Germán Rodríguez
Multilevel Models Princeton University

# A Random-Intercept Poisson Model

Rabe-Hesketh and Skrondal (2012) analyze data on lip cancer in Scotland. The data consists of the number of cancer cases observed in each of 56 counties in Scotland in 1975-80.

We also have information on the expected number of cases in each county, based on age-specific lip cancer rates for the whole of Scotland and the observed age distribution in each county.

The ratio of observed to expected counts, often times 100, is called the Standardized Mortality Ratio (SMR). For example a value of 193.2 denotes almost twice as many cases as expected.

A limitation of the SMR is that estimates for counties with small populations are very imprecise. To address this problem we will use Empirical Bayes (EB) estimates based on a random-intercept Poisson model. over-dispersion.

We can read the data from the Stata bookstore and compute the offset as the log of the expected counts

```. use https://www.stata-press.com/data/mlmus3/lips.dta, clear

. gen os = log(e)
```
```> library(foreign)
> lips\$os <- log(lips\$e)
```

We fit a random-intercept model at the county level and then produce a Bayesian estimate of the Standardized Mortality Ratio (SMR) for each county, combining the ML estimate of the mean with the posterior mean (in Stata) or mode (in R) of the random effect. For county 1 the posterior mean is 1.408 and the posterior mode is 1.456. (To obtain modes in Stata use `predict pm, ebmodes`, take logs, and subtract the linear predictor and offset.)

```. mepoisson o, offset(os) || county:

Fitting fixed-effects model:

Iteration 0:   log likelihood = -347.28421
Iteration 1:   log likelihood = -294.63904
Iteration 2:   log likelihood = -294.35162
Iteration 3:   log likelihood = -294.35155
Iteration 4:   log likelihood = -294.35155

Refining starting values:

Grid node 0:   log likelihood =  -183.5152

Fitting full model:

Iteration 0:   log likelihood =  -183.5152
Iteration 1:   log likelihood =  -181.3855
Iteration 2:   log likelihood = -181.32411
Iteration 3:   log likelihood = -181.32325
Iteration 4:   log likelihood = -181.32325

Mixed-effects Poisson regression                Number of obs     =         56
Group variable:          county                 Number of groups  =         56

Obs per group:
min =          1
avg =        1.0
max =          1

Integration method: mvaghermite                 Integration pts.  =          7

Wald chi2(0)      =          .
Log likelihood = -181.32325                     Prob > chi2       =          .
------------------------------------------------------------------------------
o ¦      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons ¦   .0803235   .1165335     0.69   0.491     -.148078     .308725
os ¦          1  (offset)
-------------+----------------------------------------------------------------
county       ¦
var(_cons)¦   .5846534    .147734                      .3562963    .9593689
------------------------------------------------------------------------------
LR test vs. Poisson model: chibar2(01) = 226.06       Prob >= chibar2 = 0.0000

. predict a, reffects
(calculating posterior means of random effects)

. gen eb = 100 * exp(_b[_cons] + a)
```
```> library(lme4)
> ri <- glmer(o ~ offset(os) + (1 | county), nAGQ = 12,
+       data = lips, family=poisson)
> a <- ranef(ri)\$county[,1]
> library(dplyr)
> lips <- mutate(lips, a = a,  eb = 100 * exp(fixef(ri) + a))
```

The next task is to plot the empirical Bayes estimates of the SMR.

An annotated do file to produce the map can be obtained from the Stata bookstore, copied to your working directory, and then run. The first thing the do file does is read the map polygons, which will replace the dataset in memory.

```. copy https://www.stata-press.com/data/mlmus3/scotmaps.do scotmaps.do, replace

. clear

.quietly  do scotmaps

. graph export scotmap.png, width(500) replace
(file scotmap.png written in PNG format)
```

We first download a file from the Stata bookstore with the map polygons. We then prepare a canvas with the right dimensions, draw the polygons for each county with a fill color to reflect the SMR, and add a legend.

```> png("scotmapr.png", width=500, height=750)
> # map
> plot(sm\$lon, sm\$lat, type = "n", axes = FALSE, xlab = "", ylab = "", asp = 1)
> counties <- unique(sm\$regid)
> colors <- c("#595959","#737373", "#8d8d8d", "#a6a6a6","#c0c0c0")[5:1]
> smrg <- cut(lips\$eb, breaks=c(0,50,80,120,200,700))
> code <- as.numeric(smrg)
> for(county in counties) {
+     cm <- filter(sm, regid == county)
+     polygon(cm\$lon, cm\$lat, col = colors[code[county]])
+ }
> # legend
> delta <- 15000
> vf <- function(x, f) f(x[!is.na(x)])
> y0 <- vf(sm\$lat, max) - 2 * delta
> x0 <- vf(sm\$lon, min)
> for(i in 1:5) {
+     polygon( c(x0,x0+delta,x0+delta,x0), c(y0, y0,y0-delta, y0-delta), col=colors[i])
+     text(x0 + 4 * delta, y0 - delta/2, levels(smrg)[i], cex=0.8)
+     y0 <- y0 - 2 * delta
+ }
> dev.off()
null device
1
```

The incidence of lip cancer is higher in coastal places, particularly in the north.

You may also want to plot the EB estimates versus the observed SMRs. You will see the usual shrinkage towards the mean, which is about 145.

Last updated 3 September 2019

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