Methodology Used to Estimate the Bayes Posterior Probability of
No Change between the 2010-2011 and 2011-2012 Small Area Estimates
Comparisons between State small area estimates displayed in Tables 1 to 26 in this document are based on the 2010 through 2012 National Surveys on Drug Use and Health (NSDUHs). The moving average State prevalence estimates for the overlapping 2010-2011 and 2011-2012 time periods were obtained from independent applications of the survey-weighted hierarchical Bayes (SWHB) methodology; that is, the 2011-2012 models were fit independently of the previously fitted 2010-2011 models. This independent analysis approach was followed because there was no desire to revise the previously published 2010-2011 estimates. New predictors were used in the 2011-2012 small area estimation (SAE) models; because of the shift in using 2010 census data in place of 2000 census data, these predictors were different from what were used in the 2010-2011 SAE models.1 The age group-specific fixed predictor variables were defined at five levels (namely, the person level, census block group level, tract level, county level, and State level). Also, each age group model had 51 State-level random effects and 300 "within-State" area-level random effects.
To estimate change in State estimates, let 
and 
denote the 2010-2011 and 2011-2012 prevalence rates, respectively, for State-s and age group-a. 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
, D
where ln denotes the natural logarithm. The p value given is computed to test the null hypothesis of no change (i.e., 
or equivalently 
). An estimate of is given by
, D
where the 
are previously published 2010-2011 State estimates2 and the 
are the 2011-2012 State estimates. To compute the variance of 
that is, 
let 
and 
, then
, D
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 variances used in calculating the previously published 2010-2011 Bayesian confidence intervals and the 2011-2012 Bayesian confidence intervals, respectively.
The correlation between and was obtained by simultaneously modeling the 2010, 2011, and 2012 NSDUH data. Note that these models used the same predictors employed in the 2011-2012 SAE models for all 3 years. This simultaneous modeling approach was adopted based on the results of the validation study3 conducted for measuring change in the 1999-2000 and 2000-2001 State estimates. For this simultaneous model, 4 age groups (12 to 17, 18 to 25, 26 to 34, and 35 or older) by 3 years (2010, 2011, and 2012), that is, 12 subpopulation-specific models, were fitted, each with its own set of fixed and random effects. In this case, the general covariance matrices for the State and within-State random effects were 12 × 12 matrices corresponding to the 12 element (age group × year) vectors of random effects. Note that the survey-weighted, Bernoulli-type log likelihood employed in the SWHB methodology was appropriate for this simultaneous model because the 12 age group × year subpopulations were nonoverlapping. The correlation [
] was approximated by the correlation calculated using the posterior distributions of 
and 
from the simultaneous model.
To calculate the p value for testing the null hypothesis of no difference (
), it is assumed that the posterior distribution of lor is normal with 
and 
. With the null value of (), 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.
