Lecture #3:
Modeling spatial
autocorrelation in normal,
binomial/logistic, and
Poisson variables:
autoregressive and spatial
filter specifications
Spatial statistics in practice
Center for Tropical Ecology and Biodiversity,
Tunghai University & Fushan Botanical Garden
Topics for today’s lecture
• Autoregressive specifications and normal curve
theory (PROC NLIN).
• Auto-binomial and auto-Poisson models: the
need for MCMC.
• Relationships between spatial autoregressive
and geostatistical models
• Spatial filtering specifications and linear and
generalized linear models (PROC GENMOD).
• Autoregressive specifications and linear mixed
models (PROC MIXED).
• Implications for space-time datasets (PROC
NLMIXED)
What is an auto- model?
Y is on both sides
of the = sign
The auto-normal
(auto-Gaussian)
model
Popular autoregressive equations
for the normal probability model
pure autocorrelation:Y  f (μ,ε)
AR:Y  ρWY  Xβ ε
SAR:Y  ρWY (I ρW)Xβ ε
CAR: Y  DY  Xβ ε, D D  (I ρC)M
T
A normality assumption usually
is added to the error term.
M is diagonal,
and often is I
spatial autoregression
The workhorse of classical statistics is
linear regression; the workhorse of
spatial statistics is nonlinear
regression.
The simultaneous autoregressive (SAR) model
Y  ρWY  (I  ρW) Xβ  ε
where
ρ
denotes the spatial autocorrelation parameter
Y  e  [ρWY  (I  ρW)Xβ]  e  ε  e
J
J
J
Georeference data preparation
• Concern #1: the normalizing factor
– Rule: probabilities must integrate/sum to 1
– Both a spatially autocorrelated and
unautocorrelated mathematical space must
satisfy this rule
• Jacobian term for Gaussian RVs – a function
of the eigenvalues of matrix W (or C)
nonsymmetric
set of
eigenvalues
symmetric
set of
eigenvalues
Calculation of the Jacobian term
Step 1
extract the eigenvalues from n-by-n matrix W
(or C)
- eigenvalues are the n solutions to the
equation det(W – λ I) = 0
- eigenvectors are the n solutions to the
equation (W - λ I)E = 0.
Step 2 (from matrix determinant)
compute 
1
n
n
 LN(1  ρλ ) ; J2 is
i
i 1
2

