Structural Equation Models for Dealing With Spatial Confounding

ABSTRACT

In regression analyses of spatially structured data, it is common practice to introduce spatially correlated random effects into the regression model to reduce or even avoid unobserved variable bias in the estimation of other covariate effects. If besides the response the covariates are also spatially correlated, the spatial effects may confound the effect of the covariates or vice versa. In this case, the model fails to identify the true covariate effect due to multicollinearity. For highly collinear continuous covariates, path analysis and structural equation modeling techniques prove to be helpful to disentangle direct covariate effects from indirect covariate effects arising from correlation with other variables. This work discusses the applicability of these techniques in regression setups, where spatial and covariate effects coincide at least partly and classical geoadditive models fail to separate these effects. Supplementary materials for this article are available online.

KEYWORDS:

• Bias reduction
• Geoadditive regression
• Path analysis
• Recursive SEM

Supplementary Materials

The online supplement for this article contains one of the simulated datasets (gradient structure, see Section 4). Additionally, we illustrate:

- how the spatial indicator variables (see Equation 3) can be constructed,

- the likelihood-based and Bayesian estimation approaches (see Section 3),

- and how the estimated effects can be visualized (similar to Figure 5).

Notes

1 We also simulated additional covariate effect sizes β₂₁. Qualitatively, the results do not change, so we restrict the presentation to the aforementioned case of β₂₁ = 3.

2 The MSE becomes substantially larger in gSEM, when there is basically no variability in the covariate beyond the spatial information. On the other hand, in these cases, it should be questioned, if confounder and covariate really represent different influences on the response anyway, depending on the application.

3 The choice of these standard errors does not affect the results of the first two scenarios qualitatively, since in these cases, no confounding is present in the data.

4 Note again, that these weakly informative priors for ${\gamma }_1$ and ${\gamma }_2$ correspond to unpenalized spatial effects from a frequentist perspective. For details on appropriate priors which incorporate the neighborhood structure, see Rue and Held (2005).

Additional information

Funding

The authors acknowledge financial support by the German Research Foundation (DFG), research training group 1644 Scaling Problems in Statistics.