In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered modeling ...forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. We propose a novel approach, spatial+, for dealing with spatial confounding when the covariate of interest is spatially dependent but not fully determined by spatial location. Using a thin plate spline model formulation we see that, in this case, the bias in covariate effect estimates is a direct result of spatial smoothing. Spatial+ reduces the sensitivity of the estimates to smoothing by replacing the covariates by their residuals after spatial dependence has been regressed away. Through asymptotic analysis we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straightforward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non‐Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.
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Regression models for spatially varying data use spatial random effects to reflect spatial correlation structure. Such random effects, however, may interfere with the covariate effect estimates and ...make them unreliable. This problem, known as spatial confounding, is complex and has only been studied for models with linear covariate effects. However, as illustrated by a forestry example in which we assess the effect of soil, climate, and topography variables on tree health, the covariate effects of interest are in practice often unknown and nonlinear. We consider, for the first time, spatial confounding in spatial models with nonlinear effects implemented in the generalised additive models (GAMs) framework. We show that spatial+, a recently developed method for alleviating confounding in the linear case, can be adapted to this setting. In practice, spatial+ can then be used both as a diagnostic tool for investigating whether covariate effect estimates are affected by spatial confounding and for correcting the estimates for the resulting bias when it is present. Supplementary materials accompanying this paper appear online.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this rejoinder, we set out some of the main points that we took from the discussions of our paper “Spatial+: A novel approach to spatial confounding.” The comments provided by the discussants ...include excellent questions and suggestions for extensions and improvements to spatial+. The discussions also highlight the growing interest in understanding spatial confounding, underpinned by the many recent contributions to the literature on this topic.
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Spatial regression models are commonly used in applied statistics to model data collected at different spatial locations. Such models use spatial random effects to account for residual spatial ...correlation in the response variable and result in fitted values that are, to some degree, smoothed across the spatial domain of the data. The main focus of this thesis is a problem known as spatial confounding, which causes covariate effect estimates in spatial models to be unreliable. By investigating the estimation theory in a commonly used spatial model formulation based on thin plate splines, we gain a deeper understanding of the problem and the existing methodology. Using this, we develop a novel and easily implementable method for avoiding spatial confounding in practice. Moreover, we include some initial analysis on spatial confounding in models with non-linear covariate effects; an area that has not yet been explored in the literature. The thesis also contains another project within the field of spatial statistics. Here, spatial modelling techniques are used to develop a method for detecting spatially coherent trends in environmental time series data. Specifically, we model river flow data from gauging stations across Great Britain. Using our methodology, we are able to verify, for the first time, a significant upward trend in flood risk over time and identify the regions with the largest trends.
With increasing concerns on the impacts of climate change, there is wide interest in understanding whether hydrometric and environmental series display any sort of trend. Many studies however, focus ...on the analysis of highly variable individual series at each measuring location. We propose a novel and straightforward approach to trend detection, modelling the test statistic for trend at each location via an areal model in which the information across measuring locations is pooled together. We exemplify the method with a detailed study of change in high flows in Great Britain. Using areal models, we detect a statistically relevant signal for a positive trend across Great Britain in the recent decades. This evidence is also found when different temporal subsets of the records are analysed. Further, the model identifies areas where the increase has been higher or lower than average, thus providing a way to prioritise intervention.
Plain language summary
With growing concerns over the potential impacts of climate change, many studies are investigating whether river extremes, such as floods, are changing. Studies based on climate change projections indicate that changes might be expected in several parts of the world, including Great Britain where floods are predicted to increase. However, studies investigating measured river flow records have mostly found inconclusive evidence of change. This does not mean that change is not happening, but finding the evidence of this change is difficult because flow records are short and very variable. In this study we suggest that river flow measuring stations on the same river will experience similar changes since they are affected by the same climate. We therefore propose to use advanced statistical models, which combine information from nearby stations and apply these model to high flows measurements in Great Britain. The analysis of data from closely located measuring stations demonstrates that flows have generally become bigger in Great Britain recently. The methods proposed in the manuscript could be easily applied to other type of data routinely measured and which might have been changing over time as a result of climate change or other drivers.
Key Points
We propose a novel approach to regional detection of trends in measured series based on areal models
We detect a clear signal that peak flows magnitudes are increasing over time in Great Britain
These changes are still found when different periods of record are analyzed, with an accelerated upward trend from 1980 onward
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The complex manifold
n
×
n+1 with symplectic form σμ = σ
n
+ μσ
n+1
, where σ
n
and σ
n+1
are normalized Fubini-Study forms, n ∈ and μ > 1 a real number, contains a natural Lagrangian sphere L
μ. We ...prove that the Dehn twist along L
μ is symplectically isotopic to the identity for all μ > 1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in
n
×
n+1 and lifts to the blow-up of
n
×
n+1 along a complex n-dimensional submanifold.
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The complex manifold ℂℙn × ℂℙn+1 with symplectic form σμ = σℂℙn + μσℂℙn+1, where σℂℙn and σℂℙn+1 are normalized Fubini–Study forms, n ∈ ℕ and μ > 1 a real number, contains a natural Lagrangian sphere ...Lμ. We prove that the Dehn twist along Lμ is symplectically isotopic to the identity for all μ > 1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in ℂℙn × ℂℙn+1 and lifts to the blow-up of ℂℙn × ℂℙn+1 along a complex n-dimensional submanifold.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
Spatial confounding is a fundamental issue in regression models for spatially
indexed data. It arises because spatial random effects, included to approximate
unmeasured spatial variation, are ...typically not independent of the covariates
in the model. This can lead to significant bias in covariate effect estimates.
Despite extensive research, it is still a topic of much confusion with
sometimes puzzling and seemingly contradictory results. In this paper we
develop a broad theoretical framework that brings mathematical clarity to the
mechanisms of spatial confounding, providing explicit and interpretable
analytical expressions for the resulting bias. From these, we see that it is a
problem directly linked to spatial smoothing, and we can identify exactly how
the features of the model and the data generation process affect the size and
occurrence of bias. We also use our framework to understand and generalise some
of the main results on spatial confounding in the past, including suggested
methods for bias adjustment. Thus, our comprehensive and mathematically
explicit approach clears up existing confusion and, indeed, demystifies the
issue of spatial confounding.
In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered ...modelling forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. The mechanism behind spatial confounding is poorly understood and methods for dealing with it are limited. We propose a novel approach, spatial+, in which collinearity is reduced by replacing the covariates in the spatial model by their residuals after spatial dependence has been regressed away. Using a thin plate spline model formulation, we recognise spatial confounding as a smoothing-induced bias identified by Rice (1986), and through asymptotic analysis of the effect estimates, we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straight-forward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non-Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.
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