Spatially Autoregressive Models for Regression Analysis

10/7/2015 DEM 7263 Fall 2015 - Spatially Autoregressive Models 1
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DEM 7263 Fall 2015 - Spatially
Autoregressive Models 1
Corey S. Sparks, Ph.D.
September 9, 2015
Introduction to Spatial Regression
Models
Up until now, we have been concerned with describing the structure of spatial data through correlational,
and the methods of exploratory spatial data analysis (http://rpubs.com/corey_sparks/105700).
Through ESDA, we examined data for patterns and using the Moran I and Local Moran I statistics, we
examined clustering of variables. Now we consider regression models for continuous outcomes. We begin
with a review of the Ordinary Least Squares model for a continuous outcome.
OLS Model
The basic OLS model is an attempt to estimate the effect of an independent variable(s) on the value of a
dependent variable. This is written as:
where y is the dependent variable that we want to model, x is the independent variable we think has an
association with y, is the model intercept, or grand mean of y, when x = 0, and is the slope
parameter that defines the strength of the linear relationship between x and y. e is the error in the model
for y that is unaccounted for by the values of x and the grand mean . The average, or expected value of
y is : , which is the linear mean function for y, conditional on x, and this gives us the
customary linear regression plot:
set.seed(1234)
x<- rnorm(100, 10, 5)
beta0<-1
beta1<-1.5
y<-beta0+beta1*x+rnorm(100, 0, 5)
plot(x, y)
abline(coef = coef(lm(y~x)), lwd=1.5)
= + ∗ +yi
β0
β1
xi
ei
β0
β1
β0
E[y|x] = + ∗β0
β1
xi

summary(lm(y~x))$coef
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.446620 1.0879494 1.329676 1.867119e-01
## x 1.473915 0.1037759 14.202863 1.585002e-25
Where, the line shows
We assume that the errors, are independent, Normally distributed and homoskdastic, with
variances .
This is the simple model with one predictor. We can easily add more predictors to the equation and
rewrite it:
So, now the mean of y is modeled with multiple x variables. We can write this relationship more compactly
using matrix notation:
Where Y is now a vector of observations of our dependent variable, X is a matrix of
independent variables, with the first column being all 1’s and e is the vector of errors for each
observation.
E[y|x] = + ∗β0
β1
xi
∼ N(0, )ei
σ2
σ2
y = + ∗ +β0
∑k
βk
xik
ei
Y = β + eX
′
n ∗ 1 n ∗ k
n ∗ 1

In matrices this looks like:
The residuals are uncorrelated, with covariance matrix =
To estimate the coefficients, we use the customary OLS estimator
this is the estimator that minimizes the residual sum of squares:
or
We can inspect the properties of the estimates by examining the residuals, or of the model. Since we
assume the data are normal, a quantile-quantile (Q-Q) plot of the residuals against the expected quantile
of the standard normal distribution should be a straight line. Formal tests of normality can also be used.
fit<-lm(y~x)
qqnorm(rstudent(fit))
qqline(rstudent(fit))
y =
⎡
⎣
⎢
⎢
⎢
⎢
y1
y2
⋮
yn
⎤
⎦
⎥
⎥
⎥
⎥
β =
⎡
⎣
⎢
⎢
⎢
⎢
β0
β1
⋮
βk
⎤
⎦
⎥
⎥
⎥
⎥
x =
⎡
⎣
⎢
⎢
⎢
⎢
1
1
1
1
x1,1
x2,1
⋮
xn,1
x1,2
x1,2
⋮
xn,2
…
…
⋮
…
x1,k
x1,k
⋮
xn,k
⎤
⎦
⎥
⎥
⎥
⎥
e =
⎡
⎣
⎢
⎢
⎢
⎢
e1
e2
⋮
en
⎤
⎦
⎥
⎥
⎥
⎥
Σ
Σ = I = ∗ =σ2
σ2
⎡
⎣
⎢
⎢
⎢
⎢
1
0
0
0
0
1
⋮
0
0
0
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0
…
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…
0
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1
⎤
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0
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0
…
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σ2
⎤
⎦
⎥
⎥
⎥
⎥
β β = ( X ( Y)X
′
)
−1
X
′
(Y − β (Y − β)X
′
)
′
X
′
(Y − (Y − )Y
̂
)
′
Y
̂
ei

shapiro.test(resid(fit))
##
## Shapiro-Wilk normality test
##
## data: resid(fit)
## W = 0.98878, p-value = 0.5677
ad.test(resid(fit))
##
## Anderson-Darling normality test
##
## data: resid(fit)
## A = 0.39859, p-value = 0.3593
lillie.test(resid(fit))

##
## Lilliefors (Kolmogorov-Smirnov) normality test
##
## data: resid(fit)
## D = 0.052017, p-value = 0.7268
We may also inspect the association between , or more appropriately the studentized/standardized
residuals, and the predictors and the dependent variable. If we see evidence of association, then
homoskedasticity is a poor assumption
par(mfrow=c(2,2))
plot(fit)
par(mfrow=c(1,1))
Model-data agreement
Do we (meaning our data) meet the statistical assumptions of our analytical models?
Always ask this of any analysis you do, if your model is wrong, your inference will also be wrong.
ei