n
n
 LN(1  ρλ )
i
i 1
Minimizing SSE
MIN OLS: 1.1486
MIN with Jacobian,
which is a weight: 1.8959
worst case scenario
0.9795
Relative plots (in z scores)
1.0542
Gaussian approximations allow an
evaluation of redundant information
effective
sample size
Houston (n=690)
% redundant
information
n*
Syracuse (n=208)
% redundant
information
n*
population density
61
66
72
15
% male
32
156
18
78
black/white ratio
63
62
57
27
% widowed
52
91
16
84
% with university
degree
70
49
43
37
% Chinese
51
93
34
48
The autobinomial/logistic
model
NOTE: a data transformation
does not exist that enables
binary 0-1 responses to conform
closely to a bell-shaped curve
Primary sources of overdispersion: binomial
extra variation [Var(Y) = np(1-p) , and >1]
• misspecification of the mean function
• nonlinear relationships & covariate
interactions
• presence of outliers
• heterogeneity or intra-unit correlation in
group data
• inter-unit spatial autocorrelation
• choosing an inappropriate probability
model to represent the variation in data
• excessive counts (especially 0s)
The auto-binomial/logistic model
• By definition, a percentage/binary response
variable is on the left-hand side of the equation,
and some spatial lagged version of this
response variable also is on the right-hand side
of the equation.
• Unlike the auto-Gaussian model, whose
normalizing constant (i.e., its Jacobian term) is
numerically tractable, here the normalizing
constant is intractable.
• A specific relationship tends to hold between the
logistic model’s intercept and autoregressive
parameters.
Pseudo-likelihood estimation
Maximum pseudo-likelihood treats areal unit
values as though they are conditionally
independent, and is equivalent to maximum
likelihood estimation when they are independent.
Each areal unit value is regressed on a
function of its surrounding areal unit values.
Statistical efficiency is lost when dependent values
are assumed to be independent.
pi  e
α βXi ρ
n
cijy j
j1
[1  e
α βXi ρ
n
cijy j
j1
]
Quasi-likelihood estimation
Maximum quasi-likelihood treats the
variation of Y values as though it is
inflated, and estimates  of the variance
term np(1-p) for the purpose of
rescaling when testing hypotheses.
This approach is equivalent to maximum
likelihood estimation when  = 1, and
most log-likelihood function asymptotic
theory transfers to the results.
Preliminary estimation (pseudo- and
quasi-likelihood) results: F/P (%)
model
intercept
SA
seSA
dispersion Deviance
binomial
auto-binomial
-1.10
-1.11
0
*****
0.89 0.001
*****
0
945.96
384.51
quasi-auto-binomial
-1.10
0.89 0.015
19.74
0.99
auto-logistic
-2.03
0.80 0.032
*****
0.93
n
n
j1
j1
Ci Y   c ij (Fj /Pj  F/P ); Ci I  25%   c ijI  25%, j
pseudo  R 2pct  0.61; pseudo  R 2I2 5 %  0.41
What is the alternative to
pseudo-likelihood?
MCMC maximum likelihood estimation!
• exploits the sufficient statistics
• based upon Markov chain transition
matrices converging to an equilibrium
• exploits marginal probabilities, and
hence can begin with pseudo-likelihood
results
• based upon simulation theory
Properties of estimators: a review
•
•
•
•
•
•
Unbiasedness
Efficiency
Consistency
Robustness
BLUE
BLUP
• Sufficiency
MCMC maximum likelihood
estimation
• MCMC denotes Markov chain Monte
Carlo
• Pseudo-likelihood works with the
conditional marginal models
• MCMC is needed to compute the
simultaneous likelihood result
• MCMC exploits the conditional
models
The theory of Markov chains was
developed by Andrei Markov at the
beginning of the 20th century.
A Markov chain is a process consisting of a finite
number of states and known probabilities, pij, of
moving from state i to state j.
Markov chain theory is based on the Ergodicity Thm:
irreducible, recurrent non-null, and aperiodic.
If a Markov chain is ergodic, then a unique steady
state distribution exists, independent of the initial
state: for transition matrix M, lim M k  M * ;
k 
P(Xt+1 = j| X0=i0, …, Xt=it) = P(Xt+1 = j| Xt=it) = tpij
Example transition matrix
convergence:
1

