**1. Introduction**

In this chapter, we discuss statistical methods for comparing multiple populations relative to one population (termed "reference"). These types of multiple comparisons commonly arise in behavioral science, for example, when multiple racial/ethnic groups are compared to non-Hispanic (NH) White smokers in terms of tobacco-use-related behaviors [1–4]. When the statistical parameter of interest is the mean difference, the most common study goal is one of the following two goals. **Goal 1** is to detect all significant mean differences among the considered populations (versus the reference population), that is, to draw an individual conclusion

regarding significance of each mean difference. **Goal 2** is to demonstrate that all mean differences among the considered ones are significant. Note that if one assessed Goal 1 and concluded that each mean difference was significant then s/he has (indirectly) assessed Goal 2 as well. Other more intricate study goals, such as the ones arising in pharmaceutical statistics which involve a hierarchical structure among the primary and secondary end points, were addressed elsewhere and are outside of the scope of this chapter [5–11].

We discuss how Goals 1 and 2 can be assessed in a study of racial and ethnic disparities, where Hispanic (H) population and five non-Hispanic populations such as American Indian/Alaska Native (AIAN), Asian (ASIAN), Black/African American (BAA), Hawaiian/Pacific Islander (HPI), and Multiracial (MULT), are compared to non-Hispanic White (W) population in terms of the mean differences.

$$\left(\mu\_{\text{AlAN}} - \mu\_W, \mu\_{\text{AlAN}} - \mu\_W, \mu\_{\text{BAA}} - \mu\_W, \mu\_H - \mu\_W, \mu\_{\text{HP}} - \mu\_W, \text{and } \mu\_{\text{MULT}} - \mu\_W, \tag{1}\right)$$

where µ *AIAN* , µ *ASIAN* , µ *BAA* , µ *<sup>H</sup>* , µ *HPI* , µ *MULT*, and µ*<sup>W</sup>* denote, respectively, the

mean responses for AIAN, ASIAN, BAA, H, HPI, MULT, and W populations. Furthermore, suppose that each positive mean difference in Eq. (1) corresponds to a significant result, for example, the first difference being positive implies that the mean response among AIANs is greater than the mean response among Ws. Then the null and alternative hypotheses corresponding to the *ith* difference, where *i* denotes AIAN, ASIAN, BAA, H, HPI, and MULT, can be stated as

$$H\_{\text{col}} : \mu\_i - \mu\_W \le \mathbf{o} \text{ and } H\_{\text{al}} : \mu\_i - \mu\_W > \mathbf{o}. \tag{2}$$

Finally, let *p i AIAN ASIAN BAA H HPI MULT <sup>i</sup>* ( , , ,, , = ) denote a p-value corresponding to testing *H*<sup>0</sup>*i* versus *Hai* . As a result, we have six pairs of hypotheses and six p-values.

Suppose the overall error rate for assessing Goal 1 (Goal 2) is fixed at α-level. Then to assess Goal 1 (to detect all significant mean differences), we should first rescale each p-value *pi* , for example, via Bonferroni, Holm, or Hochberg approaches [6, 12–14]. This rescaling is essential to control the overall error rate at the nominal α-level. For example, in our case with six null hypotheses, the p-values rescaled via Bonferroni method are given as 6 *pi* (i.e., we multiply each original p-value by six). Second, we compare each rescaled p-value with α. If *pi* ≤α , then we reject *H*<sup>0</sup>*i* and conclude that the *ith* difference is significant (positive); if *pi* >α , then we accept *H*<sup>0</sup>*i* and conclude that the *ith* difference is not significant. As a result, we draw an individual conclusion regarding significance of each mean difference. Alternatively to the above hypothesis testing, one could construct the lower 100 1 / 6 % ( −α ) confidence intervals for the mean differences in Eq. (1) and use the lower bounds to differentiate between the significant and insignificant mean differences; this approach was discussed elsewhere [15].

To assess Goal 2 (to demonstrate that all differences are significant), one can use the Min test that is an intersection-union test [16–21]. The p-value for the Min test, denoted by *p*, is simply the largest p-value among the six individual p-values:

$$p = \max\left\{ p\_{\text{AlAN}}, p\_{\text{AlAN}}, p\_{\text{BAA}}, p\_H, p\_{\text{HPI}}, p\_{\text{MULT}} \right\}. \tag{3}$$

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to 2011 [1].

