discussion in Section 7.

2. Spatial exposure models in air pollution and health studies

2.1. Model framework

Let the health effect model of interest be a simple linear regression model, , for each subject , where is a continuous health outcome, is the true unmeasured air pollution exposure at the home address of subject , and . Let be independent and identically distributed (i.i.d.) with mean 0 and variance . The goal of the analysis is to estimate , the parameter measuring the association between the health outcome and the air pollution exposure.

Let denote the -length vector of measured air pollution levels at monitor sites spread throughout the same geographic region, where . We assume spatial misalignment, where the subject address locations do not match the monitor locations. Suppose the true pollution process, , is generated by a Gaussian Random Field, and that the realizations of this process take a parametric form. Specifically, denote the length vector of parameters for the mean model as , denote the length vector of parameters for the variance model as , and let . Then a realization of one surface follows:

(2.1)

The Kriging estimator for conditional on the observed monitor data is defined as

(2.2)

We now consider two spatial exposure model scenarios in this study.

Scenario I: Universal Kriging model. Consider equation (2.2) and assume that follows a Matérn covariance with parameters for range , smoothness , and variance (see Appendix of the supplementary material available at Biostatistics online for explicit covariance function). Universal Kriging assumes a spatial correlation structure for the variance, and allows the mean model to depend linearly on a set of covariates or to be constant. For the mean model we consider two scenarios. First, we consider a constant mean model, in Scenario IA. Second, we consider a mean model that depends linearly on covariates, , where we assume that covariates, , represent spatially varying land-use characteristics. This type of model is often called a “land-use regression” model, which we refer to as Scenario 1B. Land-use covariates for air pollution models include measures such as percentages of residential land, greenspace, industry, population size, distances to major roads, and traffic intensity (Ross and others, 2007).

Scenario II: Misspecified universal Kriging model. Scenario II considers a misspecified universal Kriging model. We assume that the true exposure is generated under the universal Kriging model defined above, and that the fitted exposure model is misspecified by omitting . Thus, the misspecified exposure model is

(2.3)

where , , and , with the subscript denoting the naive parameters from the misspecified model. Here, we also assume that and are each spatially correlated, generated from their own Gaussian Processes.

2.2. Decomposition into Berkson and classical error components

We now review and extend the decomposition of exposure measurement error into Berkson and Classical components for each of these three scenarios. The mean and variance parameters can be estimated jointly via maximum likelihood. Let be the vector of maximum likelihood estimates of for the true model and let be the vector of maximum likelihood estimates of in the naive misspecified model. A general measurement error framework can be used to characterize the difference between the true unobserved exposures and the predicted exposures . The decomposition of errors into Berkson and classical measurement error components follows the development of Gryparis and others (2009) and Szpiro and others (2011), where Gryparis and others (2009) consider a Bayesian Gaussian Process model with constant mean, and Szpiro and others (2011) consider a universal Kriging model where the mean depends on several spatial covariates. We now extend this viewpoint to our models of interest defined in Section 2.1. For Scenario I, the predicted exposures, , have the form

(2.4)

The error can be decomposed into two components. The Berkson error component, , represents the difference between the true measurements and the expectation of conditional on . The classical error component, , represents the difference between the true model and the estimated model, and we refer to this classical error as estimation error. In Scenario I, this Kriging estimator fit under the correct model is the best linear unbiased predictor (Cressie, 1993).

For Scenario II, the predicted exposures, , have the form

(2.5)

Decomposing the error into Berkson and classical components,

(2.6)

Thus, there are two classical measurement error components. The first is attributed to choosing the incorrect model, and the second is due purely to estimation error of the parameters.

3. Analysis of bias in health effect estimates induced by exposure models

Now, with the measurement error framework established, we study the impact of measurement error in the predicted exposures on bias of the coefficient representing the association between air pollution exposure and the health outcome. First, we investigate bias in the case of estimation error only in Scenario I analytically, focusing on the small-sample bias properties not previously addressed in other studies. Next, we study the asymptotic bias in the case of model misspecification error in Scenario II by deriving the probability limits of the MLEs in the misspecified model and deriving the particular form of the classical error variance. Later, in Section 5, we will complement this analysis with a simulation study.

