Regression with spatially misaligned data

L. Madsen, D. Ruppert, N. S. Altman

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


We present a simple approach to the problem of estimating the regression slope parameter from spatially misaligned point data. We assume a linear regression model with errors and covariates from two independent Gaussian spatial processes where covariate and response are observed at different locations. Correlation in the covariate is exploited to predict unobserved covariates via kriging. Kriged values are used to find weighted least squares estimates of regression parameters in a 'krige-and-regress' (KR) procedure. The variance of this estimator is calculated, and a variance estimator is proposed. Because the model and assumptions make it possible to write down the joint likelihood of the data, a maximum likelihood (ML) estimator can be found. Under regularity conditions, this estimator is asymptotically normal with asymptotic variance given by the inverse information matrix, which yields a variance estimator for the ML estimator of the regression parameters. The KR and ML estimators are compared in an example using Environmental Protection Agency data and a simulation study is conducted. While the ML estimator of the slope parameter has a smaller variance than the KR estimator, the ML variance estimator is too small to be used for inference whereas the KR variance estimator gives approximately correct inference.

Original languageEnglish (US)
Pages (from-to)453-467
Number of pages15
Issue number5
StatePublished - Aug 2008

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Ecological Modeling


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