Spatial regression models

Introduction

This chapter deals with the problem of inference in (regression) models with spatial data. Inference from regression models with spatial data can be suspect. In essence this is because nearby things are similar, and it may not be fair to consider individual cases as independent (they may be pseudo-replicates). Therefore, such models need to be diagnosed before reporting them. Specifically, it is important to evaluate the for spatial autocorrelation in the residuals (as these are supposed to be independent, not correlated).

If the residuals are spatially autocorrelated, this indicates that the model is misspecified. In that case you should try to improve the model by adding (and perhaps removing) important variables. If that is not possible (either because there is no data available, or because you have no clue as to what variable to look for), you can try formulating a regression model that controls for spatial autocorrelation. We show some examples of that approach here.

Reading & aggregating data

We use California house price data from the 2000 Census.

Get the data

if (!require("rspat")) remotes::install_github("rspatial/rspat")
## Loading required package: rspat
library(rspat)
h <- spat_data('houses2000')

I have selected some variables on on housing and population. You can get more data from the American Fact Finder http://factfinder2.census.gov (among other web sites).

dim(h)
## [1] 7049   29
names(h)
##  [1] "TRACT"      "GEOID"      "label"      "houseValue" "nhousingUn"
##  [6] "recHouses"  "nMobileHom" "yearBuilt"  "nBadPlumbi" "nBadKitche"
## [11] "nRooms"     "nBedrooms"  "medHHinc"   "Population" "Males"
## [16] "Females"    "Under5"     "MedianAge"  "White"      "Black"
## [21] "AmericanIn" "Asian"      "Hispanic"   "PopInHouse" "nHousehold"
## [26] "Families"   "householdS" "familySize" "County"

These are the variables we have:

variabl e

description

nhousin gUn

number of housing units

recHous es

number of houses for recreational use

nMobile Hom

number of mobile homes

nBadPlu mbi

number of houses with incomplete plumbing

nBadKit che

number of houses with incomplete kitchens

Populat ion

total population

Males

number of males

Females

number of females

Under5

number of persons under five

White

number of persons identifying themselves as white (only)

Black

number of persons identifying themselves African-american (only)

America nIn

number of persons identifying themselves American Indian (only)

Asian

number of persons identifying themselves as American Indian (only)

Hispani c

number of persons identifying themselves as hispanic (only)

PopInHo use

number of persons living in households

nHouseh old

number of households

Familie s

number of families

houseVa lue

value of the house

yearBui lt

year house was built

nRooms

median number of rooms per house

nBedroo ms

median number of bedrooms per house

medHHin c

median household income

MedianA ge

median age of population

househo ldS

median household size

familyS ize

median family size

First some data massaging. These are values for Census tracts. I want to analyze these data at the county level. So we need to aggregate the values.

# using a tiny buffer to get a cleaner aggregation
hb <- buffer(h, 1)
values(hb) <- values(h)
hha <- aggregate(hb, "County")

Now we have the county outlines, but we also need to get the values of interest at the county level. Although it is possible to do everything in one step in the aggregate function, I prefer to do this step by step. The simplest case is where we can sum the numbers. For example for the number of houses.

d1 <- as.data.frame(h)[, c("nhousingUn", "recHouses", "nMobileHom", "nBadPlumbi",
 "nBadKitche", "Population", "Males", "Females", "Under5", "White",
 "Black", "AmericanIn", "Asian", "Hispanic", "PopInHouse", "nHousehold", "Families")]
 d1a <- aggregate(d1, list(County=h$County), sum, na.rm=TRUE)

In other cases we need to use a weighted mean. For example for houseValue. We should weight it by the number of houses (households) in each tract.

d2 <- as.data.frame(h)[, c("houseValue", "yearBuilt", "nRooms", "nBedrooms",
       "medHHinc", "MedianAge", "householdS",  "familySize")]
d2 <- cbind(d2 * h$nHousehold, hh=h$nHousehold)
d2a <- aggregate(d2, list(County=h$County), sum, na.rm=TRUE)
d2a[, 2:ncol(d2a)] <- d2a[, 2:ncol(d2a)] / d2a$hh

Combine these two groups:

d12 <- merge(d1a, d2a, by='County')

And merge the aggregated (from census tract to county level) attribute data with the aggregated polygons

hh <- merge(hha[, "County"], d12, by='County')

Let’s make some maps, at the orignal Census tract level. First the house value, using a legend with 10 intervals.

