The Price of Illicit Drugs: 1981 through the - The White House

model because each purity observation corresponds to a binomial observation. For example, if a
purchase had an observed purity of 0.8 then, if m was 1,000, we have effectively observed 800
events in 1,000 trials.
The problem is that the value of m is unknown. While a purity measurement of 0.8 is consistent
with obtaining 800 events in 1,000 trials, it is also consistent with obtaining 8 events in 10 trials,
but these two binomial observations convey different information. However, if we assume the
value of m is the same for all purchases (i.e. that the size of the small sample used in the laboratory
analysis was the same size for all purchases), any choice of m will suffice provided the variance
function incorporates a dispersion parameter. That is, estimation and inference based on an
overdispersed binomial model is invariant to the choice of m. Thus, our mean and variance
functions for purity were:
E(purity itj ) = exp( + city i + time t )/{1 + exp( + city i + time t )} (5)
V(purity itj ) = {E(purity itj )(1 - E(purity itj ))}/m (6)
Here, purity itj represents the jth observation from the ith city at the tth time period, and is the
dispersion parameter. Equation (5) specifies a logistic model, because the inverse of the mean
function is the logit function. In contrast to the linear model (3), the logistic model restricts the
mean purity to the unit interval.
This purity model contained 107 parameters (29 for the 30 cities, 77 for the 78 quarters, and one
for the intercept). Based on estimate of these parameters, we derived estimates for mean purities
for each city in each quarter. For a given quarter, the 30 city means were multiplied by their
respective DAWN weights and the sum of these contributions provided a weighted estimate for
the mean price and mean purity in the U.S. for that quarter. The national results are show in Tables
1-4, and Figures 3, 6, and 8.
6. A Model for Weights
As previously noted, the price and purity models described above provided estimates for mean
price and mean purity for each city in each quarter. For a given quarter, the 30 city means were
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