Since spatial data often display correlations amongst closely located observations (autocorrelation), we
should probably test for autocorrelation in the model residuals, as that would violate the assumptions of
the OLS model.
One method for doing this is to calculate the value of Moran’s I for the OLS residuals.
library(spdep)
library(maptools)
library(RColorBrewer)
setwd("~/Google Drive/dem7263/data/")
dat<-readShapePoly("SA_classdata.shp", proj4string=CRS("+proj=utm +zone=14 +nort
h"))
#Make a rook style weight matrix
sanb<-poly2nb(dat, queen=F)
summary(sanb)
## Neighbour list object:
## Number of regions: 235
## Number of nonzero links: 1106
## Percentage nonzero weights: 2.002716
## Average number of links: 4.706383
## Link number distribution:
##
## 1 2 3 4 5 6 7 8 9
## 4 10 30 62 66 34 24 3 2
## 4 least connected regions:
## 61 82 147 205 with 1 link
## 2 most connected regions:
## 31 55 with 9 links
salw<-nb2listw(sanb, style="W")
fit2<-lm(I(viol3yr/acs_poptot) ~ pfemhh + hwy + p5yrinmig + log(MEDHHINC), data=da
t )
dat$resid<-rstudent(fit2)
spplot(dat, "resid",at=quantile(dat$resid), col.regions=brewer.pal(n=5, "RdBu"), m
ain="Residuals from OLS Fit of Crime Rate")

lm.morantest(fit2, listw = salw, resfun = rstudent)
##
## Global Moran's I for regression residuals
##
## data:
## model: lm(formula = I(viol3yr/acs_poptot) ~ pfemhh + hwy +
## p5yrinmig + log(MEDHHINC), data = dat)
## weights: salw
##
## Moran I statistic standard deviate = 0.75475, p-value = 0.2252
## alternative hypothesis: greater
## sample estimates:
## Observed Moran's I Expectation Variance
## 0.021326432 -0.011406176 0.001880845
Which, in this case, there appears to be no clustering in the residuals, since the observed value of Moran’s
I is .021, with a z-test of 0.75, p= .225.
Extending the OLS model to accommodate spatial structure
If we now assume we measure our Y and X’s at specific spatial locations (s), so we have Y(s) and X(s).

In most analysis, the spatial location (i.e. the county or census tract) only serves to link X and Y so we can
collect our data on them, and in the subsequent analysis this spatial information is ignored that explicitly
considers the spatial relationships between the variables or the locations.
In fact, even though we measure Y(s) and X(s) what we end up analyzing X and Y, and apply the ordinary
regression methods on these data to understand the effects of X on Y.
Moreover, we could move them around in space (as long as we keep the observations together with )
and still get the same results. Such analyses have been called a-spatial. This is the kind of regression
model you are used to fitting, where we ignore any information on the locations of the observations
themselves.
However, we can extend the simple regression case to include the information on (s) and incorporate it
into our models explicitly, so they are no longer a-spatial.
There are several methods by which to incorporate the (s) locations into our models, there are several
alternatives to use on this problem:
The structured linear mixed (multi-level) model, or GLMM (generalized linear mixed model)
Spatial filtering of observations
Spatially autoregressive models
Geographically weighted regression
We will first deal with the case of the spatially autoregressive model, or SAR model, as its structure is just
a modification of the OLS model from above.
Spatially autoregressive models
We saw in the normal OLS model that some of the basic assumptions of the model are that the: 1) model
residuals are distributed as iid standard normal random variates 2) and that they have common (and
constant, meaning homoskedastic) unit variance.
Spatial data, however present a series of problems to the standard OLS regression model. These
problems are typically seen as various representations of spatial structure or dependence within the data.
The spatial structure of the data can introduce spatial dependence into both the outcome, the predictors
and the model residuals.
This can be observed as neighboring observations, both with high (or low) values (positive autocorrelation)
for either the dependent variable, the model predictors or the model residuals. We can also observe
situations where areas with high values can be surrounded by areas with low values (negative
autocorrelation).
Since the standard OLS model assumes the residuals (and the outcomes themselves) are uncorrelated, as
previous stated, the autocorrelation inherent to most spatial data introduces factors that violate the iid
distributional assumptions for the residuals, and could violate the assumption of common variance for the
OLS residuals. To account for the expected spatial association in the data, we would like a model that
accounts for the spatial structure of the data. One such way of doing this is by allowing there to be
correlation between residuals in our model, or to be correlation in the dependent variable.
yi
xi