library(RColorBrewer)
grps <- 10
brks <- quantile(h$houseValue, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(h, "houseValue", breaks=brks, col=rev(brewer.pal(grps, "RdBu")), border=NA)
lines(hh, col="white")

image0

A map of the median household income.

brks <- quantile(h$medHHinc, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(h, "medHHinc", breaks=brks, col=rev(brewer.pal(grps, "RdBu")), border=NA)
lines(hh, col="white")

image1

Basic OLS model

I now make some models with the county-level data. I first compute some new variables (that I might not all use).

hh$fBadP <- pmax(hh$nBadPlumbi, hh$nBadKitche) / hh$nhousingUn
hh$fWhite <- hh$White / hh$Population
hh$age <- 2000 - hh$yearBuilt
f1 <- houseValue ~ age +  nBedrooms
m1 <- lm(f1, data=as.data.frame(hh))
summary(m1)
##
## Call:
## lm(formula = f1, data = as.data.frame(hh))
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -222541  -67489   -6128   60509  217655
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -628578     233217  -2.695  0.00931 **
## age            12695       2480   5.119 4.05e-06 ***
## nBedrooms     191889      76756   2.500  0.01543 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 94740 on 55 degrees of freedom
## Multiple R-squared:  0.3235, Adjusted R-squared:  0.2989
## F-statistic: 13.15 on 2 and 55 DF,  p-value: 2.147e-05

Just for illustration, here is how you can do OLS with matrix algebra. First set up the data. I add a constant variable ‘1’ to X, to get an intercept.

y <- matrix(hh$houseValue)
X <- cbind(1, hh$age, hh$nBedrooms)

Then use matrix algebra

ols <- solve(t(X) %*% X) %*% t(X) %*% y
rownames(ols) <- c('intercept', 'age', 'nBedroom')
ols
##                 [,1]
## intercept -628577.95
## age         12694.75
## nBedroom   191888.89

So, according to this simple model, “age” is highly significant. The older a house, the more expensive. You pay 1,269,475 dollars more for a house that is 100 years old than a for new house! While the p-value for the number of bedrooms is not impressive, but every bedroom adds about 200,000 dollars to the value of a house.

Question 1: What would be the price be of a house built in 1999 with three bedrooms?

(the answer may surprise you),

Let’s see if the errors (model residuals) appear to be randomly distributed in space.

hh$residuals <- residuals(m1)
brks <- quantile(hh$residuals, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(hh, "residuals", breaks=brks, col=rev(brewer.pal(grps, "RdBu")))

image2

What do think? Is this a random pattern? Let’s see what Mr. Moran would say. First make a neighborhoods list. I add two links: between San Francisco and Marin County and vice versa (to consider the Golden Gate bridge).

library(spdep)
sfhh <- sf::st_as_sf(hh)
nb <- poly2nb(sfhh, snap=1/120)
nb[[21]] <- sort(as.integer(c(nb[[21]], 38)))
nb[[38]] <- sort(as.integer(c(21, nb[[38]])))
nb
## Neighbour list object:
## Number of regions: 58
## Number of nonzero links: 278
## Percentage nonzero weights: 8.263971
## Average number of links: 4.793103
par(mai=c(0,0,0,0))
plot(hh)
plot(nb, crds(centroids(hh)), col='red', lwd=2, add=TRUE)

image3

We can use the neighbour list object to get the average value for the neighbors of each polygon.

resnb <- sapply(nb, function(x) mean(hh$residuals[x]))
cor(hh$residuals, resnb)
## [1] 0.6311218
plot(hh$residuals, resnb, xlab="Residuals", ylab="Mean adjacent residuals", pch=20)
abline(lm(resnb ~ hh$residuals), lwd=2, lty=2)

image4

The residualso appear to be autocorrelated. A formal test:

lw <- nb2listw(nb)
moran.mc(hh$residuals, lw, 999)
##
##  Monte-Carlo simulation of Moran I
##
## data:  hh$residuals
## weights: lw
## number of simulations + 1: 1000
##
## statistic = 0.41428, observed rank = 1000, p-value = 0.001
## alternative hypothesis: greater

Clearly, there is spatial autocorrelation. Our model cannot be trusted. so let’s try SAR models.

