Spatial Analysis of Census Mail Response Rates: 1990–2010 - Space-Time Integration in Geography and GIScience - page 148

Tabl e 9. 3 Results from including a random effects term in the binomial model specification
containing time lagged MRRs
MRRs Lag C SSRE Lag C SSRE C SURE

Pseudo-R 2 Pseudo-R 2 Pseudo-R 2 Year Deviance Deviance Deviance

0.9296 a 2000 2.1512 0.5095 1.9023 0.5647 0.7886

2010 2.4977 0.5584 1.9617 0.6564 0.5317 0.9128

a This value assumes that the random effects terms has only 2 degrees of freedom

variance, and accounts for only a modest amount (i.e., 5-10 %) of the variability in
the MRRs (Table 9.3 ; i.e., the immediately preceding decennial map captures much
of the spatial structure effect). Its conspicuous map pattern (Fig. 9.3 a) is very similar
to that portrayed in Fig. 9.2 a. Once more, heterogeneity accounts for far more of the
variance, being second to the time lagged MRRs, and essentially further reduces the
overdispersion; this case more strongly suggests the presence of underdispersion.
The SURE component exhibits a random map pattern (Fig. 9.3 b).

9.5 Eigenvector Space-Time Filter Results

Griffith ( 2012 ) extends the eigenvector spatial filtering methodology to space-
time contexts, exploiting the space-time structure latent in the space-time Moran
Coefficient (Cliff and Ord 1981 ;Griffith 1981 ). One of his extensions is for
contemporaneous spatial dependence (i.e., a value at a given location for a particular
point in time is a function of the value at that location for the previous point in
time and the values of nearby locations for the same point in time). The space-time
structure for this situation is given by
I - 11 T =n . I T ˝ C s C C T ˝ I s / I - 11 T =n ;

(9.2)

where: ˝ denotes Kronecker product, ( I - 11 T /n) now is an nT-by-nT projection
matrix, C s is the n-by-n binary spatial weights matrix appearing in expression ( 9.1 ),
and C T is a binary T-by-T time series weights matrix, with 1s in its upper and lower
off-diagonals and 0 elsewhere.
With three points in time (i.e., 1990, 2000, and 2010) and 3,067 counties, the
space-time matrix given by expression ( 9.2 ) is 9,201-by-9,201. This matrix has
2,264 eigenfunctions with relative Moran Coefficients of at least 0.25. Of these,
the resulting eigenvector space-time filter (ESTF) comprises 723 vectors yielding
individual year Moran Coefficients of 0.86589 (1990), 0.83668 (2000), and 0.89125
(2010). This synthetic variate offers an alternative covariate to those based upon
only spatial structure (see Table 9.2 ) for predicting MRRs. When compared with
the time-lagged model specification, it offers the advantage of being able to more
fully incorporate the first time period (i.e., 1990), which is very appealing in short
time series, especially ones containing only three points in time.