Distribution

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Why not to use a normal distribution to approximate a binomial distribution

April 28, 2009 in Econometrics | No comments

Most people know that under the central limit theory claims, the distribution of the mean of a distribution will be normally distributed as the number of observations gets large.  The question is, if we have a series of discrete events that we want to approximate the distribution of the mean with a continuous distribution, should we estimate them with a normal distribution?

For instance, let us assume we have 20 observations on patient admittance to the hosptial and in 3 of those cases, the individual died.  we can use a binomial distribution to estimate the distribution of the prior as:

  •  nCr*πr(1-π) n-r. 

We can estimate π with the 3/20 = 0.15.  For our prior distribution, we could fit a normal distribution.  Using a normal distribution, however, would include values less than 0.  This is especially problematic if there is a small samples sizes (e.g., n=20).  A truncated normal would solve the problem of negative values, but eliminating one portion of the distribution will change the distribution’s mean and variance.

Another option is to use the beta distribution for the prior.  The beta distribution for the value of π is:

  • p(π) = {Γ(α + β)/[Γ(α)Γ(β)]} πα-1(1-π)β-1

If we apply Bayes’ theorm to the binomial data with a beta prior, we get:

  • p(π) ∝ πr(1-π)n-rπα-1(1-π)β-1
  • p(π) ∝ πα+r-1*(1-π)β + n-r -1

Now we have that the posterior distribution is Beta(α+r-1,β + n-r -1).  We already know r and n, and can match α and β with the methods of moments.

  • E(θ) = α/(α + β)
  • var(θ) = αβ/[(α + β)2(α + β+1)]

Now we estimate E(θ) and var(θ) with the sample moments. If 3/20 people died, then we estimate E(θ) with 3/20 = 0.15. Further, with a binomial distribution, we can estimate var(θ) with p(1-p)/n = .15*.85/20 = .00638. This means that the s(θ)=.006381/2 = .07984. Thus we can solve for α and β since we now have 2 equations and two unknowns.

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Vaccine Logistics

March 19, 2008 in HC Economist Models, Vaccinations | Permalink

Vaccination is one of the most cost effective medical treatments we have.  It is important that providers vaccines in a timely manner.

In attempt to streamline vaccine distribution systems, the CDC created Vaccine Management Business Improvement Project (VMBIP).  Instead of having providers place orders with the grantee (i.e.: state health department), and then having the grantee ship them to a local distributor, VMBIP is an attempt to reduce warehouse costs by shipping vaccines from a centralized warehouse directly to the provider.  This may save money, if the vaccines are sent in a timely manner.

My presentation at the National Immunization Conference analyzed some data from southern California providers and found that the time from the vaccine order being place to delivery increased from 1.6 work days to 13.5 workdays after VMBIP was implemented.  I received other anecdotal evidence that these delays were affecting the vaccine supply of many California providers, but I did not know how efficiently the VMBIP program was operating in other states.

I found that California’s 13.5 day delay may not be so bad compared to the rest of the country.  One nurse from Texas said that vaccines delivery could take as long as 6 weeks.  There was significant variability so that the clinic would run out of vaccines occasionally so would have to place their orders early.  Sometime the vaccines would arrive within 2 days, but since the provider had anticipated a 2-4 week delay, there was no room in the refrigerator to store the vaccine.

Another conference attendee explained to me her experience in Minnesota.  Vaccines must be stored at a certain temperature to ensure they do not spoil.  Some winter days are so cold in Minnesota that the state public health department would advise distributors not to ship on those days to insure that they would not freeze.  Under the new, centralized VMBIP system, the national warehouse–which is run by McKesson–was not sensitive to these regional variations.  Minnesota providers have received frozen vaccines since McKesson did not know about how Minnesota winters effect vaccines.  These frozen vaccines are completely useless and must be discarded.

Overall, I doubt that centralized vaccine distribution is a good model.  Wal-mart can operate a centralized distribution system because all the stores are on the same computer network, they work under the centralized location, and receive extensive logisitcs training.  Further, Wal-mart is a hierarchical organization.  On the other hand, physicians are not integrated into a public health IT database–VACMAN not withstanding.  Further, providers are well trained on medical issues, but not logistics or filling out forms.  Since vaccine distribution is not a hierarchical system, a more flexible, less centralized, system would likely be optimal.

I would like to thank all the people who attended my presentation today at the National Immunization Conference and all the helpful feedback I have received.

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