3
1
4

0

1
3

0

0


A
B
C
0.15 0.20 0.15
D
E
F
0.15 0.20 0.15
1
3
1
4
1
3
1
3
0
0
1
4
0
0
1
3
1
4
1
3
1
4
1
3

0

0

1

3
0

1

4
1

3
0
1
4
1
3
0
0
1
4
0
0
1
3
0
 34

 0.15

 0.15
 0.15

 0.15

 0.15
 0.15

0.20 0.15 0.15 0.20 0.15 

0.20 0.15 0.15 0.20 0.15 
0.20 0.15 0.15 0.20 0.15 

0.20 0.15 0.15 0.20 0.15 

0.20 0.15 0.15 0.20 0.15 
0.20 0.15 0.15 0.20 0.15 
Monte Carlo simulation is named after
the city in the Monaco principality,
because of a roulette, a simple random
number generator. The name and the
systematic development of Monte
Carlo methods date from about 1944.
The Monte Carlo method provides
approximate solutions to a variety of
mathematical problems by performing
statistical sampling experiments with a
computer using pseudo-random numbers.
MCMC has been around for about
50 years.
MCMC provides a mechanism for taking
dependent samples in situations where
regular sampling is difficult, if not
completely impossible. The standard
situation is where the normalizing
constant for a joint or a posterior
probability distribution is either too
difficult to calculate or analytically
intractable.
What is MCMC? A definition
MCMC is used to simulate from some
distribution p known only up to a constant
factor, C:
pi = Cqi
where qi is known but C is unknown and too
horrible to calculate.
MCMC begins with conditional (marginal)
distributions, and MCMC sampling outputs a
sample of parameters drawn from their joint
(posterior) distribution.
Starting with any Markov chain having
transition matrix M over the set of states i on
which p is defined, and given Xt = i, the idea is
to simulate a random variable X* with
distribution qi: qij = P(X* = j| Xt = i).
The distribution qi is called the proposal
distribution.
After a burn-in
set of simulations,
a chain converges
to an equilibrium
po = 0.5
p=0.2
Gibbs sampling is a MCMC scheme for
simulation from p where a transition kernel is
formed by the full conditional distributions of p.
• a stochastic process that returns a different result with each
execution; a method for generating a joint empirical
distribution of several variables from a set of modelled
conditional distributions for each variable when the structure
of data is too complex to implement mathematical formulae
or directly simulate.
• a recipe for producing a Markov chain that yields simulated
data that have the correct unconditional model properties,
given the conditional distributions of those variables under
study.
• its principal idea is to convert a multivariate problem into a
sequence of univariate problems, which then are iteratively
solved to produce a Markov chain.
A Gibbs sampling algorithm
(1) t = 0; set initial values 0x = (0x1, …, 0xn)’
(2) obtain new values tx = (tx1, …, txn)’ from t-1x:
tx
t-1x , …, t-1x )
~
p
(x
|{
1
1
2
n
tx , t-1x , …, t-1x )
~
p
(x
|{
2
2
1
3
n
…
tx
tx , …, tx )
~
p
(x
|{
n
1
1
n-1
(3) t = t+1; repeat step (2) until convergence.
tx
Monitoring convergence
MCMC exploits the sufficient statistics,
which should be monitored with a timeseries plot for randomness.
After removing burn-in iteration results, a
chain should be weeded (i.e., only every kth
output is retained). These weeded values
should be independent; this property can be
checked by constructing a correlogram.
Convergence of m chains can be assessed using
ANOVA: within-chain variance pooling is
legitimate when chains have converged.
Sufficient statistics for normal,
binomial, and Poisson models
A sufficient statistic (established with the
Rao-Blackwell factorization theorem) is a
statistic that captures all of the information
contained in a sample that is relevant to
the estimation of a population parameter.
n
y1
i 1
i
n
y
i 1
i
x ij , j  1, 2, ..., p
Implementation of MCMC for the
autologistic model
Y1
Y2
…
Y20
Y21
Y22
…
Y40
.
.
.
.
.
.
.
.
.
.
Y381 Y382
.
.
…
drawings
τ  0 : p i  12 (or  y i /n)
from the
i 1
binomial distribution
Yi,τ ~ binomial(n  1, p i,τ )
is the Monte Carlo
α̂  ρ̂  c y
part
n
n
ij j,τ*
p i,τ 
Y400
e
1 e
j1
α̂  ρ̂
n
 cijy j,τ*
, where τ * is a mixture of τ and τ - 1
j1
(  1)   is the Markov chain part
n
sufficient statistics : T1,τ   y i,τ  1
MCMC-MLEs are extracted
from the generated chains
i 1
T2,τ 
n
1
2
y
i 1
n
i,τ
  c ij y j,τ
j1
25,000 + 225,000/100
burn-in + weeded
MCMC results
alpha
df
F
prob
rho
F
prob
iteration
44
1.0 0.52 1.0 0.47
chain
2
0.1 0.91 0.1 0.92
interaction
88
1.0 0.56 1.0 0.54
error
6615
Some prediction comparisons
The (modified)
auto-Poisson
model
NOTE: the auto-Poisson model
can only capture negative spatial
autocorrelation
NOTE: excessive zeroes is a
serious problem with empirical
Poisson RVs
Spatial autoregression: the
auto-Poisson model
The workhorse of spatial statistical
generalized linear models is MCMC
μ
For counts, y, in the set
of integers {0, 1, 2, 3, … }
e
μ
P(Y  y ) 
E(Y)  μ
VAR(Y)  μ
LN (μ)  ρCY  Xβ
y
y!
MCMC is initiated with pseudolikelihood estimates
1
Pi  c e
-(α βXi  ρ
n
 cijy j )
j1
n
(α  βX i  ρ  c ij y j )
j1
yi
yi !
c-1 is an intractable normalizing factor
but ρ  0 : negative spatial autocorrelation
positive spatial autocorrelation can be handled
with Winsorizing, or binomial approximation
When VAR(Y) > μ
overdispersion (extra Poisson variation)
is encountered
• Detected when deviance/df > 1
• Often described as VAR(Y) = (1  ημ)μ
• Leads to the Negative Binomial model
Conceptualized
as the number of
times some
phenomenon
occurs before a
fixed number of
times (r) that it
does not occur.
 y  r  1 y
r
p (1  p)
P(Y  y )  C
 r 1 
E(Y)  r(1 - p)/p
VAR(Y)  r(1 - p)/p
2
Preliminary estimation (pseudoand quasi-likelihood) results: B/D
model
Poisson
SA
0
seSA dispersion Deviance
*****
***** 1230.20
auto-Poisson
0.02 <0.001
quasi-auto-Poisson
0.02
auto-negative binomial 0.02
0
822.23
0.006
29.2553
0.96
0.007
0.0626
1.01
n
Ci Y   cij (b j /d j )
j1
2
pseudo  R counts
 0.86; pseudo  R 2ratio  0.17


