Dear list members,
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.

If I am not wrong the R function implementing the Agresti et al. method is prop.test(). I ask an help because I have some difficulties in reading the output of that function.

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

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

So, I apply the function prop.test(), summing the values of Conditions 2 and 3):

table<-matrix(c(9,3,10,2,8,4),ncol=2,byrow=T)
rownames(table)<-c("stimulusA","stimulusB","stimulusC")
colnames(table)<-c("Condition1","Conditions2and3")

> table
          Condition1 Conditions2and3
stimulusA          9               3
stimulusB         10               2
stimulusC          8               4


prop.test(table)

>     prop.test(table)

    3-sample test for equality of proportions without continuity correction

data:  table 
X-squared = 0.8889, df = 2, p-value = 0.6412
alternative hypothesis: two.sided 
sample estimates:
   prop 1    prop 2    prop 3 
0.7500000 0.8333333 0.6666667 

Warning message:
In prop.test(table) : Chi-squared approximation may be incorrect

I don't understand where I can deduct that Condition1 is more preferred than Conditions 2 and 3. 
Should I simply look at the p-value?

The fact is that  tried with a more extreme example, but the p-value results still above 0.05:
This is the table I used:

> table2
          Condition1 Condition2
stimulusA         12          0
stimulusB         10          2
stimulusC         11          1

> table2<-matrix(c(12,0,10,2,11,1),ncol=2,byrow=T)
> rownames(table2)<-c("stimulusA","stimulusB","stimulusC")
>      colnames(table2)<-c("Condition1","Condition2")
>      prop.test(table2)

    3-sample test for equality of proportions without continuity correction

data:  table2 
X-squared = 2.1818, df = 2, p-value = 0.3359
alternative hypothesis: two.sided 
sample estimates:
   prop 1    prop 2    prop 3 
1.0000000 0.8333333 0.9166667 

Warning message:
In prop.test(table2) : Chi-squared approximation may be incorrect



Could you please enlighten me?


Thanks in advance

   
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