havior of nonrespondents. With a shortened questionnaire and monetary incentives, Caspar (1992) surveyed nearly 40 percent of the nonrespondents to the 1990 NHSDA in the Washington, D.C., area. In this survey, nonrespondents have higher prevalence rates than respondents. Whether these findings apply to all nonrespondents, not just the fraction who replied to the follow-up survey in the Washington, D.C., area, is unknown.

The Monotone Selection Assumption

Rather than impose the missing-at-random assumption, it might be sensible to assume that the prevalence rate of nonrespondents is no less than the observed rate for respondents. Arguably, given the stigma associated with use, nonrespondents have higher prevalence rates than respondents. Formally, this monotone selection assumption implies

P[yt=1|zt=1]=P[yt=1|zt=0]=1.

(4)

The lower bound results if the prevalence rate for nonrespondents equals the rate for respondents. The upper bound results if all nonrespondents consume illegal drugs. The true rate lies within these bounds.

This restriction on the prevalence rates for nonrespondents implies bounds on the population prevalence rates:

P[yt=1|zt=1]=P[yt=1]=P[yt=1|zt=1]P[zt=1]

+P[zt=0].8

(5)

Notice that the lower bound is the fraction of respondents who use illegal drugs in the past year, while the upper bound increases with the fraction of nonrespondents. The width of the bound equals {1–P[yt=1|zt=1] }P[zt=0]. Thus, the uncertainty reflected in the bounds increases with the fraction of nonrespondents, P[zt=0] and decreases with the prevalence rates of respondents. In the extreme, if all respondents are using drugs, then the monotonicity assumption implies that all nonrespondents would be using drugs as well, so that the prevalence rate would be identified.

Given the MTS assumption in Equation (4), we can also bound the trend from time (t) to time (t+j):

{P[yt+j=1|zt+j=1]–P[yt=1|zt=1]}P[z=1]

+{P[yt+j=1|zt+j=1]–1}P[z=0]=P[yt+j=1]–P[yt=1]=

{P[yt+j=1|zt+j=1]–P[yt=1|zt=1]}P[z=1]

+{1–P[yt=1|zt=1]}P[z=0]

(6)

8  

Manski and Pepper (2000) formalize the implications of this assumption.



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