Advanced Network Analysis

class: center, middle, inverse, title-slide

.title[
# Advanced Network Analysis
]
.subtitle[
## Intro to Spatial Statistics
]
.author[
### Olga Chyzh [www.olgachyzh.com]
]

---

## 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.

`$$Covid19\ cases/cap_i= \beta_0+\beta_1Urban_i+\beta_2Trump16_i+\\\beta_3medinc_i +u_i,$$`

---
## Homophily: Spatial X

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

`$$Covid19\ cases/cap_i= \beta_0+\beta_1Urban_i+\beta_2Trump16_i+\\\beta_3medinc_i+\rho\sum\limits_{j\neq i}^{N}{w_{ij}\ Trump16_j} +u_i,$$`
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.

- `$\sum\limits_{j\neq i}^{N}{w_{ij}\ Trump16_j}$` 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

`$$Covid19\ cases/cap_i= \beta_0+\beta_1Urban_i+\beta_2Trump16_i+\\\beta_3medinc_i+\rho\sum\limits_{j\neq i}^{N}{w_{ij}\ Covid19\ cases/cap_j} +u_i,$$`
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.

- `$\sum\limits_{j\neq i}^{N}{w_{ij}\ Covid19\ cases/cap_j}$` 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

`$$\textbf{y}= \rho\textbf{W}\textbf{y}+\textbf{X}\pmb{\beta}+\pmb{\epsilon},$$`
- `$\textbf{y}$` the dependent variable, is an N x 1 vector of cross sections stacked by period;

- `$\rho$` is the spatial coefficient;

- `$\textbf{W}$` is an N x N spatial-weighting matrix;

- `$\textbf{X}$` contains N observations on k independent variables

- `$\pmb{\beta}$` is a k x 1 vector of coefficients;

- `$\pmb{\epsilon}$` is an N by 1 vector of stochastic components.

---
## Spatial Y Model

`$$\begin{bmatrix}
y_{1} \\
y_{2} \\
y_{3} \\
\vdots \\
y_{N}
\end{bmatrix}
=
\rho
\begin{bmatrix}
0 & W_{12} &W_{13}& \cdots & W_{1N} \\
W_{21}& 0 &W_{23}& \cdots & W_{2N} \\
W_{31}& W_{32} &0 & \cdots & W_{3N} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
W_{N1} & W_{N2} & W_{N3} & \cdots & 0
\end{bmatrix}+\\
\begin{bmatrix}
x_{11} & x_{12} & \cdots & x_{1k} \\
x_{21} & x_{22} & \cdots & x_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
x_{N1} & x_{N2} & \cdots & x_{Nk}
\end{bmatrix}
\begin{bmatrix}
\beta_1 \\
\beta_2 \\
\vdots \\
\beta_k
\end{bmatrix}+
\begin{bmatrix}
\epsilon_1 \\
\epsilon_2 \\
\vdots \\
\epsilon_{N}
\end{bmatrix}$$`

---

## Spatial Lag Model

`$$\textbf{y}= \rho\textbf{W}\textbf{y}+\textbf{X}\pmb{\beta}+\pmb{\epsilon},$$`

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

`$$\textbf{y}= [\pmb{I_{N}}-\rho\textbf{W}_{N}]^{-1}\{\textbf{X}\pmb{\beta}+\pmb{\epsilon}\}$$`
---
## Likelihood

---
## Other Types of Space

- Ideology

- International trade

- Alliances

- Other examples?

---
class: inverse, middle, center
# Lab
---
## Example: Spatial X

```r
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

```r
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

```r
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

```r
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

```r
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

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

---
## A Bit Nicer of a Map

```r
#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

```r
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:

```r
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

```r
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

```r
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:

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

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