Uncertainty of variance component estimates in nested sampling: a case study on the field-scale spatial variability of a restored soil

We studied the variation of soil properties on a 6-ha artificial catchment constructed near Cottbus, Germany, to investigate processes of initial ecosystem genesis. We wanted to evaluate whether spatial auto-correlation patterns could be identified 3 years after site construction. Topsoil was sampled at 192 locations using a balanced nested design involving six spatial scales (0.2 to 60 m) and analysed for particle size, organic matter content, pH, soluble P and various fractions of selected metals. Variance components were estimated by residual maximum likelihood. The uncertainty of variance estimates was characterized by the Fisher Information matrix and likelihood joint confidence regions. The latter approach was used for the first time to characterize uncertainties of variance estimates in spatial nested sampling. Likelihood ratio tests showed that all variables were spatially auto-correlated but the allocation of the variance to specific spatial scales was highly uncertain. For most variables, at least one variance component could not be estimated precisely because the profile likelihood was either flat or the maximum lay on the boundary of the parameter space. Uncertainty estimates derived from the Fisher Information matrix either could not be computed or were unrealistic in these cases. Likelihood joint confidence regions gave more realistic uncertainty estimates. Joint confidence regions for accumulated variance components showed that the shape of the estimated variograms was poorly defined for most variables. Simulations indicated that poor identification of variance components might be a general problem of nested sampling surveys, which has been under-estimated in the past. Hence, our work provides some incentive for re-examining the statistical properties of the methodology.