We are familiar with the concept of autoregression amongst neighboring observations. This concept is
that a particular observation is a linear combination of its neighboring values. This autoregression
introduces dependence into the data. Instead of specifying the autoregression structure directly, we
introduce spatial autocorrelation through a global autocorrelation coefficient and a spatial proximity
measure.
There are 2 basic forms of the spatial autoregressive model: the spatial lag and the spatial error models.
Both of these models build on the basic OLS regression model: $ Y = dots X ’ + e$
Where Y is the dependent variable, X is the matrix of independent variables, is the vector of regression
parameters to be estimated from the data, and e are the model residuals, which are assumed to be
distributed as a Gaussian random variable with mean 0 and constant variance-covariance matrix .
The spatial lag model
The spatial lag model introduces autocorrelation into the regression model by lagging the dependent
variables themselves, much like in a time-series approach .
The model is specified as:
where is the autoregressive coefficient, which tells us how strong the resemblance is, on average,
between and it’s neighbors. The matrix ** W** is the spatial weight matrix, describing the spatial
network structure of the observations, like we described in the ESDA lecture.
In the lag model, we are specifying the spatial component on the dependent variable. This leads to a
spatial filtering of the variable, where they are averaged over the surrounding neighborhood defined in W,
called the spatially lagged variable.
The specification that is used most often is a spatially filtered Y variable that can then be regressed on X,
which can directly be seen in a re-expression of the OLS model as:
where the direct effect of the spatial lagging of the dependent variable is seen.
To estimate these models we can use either GeoDa or R in R we use the spdep package, and the
lagsarlm() function
The lag model is:
fit.lag<-lagsarlm(I(viol3yr/acs_poptot) ~ pfemhh + hwy + p5yrinmig + log(MEDHHIN
C), data=dat, listw = salw)
summary(fit.lag, Nagelkerke=T)
β
Σ
Y = ρWY + β + eX
′
ρ
Yi
Y = ρWY + β + eX
′

##
## Call:
## lagsarlm(formula = I(viol3yr/acs_poptot) ~ pfemhh + hwy + p5yrinmig +
## log(MEDHHINC), data = dat, listw = salw)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0635446 -0.0147641 -0.0036721 0.0090372 0.3252902
##
## Type: lag
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.3136449 0.0924307 3.3933 0.0006906
## pfemhh 0.1913535 0.0336049 5.6942 1.239e-08
## hwy 0.0075802 0.0056013 1.3533 0.1759604
## p5yrinmig 0.0794330 0.0202592 3.9208 8.824e-05
## log(MEDHHINC) -0.0337148 0.0082612 -4.0811 4.482e-05
##
## Rho: 0.034756, LR test value: 0.18517, p-value: 0.66697
## Asymptotic standard error: 0.082235
## z-value: 0.42264, p-value: 0.67256
## Wald statistic: 0.17862, p-value: 0.67256
##
## Log likelihood: 441.5604 for lag model
## ML residual variance (sigma squared): 0.0013657, (sigma: 0.036955)
## Nagelkerke pseudo-R-squared: 0.36486
## Number of observations: 235
## Number of parameters estimated: 7
## AIC: -869.12, (AIC for lm: -870.94)
## LM test for residual autocorrelation
## test value: 0.4691, p-value: 0.4934
We see that is estimated to be .034, and the likelihood ratio test shows that this is not significantly
different from 0.
The spatial error model
The spatial error model says that the autocorrelation is not in the outcome itself, but instead, any
autocorrelation is attributable to there being missing spatial covariates in the data. If these spatially
patterned covariates could be measures, the tne autocorrelation would be 0. This model is written:
This model, in effect, controls for the nuisance of correlated errors in the data that are attributable to an
inherently spatial process, or to spatial autocorrelation in the measurement errors of the measured and
possibly unmeasured variables in the model.
This model is estimated in R using errorsarlm() in the spdep library
ρ
Y = β + eX
′
e = λWe + v

fit.err<-errorsarlm(I(viol3yr/acs_poptot) ~ pfemhh + hwy + p5yrinmig + log(MEDHHIN
C), data=dat, listw = salw)
summary(fit.err, Nagelkerke=T)
##
## Call:
## errorsarlm(formula = I(viol3yr/acs_poptot) ~ pfemhh + hwy + p5yrinmig +
## log(MEDHHINC), data = dat, listw = salw)
##
## Residuals:
## -0.0650577 -0.0141501 -0.0034659 0.0092839 0.3241926
##
## Type: error
## (Intercept) 0.3138785 0.0922430 3.4027 0.0006671
## pfemhh 0.1971225 0.0340765 5.7847 7.264e-09
## hwy 0.0073140 0.0057043 1.2822 0.1997762
## p5yrinmig 0.0781620 0.0205345 3.8064 0.0001410
## log(MEDHHINC) -0.0337316 0.0082489 -4.0893 4.328e-05
##
## Lambda: 0.070725, LR test value: 0.53188, p-value: 0.46582
## z-value: 0.7507, p-value: 0.45283
##
## Log likelihood: 441.7338 for error model
## AIC: -869.47, (AIC for lm: -870.94)
We see = .071, with a p-value of .465, suggesting again that, in this case, there is no autocorrelation in
the model residuals.
We can examine the relative fits of each model by extracting the AIC values from each:
AIC(fit.lag)
## [1] -869.1208
AIC(fit.err)
## [1] -869.4675
λ