3.1. Bias analysis for Scenario I

To study the estimation error bias in Scenario I, we introduce notation for the least squares estimators. Without loss of generality we assume centered variables. Let be the function for the least squares estimate of given monitoring data, spatial covariates, and observed health outcomes. Our notation explicitly shows the dependence on and , and implicitly also depends on . Specifically, define

(3.1)

for vector , matrix , and vector . Then let denote the least squares estimate of based on the exposure model using the true parameters , so . Similarly, let denote the least squares estimate of based on the exposure model using the estimated exposure model parameters , so .

First, we note that the Berkson error component does not induce any bias in the estimate of . Hence, any bias in the estimator comes from the classical error component. Using a second-order Taylor expansion of around the approximate bias of is

(3.2)

where and . Equation (3.2) illustrates that when the number of monitors is large, then , resulting in an asymptotically unbiased estimator for . However, following a similar argument to Zimmerman and Cressie (1992), Jensen's inequality says that if is strictly concave, then

(3.3)

Thus, in practice when is small, this bias that disappears asymptotically will be present. Moreover, the Jensen's inequality argument applies even when the covariance parameters are unbiased, . Similarly, if is strictly convex, the resulting bias is upward and only linearity of yields an unbiased estimator. Generally, is a nonlinear function of the exposure model covariance parameters, and its form depends on the spatial covariance function, the distances of the monitors from each other, and the distances of the subject addresses from the monitors. Figure S1 of the supplementary material available at Biostatistics online shows an example of a particular choice of covariance matrix and set of covariates where as a concave function, and its scale suggests that the bias may be small. Thus, in practice, we expect small sample bias in due to a small number of monitors . We investigate this small sample bias via simulations in Section 5.

3.2. Bias analysis for Scenario II

Scenario II contains the additional component of error due to model misspecification, . To understand the bias induced by this component, we first consider the asymptotic behavior of the naive model parameter estimates. Following Wang and others (1998), the MLE's will be the solutions to the score equations based on the Multivariate Normal likelihood for Equation (2.3), and thus will converge in probability to the solutions of the following equations:

(3.4)

(3.5)

where is the subset of spatial covariates in the misspecified model, , and indexes the variance parameters. Solving these for yields the asymptotic relationship between and , as shown in Appendix of the supplementary material available at Biostatistics online. Closed-form solutions exist for , but for the variance parameters we derive equations which can only be solved numerically. In general, the solutions depend on the joint distribution of the correlated spatial covariates.

4. SIMEX for correlated Berkson and classical errors

The SIMEX method has been developed as a flexible method to correct for the effect of classical measurement errors on the estimation of a parameter of interest (Cook and Stefanski, 1994). SIMEX is a functional method which uses resampling techniques and places minimal assumptions on the underlying distribution of the exposures. SIMEX has two steps: a simulation (SIM) step, where simulated error is added to the mismeasured exposures in increasing amounts, and an extrapolation (EX) step, where a trend is fit to the mean of the parameter estimates over the increasing error levels and extrapolated back to the case of no error. It has been suggested that SIMEX may be suitable for several exposures with correlated classical errors when the correlations of the errors are known or estimable (Carroll and others, 2006). We now present the spatial SIMEX procedure, an extension of SIMEX that allows the classical measurement errors to be correlated over space.

4.1. Spatial SIMEX procedure

The spatial SIMEX procedure is implemented as follows. Let and be given positive integers, and let be an increasing sequence of non-negative numbers starting with .

Simulation step. For each and , we generate a pseudo-dataset of exposures, , where . Adding the error creates pseudo-datasets equal to the unbiased exposure plus an error component with a covariance of . This allows exploration of how the health effect parameter is biased as a function of increasing measurement error variance. For each and , we estimate the parameter of interest by fitting the linear health model using the pseudo-dataset. Thus, estimates the association between the pseudo-exposures and the outcome .

Extrapolation step. We obtain an estimate of for each by averaging over the simulations, . We then fit a trend to versus using a linear or quadratic model. The predicted value of this trend at is the spatial SIMEX corrected estimate of the parameter, , estimating the health effect parameter under no measurement error.

Spatial SIMEX can be implemented with a bootstrap standard error estimate, following the same general bootstrap approach used for one-dimensional SIMEX (Carroll and others, 2006). To implement the bootstrap standard error, we first estimate . Then, for bootstrap samples: (i) resample monitor locations with replacement, (ii) fit the initial exposure model to the new sample of monitoring data, (iii) predict the exposures at the health locations, and (iv) repeat the entire SIMEX procedure using these new predictions to obtain . The standard error estimate is then computed by the standard deviation of the bootstrap SIMEX estimates, .

