Simultaneous Confidence Intervals for Comparing Binomial Parameters

Abstract

To compare proportions with several independent binomial samples, we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic. It applies to a variety of measures, including the difference of proportions, odds ratio, and relative risk. For the odds ratio, a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the Bonferroni implementation of Wald or "exact" small-sample pairwise intervals. It performs well even for the problematic but practically common case in which the binomial parameters are relatively small. For the difference of proportions, the proposed method has performance comparable to a method proposed by Piegorsch (1991, Biometrics 47, 45-52).

Clinical Trial Registration in Oral Health Journals

“…Second, we calculated pairwise comparisons between pairs of proportions with correction for multiple testing, as well as the corresponding 95% confidence intervals (CIs) for the difference in proportions (Agresti et al 2008). Analyses involved use of the R software (R Foundation for Statistical Computing, Vienna, Austria).…”

The optimal method to make inferences about a linear combination of proportions

“…Historically, literature in this field has paid special attention to the case of K2 (which contains the cases with one proportion and the difference or ratio for two proportions), but there is increasing interest in the case of K>2 (Newcombe, 2001;Price and Bonett, 2004;Schaarschmidt et al 2008;Tebbs and Roths, 2008;Agresti et al, 2008;Zou et al, 2009 andMartín et al, 2010). The linear combination L may be a contrast ( i =0), in which case it is usually interesting to carry out the test for H: L=0 or to determine a confidence interval for L, or may not be ( i 0), in which case it is usually interesting to determine a CI for L; therefore this article has concentrated on the diverse procedures to carry out the test H: L= vs. K: L or to obtain a CI for L through inversion of the previous test.…”

“…When K=2, there may be several objectives: the difference between the two proportions if  1 =1 and  2 =+1 (as in Agresti and Caffo, 2000); the sum of two proportions if  1 =+1 and  2 =+1 (as in Pham-Gia and Turkkan, 1994); the ratio  of two proportions if  1 = and  2 =+1 (as in Agresti, 2003); or a linear combination of two proportions with  1 <0 (as in Phillips, 2003). Cases with K>2 are historically rather less frequent, although in recent years they have received more and more attention due to their great practical interest (Newcombe, 2001;Price and Bonett, 2004;Schaarschmidt et al 2008;Tebbs and Roths, 2008;Agresti et al, 2008;Zou et al, 2009 andMartín et al, 2010).…”

A Smooth Bootstrap Procedure towards Deriving Confidence Intervals for the Relative Risk

“…Furthermore, note that the 95% Wilson score intervals for π 1 and π 2 given 0/3 and 0/4 are ${\mathrm{CI}}_{{\pi }_1}=(0,0.69)$ and ${\mathrm{CI}}_{{\pi }_2}=(0,0.60)$, respectively. A reasonable confidence interval of RR = π 1 / π 2 would be approximated to $$\left(\frac{\text{lower}\text{bound}\text{of}{\text{CI}}_{{\pi }_1}}{\text{upper}\text{bound}\text{of}{\text{CI}}_{{\pi }_2}},\frac{\text{upper}\text{bound}\text{of}{\text{CI}}_{{\pi }_1}}{\text{lower}\text{bound}\text{of}{\text{CI}}_{{\pi }_2}}\right)$$after certain multiple testing adjustment; see more details about constructing simultaneous confidence intervals for binomial proportions in Agresti et al (2008). Towards this end, consider the third set of data with $true{\sum }_{i=1}^{n_1=12}{x}_i=1$ and $true{\sum }_{i=1}^{n_1=15}{y}_i=1$ events in each group, the 95% Wilson score intervals for π 1 and π 2 are ${\text{CI}}_{{\pi }_1}=false(0.0044,0.40false)$ and ${\text{CI}}_{{\pi }_2}=false(0.0035,0.34false)$, respectively.…”