Range, Sill, and Nugget in Spatial Models

Why These Parameters Matter

A spatial model describes how latent values at nearby locations tend to vary together. Three terms summarize different features of that variation: the range describes how quickly spatial dependence fades with distance; the sill describes the variance reached at large separation; and the nugget describes variation that does not appear as spatially correlated variation between distinct observations. Interpreting these terms correctly matters because a large nugget and a short range can look similar in sparse data, even though they imply different spatial structures.

In a GLMM, these parameters usually describe latent variation on the linear-predictor scale. They do not, by themselves, describe the covariance of the observed responses on their original scale. The response distribution and link function also matter.

A Covariance Model and Its Terminology

Suppose observation \(i\), at spatial location \(s_i\), has a latent contribution \(w(s_i)+e_i\) to its linear predictor. Let \(w(s)\) be a mean-zero, stationary spatial Gaussian field, and let \(e_i\) be an independent mean-zero latent term. A common model is

where \(g\) is the link function, \(x_i\) is the covariate vector, \(d_{ij}\) is the distance between locations, \(\sigma^2\) is the spatially structured variance, and \(\rho(d)\) is a correlation function with \(\rho(0)=1\). The independent terms \(e_i\) are also independent of the spatial field. For distinct observations, the latent covariance is

The marginal variance of one latent contribution \(w(s_i)+e_i\) is \(\sigma^2+\tau^2\). This model assumes that the spatial field has constant variance and covariance determined by separation distance, and that the \(e_i\) terms are independent with a common variance. It also assumes that the specified correlation function is appropriate for the spatial domain and that the latent effects combine additively on the linear-predictor scale.

The nugget is a model component, not a unique scientific explanation. It can represent independent observation-level latent heterogeneity, unresolved small-scale variation, or other variation not captured by the modeled spatial field. Whether it represents measurement error depends on how the model is formulated; in a non-Gaussian GLMM it should not automatically be called measurement error.

Range: How Quickly Dependence Fades

For a correlation function that approaches zero gradually, there is no finite distance at which correlation becomes exactly zero. In that setting, a reported range is usually a practical range: a distance at which correlation has fallen below a stated small threshold. The threshold must be reported or understood from the convention being used. A common convention is the distance at which correlation is about 0.05. The scale parameter in a covariance formula is not necessarily this practical range.

For the exponential correlation \(\rho(d)=\exp(-d/\phi)\), the distance where correlation is 0.05 solves \(\exp(-d/\phi)=0.05\), giving \(d=-\phi\log(0.05)\), approximately \(3\phi\). For the Gaussian correlation \(\rho(d)=\exp\{-(d/\phi)^2\}\), that threshold is \(d=\phi\sqrt{-\log(0.05)}\), approximately \(1.731\phi\). These conversions are specific to the stated parameterizations. A Matérn model’s practical range also depends on its parameterization and smoothness. Some compactly supported covariance functions instead have a finite distance beyond which covariance is exactly zero.

A practical range is a description of the fitted correlation curve, not a claim that all observations beyond that distance are independent. For example, if the correlation is 0.05 at the practical range, it is small by the chosen convention, but not exactly zero.

Sill and Nugget Through the Semivariogram

A semivariogram expresses how dissimilar values are expected to be at a given separation. For two distinct observations under the model above, define the semivariance at distance \(d\) as half the variance of their latent difference. Because each observation has total latent variance \(\sigma^2+\tau^2\), and their covariance is \(\sigma^2\rho(d)\), the variance-of-a-difference identity gives

This derivation explains the familiar variogram shape. As distinct locations approach one another, \(\rho(d)\) approaches 1, so the semivariogram approaches \(\tau^2\), not zero, when independent terms remain. This near-zero-distance limit is the nugget effect. At large distances, if \(\rho(d)\) approaches zero, the semivariogram approaches \(\tau^2+\sigma^2\), the total sill. At zero separation for the exact same observation, the difference is zero; the nugget describes differences between distinct observations, including replicates at the same location when their independent terms differ.

Interpreting the Components in a GLMM

A nugget and a spatial field contribute differently to dependence. The spatial field gives observations at distinct locations covariance \(\sigma^2\rho(d)\), which changes with distance. Independent terms \(e_i\) add variance to each linear predictor but no covariance between distinct observations. Consequently, increasing \(\tau^2\) raises the total variance and the semivariogram’s short-distance and long-distance levels, but does not increase the structured covariance at any positive separation.

For example, consider a hypothetical logistic GLMM with \(\sigma^2=0.50\), \(\tau^2=0.50\), and exponential correlation with \(\phi=4\) km. At 4 km, \(\rho(4)=\exp(-1)\approx0.3679\), so the latent covariance between distinct observations is \(0.50(0.3679)=0.1839\). Each latent contribution has variance 1.00, giving latent correlation 0.1839. At the same location, two distinct observations share the spatial field but have independent \(e_i\) terms, so their latent covariance is 0.50 and their latent correlation is 0.50. These are calculations on the linear-predictor scale; the response-scale association in a logistic model is not obtained by simply labelling these numbers as response correlations.

Diagnostics, Comparisons, and Limitations

Parameter estimates can be unreliable when the sampling design provides little information about the covariance curve. If the smallest separation among sampled locations is large, the near-zero-distance behavior and nugget may be poorly determined. If there are no observations at sufficiently large separations, the plateau and practical range may also be weakly identified. A fitted range that grows very large, a variance estimate near a boundary, or a wide interval for the nugget can signal that the data do not distinguish among plausible covariance shapes. These patterns do not prove a model is wrong, but they warrant caution.

Empirical variograms of suitable residual summaries can help reveal remaining distance-dependent structure, but GLMM residuals are not direct observations of the latent Gaussian field. Their variance and distribution depend on the response family and fitted mean. Diagnostics should therefore be interpreted alongside model-based checks and the sampling design. Replicate observations at identical or very close locations are particularly informative about the nugget, provided the model correctly represents which latent effects are shared across replicates.

A continuous-distance covariance model is not automatically preferable to an independent group effect or an adjacency-based model: the choice should reflect how locations and dependence are represented by the design. The parameters are also conditional on the chosen correlation family. Different families can fit similar covariance patterns over the observed distances yet imply different behavior outside that range. Range and nugget estimates should therefore be reported with their parameterization, units, and relevant uncertainty. If nearby observations show directional patterns, an isotropic distance-only model may be inadequate; the direction-dependent structure is a separate modeling issue.

Common Mistakes

• Calling the scale parameter the range without checking the formula. For the exponential model here, \(\phi\) is not the 0.05 practical range; that range is approximately \(3\phi\).
• Calling the nugget the total sill. The nugget is the short-distance semivariogram limit for distinct observations. The total sill is \(\tau^2+\sigma^2\).
• Assuming a nugget has one definite physical cause. It is a statistical component; the design and scientific context are needed to distinguish unresolved spatial variation from independent latent heterogeneity.
• Interpreting latent-scale correlation as response-scale correlation. In a non-Gaussian GLMM, the link and response distribution affect the relationship between latent dependence and observed outcomes.

A practical interpretation should state which correlation function and parameterization were used, whether range is a threshold-based practical range, and whether sill refers to structured variance or total semivariogram plateau. Without these details, the same parameter label can be ambiguous.