In general, the asymptotic results of and Cook and Stefanski (1994) for the unbiasedness of the point estimate apply when (i) the bias in the naive estimator is a continuous function of the measurement error variance, (ii) the measurement error variance is known, and (iii) the true extrapolant function based on the bias function is known. In practice, the measurement error variance and the true extrapolant function are often unknown. Still, even an approximate exrapolant function can help reduce bias (Carroll and others, 2006). In addition, the degree of fit to the error-inflated parameters can be assessed, and if the trend in bias is unclear, the number of SIMEX samples can be increased as well as the number of 's, . The next subsection discusses how to estimate the classical measurement error variance.

4.2. Estimation of spatial measurement error variance parameters

The simulation step of spatial SIMEX relies on generating random samples of error from the multivariate classical error distribution. Based on the derivations in our bias analysis of Section 3, the only component of error that leads to asymptotic bias is the model misspecification component of classical error. Thus, only the model misspecification variance is needed to generate these pseudo-datasets in the SIMEX procedure to asymptotically correct for the bias. Derivations in the supplementary material available at Biostatistics online show that the classical error due to model misspecification has the distribution , where

(4.1)

Equation (4.1) shows that depends on the spatial covariances of the exposures under both the true parameters and the naive parameters as well as the spatial covariance of the unobserved spatial covariate . In practice, these spatial covariances would not be known and thus would not be known. In that case, can be approximated by using external validation data from held-out monitors. We fit the spatial exposure model and predict the exposure at the held-out monitor locations. Then, we compute the difference between the predicted and observed exposures to obtain the residuals at the held-out monitor locations. We fit a spatial model to the set of residuals to estimate the total spatial covariance. In practice, we use a Matérn covariance structure for the spatial model of the residuals.

One key issue which has been discussed in previous studies is the non-identifiability of the proportion of measurement error that is Berkson versus classical in models with both Berkson and classical measurement error (Mallick and others, 2002; Li and others, 2007). While external validation data allow the estimation of the total error variance, the relative proportions of Berkson and classical errors cannot be determined. These previous studies consider the case where both the Berkson errors and the classical errors are assumed to be i.i.d. normal in one dimension. To deal with the identifiability issue in a practical application, the authors perform sensitivity analyses regarding the percentage of variance assumed to be classical versus Berkson. We take a similar approach, estimating the total spatial covariance by using external validation data. We then compute , where represents the proportion of the total spatial error attributable to classical error, with the remaining error as Berkson.

5. Simulation study

We conduct a simulation study to explore the degree of bias for the models described in Section 2. Scenario IA assumes a constant mean model, and Scenario IB assumes that the mean depends on two spatial covariates. For the spatially correlated residuals, we use a Matérn covariance function, with range and variance . We consider a smooth surface with smoothness , and a rough surface with smoothness , with examples shown in Figure S2 of the supplementary material available at Biostatistics online. Note that the rough surface does not satisfy the smoothness conditions needed in our Taylor expansion because it only has first derivatives. To generate the model misspecification scenario, we use the exposure model of Scenario IB with two spatial covariates and smooth residuals, and then fit the exposure model omitting the second spatial covariate. Our first set of simulations assumes that is known, and in later simulations we relax this assumption.

Specifically, we assume that one-third of the available monitors are held-out as a validation dataset, and we estimate using that data as described in Section 4. The estimate of also depends on the assumed proportion of classical to Berkson error. Most of the error will be classical, but the exact percentage is not identifiable. We choose classical error proportions of as realistic approximations, and we look at extremes of to examine the robustness of the spatial SIMEX estimator to the choice of .

Simulation results for Scenario I are given in Table 1. For the smooth surface, we observe negligible bias even for small numbers of monitors. The rough surface is particularly difficult to estimate because we use a small number of monitors to estimate the form of a rough surface. We see a small amount of bias in the case of Scenario IA with only 20 monitors. Interestingly, the direction of this bias here is upward, which may be an artifact of using a particularly sparse dataset and a rough surface. As expected, the Berkson error underestimates the standard error, leading to insufficient coverage of the CIs.

Table 1.

