Model Building and the Analysis of Spatial Pattern in Human Geography

Abstract

1. Introduction and Summary It is the purpose of this paper to determine how far various statistical models and methods of statistical inference have enabled the aims of geographical research to be met in the problem areas to which they have been applied. In so doing, we hope we can indicate to the statistician questions of geographical interest which cannot readily be answered by existing statistical methods; and to the geographer, some of the insights into geographical processes which may be gained from a statistical and model building approach. We would stress that we have not tried to be all inclusive in our coverage. Instead, we have tried to select some topics which best seem to convey the flavour of the kinds of things human geographers have been doing, and which will, at the same time, be of interest to statisticians on either theoretical or empirical grounds. In addition, our own interests mean that we have concentrated upon examples in human (economic and urban), rather than physical, geography, although similar approaches have been used there to a lesser degree. More general reviews are provided by Gould (1969), Berry (1971) and Wilson (1972).

To be or not to be valid in testing the significance of the slope in simple quantitative linear models with autocorrelated errors

“…The bivariate relationships with soil pH are shown in Figure 2. Numerical results are reported in Table III for: the classical t-test of the slope (procedure 1); another t-test with n À 2 df based on an estimated GLS estimator of the slope, in which the sample autocorrelation coefficients rðkÞ are replaced by Moran's I correlogram ordinates (Cliff and Ord, 1975) -this procedure is similar to procedure 5 in the Monte Carlo study; the likelihood-ratio w 2 -test (procedure 12); the REML F-test [procedure 14 in which a spherical variogram model was used to take both the autocorrelation of AR(1) type and the spatial variability at small scale into account; this spatial variability at small scale is called ''nugget effect'' in geostatistics]; the FD and FDR t-tests (procedures 15 and 16); and the modified t-test using Dutilleul's effective sample size (procedure 19) only for Transect Line Cliff.…”

“…When the sample data are positively autocorrelated in space, the classical t-test overstates the significance of the population mean (Cliff and Ord, 1975) and that of individual slopes in linear regression models (Cook and Pocock, 1983). Before the repeated measures ANOVA techniques (see Crowder and Hand, 1990 for a review), little was known about the robustness of statistical methods that assume the independence of errors against the departure from this assumption.…”

Scales of climate impacts

“…*These problems include limitations of data, ambiguity regarding the proper units for the vertical axis, and what Stommel (1965) called the 'desperate thing' of assuming statistical staLionarity in physical and biological processes that most certainly do depend on absolute locations in space and time. The situation is even worse in terrestrial contexts (see Curry and Bannister, 1974;Cliff and 0rd, 1975;Granger, 1975;Haggett, 1976). Haury et al, 1978).…”

A Biparametric Approach to Spatial Autocorrelation

“…(2) multilevel structures, where variables from different aggregation levels are incorporated in the same model (that is, the aggregation operator implies spatial dependences-cf Streitberg, 1979); and (3) spatial moving-average processes (cf Cliff and Ord, 1975;Haining, 1978a).…”