Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

Page, Garritt L.; Liu, Yanjun; He, Zhuoqiong; Sun, Donchu

doi:10.1111/sjos.12275

Cited by 29 publications

(55 citation statements)

References 23 publications

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“…In addition, if we have R = R 1 = R 2 = R 3 , then

σ_{W}^{2} = b_{31}^{2} + b_{33}^{2}

, and

Bias false(\overset{bold-italicβ}{true} false) = {((), 0 0 σ_{ν, W} false/ σ_{W}^{2})}^{boldT}

(see Section 1.1.2 of the Supporting Information for details). The latter result is similar to the one obtained by Page et al (2017) as we can write

Bias false(\overset{bold-italicβ}{true} false) = {((), 0 0 ρ_{ν, W} σ_{ν} false/ σ_{W})}^{boldT}

, because

ρ_{ν, W} = Corr false(ν false(bolds false), W false(bolds false) false) = σ_{ν, W} false/ \sqrt{σ_{ν}^{2} σ_{W}^{2}}

with σ ν , W = Cov( ν ( s ), W ( s )). And it is clear that the bias does not depend on the spatial structure of the processes nor on the variance of X j (·).…”

Section: Spatial Random Intercept Modelssupporting

confidence: 88%

“…Looking at the elements of T , we note that

Var false(ν false(bolds false) false) = σ_{ν}^{2} = b_{11}^{2}

Var false(κ false(bolds false) false) = σ_{κ}^{2} = b_{21}^{2} + b_{22}^{2}

and

Var false(W false(bolds false) false) = σ_{W}^{2} = b_{31}^{2} + b_{32}^{2} + b_{33}^{2}

. To investigate the effect of the structure in Equation 4 on the estimator

\overset{bold-italicβ}{true}

, we follow similar steps as those described in Page et al (2017).…”

Section: Spatial Random Intercept Modelsmentioning

confidence: 99%

“…Section 2 discusses the effects of confounding in a random intercept model considering unit and cluster‐level covariates. The main differences from previous analysis (Paciorek, 2010; Page et al , 2017) are that we have more than one observation per cluster (location) and they are assumed independent conditional on any cluster‐specific random effects. As there are replicates available in each cluster, we show that even when the cluster‐level random effect is assumed to be i.i.d.…”

Section: Introductionmentioning

confidence: 95%

“…To mitigate this potential problem, they propose a posterior predictive approach (RSR‐PPD), which inflates the variance of the regressor coefficients to reflect possible collinearity between fixed and random effects. Page et al (2017) follow a similar setup of Paciorek (2010) and investigate the effects of spatial confounding on the kriging predictor for observed quantities. The results suggest lower values of bias as the effective range of observable and unobservable variables varies in similar spatial scales.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Effects of Spatial Confounding in Hierarchical Models

Nobre

Schmidt

Pereira

2020

Int Statistical Rev

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Summary Usually, in spatial generalised linear models, covariates that are spatially smooth are collinear with spatial random effects. This affects the bias and precision of the regression coefficients. This is known in the spatial statistics literature as spatial confounding. We discuss the problem of confounding in the case of multilevel spatial models wherein there are multiple observations within clusters. We show that even under the standard multilevel model, which allows for independent (i.e. not spatially correlated) cluster effects, the cluster‐level fixed effects might be biased depending on the structure of the ‘true’ generating mechanism of the processes. We provide simulation studies in order to investigate the effects of confounding in the estimation of fixed effects present in random intercept models under different scenarios of confounding. One remedy to spatial confounding is restricted spatial regression wherein the spatial random effects are constrained to be orthogonal to the fixed effects of the model. We propose one way to fit a restricted spatial regression model for multilevel data and illustrate it with artificial data analyses. We also briefly describe the issue of confounding in random intercept and slope models.

