The upper bound for the trend equals the upper bound for the usage rate at time t+j minus the lower bound at time t. Thus, if the lower bound at time t exceeds the upper bound at time t+j, the fraction of users must have fallen over the period. Likewise, if the lower bound at time t+j exceeds the upper bound at time t, then the fraction of users must have risen.

Figures D.2 and D.3 display the estimated bounds from 1975 to 1997 for 12th graders from the MTF survey and for adolescents ages 12–17 from the NHSDA, respectively. For simplicity, I assume the nonresponse rate is fixed at 15 percent in the MTF survey and 25 percent in the NHSDA. I also abstract from concerns about statistical variability and instead focus on the point estimates.

Under the monotone selection assumption, data from the MTF imply that the annual prevalence rate for 12th graders lies between 29 and 40 percent in 1991 and between 42 and 51 percent in 1997.9 Thus, the data bound the level estimates to lie within about a 10-point range. Notice also that these estimates imply that the fraction of users increased in the 1990s, although the magnitude of these changes and the directions of the year-to-year variations are not revealed. In particular, from 1991 to 1997, the prevalence rate increased by at least 2 points (from 40 to 42 percent), and perhaps by as much as 22 points (from 29 to 51 percent).

The bounds displayed in Figure D.2 reveal the uncertainly implied by student nonresponse to the MTF. These bounds, however, do not reflect school nonresponse. The MTF uses a clustered sampling design whereby schools and then individuals within a school are asked to participate in the study. Each year between 30 to 50 percent of the selected schools decline to participate and are replaced by similar schools in terms of observed characteristics such as size, geographic area, urbanicity, and so forth (Johnston et al., 1998a). In 1995, for example, nearly 38 percent of schools and 16 percent of students declined to participate, so that the overall response rate for the 12th grade survey is only 52 percent (Gfroerer et al., 1997b).

To the extent that nonrespondent schools have drug usage rates that systematically differ from respondent schools, inferences drawn using the survey will be biased. With school nonresponse and replacement rates of nearly 50 percent, this is an especially important nonresponse problem. If incorporated into the bounds developed above, the data no longer even reveal the direction of the largest trends.

Data from the NHSDA imply that the annual prevalence rate of use


Using the law of total probability to weight the fraction of respondents and nonrespondents, the upper bound prevalence rates are found by assuming that all nonrespondents use drugs. In 1991, for example, the upper bound rate of 40=29*0.85+100*0.15.

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