---
title: "North Carolina SIDS data set (models)"
author: "Roger Bivand"
output:
  html_document:
    toc: true
    toc_float:
      collapsed: false
      smooth_scroll: false
    toc_depth: 2
bibliography: refs.bib
link-citations: yes
vignette: >
  %\VignetteIndexEntry{Introduction to the North Carolina SIDS data set (re-revised)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

## Introduction

This data set was presented first in @symonsetal:1983, analysed with
reference to the spatial nature of the data in @cressie+read:1985,
expanded in @cressie+chan:1989, and used in detail in @cressie:1991. It
is for the 100 counties of North Carolina, and includes counts of
numbers of live births (also non-white live births) and numbers of
sudden infant deaths, for the July 1, 1974 to June 30, 1978 and July 1,
1979 to June 30, 1984 periods. In @cressie+read:1985, a listing of
county neighbours based on shared boundaries (contiguity) is given, and
in @cressie+chan:1989, and in @cressie:1991 [pp. 386–389], a different
listing based on the criterion of distance between county seats, with a
cutoff at 30 miles. The county seat location coordinates are given in
miles in a local (unknown) coordinate reference system. The data are
also used to exemplify a range of functions in the spatial statistics
module user’s manual [@kaluznyetal:1996].

## Getting the data into R 

```{r, echo=FALSE,eval=TRUE,warning=FALSE, message=FALSE}
library(spdep)
```

We will be using the **spdep** and **spatialreg** packages, here version: `r spdep()[1]`, the **sf** package and the **tmap** package. The data from the sources
referred to above is documented in the [help page](https://nowosad.github.io/spData/reference/nc.sids.html) for the `nc.sids`
data set in **spData**. The actual data, included in a shapefile of the county boundaries for North Carolina were made available in the **maptools** package [^1]. These data are known to be geographical coordinates (longitude-latitude in decimal degrees) and are assumed to use the NAD27 datum.

```{r echo=TRUE,eval=TRUE}
library(spdep)
nc <- st_read(system.file("shapes/sids.shp", package="spData")[1], quiet=TRUE)
st_crs(nc) <- "+proj=longlat +datum=NAD27"
row.names(nc) <- as.character(nc$FIPSNO)
```


```{r echo=TRUE}
nc$ft.SID74 <- sqrt(1000)*(sqrt(nc$SID74/nc$BIR74) + sqrt((nc$SID74+1)/nc$BIR74))
nc$both <- factor(paste(nc$L_id, nc$M_id, sep=":"))
``` 

```{r, echo=FALSE}
is_tmap <- FALSE
if (require(tmap, quietly=TRUE)) is_tmap <- TRUE
is_tmap
```

We will now examine the data set reproduced from Cressie and
collaborators, included in **spData** (formerly in **spdep**), and add the neighbour relationships used in
@cressie+chan:1989 to the background map as a graph shown in Figure
\ref{plot-CC89.nb}:

```{r echo=TRUE, eval=TRUE}
gal_file <- system.file("weights/ncCC89.gal", package="spData")[1]
ncCC89 <- read.gal(gal_file, region.id=nc$FIPSNO)
```

### CAR model fitting

We will now try to replicate three of the four models fitted by
[@cressie+chan:1989] to the transformed rates variable. The first thing
to do is to try to replicate their 30 mile distance between county seats
neighbours, which almost works. From there we try to reconstruct three
of the four models they fit, concluding that we can get quite close, but
that a number of questions are raised along the way.

Building the weights is much more complicated, because they use a
combination of distance-metric and population-at-risk based weights, but
we can get quite close [see also @kaluznyetal:1996]:

```{r echo=TRUE}
sids.nhbr30.dist <- nbdists(ncCC89, cbind(nc$east, nc$north))
sids.nhbr <- listw2sn(nb2listw(ncCC89, glist=sids.nhbr30.dist, style="B", zero.policy=TRUE))
dij <- sids.nhbr[,3]
n <- nc$BIR74
el1 <- min(dij)/dij
el2 <- sqrt(n[sids.nhbr$to]/n[sids.nhbr$from])
sids.nhbr$weights <- el1*el2
sids.nhbr.listw <- sn2listw(sids.nhbr)
```

The first model (I) is a null model with just an intercept, the second
(II) includes all the 12 parcels of contiguous counties in 4 east-west
and 4 north-south bands, while the fourth (IV) includes the transformed
non-white birth-rate:

```{r echo=TRUE}
nc$ft.NWBIR74 <- sqrt(1000)*(sqrt(nc$NWBIR74/nc$BIR74) + sqrt((nc$NWBIR74+1)/nc$BIR74))
```

Cressie identifies Anson county as an outlier, and drops it from further
analysis. Because the weights are constructed in a complicated way, they
will be subsetted by dropping the row and column of the weights matrix:

```{r echo=TRUE}
lm_nc <- lm(ft.SID74 ~ 1, data=nc)
outl <- which.max(rstandard(lm_nc))
as.character(nc$NAME[outl])
``` 

```{r echo=TRUE}
W <- listw2mat(sids.nhbr.listw)
W.4 <- W[-outl, -outl]
sids.nhbr.listw.4 <- mat2listw(W.4)
nc2 <- nc[!(1:length(nc$CNTY_ID) %in% outl),]
``` 

It appears that both numerical issues (convergence in particular) and
uncertainties about the exact spatial weights matrix used make it
difficult to reproduce the results of @cressie+chan:1989, also given in
@cressie:1991. We now try to replicate them for the null weighted CAR
model (Cressie has intercept 2.838, $\hat{\theta}$ 0.833, for k=1):

```{r echo=TRUE}
library(spatialreg)
ecarIaw <- spautolm(ft.SID74 ~ 1, data=nc2, listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
summary(ecarIaw)
```

The spatial parcels model seems not to work, with Cressie's $\hat{\theta}$ 0.710, and failure in the numerical Hessian used to calculate the standard error of the spatial coefficient:

```{r echo=TRUE}
ecarIIaw <- spautolm(ft.SID74 ~ both - 1, data=nc2, listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
summary(ecarIIaw)
```

Finally, the non-white model repeats Cressie’s finding that much of the
variance of the transformed SIDS rate for 1974–8 can be accounted for by
the transformed non-white birth variable (Cressie intercept 1.644,
$\hat{b}$ 0.0346, $\hat{\theta}$ 0.640 — not significant), but the estimate of the spatial coefficient is not close here:

```{r echo=TRUE}
ecarIVaw <- spautolm(ft.SID74 ~ ft.NWBIR74, data=nc2, listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
summary(ecarIVaw)
```

```{r, eval=is_tmap, echo=TRUE}
nc2$fitIV <- fitted.values(ecarIVaw)
sf_use_s2(FALSE)
tm_shape(nc2) + tm_fill("fitIV")
```

The final figure shows the value of the log likelihood function for the null model (I):

```{r echo=TRUE}
ecarIawll <- spautolm(ft.SID74 ~ 1, data=nc2, listw=sids.nhbr.listw.4, weights=BIR74, family="CAR", llprof=seq(-0.1, 0.9020532358, length.out=100))
plot(ll ~ lambda, ecarIawll$llprof, type="l")
```

## References

[^1]: These data were taken with permission from a now-offline link:
[sal.agecon.uiuc.edu/datasets/sids.zip]; see also [GeoDa Center](https://geodacenter.github.io/data-and-lab/) for a contemporary source.