1997 National Household Survey on Drug Abuse |
Sampling Error and Confidence Intervals
To illustrate, let the proportion P_{d} represent the true prevalence rate for a particular analysis domain "d." Then the logit transformation of P_{d}, commonly referred to as the "log odds," is defined as
where "ln" denotes the natural logarithm.Letting p_{d} be the estimate of the proportion, the log odds estimate becomes
Then the lower and upper limits of L are calculated as where var(p_{d}) is the variance estimate of p_{d}, and K is the constant chosen to yield the proper level of confidence (e.g., K = 1.96 for 95% confidence limits).
Applying the inverse logit transformation to A and B above yields a confidence interval for p_{d} as follows:
where "exp" denotes the inverse log transformation. The upper and lower limits of the percentage estimate are obtained by simply multiplying the upper and lower limits of p by 100.
where
The confidence interval foris obtained by multiplying the lower and upper limits of the proportion confidence interval by. In addition, the variance ofcan be estimated as
.32
As in other publications, estimates with low precision were not reported. The criterion used for suppressing estimates was based on the relative standard error (RSE). The RSE is defined as the ratio of the standard error (SE) of the estimate over the estimate. For the 1997 NHSDA reports, the log transformation of the proportion estimate was used to calculate the RSE. Specifically, percentages and corresponding population estimates were suppressed if
or
For computational purposes, this is equivalent
to
or
where SE(p) equals the standard
error estimate of p. The log transformation of p is used
to provide a more balanced treatment of measuring the quality of small,
large, and intermediate p values. The switch to (1-p) for
p greater than 0.5 provides a symmetric suppression rule across
the range of possible p values.
32 This approach treats the estimated domain size, N hat sub d, as fixed. This is a good approximation in most cases because ratio adjustments tend to control the variability of estimates of domain sizes.
This page was last updated on December 30, 2008. |