b j /d j
B̂i   e
 j1
n

c ij
ρ̂
 LN(μ̂)
Di
 e

MCMC results
typical correlogram
25,000 + 500,000/100
burn-in + weeded
Some prediction comparisons
Geographic covariation:
n-by-n matrix V
1/2
Y  μ1  V ε,
if no spatial autocorrel ation, then V  I
1/ 2
SAR : V
 (I - ρW)  (I - ρS C)
geostatist ics : (V
-1
-1/2 T
) (V
-1/2
)  kR
autoregression works with the inverse covariance matrix &
geostatistics works with the covariance matrix itself
Relationships between the range
parameter and rho for an ideal
infinite surface
ρ̂ CAR 
ρ̂SAR 

 r  2.1825  


 6.7229 r 1.8403
0.251  B1 
1

e


0.2298



 


RESS  0.00003
2.0130



r



 0.6842 r 0.887 9
0.251  B1 
1

e


0.2896




 


RESS  0.00041

modified
Bessel function
for CAR
 d ij 


 r 
b



Bessel function
for SAR
1
 d ij 


 r 
Constructing eigenfunctions for
filtering spatial autocorrelation
out of georeferenced variables:
MC = (n/1T C1)x
YT(I – 11T/n)C (I – 11T/n)Y/ YT(I – 11T/n)Y
the eigenfunctions come from
(I –
T
11 /n)C
(I –
T
11 /n)
C versus (I – 11T/n)C(I – 11T/n) = MCM
C
1
ρ
λ1,C
λ̂ MCM 
0.011  1.005λ C
MC 
n λ1,MCM
1T C1
n λ n,MCM
T
1 C1
 MC
MCM 2.06
2.07 *
5.51
*
1.92
1.92 -0.10 -0.10 -1.21 -1.21 -2.02 -2.02
5.19
5.21
1.78
1.79 -0.13 -0.12 -1.24 -1.23 -2.08 -2.08
4.91
4.99
1.57
1.59 -0.15 -0.15 -1.33 -1.33 -2.12 -2.12
4.35
4.35
1.32
1.35 -0.29 -0.28 -1.38 -1.38 -2.15 -2.15
4.01
4.06
1.23
1.26 -0.33 -0.33 -1.44 -1.44 -2.23 -2.23
3.96
3.96
1.05
1.06 -0.49 -0.46 -1.54 -1.52 -2.24 -2.24
3.84
3.88
0.93
0.94 -0.53 -0.53 -1.56 -1.55 -2.33 -2.32
3.42
3.43
0.80
0.80 -0.59 -0.59 -1.60 -1.59 -2.40 -2.39
3.35
3.35
0.78
0.79 -0.63 -0.63 -1.64 -1.64 -2.41 -2.41
2.90
2.91
0.58
0.61 -0.80 -0.80 -1.74 -1.74 -2.43 -2.42
2.65
2.72
0.38
0.38 -0.89 -0.88 -1.84 -1.84 -2.54 -2.54
2.53
2.59
0.27
0.27 -0.92 -0.92 -1.87 -1.87 -2.62 -2.62
2.35
2.40
0.17
0.19 -0.95 -0.95 -1.90 -1.90 -2.67 -2.67
2.20
2.24
0.12
0.12 -1.07 -1.06 -1.96 -1.96 -2.70 -2.70
0.00
-1.10 -1.09 -1.98 -1.98
Eigenvectors of MCM
• (I – 11T/n) = M ensures that the eigenvector means are
0
• symmetry ensures that the eigenvectors are orthogonal
• M ensures that the eigenvectors are uncorrelated
• replacing the 1st eigenvalue with 0 inserts the intercept
vector 1 into the set of eigenvectors
• thus, the eigenvectors represent all possible
distinct (i.e., orthogonal and uncorrelated) spatial
autocorrelation map patterns for a given surface
partitioning
• Legendre and his colleagues are developing
analogous eigenfunction spatial filters based upon the
truncated distance matrix used in geostatistics
Expectations for the Moran
Coefficient for linear
regression with normal
residuals
T
1
T
n TR[( X X) X CX]
E(MC)   T
n 1 p
1 C1
σ MC
2
 T
1 C1
A spatial filtering counterpart to the
auto-normal model specification.
• Y = Ekß + ε
• b = EkTY
• Only a single regression is needed to
implement the stepwise procedure.
MAX: R2; eigenvectors selected in order of their bivariate correlations
residual spatial autocorrelation =
Selected demographic attributes
of China
# common
to MAXattribute
R2, MINMC
population density
149
(|zres| = 7.5 → 6.3)
crude fertility rate
(|zres| = 4.4 → 2.7)
% 100+ years old
(|zres| = 0.4 → 0.0)
births/deaths ratio
(|zres| = 2.7 → 0.6)
# not truly
redundant
info
(~MAX-R2, MIN-MC)
