Comparison of the 2006-2008 and 2008-2010 Substate Estimates
This discussion focuses on the methodology used to conduct statistical tests of significance for comparing National Survey on Drug Use and Health (NSDUH) substate estimates between 2006-2008 and 2008-2010. Estimates shown in Tables 1 to 23 earlier in this document are based on small area estimation (SAE) procedures, and the methodology used to compare these estimates is the same methodology used to compare the State small area estimates. For details, see Section A.13 of Appendix A in the 2009-2010 State SAE report.1
For these tables, the 2006-2008 substate small area estimates were produced using the 2008-2010 substate region definitions. Note that the substate region estimates for 2006-2008 in all States were recalculated after removing erroneous (falsified) data for Pennsylvania and Maryland.2 Hence, the 2006-2008 substate region estimates provided in these tables may not match previously published small area estimates.3 The updated 2006-2008 substate region definitions consist of the original 2006-2008 definitions for 43 States and the District of Columbia and revisions in the following 7 States: Alaska, Arkansas, California, Georgia, North Carolina, Pennsylvania, and West Virginia. Using the 2008-2010 substate region definitions on the 2006-2008 data made comparisons between the two sets of estimates possible for all regions.
Comparisons of substate estimates between 2006-2008 and 2008-2010 were made only for persons aged 12 or older except for the underage alcohol use outcome in Table 11 and the underage binge alcohol use outcome in Table 12 (where comparisons were made for persons aged 12 to 20) and for the major depressive episode outcome in Table 23 (where comparisons were made for persons aged 18 or older).
To determine whether differences between substate estimates over two time periods were statistically significant, let 
and 
denote 2006-2008 and 2008-2010 prevalence rates, respectively, for substate-s. The change between and is defined in terms of the log-odds ratio 
as opposed to the simple difference because the posterior distribution of the is closer to Gaussian than the posterior distribution of the simple difference, 
. The is defined as follows:
, 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, 
or equivalently 
. An estimate of is given by
, D
where the 
are the 2006-2008 substate estimates based on the 2008-2010 substate region definitions, presented for the first time in this document's tables, and the 
are the 2008-2010 substate estimated presented in the 2008-2010 NSDUH Substate Regions: Excel tables at https://www.samhsa.gov/data/sites/default/files/Substate2k10-NSDUHsubstateStates2010/NSDUHsubstate51StateTOC2010.htm.
To compute the variance of 
, that is, 
, let and 
and 
,then 
,
where 
denotes the covariance between 
and 
. This covariance is defined in terms of the associated correlation, as follows:
. D
Note that the 
and 
used here to calculate 
are the same variances used in calculating the 2006-2008 Bayesian confidence intervals (CIs) and the 2008-2010 CIs, respectively.
The correlation between and was obtained by simultaneously modeling the 2006-2007, 2008, and the 2009-2010 NSDUH data using a method similar to obtaining the correlation for producing State estimates of change. For details, see Section A.13 of Appendix A in the 2009-2010 report.4
To calculate the p value for testing the null hypothesis of no difference (lor = 0), it was assumed that the posterior distribution of 
is normal with 
and 
. With the null value of (lor = 0), the Bayes p value or posterior probability of no difference is 
where Z is a standard normal random variate, 
, and 
denotes the absolute value of z.