AIC(fit2)
## [1] -870.9356
Which, while slightly lower than the OLS model, show little evidence of favoring the spatial regression
models in this case.
Examination of Model Specification
To some degree, both of the SAR specifications allow us to model spatial dependence in the data. The
primary difference between them is where we model said dependence.
The lag model says that the dependence affects the dependent variable only, we can liken this to a
diffusion scenario, where your neighbors have a diffusive effect on you.
The error model says that dependence affects the residuals only. We can liken this to the missing spatially
dependent covariate situation, where, if only we could measure another really important spatially
associated predictor, we could account for the spatial dependence. But alas, we cannot, and we instead
model dependence in our errors.
These are inherently two completely different ways to think about specifying a model, and we should really
make our decision based upon how we think our process of interest operates.
That being said, this way of thinking isn’t necessarily popular among practitioners. Most practitioners want
the best fitting model, ‘nuff said. So methods have been developed that test for alternate model
specifications, to see which kind of model best summarizes the observed variation in the dependent
variable and the spatial dependence.
These are a set of so-called Lagrange Multiplier (econometrician’s jargon for a score test
(https://en.wikipedia.org/wiki/Score_test)) test. These tests compare the model fits from the OLS, spatial
error, and spatial lag models using the method of the score test.
For those who don’t remember, the score test is a test based on the relative change in the first derivative
of the likelihood function around the maximum likelihood. The particular thing here that is affecting the
value of this derivative is the autoregressive parameter, or . In the OLS model or = 0 (so both the
lag and error models simplify to OLS), but as this parameter changes, so does the likelihood for the model,
hence why the derivative of the likelihood function is used. This is all related to how the estimation
routines estimate the value of or .
Using the Lagrange Multiplier Test (LMT)
In general, you fit the OLS model to your dependent variable, then submit the OLS model fit to the LMT
testing procedure.
Then you look to see which model (spatial error, or spatial lag) has the highest value for the test.
Enter the uncertainty…
So how much bigger, you might say?
ρ λ ρ λ
ρ λ

Well, drastically bigger, if the LMT for the error model is 2500 and the LMT for the lag model is 2480, this
is NOT A BIG DIFFERENCE, only about 1%. If you see a LMT for the error model of 2500 and a LMT for
the lag model of 250, THIS IS A BIG DIFFERENCE.
So what if you don’t see a BIG DIFFERENCE, HOW DO YOU DECIDE WHICH MODEL TO USE???
Well, you could think more, but who has time for that.
The econometricians have thought up a “better” LMT test, the so-called robust LMT, robust to what I’m
not sure, but it is said that it can settle such problems of a “not so big difference” between the lag and
error model specifications.
So what do you do? In general, think about your problem before you run your analysis, should this fail you,
proceed with using the LMT, if this is inconclusive, look at the robust LMT, and choose the model which
has the larger value for this test.
More Data Examples:
San Antonio, TX mortality data

#Spatial Regression Models 1
setwd("~/Google Drive//dem7263/data")
#, proj4string=CRS("+proj=utm zone=14")
dat<-readShapePoly("SA_classdata.shp")
dat<-dat[which(dat$acs_poptot>100),]
#Create a good representative set of neighbor types
sa.nb6<-knearneigh(coordinates(dat), k=6)
sa.nb6<-knn2nb(sa.nb6)
sa.wt6<-nb2listw(sa.nb6, style="W")
sa.wt3<-nb2listw(sa.nb3,style="W")
sa.wt2<-nb2listw(sa.nb2,style="W")
sa.nbr<-poly2nb(dat, queen=F)
sa.wtr<-nb2listw(sa.nbr, zero.policy=T)
sa.nbq<-poly2nb(dat, queen=T)
sa.wtq<-nb2listw(sa.nbr, style="W", zero.policy=T)
sa.nbd<-dnearneigh(coordinates(dat), d1=0, d2=10000)
sa.wtd<-nb2listw(sa.nbd, zero.policy=T)
#create a mortality rate, 3 year average
dat$mort3<-apply(dat@data[, c("deaths09", "deaths10", "deaths11")],1,mean)
dat$mortrate<-1000*dat$mort3/dat$acs_poptot
#just a
hist(dat$mortrate)

#do some basic regression models, without spatial structure
fit.1<-lm(scale(mortrate)~scale(ppersonspo)+scale(I(viol3yr/acs_poptot))+scale(dis
sim)+scale(ppop65plus), data=dat)
summary(fit.1)

##
## Call:
## lm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(I(viol3yr/acs_poptot))
+
## scale(dissim) + scale(ppop65plus), data = dat)
##
## Residuals:
## -1.96044 -0.27804 -0.00673 0.26359 2.18006
##
## Coefficients:
## (Intercept) 1.723e-16 3.120e-02 0.000 1.0000
## scale(ppersonspo) 1.215e-01 5.047e-02 2.407 0.0169 *
## scale(I(viol3yr/acs_poptot)) 2.287e-01 3.841e-02 5.953 9.8e-09 ***
## scale(dissim) 8.467e-02 4.817e-02 1.758 0.0801 .
## scale(ppop65plus) 7.240e-01 3.594e-02 20.146 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4773 on 229 degrees of freedom
## Multiple R-squared: 0.7761, Adjusted R-squared: 0.7722
## F-statistic: 198.4 on 4 and 229 DF, p-value: < 2.2e-16
vif(fit.1)
## scale(ppersonspo) scale(I(viol3yr/acs_poptot))
## 2.605367 1.508595
## scale(dissim) scale(ppop65plus)
## 2.372665 1.320878
par(mfrow=c(2,2))
plot(fit.1)

par(mfrow=c(1,1))
#this is a test for constant variance
bptest(fit.1) #looks like have heteroskedasticity
##
## studentized Breusch-Pagan test
##
## data: fit.1
## BP = 51.088, df = 4, p-value = 2.14e-10
#extract studentized residuals from the fit, and examine them
dat$residfit1<-rstudent(fit.1)
cols<-brewer.pal(5,"RdBu")
spplot(dat,"residfit1", at=quantile(dat$residfit1), col.regions=cols, main="Residu
als from Model fit of Mortality Rate")