Simulation results for smooth and rough exposure surfaces for Scenario I with different number of monitors

Scenario	m	Exposure	Bias	Empirical SE	Model SE	MSE	Coverage
IA Smooth	20	True	0.001	0.082	0.076	0.007	94.200
IA Smooth	20		0.003	0.108	0.078	0.012	86.400
IA Smooth	20		0.004	0.111	0.080	0.012	85.200
IA Smooth	40	True	0.000	0.086	0.076	0.007	94.000
IA Smooth	40		0.001	0.088	0.076	0.008	93.000
IA Smooth	40		0.002	0.088	0.076	0.008	93.000
IB Smooth	20	True	0.001	0.044	0.043	0.002	94.990
IB Smooth	20		0.002	0.055	0.044	0.003	89.379
IB Smooth	20		0.002	0.059	0.044	0.003	87.976
IB Smooth	40	True	0.002	0.043	0.043	0.002	94.400
IB Smooth	40		0.003	0.044	0.043	0.002	93.600
IB Smooth	40		0.003	0.045	0.043	0.002	93.800
IA Rough	20	True	0.001	0.057	0.055	0.003	93.865
IA Rough	20		0.007	0.179	0.080	0.032	63.190
IA Rough	20		0.058	0.220	0.089	0.052	57.055
IA Rough	40	True	0.000	0.057	0.054	0.003	94.990
IA Rough	40		0.002	0.104	0.067	0.011	80.962
IA Rough	40		0.024	0.114	0.070	0.014	78.557
IB Rough	20	True	0.002	0.038	0.038	0.001	94.400
IB Rough	20		0.002	0.040	0.038	0.002	93.600
IB Rough	20		0.002	0.042	0.038	0.002	92.200
IB Rough	40	True	0.002	0.038	0.038	0.001	95.000
IB Rough	40		0.002	0.038	0.038	0.001	95.000
IB Rough	40		0.002	0.038	0.038	0.001	94.600

Table 2 gives simulation results for Scenario II. We observe substantial bias toward the null in the health effect parameter. Results show that spatial SIMEX corrects this bias approximately. The two extrapolation functions perform differently, where the linear extrapolation function under corrects the bias. In the simulation step of the spatial SIMEX procedure, we use the derived covariance matrix with true parameter values to sample random error to generate the pseudo-datasets. We implement the bootstrap standard error as described in Section 4.3 using 200 bootstrap resampling steps.

Table 2.

Simulation results for Scenario II, misspecified exposure model, and correction by spatial SIMEX when spatial measurement error variance is known

Scenario	m	Exposure	Bias	Empirical SE	Model SE	MSE	Coverage
II	50	True	0.000	0.036	0.034	0.001	93.6
II	50		0.000	0.036	0.034	0.001	93.6
II	50		0.203	0.182	0.039	0.074	22.2
II	50	Spatial SIMEX, linear	0.072	0.211	0.241	0.050	90.6
II	50	Spatial SIMEX, quad	0.026	0.254	0.285	0.065	91.9

Table 3 gives the results for the simulations where is estimated using the available monitoring data. We find that the spatial SIMEX procedure still works well even when approximating . The best performance is seen when is 0.80. In the extreme case assuming classical error, the spatial SIMEX procedure using the quadratic extrapolation appears to over-correct the bias. In the other extreme case assuming classical error, both the linear and quadratic extrapolations under-correct the bias. Although there is sensitivity to the choice of , all the spatial SIMEX estimates noticeably reduce the bias. There is slight under-coverage in the CI estimates across all choices of .

Table 3.

Simulation results for Scenario II, misspecified exposure model, and correction by spatial SIMEX when spatial measurement error variance parameters are estimated, for different proportions, of classical error

Scenario		Exposure	Bias	Empirical SE	Model SE	MSE	Coverage
II		True	0.001	0.036	0.034	0.001	93.6
II			0.001	0.036	0.034	0.001	93.6
II			0.200	0.180	0.039	0.072	22.4
II	1.00	Spatial SIMEX, linear	0.068	0.213	0.228	0.050	91.3
II	1.00	Spatial SIMEX, quadratic	0.067	0.322	0.336	0.108	87.2
II	0.90	Spatial SIMEX, linear	0.075	0.211	0.228	0.050	91.1
II	0.90	Spatial SIMEX, quadratic	0.044	0.308	0.331	0.097	87.8
II	0.80	Spatial SIMEX, linear	0.076	0.208	0.227	0.049	91.6
II	0.80	Spatial SIMEX, quadratic	0.028	0.294	0.324	0.087	89.1
II	0.50	Spatial SIMEX, linear	0.112	0.200	0.225	0.053	89.6
II	0.50	Spatial SIMEX, quadratic	0.058	0.247	0.304	0.064	91.3

