Data Aggregation cannot be neglected

The Effect of Aggregation On Univariate Statistics Summary The resistance of the Modifiable Area Unit Problem to analytical solution requires that it be investigated by numerical and empirical studies that have the potential to lay the foundations for analytical approaches. The use of synthetic spatial datasets, whose spatial autocorrelation, mean, and variance of individual variables, and Pearson correlation between variables, can be con- trolled greatly enhances the ability of the analyst to study the MAUP in this manner. This chapter explores the effects of spatial aggregation on the variance and three univariate spatial autocorrela- tion statistics using a synthetic 400-region dataset. The relationship between the relative change in variance and a modified version of the G statistic that was first proposed by Amrhein and Rey- nolds (1996, 1997) is explored in more detail. These results compare favourably with results gen- erated from the Lancashire dataset of Amrhein and Reynolds (1996). Introduction The Modifiable Area Unit Problem (MAUP) has been the focus of research interest for many years, with the current resurgence in interest being initiated by Openshaw and Taylor (1979) and fueled by the rapidly increasing computing power available to analysts. It is well known that the application of statistical results derived from one level of spatial resolution to a higher resolu- tion (such as census tract data being used to predict individual household information) can result in serious errors; this all too common error has been named the ecological fallacy. An ancillary effect of the enhanced computing power is the proliferation of Geographical Information Systems (GIS) and other spatial analysis tools. As the MAUP has been either ignored or written off as in- tractable in many research results, it can be expected to get short shrift by users of this software who are unaware of the subtleties of spatial data analysis. The importance of gaining an under- standing of the MAUP and how it can be taken into account in GIS software to reduce the num- bers of flawed analyses and their possibly expensive repercussions cannot be understated. Theoretical work, such as that by Arbia (1989), has shown that an analytical solution is possible, but under restrictive conditions that would seldom be found in real life situations. As a result, research into the MAUP has been primarily empirical, focusing on the effects of aggrega- tion on various statistics computed from a specific dataset. For example, Openshaw and Taylor (1979) examine correlation coefficients using an Iowa electoral dataset, Fotheringham and Wong (1991) study multiple regression parameters using Buffalo census data, Amrhein and Reynolds (1996), one of the papers in the special issue of Geographical Systems that focuses on the MAUP, and Amrhein and Reynolds (1997) study the effects of data aggregation on univariate statistics and make a tentative link between a spatial statistic and the relative change in variance. Recogni- tion of spatial patterns is a fundamental requirement for landscape ecology, and various spatial autocorrelation statistics, such as the Moran Coefficient, are often employed as a tool for this task (Jelinski and Wu, 1996; Qi and Wu, 1996); hence it is important to know how spatial statistics are affected by aggregation as well. The use of synthetic spatial datasets overcomes the difficulties inherent in publicly avail- able sets, with census data being the prime example. Possible errors in the data notwithstanding, the greatest frustration for researchers into the MAUP is that one has no control over the values of spatial autocorrelation, means, variances, or Pearson correlations between variables; one must work with the data at hand. Amrhein (1995) is the first to use synthetic datasets in the study of the MAUP by locating points randomly within a unit square, assigning them random values, im- posing various sized square grids, and aggregating the points within each square. This chapter extends this approach by employing more sophisticated synthetic datasets to explore the effects of spatial data aggregation on the weighted variance and on three commonly-used spatial autocorrelation statistics, the Moran Coefficient, the Geary Ratio, and the Getis (G) statistic. The following sec- tions discuss the method of analysis, the results, and the conclusions. Results The effects of aggregation on the variance Aggregated variance that is weighted by the number of regions ni in the M aggregated cells. A value of RCV near one (as in the first group of lines in Figure 4.2a) means that the aggregated weighted variance is much closer to zero than the original variance, while a value near zero (as in the last group of lines in Figure 4.2a) means that the new variance is very similar to the original. It can be shown that the variance of a spatially located variable can be partitioned into the sum of variances within various sub-regions and the variance of the average values of all the subregions (see Section 5.3 and Moellering and Tobler, 1973). The process of aggregation re- moves the former, so the more spatially homogeneous (i.e. positively autocorrelated) a variable is, the smaller the variance within each cell will be (on the average) and hence the less variance is lost. As the number of aggregate cells decreases (i.e. fewer, larger regions), the loss in variance obviously increases, since a greater number of values are being lost. Both of these patterns are well demonstrated in Figure 4.2a. As the number of aggregate cells decreases, the number of re- gions per cell increases on average, since the aggregation algorithm attempts to have similar num- bers of regions per cell, but does not strictly enforce this ideal. When significantly positively auto- correlated variables are aggregated, increasing the number of regions per cell increases the likeli- hood that more widely differing values will be included in each cell, so one would expect the variability of possible aggregate variance values to increase with a decrease in the numbers of cells. With negatively or near-randomly auto correlated variables, however, the tendency towards the juxtaposition of widely differing values means that as the number of regions per cell increases, the opportunity for variation in the aggregate variance values will tend to remain the same or decrease. Both of these patterns are demonstrated in Figure 4.2a. When variables of the same MC but different variances were aggregated, it was found that the variance of the original variable had no discernible impact upon the distributions of the RCV (not shown). Only the spatial organiza- tion of the variable plays a major role in the new variance.