Spatial correlation between the prevalence of transmissible spongiform diseases and British soil geochemistry

Abstract

Transmissible spongiform encephalopathies (TSEs) are a group of fatal neurological conditions affecting a number of mammals, including sheep and goats (scrapie), cows (BSE), and humans (Creutzfeldt-Jakob disease). The diseases are widely believed to be caused by the misfolding of the normal prion protein to a pathological isoform, which is thought to act as an infectious agent. Outbreaks of the disease are commonly attributed to contaminated feed and genetic susceptibility. However, the implication of copper and manganese in the pathology of the disease, and its apparent geographical clustering, have prompted suggestions of a link with trace elements in the environment. Nevertheless, studies of soils at regional scales have failed to provide evidence of an environmental risk factor. This study uses geostatistical techniques to investigate the correlations between the distribution of TSE prevalence and soil geochemical variables across the UK according to different spatial scales. A similar spatial pattern in scrapie and BSE occurrence is identified, which may be linked with increasing pH and total organic carbon, and decreasing iodine concentration. However, the pattern also resembles that of the density of dairy farming. Nevertheless, despite the low spatial resolution of the TSE data available for this study, the fact that significant correlations are detected indicates there is a possibility of a link between soil geochemistry, scrapie, and BSE. It is suggested that further investigations of the prevalence of TSE and environmental exposure to trace metals should take into account the factors affecting their bioavailability.

Appendix

A.1 Geostatistical estimation

In brief, geostatistics is a set of tools which exploits the spatial correlation structure in a dataset to make unbiased estimates of one or more variables at unsampled locations. In its simplest form, the correlation structure is assessed using the variogram γ(h), calculated by pooling pairs of data x _i into specified intervals of separation distance h. The differences between the values of the data in each pair are calculated, and for each distance category the mean of the squared subtractions is obtained:

$$ \gamma ({\mathbf{h}}) = \frac{1}{{2N({\mathbf{h}})}}\sum\limits_{i = 1}^{N({\mathbf{h}})} {[z({\mathbf{x}}_i ) - z({\mathbf{x}}_i + {\mathbf{h}})]^2 } $$

(1)

where N(h) is the number of pairs within a separation distance interval.

A model is then fitted to the experimental variogram to obtain a smooth function with which to calculate appropriate weights during the subsequent estimation procedure. Valid models which are commonly fitted to the experimental variogram include spherical, Gaussian, and exponential functions. These are characterised by a sill, which represents the covariance accounted for by the model, and a range, which signifies the extent of spatial correlation. The models can be nested, whereby correlation structures at different scales are represented. For example, a typical variogram for geochemical data may comprise the following components:

• • a nugget effect which accounts for the covariance in the data arising from measurement and sampling errors, and variation that is on a scale that is too small to be detected by the sampling density;

• • a short-scale correlation structure arising from local variations, because of factors such as land use and parent material geology;

• • a longer-scale correlation structure or trend arising from, for example, larger scale geological features and climate.

In its simplest form, geostatistical estimation is undertaken using a procedure called ordinary kriging. A neighbourhood is first specified which determines the minimum and maximum number of data points to be used for the estimations, how they should be distributed around the location, and what is the maximum separation distance. The choice of these parameters is generally determined by the amount and density of the data and the characteristics of the variogram. The kriging estimate Z* is defined as a linear combination of the neighbouring data Z _α :

$$ {\text{Z}}^* = \sum\limits_\alpha {\lambda ^\alpha {\text{Z}}_\alpha } $$

(2)

where λ _α are weights derived from the variogram model by minimising the expected estimation error. A comprehensive description of geostatistical estimation may be found in Isaaks and Srivastava (1989) and Goovaerts (1997).

While point kriging is used to estimate values at a point location, block kriging is used to compute an average value over a volume. Polygon kriging can be used to estimate average values over an irregular shape.

A.2 Linear model of coregionalisation

A useful tool for analysing regional geochemistry is the linear model of coregionalisation (LMC). This is essentially a set of variograms and cross-variograms calculated for two or more collocated variables. The cross-variograms are calculated by comparing pairs of different variables with increasing distance. It is therefore possible to detect any spatial scales over which the variables are correlated because of a common underlying factor.

The theory underpinning multivariate geostatistical analysis has been published in a number of texts (e.g. Goovaerts 1997; Wackernagel 1998). A brief outline is provided below.

Cross-variograms between two variables z _α and z _β may be calculated as follows:

$$ \gamma _{\alpha \beta } ({\mathbf{h}}) = \frac{1}{{2N({\mathbf{h}})}}\sum\limits_{i = 1}^{N({\mathbf{h}})} {[(z_\alpha ({\mathbf{x}}_i ) - z_\alpha ({\mathbf{x}}_i + {\mathbf{h}})) \cdot (z_\beta ({\mathbf{x}}_i ) - z_\beta ({\mathbf{x}}_i + {\mathbf{h}}))]} $$

(3)

As mentioned above, variogram models may be nested, so that they represent a number of processes acting at different spatial scales. Such a covariogram is a linear combination of N _s component functions g ^u(h):

$$ \gamma _{\alpha \beta } ({\mathbf{h}}) = \sum\limits_{u = 1}^{N_s } {\gamma _{\alpha \beta }^u ({\mathbf{h}}) = \sum\limits_{u = 1}^{N_s } {b_{\alpha \beta }^u g^u ({\mathbf{h}})} } $$

(4)

where the $b_{\alpha \beta }^u$ are coefficients which represent the importance of each spatial scale u to the relationships between the variables. This LMC can be expressed in matrix terms:

$$ \Gamma \left( {\mathbf{h}} \right) = \sum\limits_{u = 1}^{N_s } {{\mathbf{B}}^u g^u \left( {\mathbf{h}} \right)} $$

(5)

where Γ(h) is the p × p variogram matrix and B ^u is a positive semi-definite matrix of the coefficients $b_{\alpha \beta }^u$. A measure of the correlation between the variables Z _α and Z _β at the spatial scale u is given by:

$$ \rho _{\alpha \beta }^u = \frac{{b_{\alpha \beta }^u }}{{\sqrt {b_{\alpha \alpha }^u b_{\beta \beta }^u } }} $$

(6)

The structural correlation coefficients $\rho _{\alpha \beta }^u$ differ from the traditional product–moment correlation coefficients in that they focus on specific spatial scales, filtering out the processes operating over different distances. It may also be noted that they are derived within a probabilistic framework and not directly from the data pairs.