A major challenge in environmental epidemiology studies is usually exposure assessment, which typically relies on a combination of direct measurements of exposure concentrations, information about sources, and complex spatio-temporal models to estimate the exposure intensity field. These are then combined with information about study subjects’ locations over time to yield estimates of their individual exposures for analyzing exposure-response relationships. For both practical and theoretical reasons, these two tasks are typically performed separately. Exposure models are never perfect, so an important statistical problem becomes how to propagate the uncertainty in the exposure model into the health analysis. In the current issue, Szpiro and Paciorak (2013) derive the statistical properties of a two-stage estimator of the regression coefficient estimates for the health effects. Building on their earlier work (Szpiro et al., 2011b), they propose novel methods for measurement error correction, demonstrating asymptotic unbiasedness and valid confidence interval coverage for their corrected estimators. Simulation studies and an application to the MESA/Air study data illustrate the utility of their methods. The various substantive publications of MESA/Air study health associations have relied for the first stage on a sophisticated spatio-temporal model described by Sampson et al. (2011) and Szpiro et al. (2010). A recent extension to a nation-wide model for PM2.5 constituents showed significant associations of carotid intima thickness with organic carbon, silicon and sulphur; for Si and S, there was substantial spatial correlation, resulting in measurement-error corrected confidence intervals that were 50% wider than naive ones, but still significant (Bergen et al., 2013). Similar two-stage analyses have been described by a number of other groups (e.g., Gryparis et al., 2009; Li et al., 2013; Peng and Bell, 2010). Szpiro and Paciorek make a compelling case for the advantages of a two-stage approach in the introduction, and mention the alternative of a joint exposure and health modeling approach (e.g., Molitor et al. 2007; 2006). This issue has been widely debated and I’m rapidly coming around to their point of view. In the Molitor et al. papers, we fitted a model that might be represented schematically by the DAG shown in Figure 1. Here, the Xs represent the true but unobserved exposures of subjects i in the health effects study or of individuals or measurement locations j in a personal monitoring or exposure study. Direct measurements of exposure concentrations Zj are available only for the subset of locations or participants in the exposure study, but we assume both sets of exposures are determined by a common set of predictors W that are available for everybody in both datasets, such as proximity to traffic, spatial locations, etc., with the same vector of parameters θ (regression coefficients, spatial variance components, etc.) in both parts. Hence it seems reasonable to assume that a joint analysis of both datasets would be most efficient for estimating all the parameters of the model and would allow uncertainties in the exposure model to be propagated automatically through to the exposure-response parameters β that are of primary interest. Figure 1 Directed acyclic graph representing a joint exposure and health effects model (X = true exposure, Z = measured exposure, W = exposure determinants, Y = health outcome) Practically, this joint analysis is more complex than the two-stage analysis considered by Szpiro and Paciorek. As they point out, most epidemiologic studies investigate several different health outcomes Y, so this would require either separate joint analyses for each trait, or an even more complex multivariate outcome model. The idea of having separate exposure models for each outcome is itself rather unappealing! Also, most health associations in environmental epidemiology (and in air pollution research in particular) tend to be relatively weak, so the potential value-added by incorporating the health outcome into the exposure modeling could be minimal. More fundamentally, as these authors and others have pointed out, health effects models may be misspecified, so that feedback from the health model to the exposure model may have unintended consequences. This problem has been recognized in other areas, notably in pharmacokinetic models. For example, Lunn et al. (2009) considered an analogous problem where the “exposure model” consists of a pharmacokinetic model describing the relation between intake W and metabolite concentrations X and the “health model” is a pharmacodynamic model describing the relationship between X and some physiological response Y. In this context, primary interest might be in the parameters θ of the W-X relationship, and there is concern that misspecification of the X-Y relationship might bias the estimation of θ. To alleviate this concern, Lunn