Statistical Test for one-sample, multiple choice option from a survey I have conducted a survey where 410 people were given 14 brands. Of those 14, they were asked, "Which brand do you consider to be the "#1 Trust Brand for Protection". Respondents could only choose one brand for their answer.
I have the frequency and proportion results for the brands selected (using Stata)
      Choice           Freq.    Perc.       Cum.
      Brand 1|         70       17.07       17.07
      Brand 2|         64       15.61       32.68
      Brand 3|         62       15.12       47.80
      Brand 4|         46       11.22       59.02
      Brand 5|         27        6.59       65.61
      Brand 6|         25        6.10       71.71
      Brand 7|         24        5.85       77.56
      Brand 8|         22        5.37       82.93
      Brand 9|         18        4.39       87.32
    Brand 10 |         15        3.66       90.98
    Brand 11 |         14        3.41       94.39
    Brand 12 |          9        2.20       96.59
    Brand 13 |          9        2.20       98.78
    Brand 14 |          5        1.22      100.00

I am tasked with determining if Brand 1 is statistically significantly higher than Brand 2 (and then eventually all other individual brands).
I have made each Brand its own binomial variable (i.e. 0/1 for each brand, if selected)
Here are my constraints for identifying the appropriate statistical test:

*

*There is only one "group" (the n=410 sample). I'm not testing between any different groups

*I know that this is a one-sample test, because it's the same respondent across the variables.

*The test has to be appropriate for binomial variables (i.e. Brand 1 select 0/1).

I feel like I can't do a one-sample proportion test (such as prtest Brand1 = 0.0714 ) because I know the brands are NOT equally likely to be selected, since more popular brands are more likely to be chosen (I have a mix of popular and unknown brands).
I have followed this website (https://www.ssc.wisc.edu/sscc/pubs/sfs/sfs-prtest.htm) and essentially did prtest Brand1 == Brand2), knowing that this really isn't 2 sample. Is that proper?
I don't think a t-test is right because the brand variables (0/1) aren't independent (since they come from the same respondent). I don't think a paired t-test or McNemar is right (no before/after manipulation) or a chi-square is correct (such as tabulate Brand1 Brand2, chi2) because I don't have an expected distribution or outcome.
Can anyone help me understand what indeed is the right statistical test? I'm stuck because I don't have an expected frequency or proportion.
 A: Here is a Bayesian approach that uses a posterior-predictive simulation:

*

*Estimate a binomial parameter for all 14 brands, i.e. for each brand, 410 is the number of trials and column "Freq" is the number of successes.  Using the conventional Bayesian conjugate-prior approach, binomial parameters will be Beta distributed.


*In cases where credible intervals do not overlap, you are done:  The results are statistically significant.


*For each pair of brands where the credible intervals do overlap, run the following posterior-predictive simulation, i.e...

*

*Combine the observations for the two brands into a single aggregate category, and calculate its binomial parameter.

*Use the results to specify a posterior-predictive distribution of beta-binomial form (note this is one of the few cases where the posterior predictive distribution can be calculated in closed form).

*Now, generate random pairs of simulated observations from your beta-binomial posterior-predictive distribution.  Compare the difference (or ratio) of the simulated results to the difference (or ratio) of the observed result, and determine in what percentage of cases the simulated result is larger.  That is your measure of statistical significance.



