Dependence in Observational Data

• Individuals are nested in social networks

  • Individual decisions are influenced by their friends.

• Provinces are surrounded by other provinces

  • Provinces mimic one another's policies

• Country-level outcomes are often a result of negotiations with other countries:

  • Economic or environmental policies

Three Mechanisms for Spatial Dependence

• Common exposure---similarity in outcomes is driven by an exogenous factor that affects nearby units (the effect of earthquakes on housing prices)

• Homophily---similarity in outcomes is endogenous, units are similar because they self-select into the same outcome (e.g., partisan geo-sorting)

• Diffusion---nearby units affect each other through learning, imitation, etc (e.g., policy diffusion)

Source: van Weezel S. "On climate and conflict: Precipitation decline and communal conflict in Ethiopia and Kenya." Journal of Peace Research. 2019;56(4):514--528.

Source: Chyzh, Olga V. and R. Urbatsch. 2021. "Bean Counters: The Effect of Soy Tariffs on Change in Republican Vote Share Between the 2016 and 2018 Elections."Journal of Politics 83 (1): 415--419.

What Explains Variation in Covid-19 Cases?

Common Exposure

Neighboring counties have similar Covid-19 rates because of their underlying similarities, e.g. demographics, political ideology (anti-mask sentiment), etc.

$$Covid19cases/ca{p}_i={\beta }_0+{\beta }_{1}Urba{n}_i+{\beta }_{2}Trump{16}_i+{\beta }_{3}medin{c}_i+u_i,$$

Homophily: Spatial X

Neighboring units tend to converge on outcomes because the causal variables (anti-vaccine sentiments) cluster by neighborhood locations (partisan geo-sorting).

$$Covid19cases/ca{p}_i={\beta }_0+{\beta }_{1}Urba{n}_i+{\beta }_{2}Trump{16}_i+{\beta }_{3}medin{c}_i+\rho \sum _{j\ne i}^N{w}_{ij}Trump{16}_j+u_i,$$ where $\rho$ is the estimation parameter for spatial dependence, and $w_{ij}$ measures whether $i$ and $j$ are neighbors.

• This is a spatial-X regression.

• $\sum _{j\ne i}^N{w}_{ij}Trump{16}_j$ is a spatially lagged independent variable measuring the average Trump support in neighboring counties.

• The coefficient $\rho$ is a measure of spatial homophily.

Contiguity Matrix W

Contiguity Matrix W

##         Benton Linn Jones Iowa Johnson Cedar
## Benton       0    1     0    1       0     0
## Linn         1    0     1    0       1     1
## Jones        0    1     0    0       0     1
## Iowa         1    0     0    0       1     0
## Johnson      0    1     0    1       0     1
## Cedar        0    1     1    0       1     0

Row Standardized W

Divide by the row sum, so that each neighbor's influence decreases with the total number of neighbors.

##         Benton Linn Jones Iowa Johnson Cedar
## Benton    0.00 0.50  0.00 0.50    0.00  0.00
## Linn      0.25 0.00  0.25 0.00    0.25  0.25
## Jones     0.00 0.50  0.00 0.00    0.00  0.50
## Iowa      0.50 0.00  0.00 0.00    0.50  0.00
## Johnson   0.00 0.33  0.00 0.33    0.00  0.33
## Cedar     0.00 0.33  0.33 0.00    0.33  0.00

Diffusion: Spatial Y

$$Covid19cases/ca{p}_i={\beta }_0+{\beta }_{1}Urba{n}_i+{\beta }_{2}Trump{16}_i+{\beta }_{3}medin{c}_i+\rho \sum _{j\ne i}^N{w}_{ij}Covid19cases/ca{p}_j+u_i,$$ where $\rho$ is the estimation parameter for spatial dependence, and $w_{ij}$ measures whether $i$ and $j$ are neighbors.

• This is a spatial-Y regression.

• $\sum _{j\ne i}^N{w}_{ij}Covid19cases/ca{p}_j$ is a spatially lagged dependent variable measuring the average number of Covid-19 cases in neighboring counties.

• The coefficient $\rho$ is a measure of spatial dependence.