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“…In addition, if we have R = R 1 = R 2 = R 3 , then

σ_{W}^{2} = b_{31}^{2} + b_{33}^{2}

, and

Bias false(\overset{bold-italicβ}{true} false) = {((), 0 0 σ_{ν, W} false/ σ_{W}^{2})}^{boldT}

(see Section 1.1.2 of the Supporting Information for details). The latter result is similar to the one obtained by Page et al (2017) as we can write

Bias false(\overset{bold-italicβ}{true} false) = {((), 0 0 ρ_{ν, W} σ_{ν} false/ σ_{W})}^{boldT}

, because

ρ_{ν, W} = Corr false(ν false(bolds false), W false(bolds false) false) = σ_{ν, W} false/ \sqrt{σ_{ν}^{2} σ_{W}^{2}}

with σ ν , W = Cov( ν ( s ), W ( s )). And it is clear that the bias does not depend on the spatial structure of the processes nor on the variance of X j (·).…”

Section: Spatial Random Intercept Modelssupporting

confidence: 88%

“…Looking at the elements of T , we note that

Var false(ν false(bolds false) false) = σ_{ν}^{2} = b_{11}^{2}

Var false(κ false(bolds false) false) = σ_{κ}^{2} = b_{21}^{2} + b_{22}^{2}

and

Var false(W false(bolds false) false) = σ_{W}^{2} = b_{31}^{2} + b_{32}^{2} + b_{33}^{2}

. To investigate the effect of the structure in Equation 4 on the estimator

\overset{bold-italicβ}{true}

, we follow similar steps as those described in Page et al (2017).…”

Section: Spatial Random Intercept Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Effects of Spatial Confounding in Hierarchical Models

Nobre

Schmidt

Pereira

2020

Int Statistical Rev

View full text Add to dashboard Cite

show abstract

“…Clayton et al (1993) advocated for the inclusion of a spatially-correlated error term in hierarchical models for spatial data (Clayton and Kaldor, 1987;Besag et al, 1991) and claimed this would account for confounding bias but might result in conservative inference. Since then, the approach of adding spatially structured error terms has frequently been used in spatial models for areal data (Reich et al, 2006;Wakefield, 2007;Hodges and Reich, 2010;Hughes and Haran, 2013;Hanks et al, 2015;Page et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Selecting a Scale for Spatial Confounding Adjustment

Keller

Szpiro

2020

Journal of the Royal Statistical Society Series A: Statistics in Society

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Unmeasured, spatially-structured factors can confound associations between spatial environmental exposures and health outcomes. Adding flexible splines to a regression model is a simple approach for spatial confounding adjustment, but the spline degrees of freedom do not provide an easily interpretable spatial scale. We describe a method for quantifying the extent of spatial confounding adjustment in terms of the Euclidean distance at which variation is removed. We develop this approach for confounding adjustment with splines and using Fourier and wavelet filtering. We demonstrate differences in the spatial scales these bases can represent and provide a comparison of methods for selecting the amount of confounding adjustment. We find the best performance for selecting the amount of adjustment using an information criterion evaluated on an outcome model without exposure. We apply this method to spatial adjustment in an analysis of particulate matter and blood pressure in a cohort of United States women.

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Conditional Autoregressive (CAR) Model

Schmidt

Nobre

2018

Wiley StatsRef: Statistics Reference Online

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Conditional autoregressive (CAR) models are useful to obtain a multivariate joint distributions of a random vector based on univariate conditional specifications. These conditional specifications are based on Markovian properties such that the conditional distribution of a component of the random vector depends only on a set of neighbors. Conditional autoregressive models are particular cases of Markov random fields. CAR models have been applied in different areas of science; some examples are image analysis, epidemiology, and agriculture. Typically, Gaussian CAR specifications are used as latent structures in hierarchical models for areal level data. In this case, the region of interest is divided into a set of disjoint areas and a CAR random effect is used to account for possible correlation among observations made across the different areas.

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Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

Cited by 29 publications

References 23 publications

On the Effects of Spatial Confounding in Hierarchical Models

On the Effects of Spatial Confounding in Hierarchical Models

Selecting a Scale for Spatial Confounding Adjustment

Conditional Autoregressive (CAR) Model

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