Methodology Used to Estimate the Bayes Significance Level for the Null Hypothesis of No Change between the 2014-2016 and 2016-2018 Substate Population Percentages
This discussion focuses on the methodology used to conduct statistical tests of significance for comparing National Survey on Drug Use and Health (NSDUH) substate population percentages between 2014-2016 and 2016-2018. Estimates shown in Tables 1 to 14 earlier in this document are based on small area estimation (SAE) procedures. The methodology used to compare substate population percentages associated with these estimates is the same methodology used to compare the state population percentages between two time periods.1 All substate regions between 2014-2016 and 2016-2018 were defined in a similar manner, so the Bayes posterior probability of no change (p value) was produced for all substate regions.
Comparisons of substate population percentages between 2014-2016 and 2016-2018 were made only for individuals aged 12 or older except for the underage alcohol use outcome in Table 6 (where comparisons were made for individuals aged 12 to 20) and for five mental health outcomes on serious mental illness, any mental illness, received mental health services, serious thoughts of suicide, and major depressive episode in Tables 10 through 14, respectively (where comparisons were made for adults aged 18 or older). Note that these comparisons are available only for a subset of 13 outcomes that were defined consistently between 2014-2016 and 2016-2018.
To determine whether differences between population percentages of a substate region over two time periods were statistically significant, let 
and 
denote the 2014-2016 and 2016-2018 prevalence rates, respectively, for substate region-s. The difference 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
In this equation, the ln denotes the natural logarithm. The p value given in each of the tables in this document (i.e., Tables 1 to 14) is computed to test the null hypothesis of no difference; that is, 
or equivalently 
. An estimate of is given by
, D
where the 
are the 2014-2016 substate estimates presented in the "2014-2016 NSDUH Substate Region Estimates: Excel Tables" at https://www.samhsa.gov/data/, and the 
are the 2016-2018 substate estimates presented in the "2016-2018 NSDUH Substate Region Estimates: Excel Tables" at https://www.samhsa.gov/data/.
Let 
and 
, then an estimate of the posterior variance of is given by 
, where 
denotes the covariance between 
and 
. This covariance is defined in terms of the associated correlation, as follows:
. D
Note that 
and 
used here to calculate 
are the same posterior variances used in calculating the 2014-2016 Bayesian confidence intervals (CIs) and the 2016-2018 CIs, respectively. Thus, can be calculated using the published Bayesian CIs, 
and 
, as follows:
, D
where the 
and 
.
The correlation between and was obtained by simultaneously modeling the 2014-2018 NSDUH data using a method similar to modeling the correlations between the 2016-2017 and 2017-2018 state estimates.2
To calculate the p value for testing the null hypothesis of no difference (), it was assumed that the posterior distribution of is normal with 
and 
. The Bayesian p value or significance level for the null hypothesis of no difference () is 
, where Z is a standard normal random variate, 
, and 
denotes the absolute value of z. This Bayesian significance level (or p value) for the null value of , say 
, is defined following Rubin,3 as the posterior probability for the collection of the values that are less likely or have smaller posterior density 
than the null (no change) value . That is, 
. With the posterior distribution of approximately normal, 
is given by the above expression.
