13b_spatial regression solutions.Rmd

---
title: "13_spatial regression solutions"
author: "Brenna Kelly"
date: "2024-09-10"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


```{r}

library(tmap)
library(spdep)
library(spatialreg)
library(RColorBrewer)

```


Basic spatial regression workflow:  
- **Inspect** the data  
- Define the **spatial structure** (neighbors) and create the **weight matrix**  
- Check for spatial autocorrelation in the outcome (Moran's I)  
- **Model 0:** Create an aspatial null model  
-- Check for SA in the M0 residuals  
-- Interpret  
- **Model 1:** Create an aspatial model with covariates  
-- Check for SA in the M1 residuals  
-- Interpret  
- **Model 2:** Choose a spatial regression method  
-- Lag vs. error vs. hierarchical  
-- Check for SA in the M2 spatial  
-- Interpret  

#### Inspect the data

Read in the shapefile for Chicago communities (location: `data/comarea/ComArea_ACS14_f.shp`). We are interested in estimating the effect of poverty `Pov200P` on the preterm birth rate `PretBrth`. Plot the preterm birth rate with a color scale in the `RColorBrewer` library (using `display.brewer.all()` to choose one).

```{r}

com <- st_read("data/comarea/ComArea_ACS14_f.shp")

hist(com$PretBrth)
summary(com$PretBrth)
cor.test(com$PretBrth, com$Pov200P)

tm_shape(com) +
  tm_polygons(col = "PretBrth", palette = "RdPu", 
              style = "cont", lwd = 0)

```

#### Spatial structure  

Does it look like there is spatial autocorrelation in the outcome? To confirm this statistically using Moran's I, we first need to define the **neighbor structure** and create the **spatial weight matrix**.

```{r}

nb <- poly2nb(com)
com_nb_listw <- nb2listw(nb)

```

Visualize the spatial structure.

```{r}

coords <- st_coordinates(st_centroid(com))

plot(com$geometry)
plot(nb, coords, add = TRUE, col = "red")

```


#### SA in outcome

Calculate **Moran's I** using `moran.test` with the arguments `alternative = "two.sided"` and `randomisation = TRUE`. Compare the results against `moran.mc`, which uses a Monte Carlo method to estimate significance with improved precision.

```{r}

moran.test(com$PretBrth, com_nb_listw,
           alternative = "two.sided", 
           randomisation = TRUE)

moran.mc(com$PretBrth, com_nb_listw, nsim = 999,
         alternative = 'greater')

moran.plot(com$PretBrth, com_nb_listw, plot = TRUE,
           xlab = "Preterm Birth Rate",
           ylab = "Spatially Lagged Rate")

````

#### Null aspatial model

- Create a linear model of the preterm birth rate.  
- Plot the spatial structure of the residuals.  
- Check for SA in the residuals.  

```{r}

m0 <- lm(PretBrth ~ 1, data = com)

summary(m0)


com$res_m0 <- m0$residuals

tm_shape(com) +
  tm_polygons(col = "res_m0", palette = "Spectral", 
              style = "cont", lwd = 0)

moran.mc(com$res_m0, com_nb_listw, nsim = 999,
         alternative = 'greater')

```

#### Aspatial model with covariate  

- Scale the poverty variable for the regression.
- Create a linear model of the preterm birth rate with poverty as a covariate.  
- Check for structure in the residuals.
- Plot the spatial structure of the residuals.  
- Check for SA in the residuals.  

```{r}

com$poverty_scaled <- com$Pov200P / 10

m1 <- lm(PretBrth ~ poverty_scaled, data = com)

summary(m1)


com$res_m1 <- m1$residuals

plot(com$res_m1, m1$fitted.values)

tm_shape(com) +
  tm_polygons(col = "res_m1", palette = "Spectral", 
              style = "cont", size = 0.4, lwd = 0)

moran.mc(com$res_m1, com_nb_listw, nsim = 999)


```

#### Spatial econometric models  
- **error:** unexplained spatial variation; error which is correlated across spatial units  
- **lag:** influence of neighboring values on unit values; a diffusion process  
- **combination**

#### Spatial error model
- Create a spatial error model of the preterm birth rate with poverty as a covariate.  
- Plot the spatial structure of the residuals.  
- Check for SA in the residuals.  

```{r}

m2 = errorsarlm(PretBrth ~ poverty_scaled,
                data = com, 
                com_nb_listw)
summary(m2)

com$res_m2 <- m2$residuals

tm_shape(com) +
  tm_polygons(col = "res_m2", palette = "Spectral", style = "cont", lwd = 0)
 
moran.mc(com$res_m2, com_nb_listw, nsim = 999)

```

#### Spatial lag model
- Create a spatial lag model of the preterm birth rate with poverty as a covariate.  
- Plot the spatial structure of the residuals.  
- Check for SA in the residuals.  

```{r}

m3 = lagsarlm(PretBrth ~ poverty_scaled,
              data = com, 
              com_nb_listw)
summary(m3)

com$res_m3 <- m3$residuals

tm_shape(com) +
  tm_polygons(col = "res_m3", palette = "Spectral", style = "cont", lwd = 0)
 
moran.mc(com$res_m3, com_nb_listw, nsim = 999)

```

#### Error *and* lag
- Using `sacsarlm`, create a model which uses both lag and error structures.  
- Plot the spatial structure of the residuals.  
- Check for SA in the residuals.  

```{r}

m4 = sacsarlm(PretBrth ~ poverty_scaled,
              data = com, 
              com_nb_listw)
summary(m4)

com$res_m4 <- m4$residuals

tm_shape(com) +
  tm_polygons(col = "res_m4", palette = "Spectral", style = "cont", lwd = 0)
 
moran.mc(com$res_m4, com_nb_listw, nsim = 999)

```

#### Spatial Durbin error model  
Within neighbors, correlation exists between errors and between covariates.  

```{r}

m5 = errorsarlm(PretBrth ~ poverty_scaled,
                data = com, 
                com_nb_listw,
                etype = "emixed")
summary(m5)

com$res_m5 <- m5$residuals

tm_shape(com) +
  tm_polygons(col = "res_m5", palette = "Spectral", style = "cont", lwd = 0)
 
moran.mc(com$res_m5, com_nb_listw, nsim = 999)

```

#### Spatial Durbin lag model  
Within neighbors, correlation between outcome (diffusion) and correlation between covariates.  
Modify the `lagsarlm` ode to include the argument `type = "mixed"`. Perform a visual check and statistical test of spatial structure in the residuals.  

```{r}

m6 = lagsarlm(PretBrth ~ poverty_scaled,
              data = com, 
              com_nb_listw,
              type = "mixed")
summary(m6)

com$res_m6 <- m6$residuals

tm_shape(com) +
  tm_polygons(col = "res_m6", palette = "Spectral", style = "cont", lwd = 0)
 
moran.mc(com$res_m6, com_nb_listw, nsim = 999)

```

Which model to interpret?

```{r}

AIC(m0)
AIC(m1)
AIC(m2)
AIC(m3)
AIC(m4)
AIC(m5)
AIC(m6)

```