et al. proposed several ways of redrawing the DAG to “break” the linkage between the two submodels, while still taking advantage of a joint analysis. Further research to investigate the performance of single-stage analyses that avoid this feedback problem compared with two-stage procedures would be helpful. See for example (Zigler et al., 2013) for a similar discussion of the feedback problem in the context of Bayesian propensity score estimation in comparative effectiveness research. The fundamental framework Szpiro and Paciorek use assumes that the exposure field is fixed (but unknown) and the subjects’ and monitoring locations are random. There are compelling advantages to conceptualizing the model this way, making it robust to misspecification of the exposure model and providing a straight-forward way of estimating bias and variance through the bootstrap. However, it does seem a bit counterintuitive, since the exposure field really is random, subject to all kinds of unquantifiable influences, and a parallel world with the same generating model would experience a different field. Another of the key assumptions of the Szpiro and Paciorek approach is that the subject locations and the monitoring locations are not only drawn from the same distribution, but are also drawn independently. The latter assumption is frequently violated in studies where a subset of epidemiologic study subjects’ locations is sampled for exposure monitoring, as in the Children’s Health Study (Franklin et al., 2012; Gauderman et al., 2005). The authors comment in the Discussion that “The resampling procedure in our nonparametric bootstrap could be easily modified to reflect this type of data-generating mechanism.” It remains to be seen whether this type of sampling design produces bias of any importance if the naive analyses were used and how efficient such a modified procedure is compared with independent sampling. This suggests a further line of research into the implications for study design. Given a finite budget, what is the optimal allocation of resources between exposure measurement vs. the epidemiologic study (i.e., trade-offs between sample sizes and, for the measurement study, extent of measurements made, subject to constraints on total cost)? And for the exposure study, how should the locations be selected (Thomas, 2009, p. 184)—to represent the epidemiologic study base or distribution of cases, to maximize spatial variability, to maximize the variance of the predictors (possibly including previously existing measurements of other pollutants), to minimize the prediction error for unmeasured locations, or some other criterion? For example, suppose some measurements of pollution levels Zj and predictors Wj are already available at a set of locations j∈S1 and one wishes to select a set of additional locations S2 in such a way as to maximize the informativeness of the entire network S = S1∪S2 for predicting exposures Xi at some even larger set of unmeasured locations i∈N (e.g., the locations of a set of epidemiologic study subjects). This generally requires first an estimate of the exposure model parameters from the available measurements S1. For any given set of additional locations S2, one can then compute the expected variance of the prediction at other locations (taking these parameter estimates as the truth) and average these variances over the “demand surface” N. The challenge is then to choose the set S2 that would minimize this predicted variance (Diggle and Lophaven, 2005; Kanaroglou et al., 2005). Intuitively, one would like to add measurement locations that would most improve the predictions at those points that have the largest prediction variance under the current network and are the most influential; thus, one might want to select points that are not too close to the existing measurement locations or to each other, but close to many subjects for whom predictions will be needed. For example, one might select individual’s locations one at a time from the homes of the epidemiologic study participants that would lead to the largest reduction of this average variance, and then iteratively replace locations in the set with those outside the set where further reductions would result. In a two-stage design, rather than minimizing the average exposure prediction variance alone, what is really needed is to maximize the Fisher information for the relative risk parameter β from a model that will use these predictions. This may entail selecting measurement locations that will maximize the precision of the most influential points, such as those individuals predicted to be the most heavily exposed. Also, my own efforts (Thomas, 2007) to compare some of these different designs by simulation yielded disappointing results: it was difficult to find a design the consistently improved upon simple random sampling! Further research on spatial sampling designs would be very helpful (e.g., Chang et al., 2007).