Chi and Zhu (http://link.springer.com/article/10.1007/s11113-007-9051-8#page-1) suggest using a wide
array of neighbor specifications, then picking the one that maximizes the autocorrelation coefficient. So,
here I emulate their results:
#test for residual autocorrelation
resi<-c(lm.morantest(fit.1, listw=sa.wt2)$estimate[1],
lm.morantest(fit.1, listw=sa.wt3)$estimate[1],
lm.morantest(fit.1, listw=sa.wtd, zero.policy=T)$estimate[1],
lm.morantest(fit.1, listw=sa.wtq)$estimate[1],
lm.morantest(fit.1, listw=sa.wtr)$estimate[1])
plot(resi, type="l")

lm.morantest(fit.1, listw=sa.wt2)
##
##
## data:
## model: lm(formula = scale(mortrate) ~ scale(ppersonspo) +
## scale(I(viol3yr/acs_poptot)) + scale(dissim) + scale(ppop65plus),
## data = dat)
## weights: sa.wt2
##
## Moran I statistic standard deviate = -1.3282, p-value = 0.908
## -0.089262137 -0.010642737 0.003503515

##
##
## data:
## data = dat)
## weights: sa.wt3
##
## -0.005775111 -0.010724844 0.002386190
##
##
## data:
## data = dat)
## weights: sa.wt4
##
## 0.02812192 -0.01050301 0.00182003

##
##
## data:
## data = dat)
## weights: sa.wt5
##
## 0.04315932 -0.01029146 0.00145856
##
##
## data:
## data = dat)
## weights: sa.wt6
##
## 0.049203016 -0.010078442 0.001202519
lm.morantest(fit.1, listw=sa.wtd, zero.policy=T)

##
##
## data:
## data = dat)
## weights: sa.wtd
##
## 1.901642e-02 -6.929932e-03 9.436939e-05
lm.morantest(fit.1, listw=sa.wtq)
##
##
## data:
## data = dat)
## weights: sa.wtq
##
## 0.017905792 -0.010601820 0.001905428
lm.morantest(fit.1, listw=sa.wtr)

##
##
## data:
## data = dat)
## weights: sa.wtr
##
## 0.017905792 -0.010601820 0.001905428
#looks like we have minimal autocorrelation in our residuals, but the distance bas
ed
#weight does show significant autocorrelation
#Let's look at the local autocorrelation in our residuals
#get the values of I
dat$lmfit1<-localmoran(dat$mortrate, sa.wt5, zero.policy=T)[,1]
brks<-classIntervals(dat$lmfit1, n=5, style="quantile")
spplot(dat, "lmfit1", at=brks$brks
, col.regions=brewer.pal(5, "RdBu"), main="Local Moran Plot of Mortality Rate")

#Now we fit the spatial lag model
#The lag mode is fit with lagsarlm() in the spdep library
#we basically specify the same model as in the lm() fit above
#But we need to specify the spatial weight matrix and the type
#of lag model to fit
fit.lag<-lagsarlm(scale(mortrate)~scale(ppersonspo)+scale(I(viol3yr/acs_poptot))+s
cale(dissim)+scale(ppop65plus), data=dat, listw=sa.wt2, type="lag")
summary(fit.1)

##
## Call:
## lm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(I(viol3yr/acs_poptot))
+
## scale(dissim) + scale(ppop65plus), data = dat)
##
## Residuals:
## -1.96044 -0.27804 -0.00673 0.26359 2.18006
##
## Coefficients:
## (Intercept) 1.723e-16 3.120e-02 0.000 1.0000
## scale(ppersonspo) 1.215e-01 5.047e-02 2.407 0.0169 *
## scale(I(viol3yr/acs_poptot)) 2.287e-01 3.841e-02 5.953 9.8e-09 ***
## scale(dissim) 8.467e-02 4.817e-02 1.758 0.0801 .
## scale(ppop65plus) 7.240e-01 3.594e-02 20.146 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
summary(fit.lag, Nagelkerke=T)