# spatially
structured
(MAX-R2,
~MIN-MC)
151
71
229
105
0
145
8
20
233
119
0
Overdispersion: binomial extra
variation
• E(Y) = np and Var(Y) = np(1-p) , and >1
• tends to have little impact on regression
parameter point estimates (maximum likelihood
estimator typically is consistent, although small
sample bias might occur); but, regression
parameter standard error estimates
(variances/covariances) are underestimated
• may be reflected in the size of the deviance
statistic
• difficult to detect in binary 0-1 data
Spatial structure and generalized linear
modeling: “Poisson” regression
CBR: the spatial filter is
constructed with 199 of 561
candidate eigenvectors.
SF results
in green
SF
Poisson Negative SF negative
binomial
binomial
deviance
1377.31
1.02
1.10
mean
0.1241
0.1351
0.1308
dispersion
0
0.0933
0.0302
Pseudo-R2
0.762
0.762
0.903
(observed vs
predicted births)
Spatial structure and generalized linear
modeling: “binomial” regression
% population 100+ years old: the
spatial filter is constructed with
92 of 561 candidate
eigenvectors.
deviance
SF
Intercept
scale
Pseudo-R2
(observed vs
predicted births)
binomial
4.76
-12.0706
(0.0124)
1
0
SF binomial
1.00
-12.5000
(0.0276)
1.47
0.283
Advantages of spatial filtering
• Do not need MCMC for GLM parameter
estimation – conventional statistical theory
applies
• Uncover distinct map pattern components
of spatial autocorrelation that relate directly
to the MC
• The eigenvectors are orthogonal and
uncorrelated
• Can always calculate the necessary
eigenvectors as long as the number of
areal units does not exceed n ≈ 10,000
Interpretation of MIN-MC selections
Matrix Ek contains three disjoint eigenvector subsets:
Er, for those representing redundant locational
information; Es, for those representing spatially
structured random effects; and, Emisc, for those
being unrelated to Y. Accordingly, the pure spatial
autocorrelation model becomes
Y = µ1 + Erßr + (Esßs + e) ,
where ßr and ßs respectively are regression
coefficients defining relationships between Y and the
sets of eigenvectors Er and Es, and the term (Esßs +
e) behaves like a spatially structured random effect.
Random effects model
Y  f( Xβ  ξ, ε)
ξ is a random observation effect (differences
among individual observational units)
ε is a time-varying residual error (links to
change over time)
The composite error term is the sum of the
two.
Random effects model: normally
distributed intercept term
• ξ ~ N(0, σ ) and uncorrelated with covariates
• supports inference beyond the nonrandom
sample analyzed
• simplest is where intercept is allowed to vary
across areal units (repeated observations are
individual time series)
• The random effect variable is integrated out
(with numerical methods) of the likelihood fcn
• accounts for missing variables & within unit
correlation (commonality across time periods)
2
Random effects: mixed models
• Moving closer to a Bayesian perspective,
spatial autocorrelation can be accounted
for by introducing a (spatially structured)
random effect into a model specification.
• SAS PROC MIXED supports this
approach for linear modeling in which a
map is treated as a multivariate sample of
size 1.
• SAS PROC NLMIXED supports this
approach for generalized linear modeling.
