Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

Abstract

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients' and controls' residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.

Fig. 1

Information available for mapping topsoil heavy metal concentration and late-stage breast cancer incidence. (A) Soil field measurements collected at 359 point locations. (B) Location of 937 patient residences (data shuffled for confidentiality reasons). (C) Choropleth map of the main geological formations. (D) Choropleth map of late-stage breast-cancer incidence rate (age group 64–75 years) in three Michigan counties, by census tract, 1985–2002

Fig. 2

Maps of chromium concentration and late-stage breast-cancer incidence rate created by alternative interpolation techniques. (A, B) Ordinary kriging. (C, D) Kriging that combines both point and areal data. (E, F) Residual kriging with a choropleth trend model. The same color scale is used for each series of three maps

Fig. 3

Semivariogram models used for geostatistical interpolation. (A) Models fitted to semivariograms of chromium concentration before and after subtraction of trend estimates (i.e. residuals). Models fitted to indicator semivariograms of health outcomes before and after subtraction of trend estimates: entire model (B) and detailed view of first few lags (C)

Fig. 4

Scatter plots of areal chromium concentrations and ATA kriged incidence rates versus averages of kriging estimates within each mapping unit (i.e. geological formation or census tract). Only area-and-point kriging ensures the reproduction of areal data (C, D), while other interpolation techniques do not honor the coherence constraint

Fig. 5

Maps of weights assigned to different types of data in the interpolation of chromium concentration and late-stage cancer incidences using area-and-point kriging in Figs. 3C–D. (A, B) Point data (sum for 16 Cr observations or 32 late-stage indicators). (C, D) Kernel areal datum. (E, F) Neighboring areal data (second-order adjacency for soil data and first-order adjacency for cancer data). Open circles denote the location of point data while open triangles in (E) and (F) correspond to geographical centroids of areal data. The same color scale is used for all six maps

Fig. 6

Maps of prediction variance associated with the maps of Fig. 2. (A, B) Ordinary kriging. (C, D) Kriging that combines both point and areal data. (E, F) Residual kriging with a choropleth trend model. The same color scale is used for each series of three maps

Fig. 7

Impact of the sample size on the proportion of simulations for which an interpolation technique outperforms the other two. Cross-validation statistics include: the mean absolute error of prediction, the mean square standardized residual, the variance of the set of kriged estimates (smoothness), and the discriminatory power between early- and late-stage cases