Distinguishing spatially correlated random variation in soil from a ‘pure nugget’ process

In most spatial analysis of soil variation it is assumed that the random variation not captured by fixed effects (class means or continuous covariates) is spatially dependent. It is proposed that this should be tested formally, both to justify the kriging component in subsequent spatial prediction and as evidence of the extent to which the included fixed effects have succeeded in accounting for soil variation that is spatially dependent at the scales resolved by the soil sampling. A formal test is possible by computing the log ratio of the likelihoods for a full model with spatially dependent random effects and a null model which is pure nugget. It is shown that the sampling distribution of the log likelihood-ratio under the null model is not χ2(p) where p is the number of additional random effects parameters in the model with spatial dependence. This is because, while the null model is nested in the full model, parameters of the full model take bounding values in the null case. The sampling distribution may be computed by Monte Carlo simulations. It is shown that the power to reject the null model by the log likelihood-ratio test depends on the importance of the nugget effect in the underlying model, and on the sampling scheme. In many circumstances it may be hard to demonstrate spatial dependence. The recommended procedure was applied to some data on the organic carbon content of the topsoil and subsoil of a field in England. This was modelled either with the overall mean the only fixed effects, or with separate means for different soil map units as fixed effects. There was significant evidence for spatial dependence in the random effects at both depths when the overall mean was the only fixed effect. When map unit means were used as fixed effects there was significant, though weaker, spatial dependence in the topsoil, but the null model could not be rejected for the subsoil. This has implications for any further sampling to map organic carbon in the subsoil.

► In the linear mixed model it is desirable to test the significance of spatial dependence by the log-likelihood ratio L.
► The distribution of L in this case is a mixture of chi-squareds which must be obtained by simulation.
► The power of the test to identify spatial dependence depends, inter alia, on the sampling design.
► The approach is demonstrated with data on soil organic content.
► For subsoil including soil map units as fixed effects gave rise to random effects with no significant spatial dependence.