Department of Mathematics and Statistics, American University, Washington, DC 20016, USA.
Biostatistics. 2010 Jul;11(3):432-52. doi: 10.1093/biostatistics/kxq003. Epub 2010 Feb 15.
Frequently, exposure data are measured over time on a grid of discrete values that collectively define a functional observation. In many applications, researchers are interested in using these measurements as covariates to predict a scalar response in a regression setting, with interest focusing on the most biologically relevant time window of exposure. One example is in panel studies of the health effects of particulate matter (PM), where particle levels are measured over time. In such studies, there are many more values of the functional data than observations in the data set so that regularization of the corresponding functional regression coefficient is necessary for estimation. Additional issues in this setting are the possibility of exposure measurement error and the need to incorporate additional potential confounders, such as meteorological or co-pollutant measures, that themselves may have effects that vary over time. To accommodate all these features, we develop wavelet-based linear mixed distributed lag models that incorporate repeated measures of functional data as covariates into a linear mixed model. A Bayesian approach to model fitting uses wavelet shrinkage to regularize functional coefficients. We show that, as long as the exposure error induces fine-scale variability in the functional exposure profile and the distributed lag function representing the exposure effect varies smoothly in time, the model corrects for the exposure measurement error without further adjustment. Both these conditions are likely to hold in the environmental applications we consider. We examine properties of the method using simulations and apply the method to data from a study examining the association between PM, measured as hourly averages for 1-7 days, and markers of acute systemic inflammation. We use the method to fully control for the effects of confounding by other time-varying predictors, such as temperature and co-pollutants.
通常，暴露数据是在离散值网格上随时间测量的，这些值共同定义了一个功能观测。在许多应用中，研究人员有兴趣将这些测量值作为协变量，以在回归环境中预测标量响应，关注的焦点是暴露的最具生物学相关性的时间窗口。一个例子是在颗粒物（PM）健康影响的面板研究中，其中随着时间的推移测量颗粒水平。在这些研究中，功能数据的值比数据集中的观测值多得多，因此必须对相应的功能回归系数进行正则化，以便进行估计。在这种情况下，还存在暴露测量误差的可能性，以及需要纳入其他潜在混杂因素的问题，例如气象或共同污染物的测量值，这些因素本身可能具有随时间变化的影响。为了适应所有这些特征，我们开发了基于小波的线性混合分布滞后模型，将功能数据的重复测量作为协变量纳入线性混合模型中。模型拟合的贝叶斯方法使用小波收缩来正则化功能系数。我们表明，只要暴露误差在功能暴露曲线中引起细粒度的可变性，并且代表暴露效应的分布滞后函数随时间平滑变化，该模型就可以在无需进一步调整的情况下纠正暴露测量误差。我们考虑的环境应用中很可能满足这两个条件。我们使用模拟来检查方法的性质，并将该方法应用于研究 PM 与急性全身炎症标志物之间关联的研究数据，PM 以 1-7 天的每小时平均值来衡量。我们使用该方法完全控制了其他随时间变化的预测因子（如温度和共同污染物）的混杂影响。