*On Statistical Assessments of Racial/Ethnic Inequalities in Cigarette Purchase Price among Daily…*

there is at least one insignificant mean difference. Note that we cannot comment on the significance of an individual mean difference, because we tested whether all mean differences are significant (i.e., whether the smallest mean difference is significant). Nonetheless, the Min test is more suitable for assessing Goal 2 than Bonferroni approach or another approach proposed for assessing Goal 1. A statistical method is usually proposed for a specific problem and thus, the methods should be used accordingly: the union-intersection hypothesis (Goal 1) should be tested via Bonferroni or another union-intersection test, while the intersection-union hypothesis (Goal 2) should be tested via the Min or another

Alternatively to the Min test, we can use the Strassburger-Bretz-Hochberg (SBH) confidence interval approach as follows [23, 24]. First, we compute the lower

bounds be denoted as *LAIAN* , *LASIAN* , *LBAA* , *LH* , , *LHPI* and *LMULT* . Second, let *L*

difference is given by (*L*, . + ∞) If *L* > 0 , we conclude that all mean differences are significant, and if *L* ≤ 0 , then we conclude that there is at least one insignificant

We note that one needs to identify the appropriate statistical method to compute the individual p-values and confidence bounds. The choice depends on the study design, probability distributions, and other statistical considerations. The Min test and the SBH interval were discussed for parallel and factorial designs, where sample mean responses followed normal distributions with known variances or unknown (common) variance, as well as Binomial and several other distributions [20, 21, 23–27]. In addition, one needs to decide whether the analyses should adjust for explanatory factors, for example, sociodemographic characteristics [28–30]. Such adjustments may help reduce the effect of confounding factors and therefore, improve estimation [31, 32]. For example, Golden et al. examined how much smokers pay for a pack of cigarettes, on average, in the United States using data from the 2010–2011 Tobacco Use Supplement to the Current Population Survey (TUS-CPS) [1]. Among several design-based multiple linear regression models for the mean purchase price per pack (PPP) used in the study, one model adjusted for smokers' sociodemographic and smoking-related characteristics, cigarette purchase attributes, and the

Despite availability and benefits of the Min test and SBH interval, these methods have not received much attention in behavioral sciences. We illustrate benefits of using these methods over Bonferroni method and simplicity of applications of these methods. We consider a study of racial and ethnic disparities in cigarette purchase prices conducted to demonstrate that W daily smokers, on average, purchase cigarettes at lower prices than do AIAN, ASIAN, BAA, H, HPI, and MULT daily smokers in the United States. This goal was motivated by results of a prior study revealing that BAA, H, and ASIAN/HPI (ASIAN and HPI combined) smokers paid higher PPP, on average, relative to W smokers, in the United States in the period from 2010

) confidence intervals for the mean differences in Eq. (1). Let these

*L L L L LL L* = min , , , , , . { *AIAN ASIAN BAA H HPI MULT* } (4)

) confidence interval for the smallest mean

, then we reject all <sup>0</sup> s *H <sup>i</sup>* and conclude that all mean differences are

, then we fail to reject all <sup>0</sup> s *H <sup>i</sup>* and conclude that

*DOI: http://dx.doi.org/10.5772/intechopen.93380*

α

denote the smallest bound among these bounds, that is,

α

significant (positive); if *p* >

intersection-union test [12, 22].

Then the SBH lower 100 1 % ( −

If *p* ≤α

100 1 % ( −α

mean difference.

survey wave [1].

*On Statistical Assessments of Racial/Ethnic Inequalities in Cigarette Purchase Price among Daily… DOI: http://dx.doi.org/10.5772/intechopen.93380*

If *p* ≤α, then we reject all <sup>0</sup> s *H <sup>i</sup>* and conclude that all mean differences are significant (positive); if *p* >α , then we fail to reject all <sup>0</sup> s *H <sup>i</sup>* and conclude that there is at least one insignificant mean difference. Note that we cannot comment on the significance of an individual mean difference, because we tested whether all mean differences are significant (i.e., whether the smallest mean difference is significant). Nonetheless, the Min test is more suitable for assessing Goal 2 than Bonferroni approach or another approach proposed for assessing Goal 1. A statistical method is usually proposed for a specific problem and thus, the methods should be used accordingly: the union-intersection hypothesis (Goal 1) should be tested via Bonferroni or another union-intersection test, while the intersection-union hypothesis (Goal 2) should be tested via the Min or another intersection-union test [12, 22].

Alternatively to the Min test, we can use the Strassburger-Bretz-Hochberg (SBH) confidence interval approach as follows [23, 24]. First, we compute the lower 100 1 % ( −α ) confidence intervals for the mean differences in Eq. (1). Let these bounds be denoted as *LAIAN* , *LASIAN* , *LBAA* , *LH* , , *LHPI* and *LMULT* . Second, let *L* denote the smallest bound among these bounds, that is,

$$L = \min\left\{L\_{\text{AlAN}}, L\_{\text{ASIAN}}, L\_{\text{BAA}}, L\_{\text{H}}, L\_{\text{HPI}}, L\_{\text{MULT}}\right\}. \tag{4}$$

Then the SBH lower 100 1 % ( −α ) confidence interval for the smallest mean difference is given by (*L*, . + ∞) If *L* > 0 , we conclude that all mean differences are significant, and if *L* ≤ 0 , then we conclude that there is at least one insignificant mean difference.

