Insight into whether the sign of the trend is identified can be found by evaluating the restrictions on the temporal variation in the prevalence rates of nonrespondents that would be required to identify the direction of change. Suppose that the direction of the trend in prevalence rates for respondents is positive (negative). Then, from Equation (3) we see that the direction of change for the population must also be positive (negative) if the prevalence rate for nonrespondents cannot fall (rise) by more than the switching threshold
Thus, if it is known that the growth in usage rates for nonrespondents is less (in absolute value) than this switching threshold, the sign of the trend is identified. If, however, the trend in use for nonrespondents might exceed this threshold, the sign of the trend is ambiguous. Notice that to identify the sign of the trend, one need not know the magnitude or direction of the trend for nonrespondents. Rather, one simply needs to rule out the possibility that the trend for nonrespondents exceeds this threshold.
Intuitively, the switching threshold increases with the fraction of respondents and with the prevalence rate of respondents. As the fraction of respondents increases, the behavior of nonrespondents has less impact on the overall prevalence rates; hence the trend in usage rates among nonrespondents must be more exaggerated to switch the sign of the observed trend. Likewise, as the observed prevalence rate of use increases over time, the switching threshold for nonrespondents must also increase.
Table D.2 displays the estimated switching threshold along with the annual trends in usage rates from the MTF. The threshold for the MTF survey is computed by multiplying the observed trend by –5.7(=–85/15). Consider, for instance, the trend from 1992 to 1993. The fraction of respondents using drugs increased 3.9 points, implying a switching threshold of –22.1(=–5.7*3.9). That is, for the annual trend from 1992 to 1993 to fall, the rate of use among nonrespondents must have declined by over 22 points. The monotonicity assumption in Equation (4) does not rule out such large drops among nonrespondents. In fact, the rate of use may have fallen by as much as 69 points, from 100 percent in 1992 to 31 percent in 1993. If, however, it is known that the annual variation in the prevalence rates for nonrespondents cannot exceed 20 points, then the sign of the trend is identified.
Table D.3 displays the estimated switching threshold along with the annual trends in usage rates from the NHSDA. In the NHSDA, there are