SAS PROC MIXED and random effects:
Y=XB + Zu
• The spatially correlated errors model is
performed with PROC MIXED through the
REPEATED statement.
• The SUBJECT=INTERCEPT option specifies
that the correlation between units is
essentially between experimental units that
are different observations within the data set.
• The LOCAL option in the REPEATED
statement tells PROC MIXED to include a
nugget effect.
EXAMPLE: density of workers
across Germany’s 439 Kreises
LN(density – 23.53) ~ N
A spatial covariance structure coupled
with a random slope coefficient model
192,721 distance pairs
dmax = 9.32478
PROC MIXED output: intercept
intercept
estimate
correlation
-2log(L) nugget (partial) range
sill
5.28
(0.06)
5.01
(0.12)
5.01
(0.18)
5.01
(0.13)
5.01
(0.18)
none
1445.1
0
1.5782
0
spherical
1348.4
0.9139
0.5542
1.3801
exponential 1349.8
0.9154
0.5873
0.7824
Gaussian
1344.7
0.9858
0.5194
0.7260
power
1349.8
0.9154
0.5873
0.2786
Random intercept term
The spatial filter contains 27 (of 98) eigenvectors,
with R2 = 0.4542, P(S-Wresiduals) < 0.0001.
measure
No
covariates
Spatial
filter
Spherical
semivariogram
-2log(L)
1445.1
1179.3
1348.4
Intercept
variance
0.9631
0.2538
0.5542
Residual
variance
0.6116
0.6011
0.9139
Intercept
estimate
5.2827
(0.0599)
5.2827
(0.0443)
5.0142
(0.1210)
Generalized linear mixed models
• One drawback of spatial filtering is that as
the number of areal units increases,
the number of eigenvectors needed to
construct a spatial filter tends to
increase, resulting in asymptotics being
difficult or impossible to achieve.
• This situation can be remedied by
resorting to a space-time data set, with
time being repeated measures whose
correlation can be captured by a random
effects intercept term.
Unemployment in Germany: 1996-2002
year
year-specific eigenvectors
global
regional
local
E9, E16, E21, E25, E41, E52,
E53, E64
E89
E15, E19, E21, E34, E38, E64
E93
1998
E13, E15, E16, E19, E21,
E34, E38, E42, E52, E66
E68, E93
1999
E9, E13, E15, E16, E19, E21,
E34, E38, E42, E52, E66
E93
2000
E9, E13, E15, E16, E19, E21,
E25, E34, E38, E42, E51,
E52, E66
E93, E97
2001
E9, E12, E13, E15, E16, E19,
E34, E42, E52, E56, E65, E66
E68,
E93, E97
1996
1997
2002
E1
E1
E9, E12, E13, E15, E16, E19,
E20, E25, E38, E42, E52,
E65, E66
common
eigenvectors
global regional local
E2 - E5
E6 - E8,
E11, E18,
E24, E28,
E30, E39,
E60
E74
Unemployment in Germany:
annual spatial filters
year
# of
scale adjusted
SSPE/SSE
2
eigenvecvtors
pseudo-R
1996
24
21.98
0.5929
1.0232
1997
23
24.38
0.6425
1.0412
1998
27
23.52
0.6846
1.0438
1999
27
23.25
0.7068
1.0364
2000
30
23.83
0.7483
1.0507
2001
30
25.18
0.7683
1.0489
2002
29
26.08
0.7549
1.0459
Dark red:
very high
Light red:
high
Gray:
medium
Light green:
low
Dark green:
very low
The composite spatial filter
constructed with common vectors
former east-west
divide
year
1996
1997
1998
1999
2000
2001
2002
SF
SF residuals
MC
GR
0.67 → 0.21 0.62
0.73 → 0.20 0.66
0.76 → 0.20 0.64
0.79 → 0.21 0.61
0.83 → 0.25 0.59
0.85 → 0.27 0.57
0.85 → 0.27 0.56
1.14
0.15
Generated space-time predictions
the lack of serial
correlation information
in 1996 is conspicuous
the best fit is in
the center of the
space-time series
% urban in Puerto Rico: SF-logistic with
a spatial structured random effect