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Poisson Games and Sudden-Death Overtime

Mere Measles: A Look at Vaccination Rates in California, Part I

California is currently at the epicenter of a measles outbreak, a disease that was considered all but eradicated in the US as of a few years ago. Measles is a nasty disease; it's easily transmitted and at its worst can cause measles encephalitis, leading to brain damage or even death.

The increasing problem in the US with the measles, whooping cough and other nearly-eradicated diseases stems from a liberal personal belief exemption policy in California and other states. This wasn't a major problem until Andrew Wakefield famously and fraudulently tied autism to the MMR vaccine. This has led to thousands of unnecessary deaths as well as needless misery for thousands more. I myself caught a case of the whooping cough in San Diego a few years ago as a consequence of Wakefield's fraud. I've had several MMR vaccines over my life, but adults may still only be partially immune; this is yet another reason why a healthy level of herd immunity is so critical to maintain.

PBE rates in California may be relatively low, but the problem is that parents who are likely to seek a PBE exemption for the MMR vaccine tend to cluster, making them susceptible to outbreaks of highly infectious diseases such as the measles. As we'll see they cluster in private schools, they cluster in particular cites and they cluster in particular counties.

California makes vaccination and PBE (personal belief exemption) information available here (indexed by school code):

Immunization Levels in Child Care and Schools

California also makes student poverty information available here (also indexed by school code):

Student Poverty - FRPM Data

This is typical government data - Excel spreadsheets, bits of non-data all over the place. I've cleaned up both data sets for 2013-2014 here:

California kindergarten and poverty data for 2013-2014

Let's start with a basic nested model using only the vaccination data to examine the PBE rate for public vs private schools:

We get these nice ANOVA results that indicates the public and private status of schools does indeed add to our PBE model.

What's the estimated impact of public vs private? This is a nice illustration of why the odds ratio is so easy to use in logistic models with logit links. Here our the fixed effect estimates:

To get PBE rate estimates for average public and private schools, take the exponential of the estimates to get the odds ratio for the estimate. \begin{align*}
\mathrm{Intercept} &= e^{-3.113} = 0.044\\
\mathrm{public} &= e^{-0.606} = 0.546\\
\mathrm{private} &= e^{0} = 1.00
The nice thing about odds ratios is that they simply multiply. What's our estimate for the odds ratio for the PBE rate of a public school? It's \(0.044\times 0.546 = 0.024\); for a private school it's \(0.044\times 1.00 = 0.044\). Translating to PBE rates we get \(\frac{0.024}{1+0.024} = 0.023\) for public schools and \(\frac{0.044}{1+0.044} = 0.042\) for private schools.

Of course, the odds ratios for counties vary quite a bit, as do the odds ratios for cities within each county. You can see the \(\log({\mathrm{odds}})\) values for all of California's counties here and cities here.

For another example, let's do a private school in Beverly Hills, which is in Los Angeles County. The odds ratio for Beverly Hills is \(3.875\); for Los Angeles County it's 0.625. Chaining as before, we get the overall estimate PBE odds ratio for a private school in Beverly Hills, Los Angeles County to be \(0.044\times 1.00\times 0.625\times 3.875 = 0.107\). Translated to an estimated PBE rate this is \(\frac{0.107}{1+0.107} = 0.096\). Thus, we'd expect about 10% of the children in a private Beverly Hills kindergarten to be unvaccinated due to personal belief exemptions. This is well below the desired herd immunity level for measles of 95%.

In the next part I'll show how to merge the information in California's vaccination data and poverty data to examine the role of other factors in the clustering of unvaccinated children. As an example, charter schools in California tend to have even higher rates of personal belief exemptions than private schools.


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