Class, 11/21

Today we had a visit from Dr. Turner Osler, who is a medical researcher at the UVM medical school.

Just to touch on several of the important topics. You will have read the Scientific American article by Efron and Morris. The point here is that a better estimate of the individual items in a collection of items (baseball batting averages, proportion of toxoplasmosis patients in a city, proportion of mortalities in a burn unit) by combining them in a way that uses some of the information from all of the different items to get better estimates of the individual items. So if we observe the batting averages of a number of batters after they have been at-bat 40 times (early in the season), we can get a better overall estimate of the final batting averages of the players at the end of the season by using a formula like:


z_i=Y + c(y_i-Y)

where y_i is the individual item (batting average, etc.), Y is the average of all of them, and z_i is our best estimate of the true value. The number c is given by a formula, which is in the notes that Dr. Osler handed out. The important thing is that if c=1, then z_i=y_i, and if c=0 then z_i=Y. The smaller c, the more the estimates are "shrunk" towards Y. So the ideal estimator is a combination of Y and y_i, a weighted average of the two. (This is the so-called Efron-Morris estimator).

I remarked at the end of the class that the notion of "best" in this context is actually gotten by assuming a particular loss function that says that the farther away the true values are from the estimated value (in an "average" sense), the bigger the loss, so we try to choose c so as to minimize the overall loss. When you do this, the value of c in the handout pops out.

Dr. Osler is using these ideas to solve the following problem: we can estimate the proportion of "bad outcomes" in a burn unit, for example, by dividing the number of patients that die by the total number of patients. But this is like the batting average, it is based on a relatively small number of patients in many cases. But we can use the ideas of the Efron-Morris article to improve the estimates for the individual hospitals. And we learn that many of the hospitals that seemed at first glance to have excessively high mortality, probably do not. There was one hospital that looked suspicious.

Dr. Osler also discussed a "beta" prior, which we have been using without giving it that name. It is anything of the form p^a(1-p)^b for constants a and b. This will be a "bell-shaped curve," whose actual shape can be varied quite widely by choosing the constants a and b to match what we want. When you multiply this prior by the likelihood pⁿ(1-p)^N-n, which you get if n patients out of N die, you get a posterior that is also beta: p^a+n(1-p)^b+N-n. We've been doing this with spreadsheets, which are an approximation based on evaluating this formula at particular points between 0 and 1 (we used p=0.05, 0.15, 0.25,...,0.95 in class). As I remarked, the more points you use in the spreadsheet, the more accurate the results. We used 10 points because that's something easily done in class. But you could use 100 points, or 1000 of them to get more accurate answers. With a spreadsheet, it's just copying the formula down. But what Dr. Osler is doing is something we've been doing essentially for the entire semester. For example, when we used a non-uniform prior in the polling (voting) example, we were doing this.

Have a happy Thanksgiving!