Estimation in Spatial Models with Censored Response

Thais S. Barbosa
Víctor H. Lachos
Larissa A. Matos
Marcos O. Prates

Spatial environmental data can be subject to some upper and lower limits of detection (LOD), below orabove which the measures are not quantifiable. As a result, the responses are either left or right censored.Historically, the most common practice for analysis of such data has been to replace the censored observa-tions with some function of the limit of detection (LOD/2, 2LOD), or through data augmentation, by usingMarkov chain Monte Carlo methods. In this paper, we propose an exact estimation procedure to obtain themaximum likelihood estimates of the fixed effects and variance components, using a stochastic approxi-mation of the EM algorithm, the SAEM algorithm (Delyon et al., 1999). This approach permits easy andfast estimation of the parameters of spatial linear models when censoring is present. As a byproduct, pre-dictions of unobservable values of the response variable are possible. The proposed algorithm is appliedto a spatial dataset of depths of a geological horizon that contains both left- and right-censored data. Wealso use simulation to investigate the small sample properties of predictions and parameter estimates andthe robustness of the SAEM algorithm. In this simulation study comparisons are made between inferencesbased on the censored data and inferences based on complete data obtained by a crude/ad hoc imputationmethod (LOD/2, 2LOD). The results show that differences in inference between the two approaches can besubstantial.

Censored data
Geostatistical data
SAEM Algorithm
Limit of Detection (LOD)