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An assessment of coefficient accuracy in linear regression models with spatially varying coefficients

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Abstract

The realization in the statistical and geographical sciences that a relationship between an explanatory variable and a response variable in a linear regression model is not always constant across a study area has led to the development of regression models that allow for spatially varying coefficients. Two competing models of this type are geographically weighted regression (GWR) and Bayesian regression models with spatially varying coefficient processes (SVCP). In the application of these spatially varying coefficient models, marginal inference on the regression coefficient spatial processes is typically of primary interest. In light of this fact, there is a need to assess the validity of such marginal inferences, since these inferences may be misleading in the presence of explanatory variable collinearity. In this paper, we present the results of a simulation study designed to evaluate the sensitivity of the spatially varying coefficients in the competing models to various levels of collinearity. The simulation study results show that the Bayesian regression model produces more accurate inferences on the regression coefficients than does GWR. In addition, the Bayesian regression model is overall fairly robust in terms of marginal coefficient inference to moderate levels of collinearity, and degrades less substantially than GWR with strong collinearity.

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Correspondence to David C. Wheeler.

Appendix

Appendix

As mentioned in the paper, the posterior distribution of a parameter is proportional to the product of the likelihood of the data given the parameter and the priors for the parameters. The posterior can be expressed as the full conditional up to a normalizing constant. The full conditional for a parameter is the distribution for a parameter given the other model parameters. If the distribution of a full conditional for a parameter is recognizable, one can sample directly from it in a Gibbs sampler. However, if the distribution of the full conditional is not recognizable, one can sample from it using a Metropolis–Hastings algorithm or slice sampling.

In order to perform inference on the model parameters, we must write the posterior distribution for each unknown parameter using the likelihood. The derivation of the full conditional distributions in this paper utilizes two versions of the likelihood. The likelihood for the SVCP model with Y as defined in Eq. 7 of the paper text is

$$ L({\varvec{\upmu}}_{{\varvec{\upbeta}}}, {\varvec{\upbeta}},\tau ^{2}, \phi, {\mathbf{T}};{\mathbf{y}}) = {\left| {\tau ^{2} {\mathbf{I}}} \right|}^{{- 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \times \exp \left\{- \frac{1}{2}({\mathbf{y}} - {\mathbf{X}}{\varvec{\upbeta}})^{T} (\tau ^{2} {\mathbf{I}})^{{- 1}} ({\mathbf{y}} - {\mathbf{X}}{\varvec{\upbeta}})\right\}. $$
(1)

We can integrate this likelihood with respect to \({{\varvec{\upbeta}}}\) to reduce the autocorrelation in the Markov chain. The likelihood with integrating over \({{\varvec{\upbeta}}}\) is

$$ L^{*} ({\varvec{\upmu}}_{{\varvec{\upbeta}}}, \tau ^{2}, \phi, {\mathbf{T}};{\mathbf{y}}) = {\left| {\varvec{\Uppsi}} \right|}^{{- 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \times \exp \left\{- \frac{1}{2}({\mathbf{y}} - {\mathbf{X}}^{*} {\varvec{\upmu}}_{{\varvec{\upbeta}}})^{T} ({\varvec{\Uppsi}})^{{- 1}} ({\mathbf{y}} - {\mathbf{X}}^{*} {\varvec{\upmu}}_{{\varvec{\upbeta}}})\right\}, $$
(2)

where \({{\varvec{\Uppsi}} = ({\mathbf{X}}({\mathbf{H}}(\phi) \otimes {\mathbf{T}}){\mathbf{X}}^{T} + \tau ^{2} {\mathbf{I}})}\) and X * is the n ×  p matrix of covariates. As an example of the use of both likelihoods, the full conditional distribution for \({{\varvec{\upmu}}_{{\varvec{\upbeta}}} }\) is derived using the likelihood integrated over \({{\varvec{\upbeta}}, L^{*},}\) and the full conditional distribution for τ2 is derived using the likelihood L.

