Abstract

Spline surfaces are often used to capture spatial variability sources in linear mixed-effects models, without imposing a parametric covariance structure on the random effects. However, including a spline component in a semiparametric model may change the estimated regression coefficients, a problem analogous to spatial confounding in spatially correlated random effects. Our research aims to investigate such effects in spline-based semiparametric regression for spatial data. We discuss estimators' behavior under the traditional spatial linear regression, how the estimates change in spatial confounding-like situations, and how selecting a proper tuning parameter for the spline can help reduce bias.

Keywords

semiparametric regression, spatial interaction, spatial statistics

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Comments to Editor

Thank you for accepting the paper. We have made changes to reflect the referee's comments, which we address in a point-by-point basis below. (1) The intro was presented nicely and was an excellent review. My only comment on that would be to discuss Hughes and Haran in a bit more detail as I believe it is the most practical of all of the ways to mitigate confounding (for areal data). Thank you. We have now included a bit more detail about Hughes and Haran at the end of the second paragraph of the introduction. (2) Although the simulation study is illustrative for the case where the spatial process is stationary, it isn't very realistic to assume that if covariates were missing from the analysis that they would be from a stationary process. It would be helpful to see some simulations under various types of nonstationarity (or, at least a discussion of those results given potential space limitations). Thank you. We added two nonstationary cases in a new section (Section 4.3), extending the simulation results in Section 4.2. While the scope is somewhat limited, we anticipate that splines will help mitigate spatial confounding as long as the spatial process is varying smoothly. (3) It would also be interesting to see the effects of potential confounding when the measurement (nugget) variance is at different levels (i.e., how does it behave relative to different SNRs)...again, this is much more likely to be a concern in real world problems. Thank you for the suggestion. We extended Section 4.2 to include different signal to noise ratios, in the new Figure 10. (4) Lastly, I would have liked to see more discussion/simulations to show how well the new algorithm works. Thank you. This is a work in process – we are still investigating other choices of penalties. We have added a comment in the discussion section to clarify this.

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Jan 1st, 1:04 AM

On fixed effects estimation in spline-based semiparametric regression for spatial data

Spline surfaces are often used to capture spatial variability sources in linear mixed-effects models, without imposing a parametric covariance structure on the random effects. However, including a spline component in a semiparametric model may change the estimated regression coefficients, a problem analogous to spatial confounding in spatially correlated random effects. Our research aims to investigate such effects in spline-based semiparametric regression for spatial data. We discuss estimators' behavior under the traditional spatial linear regression, how the estimates change in spatial confounding-like situations, and how selecting a proper tuning parameter for the spline can help reduce bias.