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. 2017 Nov 9;20(1):1–16. doi: 10.1093/biostatistics/kxx058

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© The Author 2017. Published by Oxford University Press.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Fig. 1. — The figure represents the developed EPS framework. The left hand side corresponds to (2.1) and (2.2): the area level confounders are estimated from the individual level confounders (2.1) and the EPS is estimated (2.2). Note that this model is only specified on the areas. EPS is represented by a circle as a latent quantity (not observed). The right hand side presents the EPS imputation and adjustment and is specified on : at the top the EPS is either obtained from (2.2) for and included as an observed quantity or missing for , imputed through the relationship with , , and the spatial structure (2.3) and included as a latent quantity. Thus the EPS is represented here as both a square and a circle as the combination of observed and latent quantities. At the bottom, the estimated and imputed EPS is included in the analysis model (corresponding to (2.4)) to provide confounder adjustment when assessing the effect of on . Note that the right hand side is separated from the left hand side as the EPS estimation is carried out disjointly from the imputation and adjustment (no feedback from the latter to the former). At the same time the imputation and adjustment are jointly carried out so that feedback is allowed from to influence the EPS imputation.

Inline graphic — The figure represents the developed EPS framework. The left hand side corresponds to (2.1) and (2.2): the area level confounders are estimated from the individual level confounders (2.1) and the EPS is estimated (2.2). Note that this model is only specified on the areas. EPS is represented by a circle as a latent quantity (not observed). The right hand side presents the EPS imputation and adjustment and is specified on : at the top the EPS is either obtained from (2.2) for and included as an observed quantity or missing for , imputed through the relationship with , , and the spatial structure (2.3) and included as a latent quantity. Thus the EPS is represented here as both a square and a circle as the combination of observed and latent quantities. At the bottom, the estimated and imputed EPS is included in the analysis model (corresponding to (2.4)) to provide confounder adjustment when assessing the effect of on . Note that the right hand side is separated from the left hand side as the EPS estimation is carried out disjointly from the imputation and adjustment (no feedback from the latter to the former). At the same time the imputation and adjustment are jointly carried out so that feedback is allowed from to influence the EPS imputation.