We first derive the full conditionals that are recognizable distributions using the likelihood and priors. The full conditional for the error variance is

$$ [\tau ^{2} |{\varvec{\upbeta}};{\mathbf{y}}]\sim L \times p(\tau ^{2}) = IG\left(a + n/2,b + \frac{1}{2}({\mathbf{y}} - {\mathbf{X\beta}})^{T} ({\mathbf{y}} - {\mathbf{X\beta}})\right). $$
(3)

The full conditional for the coefficient covariance matrix at any location is

$$ [{\mathbf{T}}|{\varvec{\upmu}}_{{\varvec{\upbeta}}}, {\varvec{\upbeta}},\phi ;{\mathbf{y}}]\sim L \times p({\mathbf{T}}) \times p({\varvec{\upbeta}}) = IW\left(v + n,{\sum\limits_i {{\sum\limits_j {({\mathbf{H}}^{{- 1}} (\phi)}}}})_{{ij}} ({\varvec{\upbeta}}(s_{j}) - {\varvec{\upmu}}_{{\varvec{\upbeta}}})({\varvec{\upbeta}}(s_{i}) - {\varvec{\upmu}}_{{\varvec{\upbeta}}})^{T} + {\varvec{\Upomega}}\right), $$
(4)

where \({{\varvec{\upbeta}} = (\beta (s_{1}),\beta (s_{2}), \ldots, \beta (s_{n}))^{T}}\) and \({{\varvec{\upmu}}_{{\varvec{\upbeta}}} = (\mu _{{\beta _{1}}}, \mu _{{\beta _{2}}}, \ldots, \mu _{{\beta _{p}}})^{T} .}\) The full conditional for the coefficient means is

$$ [{\varvec{\upmu}}_{{\varvec{\upbeta}}} |{\mathbf{T}},\tau ^{2}, \phi ;{\mathbf{y}}]\sim L^{*} \times p({\varvec{\upmu}}_{{\varvec{\upbeta}}}) = N({\mathbf{m}},{\mathbf{S}}), $$
(5)

where \({{{\bf S}} = [(\sigma^{2} {{\bf I}})^{{- 1}} + {{\bf X}}^{{*^{T}}} {\varvec{\Uppsi}}^{{- 1}} {{\bf X}}^{*} ]^{{- 1}} }\) and \({{{\bf m}} = {{\bf S}}({{\bf X}}^{{*^{T}}} {\varvec{\Uppsi}}^{{- 1}} {{\bf y}} + (\sigma ^{2} {{\bf I}})^{{- 1}} {\varvec{\upmu}}).}\) The full conditional distribution for \({{\varvec{\upbeta}}}\) is

$$ [{\varvec{\upbeta}}|{\varvec{\upmu}}_{{\varvec{\upbeta}}}, \phi, {\mathbf{T}},\tau ^{2} ;{\mathbf{y}}]\sim L \times p({\varvec{\upbeta}}) = N({\mathbf{AC}},{\mathbf{A}}), $$
(6)

where A =  (X T X2H − 1 (φ) ⊗ T − 1)− 1 and \({{{\bf C}} = {{\bf X}}^{T} {{\bf y}}/\tau ^{2} + ({{\bf H}}^{{- 1}} (\phi) \otimes {{\bf T}}^{{- 1}})({{\bf 1}} \otimes {\varvec{\upmu}}_{{\varvec{\upbeta}}}).}\)

The density of the unnormalized full conditional of φ is not a recognizable distribution and is

$$ \begin{aligned} p(\phi|{\mathbf{T}},{\varvec{\upbeta}},{\varvec{\upmu}}_{\beta} ;{\mathbf{y}}) \propto L \times p(\phi) \times p({\varvec{\upbeta}})\sim {\left| {{\mathbf{H}}(\phi) \otimes {\mathbf{T}}} \right|}^{{- 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \times \exp \left\{- \frac{1}{2}({\varvec{\upbeta}} - ({\mathbf{1}} \otimes {\varvec{\upmu}}_{{\varvec{\upbeta}}}))^{T} ({\mathbf{H}}(\phi) \otimes {\mathbf{T}})^{{- 1}} ({\varvec{\upbeta}} - ({\mathbf{1}} \otimes {\varvec{\upmu}}_{{\varvec{\upbeta}}}))\right\} \\ \times \phi^{{\alpha - 1}} \exp (- \lambda \phi). \\ \end{aligned} $$
(7)

This density is used in the Metropolis–Hastings algorithm to estimate φ.

The parameterization of the gamma distribution used in this paper is

$$ [\phi ] \propto \phi ^{{\alpha - 1}} \exp (- \lambda \phi), $$
(8)

and the parameterization of the inverse Wishart distribution used in this paper is

$$ [{\mathbf{T}}] \propto |{\mathbf{T}}|^{{- (v + p + 1)/2}} \exp \left(- \frac{1}{2}trace{{\varvec{\Upomega}}{\bf T}}^{{- 1}}\right). $$
(9)

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Wheeler, D.C., Calder, C.A. An assessment of coefficient accuracy in linear regression models with spatially varying coefficients. J Geograph Syst 9, 145–166 (2007). https://doi.org/10.1007/s10109-006-0040-y

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