Spatial measurement errors in the field of spatial epidemiology
Covariate-Outcome spatially misaligned measurement errors
Covariate-Outcome spatial misaligned measurement errors comes from the process of alignment between covariate and outcome, which is mainly caused by the inconsistent measurement or usage of location data. The outcome and the covariates are often observed at different locations or aggregated over different geographical units. Such data are said to be spatially misaligned. For example, a point-to-point misalignment problem arises when relating air quality measurements, observed at monitors (points), and birth weights observed at the residential locations of the mothers (different points). In many spatial epidemiologic studies, the locations of covariates and health outcomes do not match.
Standard regression methods cannot be applied to such misaligned data. To overcome this problem, several methods have been proposed. Most approaches involve directly using predictions from statistical exposure models that incorporate spatial structure [73–75] such as kriging and its extensions [76], Gaussian process (GP) modeling and Bayesian smoothing [77, 78], penalized regression splines [73, 79], and kernel smoothing [80], among others. The most common strategy is to employ one of the previously mentioned models to predict covariates at locations with outcomes and then estimate a regression parameter of interest using the predicted covariates. For example, Higgins et al. [81] used polynomial regression to generate covariate predictions when outcomes and covariates were misaligned. Waller and Gotway [82] used kriging to predict exposures and used resampling to account for the uncertainty in using the predictions in place of the true values. Kunzli et al. [74] assigned exposure values for subject-specific locations derived from a geostatistical model and used weighted least squares in the subsequent health effects model with the weights specified as the inverse of the standard errors (SEs) from the exposure kriging model. For a similar problem, Madsen et al. [83] considered both a generalized least squares (GLS) estimator with a bootstrap type variance estimator and a maximum likelihood approach that jointly fits the exposure and health models.
Such covariate predictions contain measurement errors since the predicted values will not equal the true exposures [84, 85]. Using predictions rather than true exposures in health modeling introduces two sources of measurement error-Berkson–like errors arising from smoothing the true covariate surface and classical–like errors coming from estimating the covariate model parameters, which have been widely discussed in environmental epidemiology [86]. For example, when assessing health effects of particulate matter (PM) constituents, it is necessary to effectively estimate exposure and to account for exposure errors induced by spatial misalignment to avoid bias [87]. After characterizing the spatial misalignment using geostatistical methods, Goldman et al. [88] found that errors due to spatial misalignment resulted in average risk ratio reductions of 16 % for secondary pollutants (O3, PM2.5 sulfate, nitrate and ammonium) and between 43 and 68 % for primary pollutants (NOx, NO2, SO2, CO, PM2.5 elemental carbon) while pollutants of mixed origin (PM10, PM2.5, PM2.5 organic carbon) had intermediate impacts. Sheppard et al. [89] found that measurement errors resulting from spatial misalignment led to an attenuation of acute health effect estimates of 7.7 % for PM2.5 mass. Ong et al. [90] found that relative to geographically corrected data, spatial misaligned information produced a modest bias in the aggregated number of facilities at risk but generated a substantial number of false positives and negatives.
Some methods to correct for spatial misalignment have been proposed. Lopiano et al. [91] developed an approach for an REML-based estimated generalized least squares (RBEGLS) estimator accounting for the misalignment error structure and estimating covariance parameters using likelihood-based methods. They also provide insights into when it is important to fully account for the covariance structure induced from the different error sources. These researchers also developed another pseudo-penalized quasi-likelihood algorithm to account for spatial misaligned errors and showed by simulation that the method performs well in terms of coverage for 95 % confidence intervals [92]. Szpiro et al. [93] characterized the measurement errors by decomposing it into Berkson-like and classical-like components and developed two correction approaches of the parametric bootstrap and the “parameter bootstrap” [86] and also proposed another robust method for the spatially misaligned errors to correct finite-sample bias and correctly estimate standard errors. Gryparis et al. [84] developed a generalized linear model framework for spatial misaligned measurement error modeling; Chang et al. [94] addressed the challenge of exposure measurement errors due to spatial misalignment through measurement error modeling and developed a Bayesian framework to fully account for uncertainty in the estimation of model parameters. Bayesian hierarchical models which account for uncertainties due to spatial misalignment were also applied to spatial correlated exposures measured with errors by Smith et al. [95]. and Molitor et al. [96].
In the presence of Covariate-Outcome spatial misaligned measurement errors, the covariate is not accurate (?
6 ? 0, ?
7 ? 0, ?
8 ? 0). The mismeasured locations and the outcome (Y) are only affected by themselves (?
0 ? 1, ?
4 ? 1). The observed data takes the form (L, Y, X).