This example shows why we need a model-based (Bayesian) estimate of cancer rates across many U.S. counties, and how a common informative prior can stabilize noisy rates from small counties. It also serves as a gentle introduction to hierarchical thinking.

1. The puzzle from the maps

Two maps were shown:

  • One map shaded the top 10% of U.S. counties in kidney/ureter cancer mortality (white males, 1980–1989).
  • Another map shaded the bottom 10% of counties.

Surprisingly, both maps highlighted many counties in the middle of the U.S. (Great Plains), not the coasts.

At first glance you might think: “Maybe the Great Plains has special environmental or medical issues.”
But the second map (lowest rates) shows the same region, so that explanation doesn’t hold.

Real reason: many of those counties have very small populations. With small nnn, even a single death can make the rate look high, and zero deaths makes it look extremely low. So small-population counties are overrepresented at both extremes.

So the pattern is not about geography — it’s about sample size variation.

2. A Poisson model for each county

To fix this, the authors propose a simple Bayesian model for each county $j$:

$y_j \sim \text{Poisson}(10 n_j \, θ_j) \tag{1}$

where:

  • $y_j$​: number of kidney cancer deaths in county $j$ over 10 years
  • $n_j$​: county population
  • $θ_j$​: underlying true cancer death rate (per person per year) for county $j$
  • factor 10 is there because we observed 10 years → exposure is $10 n_j$

The maps were really plotting the raw rate:

$\frac{y_j}{10 n_j}$

but that raw rate is very noisy when $n_j$​ is small.

Here, unlike the simple Poisson model earlier, each county has its own parameter $θ_j$​. So we are doing many parallel inferences with a common prior.

3. Choosing an informative prior

They choose a Gamma prior for each county’s rate:

$θ_j \sim \text{Gamma}(\alpha, \beta)$

For this problem they argue that $θ_j \sim \text{Gamma}(20,\, 430{,}000)$

is reasonable for U.S. kidney cancer mortality in that period.

Recall our parameterization:

$p(θ) \propto θ^{\alpha – 1} e^{-\beta θ}$

then

$E(θ) = \frac{\alpha}{\beta}, \quad \text{sd}(θ) = \frac{\sqrt{\alpha}}{\beta}$

So with $\alpha = 20$, $\beta = 430{,}000$:

$E(θ) = \frac{20}{430{,}000} = 4.65 \times 10^{-5}$

This is the national-level prior rate (deaths per person per year).

This is a single prior shared by all counties — that’s the “large set of inferences with a common prior” part.

4. Posterior for each county

Because Poisson–Gamma is conjugate, the posterior for county $j$ is:

$θ_j \mid y_j \sim \text{Gamma}\bigl(20 + y_j,\; 430{,}000 + 10 n_j\bigr)$

From that we get:

$E(θ_j \mid y_j) = \frac{20 + y_j}{430{,}000 + 10 n_j}$

$\operatorname{Var}(θ_j \mid y_j) = \frac{20 + y_j}{(430{,}000 + 10 n_j)^2}$

Key point: the posterior mean is a weighted average of

  • the raw rate $\dfrac{y_j}{10 n_j}$, and
  • the prior mean $\dfrac{20}{430{,}000} = 4.65 \times 10^{-5}$

When $n_j$​ is small → denominator is small → prior term dominates.
When $n_j$ is large → data term dominates.

So this is classic Bayesian shrinkage toward a reasonable national rate.

5. Small county vs large county

(a) Small county: $n_j = 1000$

Exposure over 10 years = $10 \times 1000 = 10{,}000$.

  • If $y_j = 0$:
    • raw rate = 0
    • posterior mean:
      • $E(θ_j \mid y_j=0) = \frac{20}{430{,}000 + 10{,}000} = \frac{20}{440{,}000} \approx 4.55 \times 10^{-5}$
      • So we do not conclude “cancer rate is 0.” We pull it back to a realistic value.
  • If $y_j = 1$:
    • raw rate = $\dfrac{1}{10{,}000} = 10^{-4}$ (which looks high!)
    • posterior mean:
      • $E(θ_j \mid y_j=1) = \frac{21}{440{,}000} \approx 4.77 \times 10^{-5}$
      • Much closer to the prior than to the raw rate.
  • If $y_j = 2$:
    • raw rate = $2 \times 10^{-4}$ (extremely high)
    • posterior mean:
      • $E(θ_j \mid y_j=2) = \frac{22}{440{,}000} = 5.00 \times 10^{-5}$
      • Still only slightly above the prior.

Conclusion: for small counties, the data are too thin → the prior dominates → we avoid silly extremes.

They also note: if you simulate from the predictive distribution (Poisson with Gamma prior → Negative Binomial), for such a small county most simulated $y_j$ are 0 or 1 anyway, so those extreme raw rates are expected.

(b) Large county: $n_j = 1{,}000{,}000$

Exposure over 10 years = $10 \times 1{,}000{,}000 = 10^{7}$.

Simulating from the same prior+Poisson setup gives counts like 393–545 deaths over 10 years. Then:

  • raw rate for 393 deaths:
    • $\frac{393}{10{,}000{,}000} = 3.93 \times 10^{-5}$
    • posterior mean:
      • $\frac{20 + 393}{430{,}000 + 10{,}000{,}000} \approx 3.96 \times 10^{-5}$
  • raw rate for 545 deaths:
    • $5.45 \times 10^{-5}$ posterior mean:
      • $\approx 5.41 \times 10^{-5}$

Now the data dominate — prior barely moves the estimate. That’s exactly what we want.

6. Why the Bayesian estimates look nicer

In the Poisson model $\frac{y_j}{10 n_j}$

has variance roughly proportional to $1 / n_j$​. So small $n_j$​ → very noisy rates → those are the extreme points in below.

But the Bayesian posterior means $E(θ_j \mid y_j) = \frac{20 + y_j}{430{,}000 + 10 n_j}$

are much less variable across counties, especially for the small ones. That’s what below image is showing:

the “Bayes-adjusted” rates don’t jump wildly for low-population counties, and the posterior intervals are wider for small counties (they contain less information).

So the model automatically encodes: “small counties: be cautious; large counties: trust the data.”

7. Where did that $\text{Gamma}(20, 430{,}000)$ come from?

They didn’t pick it from thin air. Rough idea:

  1. Under the model, $y_j \sim \text{NegBin}\left(\alpha,\; \frac{\beta}{10 n_j}\right)$ because Poisson with Gamma prior ⇒ Negative Binomial predictive.
  2. The Negative Binomial has known mean and variance (they derive):
    • $E(y_j) = 10 n_j \frac{\alpha}{\beta}$
    • $\operatorname{Var}(y_j) = 10 n_j \frac{\alpha}{\beta} + (10 n_j)^2 \frac{\alpha}{\beta^2}$
  3. They then match these theoretical moments to the empirical mean and variance of the observed county rates (after age adjustment), solving for $\alpha, \beta$.

This is a kind of moment matching — not fully Bayesian, but a practical shortcut when you want a single prior that’s reasonable for all counties.

The “right” way is to make $\alpha, \beta$ themselves unknown and estimate them → hierarchical model.

8. What this example teaches

  • A map of raw rates can be misleading when sample sizes differ.
  • A common informative prior pulls extreme small-area rates toward a plausible national rate.
  • The amount of shrinkage is data-size dependent (automatic in the math).
  • Poisson + Gamma is a powerful small-area model.
  • Moment matching to pick the Gamma prior is OK as an approximation, but hierarchical Bayes is better.
  • This is exactly the kind of problem where partial pooling (hierarchical modeling) is necessary — many similar units, each with weak data.