##
## Call:
## lagsarlm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(I(viol3yr/acs_po
ptot)) +
## scale(dissim) + scale(ppop65plus), data = dat, listw = sa.wt2,
## type = "lag")
##
## Residuals:
## -1.91953751 -0.28320690 0.00039592 0.25188870 2.28912610
##
## Type: lag
## (Intercept) -0.0035113 0.0307209 -0.1143 0.9090
## scale(ppersonspo) 0.1047111 0.0509073 2.0569 0.0397
## scale(I(viol3yr/acs_poptot)) 0.2235749 0.0379480 5.8916 3.825e-09
## scale(dissim) 0.0758260 0.0476155 1.5925 0.1113
## scale(ppop65plus) 0.7019724 0.0376957 18.6221 < 2.2e-16
##
## Rho: 0.066138, LR test value: 2.1119, p-value: 0.14616
## z-value: 1.5104, p-value: 0.13095
##
## Log likelihood: -155.394 for lag model
## AIC: 324.79, (AIC for lm: 324.9)
## test value: 9.0458, p-value: 0.002633
bptest.sarlm(fit.lag)
##
##
## data:
## BP = 53.263, df = 4, p-value = 7.506e-11
#robust SE's for the spatial model
library(sandwich)
lm.target <- lm(fit.lag$tary ~ fit.lag$tarX - 1)
coeftest(lm.target, vcov.=vcovHC(lm.target, type="HC0"))

##
## t test of coefficients:
##
## Estimate Std. Error t value
## fit.lag$tarXx(Intercept) -0.0035113 0.0307070 -0.1143
## fit.lag$tarXxscale(ppersonspo) 0.1047111 0.0790642 1.3244
## fit.lag$tarXxscale(I(viol3yr/acs_poptot)) 0.2235749 0.1140534 1.9603
## fit.lag$tarXxscale(dissim) 0.0758260 0.0498978 1.5196
## fit.lag$tarXxscale(ppop65plus) 0.7019724 0.0617274 11.3721
## Pr(>|t|)
## fit.lag$tarXx(Intercept) 0.90906
## fit.lag$tarXxscale(ppersonspo) 0.18670
## fit.lag$tarXxscale(I(viol3yr/acs_poptot)) 0.05118 .
## fit.lag$tarXxscale(dissim) 0.12998
## fit.lag$tarXxscale(ppop65plus) < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Next we fit the spatial error model
fit.err<-errorsarlm(scale(mortrate)~scale(ppersonspo)+scale(I(viol3yr/acs_popto
t))+scale(dissim)+scale(ppop65plus), data=dat, listw=sa.wt2)
summary(fit.err, Nagelkerke=T)

##
## Call:
## errorsarlm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(I(viol3yr/ac
s_poptot)) +
## scale(dissim) + scale(ppop65plus), data = dat, listw = sa.wt2)
##
## Residuals:
## -1.9126219 -0.2695111 -0.0030349 0.2680765 2.1121586
##
## Type: error
## (Intercept) 0.0029538 0.0277141 0.1066 0.91512
## scale(ppersonspo) 0.1202146 0.0484444 2.4815 0.01308
## scale(I(viol3yr/acs_poptot)) 0.2303560 0.0371271 6.2045 5.486e-10
## scale(dissim) 0.0870377 0.0458697 1.8975 0.05776
## scale(ppop65plus) 0.7350826 0.0340110 21.6131 < 2.2e-16
##
## Lambda: -0.10644, LR test value: 2.2391, p-value: 0.13456
## z-value: -1.4877, p-value: 0.13684
##
## Log likelihood: -155.3304 for error model
## AIC: 324.66, (AIC for lm: 324.9)
#As a pretty good indicator of which model is best, look at the AIC of each
AIC(fit.1)
## [1] 324.8999
AIC(fit.lag)
## [1] 324.788
AIC(fit.err)
## [1] 324.6608
Larger data example US counties

This example shows a lot more in terms of spatial effects.
spdat<-readShapePoly("~/Google Drive/dem7263/data/usdata_mort.shp")
#Create a good representative set of neighbor types
us.nb6<-knearneigh(coordinates(spdat), k=6)
us.nb6<-knn2nb(us.nb6)
us.wt6<-nb2listw(us.nb6, style="W")
us.wt3<-nb2listw(us.nb3,style="W")
us.wt2<-nb2listw(us.nb2,style="W")
us.nbr<-poly2nb(spdat, queen=F)
us.wtr<-nb2listw(us.nbr, zero.policy=T)
us.nbq<-poly2nb(spdat, queen=T)
us.wtq<-nb2listw(us.nbr, style="W", zero.policy=T)
hist(spdat$mortrate)

spplot(spdat,"mortrate", at=quantile(spdat$mortrate), col.regions=brewer.pal(n=5,
"Reds"), main="Spatial Distribution of US Mortality Rate")

#do some basic regression models, without spatial structure
fit.1.us<-lm(scale(mortrate)~scale(ppersonspo)+scale(p65plus)+scale(pblack_1)+scal
e(phisp)+factor(RUCC), spdat)
summary(fit.1.us)

