# Simultaneous confidence interval for multiple proportions in R

I want to perform in R the analysis "simultaneous confidence interval for multiple proportions", as illustrated in the article of Agresti et al. (2008) "Simultaneous confidence intervals for comparing binomial parameter", Biometrics 64, 1270-1275.

Unfortunately I did not succed to find any R example, and I have no idea how to implement this process in R.

As a case study, I need to apply such analysis on the following simple prolbem:

I did an experiment in which 12 participants had to choose between 3 conditions when provided with 3 stimuli.

Stimulus  Condition1  Condition2 Condition 3
A            9          1          2
B           10          2          0
C            8          2          2


I want to prove that it is not by chance that Condition 1 is preferred rather than the other two conditions.

Can you please give me an example in R on how to perform such analysis?

In addition I would need to know how would you report such analysis in a paper.

I just write a function for the Simultaneous Agresti-Caffo Confidence Interval based on Agresti et al. (2008) "Simultaneous confidence intervals for comparing binomial parameter", Biometrics 64, 1270-1275. I hope this helps

ac.sci = function (n1, x1, n2, x2, cl=0.95, k){
p1 = x1/n1; p2 = x2/n2
p1.tilde = (x1+1)/(n1+2)
p2.tilde = (x2+1)/(n2+2)
Q = qtukey(cl, nmeans=k, df=Inf)
se = sqrt((p1.tilde*(1-p1.tilde)/(n1+2))+(p2.tilde*(1-p2.tilde)/(n2+2)))
L = (p1.tilde-p2.tilde) - Q*se/sqrt(2)
U = (p1.tilde-p2.tilde) + Q*se/sqrt(2)
result = matrix(data=c(p1-p2, L, U), nrow=1, ncol=3)
colnames(result) = c("Diff.P.hats", "Lower", "Upper")
print(result)}


Example 1: Condition1 vs. Condition2 given A, where n1 = sum of Stimulus A + B + C of Condition1, and x1 = frequencies of Condition1 given Stimulus A. Similar to n2 and x2 for Condition2. ac.sci(n1 = 27, x1 = 9, n2 = 5, x2 = 1, k=3) the 95% CI = (-0.391, 0.509), since 95% CI contains 0 thus, we can conclude that Condition 1 is not statistically significant different from Condition 2 at a 5% level.

Example2: A vs. B given conditions, where n1 = sum of conditions 1 + 2 + 3 of A, and x1 = frequencies of A given Condition1. Similar to n2 and x2 for B.

ac.sci(n1 = 12, x1 = 9, n2 = 10, x2 = 12, cl=0.95, k = 3)
the 95% CI = (-0.565, -0.172). Since 95% simultaneous Agresti-Caffo CI does not contain 0. Thus, we can conclude that A and B are different.

• Hi, thanks a lot! Regarding the first example why do you check the frequencies of Condition1 given Stimulus A? I am interested in all the stimuli, not only in the effect of stimulus A. My goal is to know if Condition 1 is better than Condition2 and Condition3, GLOBALLY. Please enlighten me! – L_T Dec 10 '11 at 12:50
• In addition, is prop.test the right function implementing the Simultaneous Agresti-Caffo Confidence Interval method? In what does it differ from your implementation? – L_T Dec 10 '11 at 12:51
• You have three stimulus and these will be an effect on your conditions. You should compare your condition 1 vs. 2 based on Stimulus A, B or C. Do the same thing for Condition 1 vs. 3 and 2 vs. 3. – Tu.2 Dec 10 '11 at 21:13