Exploring a Relationship between Aggregate and Individual Levels Spatial Data through Semivariogram Models

Article excerpt

Analysis of social data is frequently done using aggregate-level data. There may not be a direct interest in spatial relationships in the data, but the presence of spatial interdependence may still need to be taken into account. This article explores the aggregation effect from a spatial perspective by assuming nonzero covariance for individual data from two different groups. We investigate the bias associated with aggregate-level data for semivariogram analysis. We show that the bias mainly arises from the average of the semivariogram within the groups. It is also shown how aggregated-level data may be used to estimate parameters of an individual-level semivariogram model. A nonlinear regression method is proposed to carry out this estimation procedure and a simulation is done to clarify the results.

Introduction

The semivariogram is often used as part of a kriging analysis, which is a method used to estimate point values and construct the estimated surface map over a study region based on a few observation points (Cressie 1991; Carrat and Valleron 1992). These points sometimes represent aggregated values. Carrat and Valleron (1992) discussed such an application in epidemiological mapping of influenza-like illness in France. The estimated surface map was used to make inferences for all the points within the region.

Aggregation implies that variation at the individual level is lost, as a result of the transformation from individual to aggregate data. Two key problems associated with analyzing aggregate data are the modifiable areal unit problem (MAUP) and the ecological fallacy (see Wong 1996). Cressie (1996) notes that the MAUP cannot be resolved until the spatial aspect is incorporated into the problem formulation. Amrhein and Reynolds (1996) further discuss a relation between aggregation effects and spatial effects. The aggregation effect is the difference in results from analysis of a data set at two different spatial scales, for example, between individual- and aggregate-level data.

The aim of the study is to explore the aggregation effect on data analysis by incorporating spatial dependency among observations and how aggregate-level data can be used to infer individual-level data. An exploratory method is considered and a semivariogram model is applied to implement the proposed method.

Problem identification

Problems arise when the data are available in aggregated form but interferences concerning individual-level relationships are required. If spatial dependency is expected at the individual level, then individual-level data should be used to define the model. But sometimes this level of data is not available. For example, Carrat and Valleron (1992) develop a mapping of an influenza-like illness epidemic in France using sampled group-level data. The resulting inference is specific to the group-level data used. If the target of interference is individual level, then the result may not be appropriate.

Steel, Holt, and Tranmer (1996) investigated the aggregation effects on statistical analysis from a nonspatial point of view by assuming zero covariance of the values of individuals from two different groups. In this article we allow for the existence of correlations between two individuals from two different groups, by considering spatial correlation.

Steel, Holt, and Tranmer (1996) noted that the aggregation will affect variances because geographic areas do not contain random groupings of people or households. Further they noted that the population structure must be incorporated into the statistical model corresponding to the analysis if aggregation effects are to be understood. This idea was extended by Pawitan (2001) for analyzing aggregated areal-level data.

An initial approach to examining a spatial model of interrelationship between individuals is through the construction of a semivariogram model. …

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