# One-subject categorical frequency data

#### Ben

##### New Member
Hello all.
I hope this is a simple question.

Here's the basic idea:
On each trial a child gets a choice of 12 types of candy.
She can only pick one.
After she eats it, she gets the same choice with the same 12 types of candy.

She completes 20 trials a day.
She works for 10 days.

What type(s) of candy does she prefer?

My understanding:
I would like to know for each category (type of candy) whether or not it was preferred above chance.
I am under the impression that I cannot do a normal one-sample t-test for each one because the categories are dependent on each other.

I am also under the impression that I cannot do a chi-square test because the frequency scores are not from individual subjects and thus violate the assumption of independence.
Also, I think chi-square would only tell me if the distribution of choices were evenly distributed, but would not tell me which particular category (or categories) were preferred.

I'm sure there must be a simple statistical test for such a simple food preference task, but I don't know what it is.
I hope somebody here can help me out.

Thank you very much in advance.

--Ben

#### JohnM

##### TS Contributor
I'm thinking a chi-square test of "goodness-of-fit" would work here.

If there are 12 types, and each child has 200 "trials," then if the choices were random, you would expect each type to be chosen 200/12 = 16.67 times, on average (i.e., if they were choosing the types out of a bag without being able to see them) - this would be a uniform distribution.

Just test the actual frequencies of the choices vs. the uniform distribution...

#### Ben

##### New Member

I thought the chi-square test would only give me information about whether or not the distribution of frequencies was evenly distributed.
Are you saying that it will also tell me which particular choice or choices were preferred above chance?
Or is there some sort of post-hoc/planned comparison test that I have to use after the chi-square?

#### JohnM

##### TS Contributor
What you say about the goodness-of-fit test is true. Now that I've read your post more carefully, it's clear that you want more than just a general comparison between the actual choice distribution and the uniform distribution.

You could compare individual choice categories (% of time they're selected) vs the uniform (random) % of 12/200 = 0.06. This would be done as a test of the actual proportion vs a standard of 0.06.

This link explains how to do the test:

http://davidmlane.com/hyperstat/B71928.html

#### Ben

##### New Member
Thanks a lot for the great link.

A comparison between the percentage chosen and random chance for each category would be a great solution.

Does it matter that the score in one category is dependent on all the other categories (i.e. - because there is only one individual with a fixed number of choices, an increase in frequency for one category will cause a decrease in one or more other categories)?

#### JohnM

##### TS Contributor
I don't think so - the same problem applies to the "grab bag."