Spatial lag model

Here I show a how to do spatial regression with a spatial lag model (lagsarlm), using the spatialreg package.

library(spatialreg )
m1s <- lagsarlm(f1, data=as.data.frame(hh), lw, tol.solve=1.0e-30)
summary(m1s)
##
## Call:lagsarlm(formula = f1, data = as.data.frame(hh), listw = lw,
##     tol.solve = 1e-30)
##
## Residuals:
##       Min        1Q    Median        3Q       Max
## -108145.2  -49816.3   -1316.3   44604.9  171536.0
##
## Type: lag
## Coefficients: (asymptotic standard errors)
##              Estimate Std. Error z value Pr(>|z|)
## (Intercept) -418674.1   153693.6 -2.7241 0.006448
## age            5533.6     1698.2  3.2584 0.001120
## nBedrooms    127912.8    50859.7  2.5150 0.011903
##
## Rho: 0.77413, LR test value: 34.761, p-value: 3.7282e-09
## Asymptotic standard error: 0.08125
##     z-value: 9.5277, p-value: < 2.22e-16
## Wald statistic: 90.778, p-value: < 2.22e-16
##
## Log likelihood: -727.9964 for lag model
## ML residual variance (sigma squared): 3871700000, (sigma: 62223)
## Number of observations: 58
## Number of parameters estimated: 5
## AIC: 1466, (AIC for lm: 1498.8)
## LM test for residual autocorrelation
## test value: 0.12431, p-value: 0.72441
hh$residuals <- residuals(m1s)
moran.mc(hh$residuals, lw, 999)
##
##  Monte-Carlo simulation of Moran I
##
## data:  hh$residuals
## weights: lw
## number of simulations + 1: 1000
##
## statistic = -0.016, observed rank = 513, p-value = 0.487
## alternative hypothesis: greater
brks <- quantile(hh$residuals, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(hh, "residuals", breaks=brks, col=rev(brewer.pal(grps, "RdBu")))

image5

Spatial error model

And now with a “Spatial error” (or spatial moving average) models (errorsarlm). Note the use of the lw argument.

m1e <- errorsarlm(f1, data=as.data.frame(hh), lw, tol.solve=1.0e-30)
summary(m1e)
##
## Call:errorsarlm(formula = f1, data = as.data.frame(hh), listw = lw,
##     tol.solve = 1e-30)
##
## Residuals:
##       Min        1Q    Median        3Q       Max
## -100640.7  -47783.1   -2364.5   44180.6  181876.5
##
## Type: error
## Coefficients: (asymptotic standard errors)
##              Estimate Std. Error z value Pr(>|z|)
## (Intercept) -185443.8   180133.7 -1.0295  0.30325
## age            4313.6     2214.9  1.9475  0.05147
## nBedrooms    117864.5    51564.7  2.2858  0.02227
##
## Lambda: 0.82151, LR test value: 29.781, p-value: 4.8373e-08
## Asymptotic standard error: 0.071111
##     z-value: 11.552, p-value: < 2.22e-16
## Wald statistic: 133.46, p-value: < 2.22e-16
##
## Log likelihood: -730.4863 for error model
## ML residual variance (sigma squared): 4.07e+09, (sigma: 63797)
## Number of observations: 58
## Number of parameters estimated: 5
## AIC: 1471, (AIC for lm: 1498.8)
hh$residuals <- residuals(m1e)
moran.mc(hh$residuals, lw, 999)
##
##  Monte-Carlo simulation of Moran I
##
## data:  hh$residuals
## weights: lw
## number of simulations + 1: 1000
##
## statistic = 0.039033, observed rank = 748, p-value = 0.252
## alternative hypothesis: greater
brks <- quantile(hh$residuals, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(hh, "residuals", breaks=brks, col=rev(brewer.pal(grps, "RdBu")))

image6

Are the residuals spatially autocorrelated for either of these models? Let’s plot them for the spatial error model.

brks <- quantile(hh$residuals, 0:(grps-1)/(grps-1), na.rm=TRUE)
plot(hh, "residuals", breaks=brks, col=rev(brewer.pal(grps, "RdBu")))

image7

Questions

Question 2: The last two maps still seem to show a lot of spatial autocorrelation. But according to the tests there is none. Now why might that be?

Question 3: One of the most important, or perhaps THE most important aspect of modeling is variable selection. A misspecified model is never going to be any good, no matter how much you do to, e.g., correct for spatial autocorrelation.

  1. Which variables would you choose from the list?

  2. Which new variables could you propose to create from the variables in the list.

  3. Which other variables could you add, created from the geometries/location (perhaps other geographic data).

  4. add a lot of variables and use stepAIC to select an ‘optimal’ OLS model

  5. check for spatial autocorrelation in the residuals of that model