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空间暴露模型中的测量误差：研究设计意义

环境流行病学研究的一个主要挑战通常是暴露评估，它通常依赖于暴露浓度的直接测量、有关来源的信息和复杂的时空模型的组合来估计暴露强度场。然后将这些信息与有关研究对象随时间推移的位置信息相结合，以得出他们个人暴露的估计值，以分析暴露-反应关系。出于实际和理论上的原因，这两项任务通常是分开执行的。暴露模型从来都不是完美的，因此一个重要的统计问题就变成了如何将暴露模型中的不确定性传播到健康分析中。在当前的问题中，Szpiro 和 Paciorak (2013) 推导出了健康影响回归系数估计值的两阶段估计量的统计特性。在他们早期工作的基础上（Szpiro 等人，2011b），他们提出了测量误差校正的新方法，证明了其校正估计量的渐近无偏性和有效的置信区间覆盖率。模拟研究和对 MESA/Air 研究数据的应用说明了他们方法的实用性。MESA/Air 研究健康协会的各种实质性出版物在第一阶段依赖于 Sampson 等人描述的复杂时空模型。(2011) 和 Szpiro 等人。(2010)。最近对 PM2.5 成分的全国模型的扩展表明，颈动脉内膜厚度与有机碳、硅和硫有显着关联；对于 Si 和 S，存在显着的空间相关性，导致测量误差校正的置信区间比原始置信区间宽 50%，但仍然显着（Bergen 等，2013）。许多其他小组已经描述了类似的两阶段分析（例如，Gryparis 等，2009；Li 等，2013；Peng 和 Bell，2010）。Szpiro 和 Paciorek 在引言中为两阶段方法的优势提供了令人信服的案例，并提到了联合暴露和健康建模方法的替代方案（例如，Molitor 等人，2007 年；2006 年）。这个问题引起了广泛的争论，我很快就同意他们的观点。在 Molitor 等人。论文中，我们拟合了一个模型，该模型可以用图 1 所示的 DAG 示意性表示。这里，Xs 代表健康影响研究中的受试者 i 或个人监测或暴露研究中的个体或测量位置 j 的真实但未观察到的暴露。暴露浓度 Zj 的直接测量仅适用于暴露研究中的位置子集或参与者，但我们假设两组暴露都由一组共同的预测变量 W 确定，这些预测变量对两个数据集中的每个人都可用，例如接近交通、空间位置等，在这两个部分具有相同的参数向量 θ（回归系数、空间方差分量等）。因此，假设两个数据集的联合分析对于估计模型的所有参数是最有效的，并且允许暴露模型中的不确定性自动传播到主要感兴趣的暴露-响应参数β，这似乎是合理的. 图 1 表示联合暴露和健康影响模型的有向无环图（X = 真实暴露，Z = 测量暴露，W = 暴露决定因素，Y = 健康结果） 实际上，这种联合分析比所考虑的两阶段分析更复杂斯皮罗和帕乔雷克。正如他们所指出的，大多数流行病学研究调查了几种不同的健康结果 Y，因此这需要对每个特征进行单独的联合分析，或者需要一个更复杂的多变量结果模型。为每个结果拥有单独的暴露模型的想法本身就很不吸引人！此外，环境流行病学（尤其是空气污染研究）中的大多数健康协会往往相对薄弱，因此将健康结果纳入暴露模型的潜在附加值可能很小。更重要的是，正如这些作者和其他人所指出的，健康影响模型可能被错误指定，因此从健康模型到暴露模型的反馈可能会产生意想不到的后果。这个问题在其他领域已经被认识到，特别是在药代动力学模型中。例如，Lunn 等人。(2009) 考虑了一个类似的问题，其中“暴露模型”由描述摄入 W 和代谢物浓度 X 之间关系的药代动力学模型组成，“健康模型”是描述 X 与某些生理反应 Y 之间关系的药效模型。在这种情况下，主要感兴趣的可能是 WX 关系的参数 θ，并且担心 XY 关系的错误指定可能会使 θ 的估计产生偏差。为了减轻这种担忧，Lunn 等人。提出了几种重新绘制 DAG 的方法，以“打破”两个子模型之间的联系，同时仍然利用联合分析。