Spatial Y Model

$$\text{y}=\rho \text{W}\text{y}+\text{X}\beta \beta +ϵϵ,$$

• $\text{y}$ the dependent variable, is an N x 1 vector of cross sections stacked by period;

• $\rho$ is the spatial coefficient;

• $\text{W}$ is an N x N spatial-weighting matrix;

• $\text{X}$ contains N observations on k independent variables

• $\beta \beta$ is a k x 1 vector of coefficients;

• $ϵϵ$ is an N by 1 vector of stochastic components.

Spatial Y Model

$$\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ ⋮\\ y_N\end{array}\right]=\rho \left[\begin{array}{ccccc}0& W_{12}& W_{13}& \cdots & W_{1N}\\ W_{21}& 0& W_{23}& \cdots & W_{2N}\\ W_{31}& W_{32}& 0& \cdots & W_{3N}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ W_{N1}& W_{N2}& W_{N3}& \cdots & 0\end{array}\right]+\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1k}\\ x_{21}& x_{22}& \cdots & x_{2k}\\ ⋮& ⋮& \ddots & ⋮\\ x_{N1}& x_{N2}& \cdots & x_{Nk}\end{array}\right]\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ ⋮\\ {\beta }_k\end{array}\right]+\left[\begin{array}{c}{ϵ}_1\\ {ϵ}_2\\ ⋮\\ {ϵ}_N\end{array}\right]$$

Spatial Lag Model

$$\text{y}=\rho \text{W}\text{y}+\text{X}\beta \beta +ϵϵ,$$

By re-arranging, can isolate y on the left-hand side:

$$\text{y}=[I_N{I}_N-\rho {\text{W}}_N{]}^{-1}\{\text{X}\beta \beta +ϵϵ\}$$

Likelihood

Other Types of Space

• Ideology

• International trade

• Alliances

• Other examples?

Example: Spatial X

mydata<-read.csv("./data/covid_data.csv", header=TRUE) 
mydata$trumpmarg[is.na(mydata$trumpmarg)]<-0
contigmat<-read.table("data/contigmat.txt") |> as.matrix()
contigmat1<-contigmat/apply(contigmat,1,sum) #row-standardize
mydata$W_trumpmarg<-contigmat1%*%mydata$trumpmarg
m1<-lm(data=mydata, cases_pc~urb2010+trumpmarg+medinc1317)
m2<-lm(data=mydata, cases_pc~urb2010+trumpmarg+medinc1317+W_trumpmarg)

Spatial Regression

library(spdep)
library(spatialreg)
contigmat<-read.table("./data/contigmat.txt") 
contigmat<-as.matrix(contigmat)
W1<-mat2listw(contigmat, row.names = NULL, style="W", zero.policy = TRUE)
summary(W1$neighbours)
W2<-nb2listw(W1$neighbours, glist=NULL, style="W", zero.policy=TRUE)
m3 <- lagsarlm(data=mydata, cases_pc~log(totpop1317)+urb2010+trumpmarg+medinc1317, W2, zero.policy=TRUE)
summary(m3)
saveRDS(m3,"m3.RDS")

Interpretation

Set up a hypothetical scenario:

• Expected change in Covid-19 cases that would result from increasing urbanization in Johnson county, IA

names<-c("benton","cedar","iowa","johnson","jones","linn")
mymat<-matrix(c(0,0,1,0,0,1,
                0,0,0,1,1,1,
                1,0,0,1,0,0,
                0,1,1,0,0,1,
                0,1,0,0,0,1,
                1,1,0,1,1,0),nrow=6,ncol=6)
dimnames(mymat)<-list(names,names)
mymat<-round(mymat/apply(mymat,1,sum),2)
d<-dplyr::filter(mydata, state=="IA" & county %in% names)

Set up A Comparison by Shocking One of the Units on X

m3<- readRDS("m3.RDS")
I<- diag(6)
X0<-as.matrix(cbind(1,log(d$totpop1317), d$urb2010, d$trumpmarg, d$medinc1317))
urb<-d$urb2010
urb[4]<-1
X1<-as.matrix(cbind(1,log(d$totpop1317), urb, d$trumpmarg, d$medinc1317))
A<-solve(I-coef(m3)[1]*mymat)
mycoef<-as.matrix(coef(m3))
Yhat0<- A%*%(X0%*%mycoef)
Yhat1<- A%*%(X1%*%mycoef)
Y_ch<-Yhat1-Yhat0
sim<- cbind.data.frame(names,Y_ch)

Visualize the Effect

Your Turn 1

Suppose you want to test whether variable urb2010 is spatially clustered.