##
## Call:
## lm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(p65plus) +
## scale(pblack_1) + scale(phisp) + factor(RUCC), data = spdat)
##
## Residuals:
## -3.5810 -0.4280 0.0216 0.4534 4.2606
##
## Coefficients:
## (Intercept) -0.16408 0.06218 -2.639 0.00836 **
## scale(ppersonspo) 0.60632 0.01789 33.893 < 2e-16 ***
## scale(p65plus) -0.04085 0.01582 -2.582 0.00987 **
## scale(pblack_1) 0.10730 0.01641 6.538 7.29e-11 ***
## scale(phisp) -0.28734 0.01484 -19.358 < 2e-16 ***
## factor(RUCC)1 0.41534 0.08731 4.757 2.05e-06 ***
## factor(RUCC)2 0.29579 0.07226 4.094 4.36e-05 ***
## factor(RUCC)3 0.11800 0.07985 1.478 0.13955
## factor(RUCC)4 0.23900 0.08845 2.702 0.00693 **
## factor(RUCC)5 0.13588 0.09502 1.430 0.15282
## factor(RUCC)6 0.41615 0.06901 6.030 1.83e-09 ***
## factor(RUCC)7 0.17107 0.07097 2.411 0.01599 *
## factor(RUCC)8 0.11620 0.07880 1.475 0.14040
## factor(RUCC)9 -0.20337 0.07654 -2.657 0.00793 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
vif(fit.1.us)
## GVIF Df GVIF^(1/(2*Df))
## scale(ppersonspo) 1.786082 1 1.336444
## scale(p65plus) 1.397321 1 1.182083
## scale(pblack_1) 1.503301 1 1.226092
## scale(phisp) 1.229648 1 1.108895
## factor(RUCC) 1.724759 9 1.030746
par(mfrow=c(2,2))
plot(fit.1.us)

par(mfrow=c(1,1))
#this is a test for constant variance
bptest(fit.1.us) #looks like have heteroskedasticity
##
##
## data: fit.1.us
## BP = 249.87, df = 13, p-value < 2.2e-16

#extract studentized residuals from the fit, and examine them
spdat$residfit1<-rstudent(fit.1.us)
cols<-brewer.pal(5,"RdBu")
spplot(spdat,"residfit1", at=quantile(spdat$residfit1), col.regions=cols, main="Re
siduals from Model fit of US Mortality Rate")

#test for residual autocorrelation
resi<-c(lm.morantest(fit.1.us, listw=us.wt2)$estimate[1],
lm.morantest(fit.1.us, listw=us.wt3)$estimate[1],
lm.morantest(fit.1.us, listw=us.wtq,zero.policy=T)$estimate[1],
lm.morantest(fit.1.us, listw=us.wtr,zero.policy=T)$estimate[1])
plot(resi, type="l")
lm.morantest(fit.1.us, listw=us.wt2)

##
##
## data:
## scale(p65plus) + scale(pblack_1) + scale(phisp) + factor(RUCC),
## data = spdat)
## weights: us.wt2
##
## Moran I statistic standard deviate = 23.756, p-value < 2.2e-16
## 0.3965422192 -0.0018662504 0.0002812589
##
##
## data:
## data = spdat)
## weights: us.wt3
##
## 0.3879364703 -0.0018143292 0.0001956793

##
##
## data:
## data = spdat)
## weights: us.wt4
##
## 0.3798405687 -0.0017734270 0.0001504971
##
##
## data:
## data = spdat)
## weights: us.wt5
##
## 0.3625191247 -0.0017199894 0.0001212718

##
##
## data:
## data = spdat)
## weights: us.wt6
##
## 0.3600763582 -0.0016700396 0.0001014426
lm.morantest(fit.1.us, listw=us.wtq, zero.policy=T)
##
##
## data:
## data = spdat)
## weights: us.wtq
##
## 0.3693002497 -0.0016900532 0.0001284917
lm.morantest(fit.1.us, listw=us.wtr, zero.policy=T)

##
##
## data:
## data = spdat)
## weights: us.wtr
##
## 0.3693002497 -0.0016900532 0.0001284917
#Now we fit the spatial lag model
#The lag mode is fit with lagsarlm() in the spdep library
#we basically specify the same model as in the lm() fit above
#But we need to specify the spatial weight matrix and the type
#of lag model to fit
fit.lag.us<-lagsarlm(scale(mortrate)~scale(ppersonspo)+scale(p65plus)+scale(pblac
k_1)+scale(phisp)+factor(RUCC), spdat, listw=us.wt2, type="lag")
summary(fit.lag.us, Nagelkerke=T)

##
## Call:
## lagsarlm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(p65plus) +
## scale(pblack_1) + scale(phisp) + factor(RUCC), data = spdat,
## listw = us.wt2, type = "lag")
##
## Residuals:
## -3.435066 -0.365396 0.016386 0.379149 4.181996
##
## Type: lag
## (Intercept) -0.0601089 0.0539296 -1.1146 0.2650300
## scale(ppersonspo) 0.4383725 0.0166361 26.3507 < 2.2e-16
## scale(p65plus) -0.0067939 0.0137105 -0.4955 0.6202305
## scale(pblack_1) 0.0555812 0.0144317 3.8513 0.0001175
## scale(phisp) -0.1891732 0.0132557 -14.2711 < 2.2e-16
## factor(RUCC)1 0.2866229 0.0756754 3.7875 0.0001522
## factor(RUCC)2 0.1402669 0.0627174 2.2365 0.0253196
## factor(RUCC)3 0.0146097 0.0692250 0.2110 0.8328515
## factor(RUCC)4 0.1496221 0.0766327 1.9525 0.0508838
## factor(RUCC)5 0.0759998 0.0823307 0.9231 0.3559531
## factor(RUCC)6 0.2510359 0.0599005 4.1909 2.779e-05
## factor(RUCC)7 0.0760537 0.0615150 1.2363 0.2163308
## factor(RUCC)8 0.0063996 0.0682837 0.0937 0.9253308
## factor(RUCC)9 -0.2506134 0.0662793 -3.7812 0.0001561
##
## Rho: 0.39892, LR test value: 679.88, p-value: < 2.22e-16
## z-value: 27.075, p-value: < 2.22e-16
## Wald statistic: 733.08, p-value: < 2.22e-16
##
## Log likelihood: -3086.33 for lag model
## AIC: 6204.7, (AIC for lm: 6882.5)
## test value: 33.636, p-value: 6.6459e-09
bptest.sarlm(fit.lag.us)