与两阶段程序相比，进一步研究单阶段分析的性能以避免这种反馈问题将是有帮助的。参见例如 (Zigler et al., 2013）在比较有效性研究中对贝叶斯倾向评分估计背景下的反馈问题进行了类似的讨论。Szpiro 和 Paciorek 使用的基本框架假设曝光场是固定的（但未知），并且受试者和监测位置是随机的。以这种方式对模型进行概念化具有令人信服的优势，使其对暴露模型的错误指定具有鲁棒性，并提供一种通过引导程序估计偏差和方差的直接方法。然而，这似乎有点违反直觉，因为曝光场确实是随机的，受到各种无法量化的影响，具有相同生成模型的平行世界将经历不同的场。Szpiro 和 Paciorek 方法的另一个关键假设是，对象位置和监测位置不仅来自同一分布，而且也是独立绘制的。在儿童健康研究（Franklin 等人，2012 年；Gauderman 等人，2005 年）中，在对流行病学研究对象的子集进行采样以进行暴露监测的研究中经常违反后一种假设。作者在讨论中评论说“我们的非参数引导程序中的重采样程序可以很容易地修改以反映这种类型的数据生成机制。” 如果使用朴素分析，这种类型的抽样设计是否会产生任何重要的偏差，以及这种修改后的程序与独立抽样相比的效率如何，还有待观察。这表明对研究设计的影响进行进一步研究。在预算有限的情况下，暴露测量与流行病学研究之间的最佳资源分配是什么（即样本大小与测量研究的测量范围之间的权衡，受总成本的限制）？对于暴露研究，应如何选择位置（Thomas，2009，第 184 页）——代表流行病学研究基础或病例分布，最大化空间变异性，最大化预测因子的方差（可能包括先前存在的其他污染物的测量），以最小化未测量位置的预测误差，或其他一些标准？例如，假设一些污染水平的测量值 Zj 和预测变量 Wj 在一组位置 j∈S1 上已经可用，并且希望选择一组额外的位置 S2 以最大化整个网络的信息量 S = S1∪S2用于预测在更大的一组未测量位置 i∈N（例如，一组流行病学研究对象的位置）的暴露量 Xi。这通常需要首先从可用的测量值 S1 估计暴露模型参数。对于任何给定的一组附加位置 S2，然后可以计算其他位置的预测的预期方差（将这些参数估计作为事实）并在“需求面”N 上对这些方差进行平均。然后挑战是选择设置 S2 可以最小化这个预测方差（Diggle 和 Lophaven，2005 年；Kanaroglou 等人，2005 年）。直观上，人们希望在当前网络下预测方差最大且影响最大的那些点上添加最能改善预测的测量位置；因此，人们可能希望选择距离现有测量位置或彼此不太近，但靠近许多需要预测的对象的点。例如，可以从流行病学研究参与者的家中一次选择一个个人的位置，这将导致该平均方差的最大减少，然后迭代地将集合中的位置替换为将导致进一步减少的集合之外的位置. 在两阶段设计中，而不是单独最小化平均曝光预测方差，真正需要的是从将使用这些预测的模型中最大化相对风险参数 β 的 Fisher 信息。这可能需要选择能够最大限度地提高最有影响力点的精度的测量位置，例如那些预测为最严重暴露的个体。此外，我自己的努力 (Thomas, 2007) 通过模拟比较这些不同的设计中的一些产生了令人失望的结果：很难找到一种在简单随机抽样的基础上持续改进的设计！对空间抽样设计的进一步研究将非常有帮助（例如，Chang 等，2007）。例如那些被预测为最严重暴露的人。此外，我自己的努力 (Thomas, 2007) 通过模拟比较其中一些不同的设计产生了令人失望的结果：很难找到一种在简单随机抽样的基础上持续改进的设计！对空间抽样设计的进一步研究将非常有帮助（例如，Chang 等人，2007 年）。例如那些被预测为最严重暴露的人。此外，我自己的努力 (Thomas, 2007) 通过模拟比较其中一些不同的设计产生了令人失望的结果：很难找到一种在简单随机抽样的基础上持续改进的设计！对空间抽样设计的进一步研究将非常有帮助（例如，Chang 等，2007）。