Using Booking Data to Model Drug User Arrest Rates: A Preliminary to Estimating the Prevalence of Chronic Drug Use

Abstract

Public policy is often concerned with the size and characteristics of special populations that are difficult to reach in household surveys. Chronic drug users, who often live outside conventional households, provide the illustration motivating this paper. An alternative to household surveys is to question chronic drug users where they congregate—jails, treatment programs, and shelters, for example. Using such opportunistic data for prevalence estimation raises difficult problems for statistical inference: Study subjects who arrive at the collection points cannot be deemed a random sample of the general population. However, if we could estimate the rates at which chronic drug users arrive at the collection points, then we could use those estimates to weight the sample to represent the population. This paper presents a modified Poisson mixture model used to estimate the stochastic process that accounts for how chronic drug users get arrested. It uses that model to estimate arrest rates for 38 counties using up to sixteen quarters of data from the Arrestee Drug Abuse Monitoring survey.

Appendix: The Distribution of γ

Equation (7) gives us the distribution of γ for “arrested” CDUs. The mean arrest rate for arrested CDUs will be larger than the mean arrest rate for CDUs in the county. To see this, follow (7A) through (7D) to compute the mean arrest rate for arrested CDUs as:

$$E\left(\gamma\vert {\rm arrest}\right)=\int\limits_{\gamma=0}^{\infty} \gamma g\left(\gamma\vert {\rm arrested}\right)d\gamma $$

(7A)

Substituting (7) from the main text into (7A) gives:

$$ E\left(\gamma\vert {\rm arrest}\right)=\frac{\int\limits_{\gamma=0}^{\infty} \gamma^{2}f\left(\gamma\right)d\gamma}{e^{\left(1/2\right)\sigma^{2}}} =\frac{\int\limits_{\gamma=0}^{\infty}{\gamma^{2}f \left(\gamma\right)d\gamma}}{E\left(\gamma\right)} $$

(7B)

Because γ is distributed as lognormal in the general population, its variance can be written:

$$ {\rm VAR}\left(\gamma\right)=e^{\sigma^{2}}\left(e^{\sigma^{2}}-1\right) =\int\limits_{\gamma=0}^{\infty}\gamma^{2}f\left(\gamma\right)d\gamma -E\left(\gamma\right)^{2} $$

(7C)

The first part of the equality is definitional: this is the variance for a lognormal random variable. The second part of the equality comes from writing the variance of a variable Y as

$$ E\left\lfloor\left(Y-\bar{Y}\right)^{2}\right\rfloor=E\left[Y^{2}\right] -2E\left[Y\right]\bar{Y}+\bar{Y}^{2}=E\left[Y^{2}\right]-\bar{Y}^{2} $$

.

Solving (7C) for $\int\gamma^{2}f(\gamma)d\gamma$, and substituting the results into (7B), yields the first term on the right of the equality sign in (7D). Because γ is lognormal, we substitute $E\left[\gamma\right]=e^{\left(1/2\right)\sigma^{2}}$ into the numerator and denominator of (7D) to yield the second term on the right of the equality. The final simplification follows from some algebraic manipulation, which recognizes that $e^{x}e^{y}=e^{x+y}$.

$$ E\left(\gamma\vert {\rm arrest}\right)=\frac{e^{\sigma^{2}} \left(e^{\sigma^{2}}-1\right)+E\left(\gamma\right)^{2}} {E\left(\gamma\right)}=\frac{e^{\sigma^{2}}\left(e^{\sigma^{2}}-1\right) +e^{\sigma^{2}}}{e^{\left(1/2\right)\sigma^{2}}} =e^{\left(3/2\right)\sigma^{2}} $$

(7D)

Figure 3, which represents a stylized version of drug user arrest rates, helps cement ideas. The picture segment on the left shows, for small γ, f(γ) as the higher curve and g(γ| arrest) as the lower curve. Inspection of the curves shows that the expected value of γ is higher for a sample of arrested CDUs than it is for the population of CDUs in the county. The picture segment on the right shows, for small N, P(N| X) as the higher curve (where N = 0) and P(N| X;arrest) as the lower curve. As would be expected, the probability of zero arrests during the window period is lower for arrested CDUs than it is for the population of CDUs. For a random sample drawn from the community, the observable data would correspond to f(γ) and P(N| X), but for the sample of arrestees, the observable data conform to g(γ| arrest) and P(N| X;arrest). Given use of the arrestee sample to estimate β and σ, estimation requires likelihood based on these conditional distributions.

Fig. 3

Distribution of γ and Arrest Rates (a) f(γ) and g(γ|arrest) (b) P(N|X) and P(N|X;arrest). Notes: In panel a, the higher curve (at low values of the arrest rateγ) represents the distribution of γ for CDUs in the county and the lower curve represents the distribution of γ for CDUs who are arrested. In panel b, the higher curve (at low values of the arrest rate γ) represents the distribution of arrests for CDUs in the county and the lower curve represents the distribution for CDUs who are arrested