Dimension Reduction and Alleviation of Confounding for Spatial Generalized Linear Mixed Models

John Hughes; Murali Haran

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Dimension Reduction and Alleviation of Confounding for Spatial Generalized Linear Mixed Models

John Hughes, Murali Haran

arXiv:1011.6649v1Nov 30, 2010stat.MEmath.ST

TL;DR

Spatial generalized linear mixed models are flexible for non-Gaussian spatial data but suffer from confounding, variance inflation, and computational difficulty from high-dimensional random effects. The paper develops a graph-based reparameterization that mitigates confounding, restricts dependence to positive patterns, and reduces random-effect dimension. Across Bernoulli, Poisson, and Gaussian settings, it reports better regression inference and more efficient MCMC, with no more than 0.1n random effects generally appearing sufficient.

Problem
SGLMMs have uninterpretable and variance-inflated regression parameters under spatial confounding, while high-dimensional random effects make Bayesian inference computationally challenging.
Method
The paper reparameterizes the areal SGLMM using graph-based geometry and a Moran operator to remove confounded components, retain positive spatial dependence, and reduce random-effect dimension.
Results
Across Bernoulli, Poisson, and Gaussian models, the approach yielded better regression inference and considerable MCMC efficiency gains; it appears to require no more than 0.1n random effects.
Takeaways & Limitations
The reduced model makes relatively rapid analysis of areal datasets feasible, including datasets previously considered too large for the areal SGLMM.
Takeaways & Limitations
The RHZ alternative permits negative spatial dependence, whereas the proposed model restricts random effects to positive spatial dependence patterns.

Abstract

from arXiv · show

Non-gaussian spatial data are very common in many disciplines. For instance, count data are common in disease mapping, and binary data are common in ecology. When fitting spatial regressions for such data, one needs to account for dependence to ensure reliable inference for the regression coefficients. The spatial generalized linear mixed model (SGLMM) offers a very popular and flexible approach to modeling such data, but the SGLMM suffers from three major shortcomings: (1) uninterpretability of parameters due to spatial confounding, (2) variance inflation due to spatial confounding, and (3) high-dimensional spatial random effects that make fully Bayesian inference for such models computationally challenging. We propose a new parameterization of the SGLMM that alleviates spatial confounding and speeds computation by greatly reducing the dimension of the spatial random effects. We illustrate the application of our approach to simulated binary, count, and Gaussian spatial datasets, and to a large infant mortality dataset.

1. Introduction

The SGLMM is a flexible, widely used areal model, but spatial confounding and high-dimensional latent effects limit parameter interpretation, inference, and computation. The paper proposes a geometry-based model addressing these issues while restricting dependence to positive patterns.

SGLMMs suffer from parameter uninterpretability and variance inflation from spatial confounding, plus computational challenges from high-dimensional latent variables.
RHZ alleviates confounding by introducing synthetic predictors orthogonal to the fixed-effects predictors, but does not account for the underlying graph.
The RHZ model permits random-effect structure corresponding to negative spatial dependence, or repulsion, which is not expected in typical applications.
The proposed model uses the natural geometry of areal models to mitigate confounding, permit only positive spatial dependence, and dramatically reduce random-effect dimension.
The authors describe their approach as the first principled dimension-reduction method for areal models while improving regression inference.
The paper presents simulations for binary, count, and Gaussian spatial data and applies the sparse model to a large US infant mortality dataset.

2. The Traditional SGLMM

The traditional areal SGLMM models observations through a link-transformed linear predictor with fixed effects and spatial random effects, whose dependence is represented by a CAR/GMRF prior based on the graph. Its intrinsic formulation is improper and therefore restricts analysis to Bayesian inference.

