# Appropriate use of the binomial test to compare expected and actual selection frequencies?

We have experimental data of food selection frequencies among a population of primates across a series of 15 consecutive trials. Individually in each trial, each subject was presented with two food items simultaneously (Food A and Food B) and their selection was noted. Only one of the two foods could be selected, although rejections (immediate drops without consumption) and null selections (nothing selected) were recorded. Therefore, we considered four outcomes: Food A, Food B, rejection, null.

Our null hypothesis is that the each food type would each be selected half the time. I was thinking of using a binomial test to compare actual frequency of Food A to the expected frequency of Food A, and doing the same for Food B, in order to see if there is a significant difference between actual and expected frequency for each. In doing so, I would treat the selection of the food of interest as a success (e.g., Food A), and any other behavior (reject, null, or selection of other food) as a failure (e.g., not Food A). In this way, we have a dichotomous outcome and the use of the binomial test seems to make sense.

Question 1: Does this seem like an acceptable approach? Or is this an inappropriate use of the binomial test because I would be treating null, rejects, and the "other" food selection as a single outcome? If not appropriate, is there a different statistical test I could use to compare expected selection frequency of each food to its actual frequency?

Question 2: My sample contains 10 primates. Is it appropriate to aggregate selection frequency across subjects when conducting the binomial test?

• Yes, a binomial test would be suitable in this situation. In the case of nulls and rejects, you should eliminate those observations from the sample.
– Remy
Commented Mar 23, 2018 at 20:27
• Agree about elimination of 'null' and 'reject' responses. Details of a possible binomial test in my Answer. Commented Mar 23, 2018 at 22:58

Unless 'null' and 'reject' occur frequently, it seems better to drop those responses and make the question about a binary choice between A and B.

Say you have 12 subjects who gave unequivocal A or B answers. Let $X$ be the number of subjects out of 12 that selected A. Then the only responses that reflect a significant difference are $X = 0, 1, 2$ indicating a preference for B and $X = 10, 11, 12$ indicating a preference for A.

Under the null hypothesis that A and B are equally likely to be chosen $P(3 \le X \le 9) \approx 0.96.$ So the significance level of the procedure I describe is about 4%.

The computation in R statistical software is:

sum(dbinom(3:9, 12, .5))
## 0.9614258


Note: For your second question, with $n = 10$ definitive selections of A or B, only results $X = 0,1$ and $X = 9,10$ lead to rejection. The logic is similar. The significance level is about 2%. [Including 3 or 8 as significant results would lead to a significance level above 10%, which does not seem persuasive.]