**4. Conclusion**

In our study, results of the Min test (and SBH interval) were different from the results of the Bonferroni method. Specifically, using the Min test (and SBH

*Recent Advances in Numerical Simulations*

interval reported in the output.

**Figure 1.**

*difference is \$0.61.*

from 2010–2011 to 2014–2015.

"upper," and "alpha = 0.05" options) when fitting the model using SAS software. Alternatively, we could use the *lsmeans* statement (with "adj = bon," "cl," and "alpha = 0.1" options), and select the comparisons of interest out of all 21 pair-wise comparisons reported and note the lower bound of the two-sided 90% confidence

*Individual lower 95% confidence intervals for the mean price per pack differences relative to non-Hispanic (NH) White daily smokers; the lower number corresponds to the lower bound and the upper number corresponds to the point estimate for the mean difference. For example, AIAN daily smokers, on average, pay at least \$0.39 more per pack of cigarettes than do NH White daily smokers, and the point estimate for the* 

**2.2 Demonstrating the study goal via the min test and SBH confidence interval**

The p-value for the Min test is *p* = 0.0087 , indicating that at 5% significance level we reject the null hypothesis in favor of the alternative. The corresponding SBH lower 95% confidence interval bound for the mean PPP difference is \$0.08 (see **Figure 1**). Therefore, all six racial/ethnic groups of daily smokers paid, on average, higher PPP relative to W daily smokers in the United States in the periods

If instead of the Min test we used the Bonferroni approach, then the adjusted p-values would be less than 0.0006 for four comparisons (AIAN versus W, ASIAN versus W, BAA versus W, and H versus W), 0.0012 for one comparison (HPI versus W), and 0.0522 for one comparison (MULT versus W). Therefore, we would conclude that only AIAN, ASIAN, BAA, H, and HPI daily smokers pay higher PPP, on average, than do W daily smokers; and would fail to demonstrate that all six considered racial/ ethnic groups of daily smokers pay higher PPP, on average, relative to W daily smokers.

The choice of the reference group as "W daily smokers" was based on the study goal and prior studies of cigarette purchasing behaviors of smokers [1, 33]. The

**170**

**3. Discussion**

interval), we demonstrated that all six racial/ethnic groups of daily smokers paid, on average, higher PPP relative to W daily smokers in the United States in the periods from 2010–2011 to 2014–2015. However, using the Bonferroni method, we failed to demonstrate this claim. This discrepancy highlights the importance of choosing the appropriate statistical method for assessing the minimum among multiple mean differences (relative to one reference population). Availability of the "Min test" and "SBH interval" procedures in survey packages would help facilitate application of these methods in behavioral research.