In practice, the true exposure surface may not exhibit a Gaussian distribution. To explore the performance of spatial SIMEX when the Gaussian assumption is not satisfied, we considered simulation settings where the spatial covariates were generated from spatial log–normal distributions, with results given in Table S1 of the supplementary material available at Biostatistics online. Overall, results are as expected, where spatial SIMEX corrects adequately for bias or may over-correct, yielding effect estimates that could have slight upward bias. The standard errors in this scenario are over-estimated, leading to CIs that are wider than necessary and coverage 99%.

6. Data Example: Association between air pollution and low birthweight

We applied our spatial SIMEX method to a study of birthweight and particulate matter exposure during pregnancy in Massachusetts. The objective of the study was to estimate the association between birthweight and PM exposure during the second and third trimesters. The study population included all singleton live births in Massachusetts from the Massachusetts Birth Registry during 2008 (January 1 to December 31), a total of 70 340 births. Individual-level data on the mother and baby come from the Massachusetts Birth Registry. Confounders in the health model include maternal age, gestational age, number of cigarettes smoked during and before pregnancy, chronic conditions of mother or conditions of pregnancy (lung disease, hypertension, gestational diabetes, and non-gestational diabetes), and socioeconomic measures (mother's race, mother's years of education, and the Kotelchuck index of adequacy of prenatal care utilization). Area-level socioeconomic status is controlled by census-tract median household income using data from the United States Census Bureau of 2000 for each census tract in Massachusetts. These covariates are consistent with the published literature on birthweight and particulate matter (Dadvand and others, 2013). Some studies also adjust for co-pollutant exposures, such as ozone, although the need for this may vary by region, where studies in the northeast have found similar effect sizes after this adjustment (Bell and others, 2007).

PM measurements during 2007 and 2008 were obtained from 40 monitoring sites in Massachusetts as part of the Environmental Protection Agency and Interagency Monitoring of Protected Visual Environments monitoring networks (Kloog and others, 2011). The residential address of each mother at a time of birth was geocoded as described in Kloog and others (2012a). To predict PM at the mother's home address for each birth, we assume a universal Kriging model with Matérn residuals. The mean function for the Kriging model includes a linear trend for three land-use covariates: distance to primary highway, distance to known particulate matter emission source, and average traffic density, as described in Kloog and others (2011). Separate models are fit for each month using the monthly average PM concentrations at the monitoring sites during 2007 and 2008. Exposures during the second and third trimesters of pregnancy are estimated by averaging the monthly PM concentrations prior to the delivery date.

We fit linear health effect models for each exposure of interest, second trimester PM and third trimester PM, adjusting for confounders. This model yields a naive effect estimate that is not corrected for measurement error. We then apply our proposed spatial SIMEX correction method using a quadratic extrapolation function and assuming that . Further description of our implementation of spatial SIMEX for this application is given in the supplementary material available at Biostatistics online.

We find negative associations between birthweight and each PM exposure without correcting for measurement error, and the estimated effect size is larger when we apply spatial SIMEX. Specifically, the change in birthweight per second trimester PM exposure is estimated to be 5.04 g, CI , without accounting for measurement error. When corrected by spatial SIMEX, this association is estimated to be 7.90 g per PM exposure in the second trimester, CI . For the third trimester, the change in birthweight per PM is estimated to be 3.49 g, CI , without any measurement error correction. Applying spatial SIMEX, we estimate an association of 4.91 g per third trimester PM exposure, CI . Figure S4 and Table S2 of the supplementary material available at Biostatistics online, we report results from sensitivity analyses varying the assumed percentage of classical error and using both linear and quadratic extrapolation functions.

7. Discussion and conclusions

In this paper, we have conducted a bias analysis of several key scenarios in exposure modeling of air pollution. We have shown that when the exposure model is misspecified by omitting an important covariate, notable downward bias in the health effect estimate can occur in addition to the underestimation of the standard errors. We have proposed a new spatial SIMEX approach to adjust for bias and standard error estimation in the presence of model misspecification. We shown that this bias due to exposure model misspecification can be approximately corrected by spatial SIMEX. We have also shown analytically and via simulation the presence of small-sample bias due to estimation error in the case of a correctly specified exposure model, although the degree of bias in practice is typically negligible. Hence, this work has demonstrated that with respect to bias, model misspecification is a much bigger problem than parameter estimation.