##
##
## data:
## BP = 238.13, df = 13, p-value < 2.2e-16
#Robust SE's
lm.target.us <- lm(fit.lag.us$tary ~ fit.lag.us$tarX - 1)
coeftest(lm.target.us, vcov.=vcovHC(lm.target.us, type="HC0"))
##
## t test of coefficients:
##
## fit.lag.us$tarXx(Intercept) -0.0601089 0.0477044 -1.2600 0.2077555
## fit.lag.us$tarXxscale(ppersonspo) 0.4383725 0.0250784 17.4801 < 2.2e-16
## fit.lag.us$tarXxscale(p65plus) -0.0067939 0.0176425 -0.3851 0.7002021
## fit.lag.us$tarXxscale(pblack_1) 0.0555812 0.0186951 2.9730 0.0029718
## fit.lag.us$tarXxscale(phisp) -0.1891732 0.0168362 -11.2361 < 2.2e-16
## fit.lag.us$tarXxfactor(RUCC)1 0.2866229 0.0597309 4.7986 1.675e-06
## fit.lag.us$tarXxfactor(RUCC)2 0.1402669 0.0501894 2.7948 0.0052264
## fit.lag.us$tarXxfactor(RUCC)6 0.2510359 0.0526992 4.7636 1.991e-06
## fit.lag.us$tarXxfactor(RUCC)9 -0.2506134 0.0664019 -3.7742 0.0001636
##
## fit.lag.us$tarXx(Intercept)
## fit.lag.us$tarXxscale(ppersonspo) ***
## fit.lag.us$tarXxscale(p65plus)
## fit.lag.us$tarXxscale(pblack_1) **
## fit.lag.us$tarXxscale(phisp) ***
## fit.lag.us$tarXxfactor(RUCC)1 ***
## fit.lag.us$tarXxfactor(RUCC)2 **
## fit.lag.us$tarXxfactor(RUCC)3
## fit.lag.us$tarXxfactor(RUCC)4 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Next we fit the spatial error model
fit.err.us<-errorsarlm(scale(mortrate)~scale(ppersonspo)+scale(p65plus)+scale(pbla
ck_1)+scale(phisp)+factor(RUCC), spdat, listw=us.wt2)
summary(fit.err.us, Nagelkerke=T)
##
## Call:
## errorsarlm(formula = scale(mortrate) ~ scale(ppersonspo) + scale(p65plus) +
## scale(pblack_1) + scale(phisp) + factor(RUCC), data = spdat,
## listw = us.wt2)
##
## Residuals:
## -3.246243 -0.363446 0.018518 0.378347 4.269437
##
## Type: error
## (Intercept) -0.1519258 0.0718058 -2.1158 0.0343630
## scale(ppersonspo) 0.4956221 0.0192362 25.7651 < 2.2e-16
## scale(p65plus) -0.0050401 0.0163231 -0.3088 0.7574975
## scale(pblack_1) 0.1454472 0.0200302 7.2614 3.830e-13
## scale(phisp) -0.2201678 0.0193200 -11.3959 < 2.2e-16
## factor(RUCC)1 0.2983426 0.0802941 3.7156 0.0002027
## factor(RUCC)2 0.1978474 0.0802418 2.4656 0.0136768
## factor(RUCC)3 0.1136176 0.0837459 1.3567 0.1748783
## factor(RUCC)4 0.2271767 0.0877477 2.5890 0.0096261
## factor(RUCC)5 0.2166439 0.0959520 2.2578 0.0239559
## factor(RUCC)6 0.3282910 0.0763724 4.2986 1.719e-05
## factor(RUCC)7 0.1870157 0.0796173 2.3489 0.0188272
## factor(RUCC)8 0.0798636 0.0828415 0.9641 0.3350191
## factor(RUCC)9 -0.1286435 0.0843255 -1.5256 0.1271197
##
## Lambda: 0.43168, LR test value: 582.46, p-value: < 2.22e-16
## z-value: 26.685, p-value: < 2.22e-16
## Wald statistic: 712.11, p-value: < 2.22e-16
##
## Log likelihood: -3135.038 for error model
## AIC: 6302.1, (AIC for lm: 6882.5)
#As a pretty good indicator of which model is best, look at the AIC of each
AIC(fit.1)

## [1] 324.8999
AIC(fit.lag.us)
## [1] 6204.659
AIC(fit.err.us)
## [1] 6302.075

Spatially Autoregressive Models for Regression Analysis

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