### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 453-467 |

Number of pages | 15 |

Journal | Environmetrics |

Volume | 19 |

Issue number | 5 |

DOIs | |

State | Published - Aug 1 2008 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Ecological Modeling

### Cite this

*Environmetrics*,

*19*(5), 453-467. https://doi.org/10.1002/env.888

}

*Environmetrics*, vol. 19, no. 5, pp. 453-467. https://doi.org/10.1002/env.888

**Regression with spatially misaligned data.** / Madsen, L.; Ruppert, D.; Altman, N. S.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Regression with spatially misaligned data

AU - Madsen, L.

AU - Ruppert, D.

AU - Altman, N. S.

PY - 2008/8/1

Y1 - 2008/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=47549087917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47549087917&partnerID=8YFLogxK

U2 - 10.1002/env.888

DO - 10.1002/env.888

M3 - Article

AN - SCOPUS:47549087917

VL - 19

SP - 453

EP - 467

JO - Environmetrics

JF - Environmetrics

SN - 1180-4009

IS - 5

ER -