We note that one needs to identify the appropriate statistical method to compute the individual p-values and confidence bounds. The choice depends on the study design, probability distributions, and other statistical considerations. The Min test and the SBH interval were discussed for parallel and factorial designs, where sample mean responses followed normal distributions with known variances or unknown (common) variance, as well as Binomial and several other distributions [20, 21, 23–27]. In addition, one needs to decide whether the analyses should adjust for explanatory factors, for example, sociodemographic characteristics [28–30]. Such adjustments may help reduce the effect of confounding factors and therefore, improve estimation [31, 32]. For example, Golden et al. examined how much smokers pay for a pack of cigarettes, on average, in the United States using data from the 2010–2011 Tobacco Use Supplement to the Current Population Survey (TUS-CPS) [1]. Among several design-based multiple linear regression models for the mean purchase price per pack (PPP) used in the study, one model adjusted for smokers' sociodemographic and smoking-related characteristics, cigarette purchase attributes, and the survey wave [1].

Despite availability and benefits of the Min test and SBH interval, these methods have not received much attention in behavioral sciences. We illustrate benefits of using these methods over Bonferroni method and simplicity of applications of these methods. We consider a study of racial and ethnic disparities in cigarette purchase prices conducted to demonstrate that W daily smokers, on average, purchase cigarettes at lower prices than do AIAN, ASIAN, BAA, H, HPI, and MULT daily smokers in the United States. This goal was motivated by results of a prior study revealing that BAA, H, and ASIAN/HPI (ASIAN and HPI combined) smokers paid higher PPP, on average, relative to W smokers, in the United States in the period from 2010 to 2011 [1].

*Recent Advances in Numerical Simulations*

outside of the scope of this chapter [5–11].

µ

eses and six p-values.

lower 100 1 / 6 % ( −α

where µ *AIAN* , µ*ASIAN* ,

 µµ

regarding significance of each mean difference. **Goal 2** is to demonstrate that all mean differences among the considered ones are significant. Note that if one assessed Goal 1 and concluded that each mean difference was significant then s/he has (indirectly) assessed Goal 2 as well. Other more intricate study goals, such as the ones arising in pharmaceutical statistics which involve a hierarchical structure among the primary and secondary end points, were addressed elsewhere and are

non-Hispanic White (W) population in terms of the mean differences.

 µµ

mean responses for AIAN, ASIAN, BAA, H, HPI, MULT, and W populations. Furthermore, suppose that each positive mean difference in Eq. (1) corresponds to a significant result, for example, the first difference being positive implies that the mean response among AIANs is greater than the mean response among Ws. Then the null and alternative hypotheses corresponding to the *ith* difference, where *i*

*H H* <sup>0</sup>*ii W* : 0 and : 0.

corresponding to testing *H*<sup>0</sup>*i* versus *Hai* . As a result, we have six pairs of hypoth-

rescale each p-value *pi* , for example, via Bonferroni, Holm, or Hochberg

Suppose the overall error rate for assessing Goal 1 (Goal 2) is fixed at α-level. Then to assess Goal 1 (to detect all significant mean differences), we should first

approaches [6, 12–14]. This rescaling is essential to control the overall error rate at the nominal α-level. For example, in our case with six null hypotheses, the p-values rescaled via Bonferroni method are given as 6 *pi* (i.e., we multiply each original p-value by six). Second, we compare each rescaled p-value with α. If *pi* ≤

 , then we accept *H*<sup>0</sup>*i* and conclude that the *ith* difference is not significant. As a result, we draw an individual conclusion regarding significance of each mean difference. Alternatively to the above hypothesis testing, one could construct the

To assess Goal 2 (to demonstrate that all differences are significant), one can use the Min test that is an intersection-union test [16–21]. The p-value for the Min test, denoted by *p*, is simply the largest p-value among the six individual p-values:

) confidence intervals for the mean differences in Eq. (1) and

*p p p p pp p* = max , , , , , . { *AIAN ASIAN BAA H HPI MULT* } (3)

we reject *H*<sup>0</sup>*i* and conclude that the *ith* difference is significant (positive); if

use the lower bounds to differentiate between the significant and insignificant

mean differences; this approach was discussed elsewhere [15].

−≤ −> *ai i W*

( , , ,, , = ) denote a p-value

µ µ

denotes AIAN, ASIAN, BAA, H, HPI, and MULT, can be stated as

µ µ

Finally, let *p i AIAN ASIAN BAA H HPI MULT <sup>i</sup>*

 µµ

µ *BAA* , µ *<sup>H</sup>* , µ *HPI* , µ

We discuss how Goals 1 and 2 can be assessed in a study of racial and ethnic disparities, where Hispanic (H) population and five non-Hispanic populations such as American Indian/Alaska Native (AIAN), Asian (ASIAN), Black/African American (BAA), Hawaiian/Pacific Islander (HPI), and Multiracial (MULT), are compared to

> µµ

*MULT*, and

− − −− − − , , , , ,and , *AIAN W ASIAN W BAA W H W HPI W MULT W* (1)

 µ

µ

 µ  µ

*<sup>W</sup>* denote, respectively, the

(2)

α, then

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*pi* >α