1. Calculate a measure of the average urbanization in neighboring states.

2. Estimate a model that accounts for clustering in urbanization.

3. Is the effect of neighbor's urbanization positive or negative?

4. Is this effect statistically significant?

Your Turn 2

Suppose you want to test whether variable votech (the change in Republican vote share between the 2016 and 2018 Congressional election) is spatially clustered.

1. Calculate a measure of the average change in Republican vote share in neighboring states.

2. Estimate a model of votech as a function of urb2010, medinc1317, perc_HS_GED, perclatino1317 and trumpmarg.

3. Estimate the same model plus a the average change in Republican vote share in neighboring states.

Making Maps

library(tidyverse)
library(mapproj)
library(maps)
library(mapdata)
states <- map_data("state")
head(states)

##        long      lat group order  region subregion
## 1 -87.46201 30.38968     1     1 alabama      <NA>
## 2 -87.48493 30.37249     1     2 alabama      <NA>
## 3 -87.52503 30.37249     1     3 alabama      <NA>
## 4 -87.53076 30.33239     1     4 alabama      <NA>
## 5 -87.57087 30.32665     1     5 alabama      <NA>
## 6 -87.58806 30.32665     1     6 alabama      <NA>

What You Need

• Latitude/longitude points for all map boundaries

• Need to know to which boundary/state lat/long points belong

• Need to know the order to connect points within each group

A Basin (Rather Hideous) Map

library(ggplot2)
ggplot() +  geom_path(data=states, aes(x=long, y=lat, group=group),color="black", size=.5)

A Bit Nicer of a Map

#Set theme options:
theme_set(theme_grey() + theme(axis.text=element_blank(),
                               axis.ticks=element_blank(),
                               axis.title.x=element_blank(),
                               axis.title.y=element_blank(),
                               panel.grid.major = element_blank(),
                               panel.grid.minor = element_blank(),
                               panel.border = element_blank(),
                               panel.background = element_blank(),
                               legend.position="none"))
ggplot() +  geom_path(data=states, aes(x=long, y=lat, group=group),color="black", size=.5)+ coord_map()

Polygon instead of Path

ggplot() +  geom_polygon(data=states, aes(x=long, y=lat, group=group),color="black", size=.5)+ coord_map()

Incorporate Information About States

• Add other geographic information (e.g., counties) by adding geometric layers to the plot

• Add non-geographic information by altering the fill color for each state

  • Use geom = "polygon" to treat states as solid shapes to add color

  • Incorporate numeric information using color shade or intensity

  • Incorporate categorical informaion using color hue

Categorical Information Using Hue

If a categorical variable is assigned as the fill color then ggplot will assign different hues for each category.

Let’s load in a state regions dataset:

statereg<- read.csv("./data/statereg.csv")
head(statereg)

##        State StateGroups
## 1 california        West
## 2     nevada        West
## 3     oregon        West
## 4 washington        West
## 5      idaho        West
## 6    montana        West

Join the Data

states.class.map <- left_join(states, statereg, by = c("region" = "State"))
head(states.class.map)

##        long      lat group order  region subregion StateGroups
## 1 -87.46201 30.38968     1     1 alabama      <NA>       South
## 2 -87.48493 30.37249     1     2 alabama      <NA>       South
## 3 -87.52503 30.37249     1     3 alabama      <NA>       South
## 4 -87.53076 30.33239     1     4 alabama      <NA>       South
## 5 -87.57087 30.32665     1     5 alabama      <NA>       South
## 6 -87.58806 30.32665     1     6 alabama      <NA>       South

Plot the Regions

ggplot() +  geom_polygon(data=states.class.map, aes(x=long, y=lat, group=group, fill = StateGroups), colour = I("black"))+ coord_map()+theme(legend.position="bottom")

Your Turn

Use color to show the expected change in Covid-19 cases that result from increasing urbanization in Johnson county, IA on a map.

Your Turn (Advanced)

1. Read in the animal.csv data:

animal <- read.csv("./data/animal.csv")

1. Plot the location of animal sightings on a map of the region
2. On this plot, try to color points by class of animal and/or status of animal
3. Advanced: Could we indicate time somehow?