Previous research in this area has suggested that the plug-in estimator typically induces little bias, and authors have advocated for using this estimator for the point estimate and adjusting the standard errors to account for the additional variability in using the exposure predictions (Szpiro and others, 2011; Madsen and others, 2008; Lopiano and others, 2013; Gryparis and others, 2009). However, those papers investigate bias in simulation studies primarily by fitting the correct exposure model used to generate the data. Our findings in Section 3 for the bias of exposure model Scenario I are consistent with these previous studies, as we also found in simulations that the degree of bias is small when the correct exposure model is specified. Our findings are also consistent with a recent study investigating exposure model misspecification via simulation, which illustrated real examples of poor-fitting Kriging exposure models that induced bias in health effect estimates (Alexeeff and others, 2014). In practice, the underlying exposure model that generates air pollution levels in any given region is not exactly known. In addition, current approaches for correcting the standard errors of estimates also rely on the assumption that the exposure model is correctly specified (Szpiro and others, 2011; Madsen and others, 2008). We have approached this problem using both analytical methods and simulation studies, and we have presented a more thorough bias analysis than what has been considered in previous work. In particular, we have extended existing work to the case of model misspecification, which is important since exposure models are typically complex and no single statistical model is likely to be correct.

This work also points to practical considerations in the implementation of spatial SIMEX. As in other SIMEX procedures, the simulated re-measurements are generated using a known classical error variance. In Section 3, we derived the particular form of the classical error variance for Scenario II, but in practice the exact classical error variance would not be known. External validation data are typically needed to estimate the measurement error variance, as we hold out a set of monitors in our birthweight application in Section 6. However, external validation data only allow the estimation of the total spatial error, while the relative proportions of Berkson and classical errors are not identifiable (Mallick and others, 2002; Li and others, 2007). We follow the approach of Li and others (2007) by varying our choice of in sensitivity analyses.

Our work points several areas of future research. First, we suggest further investigation of model misspecification in land-use regression and Kriging models for air pollution exposure, for example the case of omitting a key confounder in the exposure model. Our work here considers only one misspecification scenario, which induces notable bias. Second, we suggest further study of the practical implementation of spatial SIMEX, including methods for estimating the classical error variance given the issue of identifiability, as well as the robustness of spatial SIMEX to incorrect estimation of the classical error variance.

One advantage of SIMEX is that the general methodology can be adapted to cases in which the measurement error biases cannot be derived in closed-form, including logistic regression and Poisson regression (Carroll and others, 2006). SIMEX can also be extended to the multi-pollutant setting in which more than one exposure covariate is measured with measurement error (Carroll and others, 2006). In the multi-pollutant case, we can use the same cross-validation procedure to estimate prediction errors for all pollutants at each location. We can then fit a covariance structure model for these spatially correlated multivariate errors, such as the Kronecker product of a multi-pollutant covariance matrix for prediction errors measured at the location and parametric (e.g. exponential) spatial correlation structures for predictions errors for a given pollutant measured at different locations. Therefore, our proposed spatial SIMEX approach would be applicable to the multi-pollutant setting as long as all the pollutants are jointly measured. In addition, SIMEX can be adapted when the measurement error itself follows a different form, for example multiplicative log-Gaussian errors (Eckert and others, 1997).

This work examines aspects of exposure modeling of air pollution for health effect studies and provides some insight into the role of estimation error and model misspecification in the estimation of health effects. Understanding these impacts when constructing land-use regression and Kriging models is of fundamental importance to studies of air pollution and health. In particular, these results should be taken into account when interpreting the results of air pollution epidemiology studies that use land-use regression and Kriging models for exposure estimation. The spatial SIMEX procedure provides one possible measurement error correction strategy which may be beneficial to correct for bias induced by model misspecification.

Supplementary material

Supplementary material is available at http://biostatistics.oxfordjournals.org.

Funding

This work was supported by USEPA grant 834798 and NIH grants from the National Institute of Environmental Health Sciences (R01-ES020871, T32-ES007142) and the National Cancer Institute (P01-CA134294). This publication's contents are solely the responsibility of the grantee and do not necessarily represent the official views of the US EPA. Carroll's research was supported by National Cancer Institute grant R37-CA057030.