Spatial Analysis of Census Mail Response Rates: 1990–2010 - Space-Time Integration in Geography and GIScience - page 145

A random effects term captures both spatially structure and spatially unstructured
variation. Once estimated, it can be decomposed into these two components,
dramatically simplifying an eigenvector spatial filtering analysis that holds spatial
structure constant across time. Otherwise, the space-time structure can be posited
as either lagged or contemporaneous and estimated simultaneously with a spatially
unstructured random effects term.

9.3 Pure Spatial Effects: Initial Eigenvector Spatial

Filtering Results

Estimation of a generalized linear mixed model (using SAS NLMIXED) for the
3,067-by-3 dataset results in a random effects term that accounts for considerable
geographic variation in MRRs across counties. This term has a mean of 0.00281,
a variance of 0.15047, and a statistically significant Kolmogorov-Smirnov test
statistics for the null hypothesis that it conforms to a bell-shaped curve. In other
words, this term deviates from the assumptions of a zero mean and normality.
The number of positive spatial autocorrelation eigenvectors is 770, and the
number of negative spatial autocorrelation eigenvectors is 1,057. The random
effects term yields no evidence of negative spatial autocorrelation. In contrast,
the spatially structured component of the random effects term comprises 48
positive spatial autocorrelation eigenvectors that account for roughly 47.3 % of
its variance. 2 The resulting eigenvector spatial filter (Fig. 9.2 ), whose Moran
Coefficient (MC SF ) is 1.00774 (indicating marked positive spatial autocorrelation;
MC SF /MC max D 1.00774/1.08718 D 0.92693), reduces excess binomial variance,
and accounts for about one-third of the variability in the MRRs (Table 9.2 ). Its
conspicuous map pattern (Fig. 9.2 a) reveals a stark differentiation between the south
and Rocky Mountains regions, on the one hand, and the mid-west and mid-Atlantic
regions, on the other hand. Heterogeneity accounts for at least twice as much
variance, and essentially further reduces the overdispersion nearer to its expected
level of 1 (although the 1990 and 2000 MRRs may contain some underdispersion
now). The SURE component exhibits a random map pattern (Fig. 9.2 b).

9.4 Spatial Effects in the Presence of a Time Lag

The random effects term in the presence of time lagged MRRs has a mean
of 0.00278, a variance of 0.04531, and exhibits significant deviation from
a normal distribution. Again the random effects term yields no evidence of

2 A stepwise linear regression was employed that used the Bonferroni adjusted selection signifi-
cance level of 0.10/770.