A challenge for association studies in environmental statistics is that we cannot directly measure the exposure at every location where there is outcome data. Modern Geographic Information System (GIS) technology makes it feasible to sample environmental exposures and then to predict exposures at unmonitored locations using a statistical model such as universal kriging that exploits dependence on GIS covariates and incorporates spatial smoothing (Cressie, 1993
). The overall strategy is to use predicted exposures in place of the true exposures at locations with outcome data in order to estimate the parameter of interest in a regression model. The problem that we address in this paper is how to ensure valid inference in light of the resulting measurement error.
An example application in environmental epidemiology is evaluating the relationship between exposure to ambient air pollution and adverse health outcomes. Many studies have documented adverse effects of air pollution e.g. (Dockery and others, 1993; Samet2000; Pope2002), and recent studies emphasize the importance of using predicted individual air pollution exposures to account for spatial variability within urban areas (Jerrett and others, 2005b; Kunzli2005; Gryparis2007; Szpiro2009). Other environmental applications that do not involve human health effects are analogous from a statistical perspective. An example that we will return to later in this paper involves assessing the relationship between stream water quality and nearby watershed land cover (Madsen and others, 2008; Herlihy1998).
Various methods have been employed for predicting exposures, including nearest neighbor interpolation (Miller and others, 2007), regression based on GIS covariates (Brauer and others, 2003; Jerrett2005a), interpolation by a geostatistical method such as kriging (Jerrett and others, 2005b; Kunzli2005), and semi-parametric smoothing (Gryparis and others, 2007; Kunzli2005). All these methods result in measurement error that does not fit into the standard categories of classical or Berkson error (Carroll and others, 2006). In this paper, we focus on universal kriging.
Kim and others (2009) have shown that using predicted exposures from kriging performs better than nearest neighbor interpolation but significant errors may remain resulting in confidence intervals that do not provide correct coverage. Gryparis and others (2009) review the relevant measurement error literature and compare several correction strategies in a simulation study, and Madsen and others (2008) apply a version of the parametric bootstrap to obtain corrected standard errors (SEs).
The parametric bootstrap is effective, but it is computationally intensive since it requires solving a nonlinear optimization problem to estimate the exposure model parameters in each bootstrap sample. For a universal kriging exposure model with 450 monitors (as in the examples considered here), each nonlinear optimization takes 30–60 s on an Intel Xeon processor running at 2.33 GHz, so a parametric bootstrap with only 100 samples would take approximately 1 h. This is uncomfortably long for routine usage, but it is feasible if the bootstrap is employed judiciously. If we consider, instead, a more complex spatiotemporal model of the kind being used in modern air pollution studies (Szpiro and others, 2010), the time required for a single optimization is an hour or more, so a full parametric bootstrap is essentially impractical unless its use is restricted to a very limited number of definitive analyses.
We describe a new method termed the “parameter bootstrap” that is a less computationally demanding approximation to the parametric bootstrap. The parameter bootstrap is consistent with a decomposition of the measurement error into 2 approximately independent components, one of which is similar to Berkson error (“Berkson-like”) and the other of which is similar to classical measurement error (“classical-like”). We develop our methodology in a setting where we use universal kriging to predict the exposure and where we model the association of interest with linear regression, including the possibility of spatially correlated residuals. The methodology extends easily to more complex spatiotemporal exposure models that generalize universal kriging (Banerjee and others, 2004), (Szpiro and others, 2010).
In Section 2, we introduce notation and formally set out the problem. In Section 3, we characterize the measurement error by decomposing it into Berkson-like and classical-like components, and in Section 4, we define the parametric and parameter bootstraps and briefly review 2 alternative strategies that have been proposed in recently published papers. In Section 5, we illustrate our methodology in a simulation study and compare it to other methods, and in Section 6, we consider an example with publicly available stream data from the Environmental Protection Agency (EPA). We conclude in Section 7 with a discussion.