Mitigating the impact of measurement error when using penalized regression to model exposure in two-stage air pollution epidemiology studies

Multipollutant Measurement Error in Air Pollution Epidemiology Studies Arising from Predicting Exposures with Penalized Regression Splines

“…The ε k could be heteroscedastic, and are not assumed to be normally distributed. As in Szpiro and Paciorek (2013) and Bergen and Szpiro (2015), the subject-specific covariates z k are defined as $${\mathbf{z}}_k=\mathrm{bold\Theta }false({\mathbf{s}}_{k}false)+{\mathit{\zeta }}_k,$$ where Θ ( s k ) = (Θ k 1 , …, Θ km ) T represents the spatially structured components of the subject-specific covariates (e.g. body mass index, socioeconomic status, or thin plate splines used to adjust for unmeasured spatial confounding), and the ζ k = (ζ k 1 , …, ζ km ) are random m –vectors, independent between subjects and independent of η k , that do not depend on space.…”

“…Bergen and Szpiro (2015) show how each model yields definitions of the spatial bases and the penalty matrices D j , and show an explicit connection between penalized regression and mixed effects models. This translation motivates selecting the λ j via restricted maximum likelihood (REML).…”

“…Our assumed data generating mechanisms follow those described in Szpiro and Paciorek (2013) and Bergen and Szpiro (2015). We describe our methodology in the context of our case study, emphasizing that extension to higher-dimension pollutant vectors is easily accomplished at the cost of notational simplicity.…”

“…In this context Π is a 2 × 2 identity matrix, and Λ is block diagonal with elements λ j D j , where the D j are ( p j + q j ) × ( p j + q j ) square matrices with penalty parameters λ j penalizing the coefficients of { p j (·), q j (·)} (see Bergen and Szpiro (2015) for more details). Then γ̂ can be expressed as $$\hat{\mathit{\gamma }}=\underset{\mathit{\theta }}{\text{argmin}}true\{true(\sum _{i:{\mathbf{s}}_i^{*}\in {𝒮}_0^{*}\cup {𝒮}_1^{*}}{false(x_{i1}-{\mathbf{r}}_1{false({\mathbf{s}}_i^{*}false)}^T\mathrm{bold-italic\theta }false)}^2+{\lambda }_1{\mathit{\theta }}^T{D}_1\mathrm{bold-italic\theta }true)+true(\sum _{i:{\mathbf{s}}_i^{*}\in {𝒮}_0^{*}\cup {𝒮}_2^{*}}{false(x_{i2}-{\mathbf{r}}_2{false({\mathbf{s}}_i^{*}false)}^T\mathrm{bold-italic\theta }false)}^2+{\lambda }_2{\mathit{\theta }}^T{D}_2\mathrm{bold-italic\theta }true)true\}=\underset{\mathit{\theta }}{\text{argmin}}false(S{S}_{1}false(\mathrm{bold-italic\theta }false)+S{S}_{2}false(\mathrm{bold-italic\theta }false)false).$$ Because of the form of the r j , minimizing this equation with respect to θ is equivalent to separately minimizing SS 1 ( θ ) with respect to the first ( p 1 + q 1 ) elements of θ and SS 2 ( θ ) with respect to the last ( p 2 + q 2 ) elements of θ .…”

“…We extend the methods of Bergen and Szpiro (2015) to the multi-pollutant case, characterizing multi-pollutant measurement error when using penalized regression splines to predict exposures at health subject locations. Although our methods apply to a general class of penalized regression models, our treatment focuses on fitting separate penalized regression models to each pollutant.…”