Areal SGLMMs represent areas as vertices of an undirected graph, with edges encoding proximity and adjacency matrix A encoding neighborhood structure.
The first-stage model links the conditional expectation of each response to fixed effects Xiβ and a spatial random effect Wi.
The same SGLMM framework supports canonical link functions for binary, count, and normal spatial data.
The spatial random-effects field follows a conditionally autoregressive or Gaussian Markov random field prior.
The precision matrix Q encodes conditional neighborhood dependence and prior uncertainty through graph vertex degrees.
Because the intrinsic model has singular precision matrix Q and is improper, the traditional formulation restricts analysis to Bayesian methods unless a proper GMRF is substituted.

3. Spatial Confounding

Spatial confounding arises when spatial random-effect predictors overlap the fixed-effects design, biasing and inflating uncertainty for regression coefficients. Orthogonalizing the spatial component addresses this issue, while the paper connects the problem to analogous centered and uncentered automodel formulations.

The automodel is introduced as a related Markov random field model that represents dependence directly rather than hierarchically.
In the automodel, positive η denotes attractive dependence and negative η denotes repulsive dependence.
Centering the autocovariate by the independence expectation makes η interpretable as an observation’s reactivity to neighbors conditional on the regression component.
The uncentered automodel has conceptual and spatial confounding because its autocovariate is difficult to interpret and η and β tend to be strongly correlated.
The traditional SGLMM is likewise confounded because its random effects implicitly include predictors K collinear with X, biasing β’s posterior and inflating its variance.
RHZ removes the offending K component, producing smoothing orthogonal to fixed effects that adjusts posterior variance without inflating it while leaving the posterior mean unchanged.

4. A Sparse Reparameterization of the Areal SGLMM

The paper replaces RHZ’s geometry with a Moran-operator basis that fits spatial clustering residual to X while retaining the graph structure. A sparse selection of positive-dependence eigenvectors reduces the random-effects dimension.

Sparse reparameterization: The proposed model restricts random-effects patterns to positive spatial dependence, excluding repulsion, and uses far fewer than n random effects.The model assumes M contains q ≪ n Moran eigenvectors, discarding inconsequential positive, independent, or negative-dependence patterns.
Moran basis: The Moran operator P⊥AP⊥ defines eigenvectors representing distinct spatial-clustering patterns residual to X and accounting for the underlying graph G.Positive and negative eigenvalues correspond to positive and negative spatial dependence, respectively.
Illustration: Figure 1 compares selected RHZ and Moran eigenvectors on a 30 × 30 lattice to illustrate the Moran basis’s suitability for fitting small-scale structure.The displayed eigenvectors are 7, 13, and 42 from each basis.
Sparse reparameterization: Replacing L with M in the RHZ model yields the sparse reparameterization used for the first-stage spatial model.The sparse model is more closely analogous to the centered automodel because it accounts for X and the underlying graph.
Moran basis: Unlike RHZ, the Moran basis accounts for both the fixed-effects design X and the underlying graph when fitting residual spatial structure.RHZ accounts for X but not G, whereas the sparse SGLMM accounts for both.

5. Dimension Reduction for Spatial Models

Dimension reduction addresses the computational cost of spatial mixed models, where high-dimensional random effects can mix slowly and require expensive matrix operations. The sparse model makes quadratic-form evaluation potentially constant-time.

Computational motivation: Dense Gaussian-process fitting requires Cholesky decomposition with time complexity Θ(n3), while areal mixed models also involve expensive matrix operations.The Gaussian-process complexity arises from repeated operations involving the inverse covariance matrix H(φ).
Simulation setup: Figure 2 displays the large-scale simulation structure Xβ used in the study.The simulation design uses spatially varying large-scale structure before modeling residual spatial effects.
Computational motivation: Univariate Metropolis-Hastings updates for W mix slowly because the random-effects components exhibit strong a posteriori dependence.Block updates can improve mixing but require challenging proposals and increase running time per iteration.
Sparse computation: For the RHZ and sparse models, practically a posteriori uncorrelated random effects permit efficient spherical normal proposals.Their computational bottleneck is evaluating a quadratic form rather than handling strongly dependent random-effects updates.
Sparse computation: The RHZ quadratic-form evaluation has time complexity Θ(n), whereas the sparse model’s corresponding evaluation can be Θ(1).This difference makes the sparse model applicable to very large datasets.

