Testing the Difference between the Highest and Lowest Prevalence Rates in Substate Regions within Each State Based on Data Collected from the 2004-2006 National Surveys on Drug Use and Health

To test the difference between the prevalence rates of two substate regions in a particular State, say region 1 (with the lowest estimate) and region 2 (with the highest estimate), let π₁ and π₂ denote the respective prevalence rates. For testing the hypothesis π₂ = π₁ (or π₂ – π₁ = 0), the difference between π₁ and π₂ is defined in terms of the log-odds ratio (lor) as opposed to the simple difference because the posterior distribution of the lor is closer to a Gaussian distribution than the posterior distribution of the simple difference (π₂ – π₁).

The method used here to test the hypothesis of no difference between two substate regions, based on their 2004-2006 substate small area estimates, is similar to the method used to test the hypothesis of no change between the 2004-2005 and 2005-2006 State small area estimates in the 2005-2006 State small area estimation (SAE) report. For details, see Section A.10 of Appendix A in Hughes, Sathe, and Spagnola (2008).

The lor is defined as follows:

lor = log{[π₂/(1 – π₂)] ÷ [π₁/(1 – π₁)]}.

The above hypothesis, π₂ – π₁ = 0, is now transformed to testing lor = 0. To compute the posterior variance of lor, we define the lor for each of the 1,250 Markov Chain Monte Carlo (MCMC) cycles as follows:

l_i = log{[p_2i/(1 – p_2i)] ÷ [p_1i/(1 – p_1i)]},

where i = 1, 2,…,1,250.

In the definition of l_i above, p_2i and p_1i denote, respectively, the prevalence rates of regions 2 and 1 for the i-th MCMC cycle. An estimate of lor is given by,

= log{[p₂ /(1 – p₂)] ÷ [p₁ /(1 – p₁)]},

where and The variance of is approximated by where

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 = and variance = v.

With the null value of lor = 0, the Bayes p value or posterior probability of no difference is given by p value = 2*P[Z ≥ abs(z)], where Z is a standard normal random variate, and abs(z) denotes the absolute value of z.

Note that the unbenchmarked estimates of the prevalence rates of the 2004-2006 substate regions (Office of Applied Studies, 2008) were used in this significance testing. Testing π₂ = π₁ is similar to testing θ₂ = θ₁, where θ₂ = π₂ + Δ and θ₁ = π₁ + Δ are the respective benchmarked estimates. For details on benchmarking, see Section A.4 of Appendix A in Hughes et al. (2008).

For all States that had aggregate regions defined (e.g., New York and Texas), the prevalence rates for aggregate regions were calculated by aggregating the corresponding substate region estimates. Hence, the aggregate region-level estimates are more centered, unlike the substate region estimates. For this reason, aggregate regions were excluded from the significance testing of the difference between the highest and the lowest prevalence estimates in substate regions within a State.

References

Hughes, A., Sathe, N., & Spagnola, K. (2008). State estimates of substance use from the 2005-2006 National Surveys on Drug Use and Health (DHHS Publication No. SMA 08-4311, NSDUH Series H-33). Rockville, MD: Substance Abuse and Mental Health Services Administration, Office of Applied Studies. [Available at http://samhsa.gov/data/states.htm]

Office of Applied Studies. (2008, June). Substate estimates from the 2004-2006 National Surveys on Drug Use and Health. Rockville, MD: Substance Abuse and Mental Health Services Administration. [Available at http://samhsa.gov/data/substate.cfm]