13 Aspirin: Shrinkage (or “borrowing strength”) via hierarchical modeling (aspirin)
The following data come from a meta-analysis of heart attack data. Each observation is the results of a study of survivorship following a heart attack (myocardial infarction). In each study, some victims were given aspirin immediately following their heart attack, while some vicitims were not. The observed values of y are the differences in mean survivorship observed in each study, with the other piece of data, the standard deviations, reflecting the relative sizes of the two groups in each study (i.e., although the data are binomial, given the large number of observations per study a normal approximation is valid and reduces each study’s data to the observed treatement effect and a standard deviation). The goal of the meta-analysis is to synthesize the six studies, so as to arrive at an overall conclusion regarding the effects of aspirin on survivorship following a heart attack.
This is an extremely simple example of hierarchical modeling. Via the exchangeability assumption (i.e., the study-specific means have a common prior), the studies “borrow strength” from one another, introducing some bias (each study’s mean qi is shrunk back towards the common mean), but with the benefit of gaining precision (smaller variance). We also gain a better estimate of the overall effect of aspirin on survivorship after heart attack than we would get from naively pooling the studies.
These data and the meta-analysis is discussed at length in Draper (1992).
model{ for(i in 1:6){ ## loop over studies theta[i] ~ dnorm(mu,tau); ## prior for each study v[i] <- pow(sd[i],2); ## convert each study’s se to var precision[i] <- 1/v[i]; ## convert var to precision y[i] ~ dnorm(theta[i],precision[i]); ## model for each study b[i] <- v[i]/(v[i] + sigma2) ## shrinkage (auxilary quantity) } mu ~ dnorm(0.0, .001); ## prior for the common mean tau ~ dgamma(.01, .01); ## “between-study” precision, prior sigma2 <- 1/tau; ## convert precision to variance good <- step(mu); ## E(good) = Pr(mu>0 | data) }
13.1 data
list(y=c(2.77,2.50,1.84,2.56,2.31,-1.15), sd=c(1.65,1.31,2.34,1.67,1.98,0.90))
13.2 initial values
list(mu=0,tau=1)
Results: The boost in survivorship is estimated at 1.32 percentage points with a standard deviation of .93; the posterior mean of “good” yields a Bayesian p-value for this overall treatment effect, the (posterior) probability that aspirin does not increase survivorship is 1 - .9447 = .0553. Note that a classical analysis that simply pooled the studies yields an average treatment effect of .86 and a standard error of .59 (z = 1.47, p = .072).
node mean sd MC error 2.5% median 97.5% start sample
b[1] 0.6661 0.2461 0.007897 0.1663 0.6882 0.9953 1001 10000
b[2] 0.5894 0.2701 0.00917 0.1117 0.5818 0.9926 1001 10000
b[3] 0.7712 0.1995 0.005867 0.2863 0.8161 0.9977 1001 10000
b[4] 0.67 0.2447 0.007828 0.1696 0.6933 0.9954 1001 10000
b[5] 0.7232 0.2229 0.006835 0.2231 0.7606 0.9967 1001 10000
b[6] 0.4634 0.2923 0.0108 0.05601 0.3963 0.9844 1001 10000
good 0.9447 0.2286 0.004739 0.0 1.0 1.0 1001 10000
mu 1.316 0.9346 0.02277 -0.3015 1.263 3.304 1001 10000
sigma2 2.612 6.079 0.1084 0.01284 1.234 13.65 1001 10000
theta[1] 1.736 1.177 0.02922 -0.2638 1.603 4.352 1001 10000
theta[2] 1.738 1.065 0.02805 -0.06121 1.62 4.037 1001 10000
theta[3] 1.375 1.29 0.027 -1.053 1.269 4.246 1001 10000
theta[4] 1.655 1.176 0.02757 -0.3317 1.523 4.292 1001 10000
theta[5] 1.538 1.247 0.02587 -0.6295 1.405 4.325 1001 10000
theta[6] -0.07534 0.9488 0.02355 -1.985 -0.02785 1.636 1001 10000References
Draper, IDavid. 1992. Combining Information: Statistical Issues and Opportunities for Research. Contemporary Statistics. National Academy Press. https://books.google.com/books?id=l0ArAAAAYAAJ.