Measuring Change in Substate Estimates between 2002-2004 and 2004-2006
In this National Survey on Drug Use and Health (NSDUH) document,1 estimates of change between the substate small area estimates are based on the same methodology used to produce estimates of change between the State estimates. For details, see Section A.10 of Appendix A in the 2005-2006 small area estimation (SAE) report (Hughes, Sathe, & Spagnola, 2008). However, because some substate regions within 10 States were defined differently for the two time periods (i.e., 2002-2004 and 2004-2006), estimates of change (and their associated Bayes posterior probability of no change p value) could not be produced for these substate regions. For details regarding which substate regions changed definitions, see Section D of the report on substate estimates from the 2004-2006 NSDUHs (OAS, 2008).
Estimates of change in this document are produced only for persons aged 12 or older except for underage alcohol use and underage binge alcohol use outcomes where estimates of change are produced for those aged 12 to 20. Also note that estimates of change are not produced for two outcomes, serious psychological distress (SPD) and major depressive episode (MDE), because SPD was not defined consistently across the two time periods and MDE was not defined in 2002 and 2003. For details, see Appendix A of the 2004-2005 State SAE report (Wright, Sathe, & Spagnola, 2007).
To estimate change in substate estimates, let πs(1) and πs(2) denote 2002-2004 and 2004-2006 prevalence rates, respectively, for substate-s. The change between πs(1) and πs(2) is defined in terms of the log-odds ratio lors as opposed to the simple difference because the posterior distribution of the lors is closer to Gaussian than the posterior distribution of the simple difference, (πs(2) – πs(1)). The lors is defined as
, D
where ln denotes the natural logarithm.
The p value given in the change tables is computed to test the null hypothesis of no change; that is, πs(2) = πs(1) or equivalently lors = 0. An estimate of lors is given by
, D
where the ps(1) are the previously published 2002-2004 substate estimates presented in Section C of the 2002-2004 substate SAE report (OAS, 2006) and the ps(2) are the 2004-2006 substate estimates presented in Section C of the 2004-2006 substate SAE report (OAS, 2008).
To compute the variance of 
s, that is, v(s), let 
and 
, then
, D
where cov[ln(
1), ln(2)] denotes the covariance between ln(1) and ln(2). This covariance is defined in terms of the associated correlation, as follows:
. D
Note that the v[ln(1)] and v[ln(2)] used here to calculate v(s) are the same variances used in calculating the previously published 2002-2004 prediction intervals (PIs) and the 2004-2006 PIs, respectively.
The correlation between ln(1) and ln(2) was obtained by simultaneously modeling the 2002-2003, 2004, and the 2005-2006 NSDUH data using a method similar to obtaining the correlation for producing State estimates of change. For details, see Section A.10 of Appendix A in the 2005-2006 report (Hughes et al., 2008).
To calculate the p value for testing the null hypothesis of no difference (lor = 0), it is assumed that the posterior distribution of lor is normal with mean = s and variance = v(s). With the null value of lor = 0, the Bayes p value or posterior probability of no difference is p value = 2*P [Z ≥ abs(z)], where Z is a standard normal random variate, 
, and abs(z) denotes the absolute value of z.