6. Simulation Study

The simulation study compares spatial models for binary, count, and Gaussian data, evaluating regression inference, fit, dependence, and computational dimension reduction. Across datasets, the proposed sparse model generally preserves inference while substantially reducing random-effect dimension.

Study design: The study applies classical GLM, centered autologistic, traditional SGLMM, RHZ, and sparse SGLMMs to simulated binary, count, and Gaussian datasets.Binary and count data use a 30×30 lattice; Gaussian data use a 20×20 lattice.
Binary data: For binary data, RHZ and the true sparse model produce similar estimates and intervals, while the traditional SGLMM is biased with confidence intervals over four times wider.The RHZ and true-model intervals cover the true β values but exclude 0.
Binary data: For binary data, reducing the sparse model from 400 to 25 eigenvectors produces the expected bias-variance tradeoff, with little deterioration until 25 eigenvectors remain.Twenty-five eigenvectors represent a 97% reduction in parameters relative to the traditional and RHZ models.
Count data: For count data, all spatial models cover the true β values, whereas standard Poisson regression gives erroneous inference for β2 and the traditional SGLMM again has much wider intervals.The traditional SGLMM nevertheless gives a good point estimate for this dataset.
Gaussian data: For Gaussian data, RHZ and sparse models give narrower intervals than ordinary linear regression, while the traditional SGLMM poorly estimates β and inflates variance enough to cause a Type II error.Gaussian-data computation is slower because Gibbs updates require an expensive Cholesky decomposition.
Dimension reduction: A dimension-reduction study finds that 50–100 eigenvectors should suffice for most analyses, while approximately 10% of square-lattice eigenvectors is a more conservative choice.Using 50 eigenvectors did not relocate any confidence interval enough to miss the true parameter value.

7. Application to US Infant Mortality Data

The sparse Poisson SGLMM was applied to infant mortality rates across 3,071 US counties using county-level covariates and 100 spatial random effects. All predictors were significant, and the fitted spatial variance parameter was much smaller than its prior mean.

Data: The analysis modeled infant mortality across 3,071 contiguous-US counties using deaths-to-births rates averaged over 2002–2004.Darker map circles indicate higher empirical infant mortality rates.
Model: The sparse Poisson SGLMM included birth weight, race and ethnicity, income inequality, social affluence, and residential stability covariates.The model used deaths and live births alongside six county-level predictors and set dim(δS) = 100.
Results: All predictors were found significant in the fitted infant mortality model.The results are reported in Table 6.
Results: The posterior mean for τ was much smaller than its prior mean of 1,000, contraindicating a nonspatial analysis of these data.This indicates that the fitted model retained a substantial spatial component relative to the specified prior scale.
Results: Five eigenvectors were found to be significant predictors in the spatial component.These eigenvectors form part of the reduced spatial representation used in the analysis.

8. Discussion

The paper’s reparameterized areal SGLMM addresses spatial confounding while restricting random-effect dependence to positive patterns and reducing random-effect dimension. Across Bernoulli, Poisson, and Gaussian models, it improved regression inference and MCMC efficiency, with at most 0.1n random effects appearing sufficient.

Contributions: The framework reparameterizes the areal SGLMM to alleviate spatial confounding while permitting only positive spatial dependence.It accounts for covariates of interest and the underlying graph.
Contributions: Accounting for the underlying graph enables a dramatic reduction in the dimension of the spatial random effects.The reduced dimension is intended to speed computation for large areal datasets.
Evidence: Across Bernoulli, Poisson, and Gaussian first-stage models, the approach produced better regression inference and considerable MCMC computational-efficiency gains.The performance study covered three commonly used first-stage model families.
Evidence: A follow-up simulation study found that a small fixed number of random effects was sufficient across datasets.More conservatively, the sparse model appeared to require no more than 0.1n random effects for good regression and prediction performance.