Actually I am preparing a paper in which I am using your approach of treating a response on a likert item as if it is the overt aggregate of a covert series of binomial trials.
In my paper the binomial distribution is used in order to explain the shape
of the observed frequency distributions. The rationale behind this approach is given by two assumptions. In many applets, showing how the binomial distribution comes into existence, one has repeated independent Bernoulli trials by a single ball falling through an array of pins. Each time a ball falls onto a pin, it will bounce to the right (i.e. a success) with probability p or to the left (i.e. a failure) with probability 1-p. After the ball falls through the array, it lands in a bin labeled by the corresponding number of successes. In my paper the process of decision making is also seen as a series of repeated independent Bernoulli trials in which, at each trial, the subject decide to agree or not to agree to the statement in question. The two assumptions read as follows.
(i) At each independent Bernoulli trial the subject makes a decision to agree with probability p or not to agree (disagree) with probabiliity 1-p.
(ii) If five categories of response are available for the statement, then the number of times a Bernoulli decision is made regarding the decision to agree or not to agree (disagree) is equal to 4 (5-1).
The final choice for a specific response category is given by the following rules.
If in all (four) cases a Bernoulli decision of agreement is made, then the response 'strongly agree' will be given.
If in three cases a Bernoulli decision of agreement is made, then the response 'agree' will be given.
If in two cases a Bernoulli decision of agreement is made, then the response 'undecided' will be given.
If in only one case a Bernoulli decision of agreement is made, then the response 'disagree' will be given.
If in no case a Bernoulli decision of agreement is made, then the response 'stronglly disagree' will be given.
A similar reasoning can be given using 'disagree' decisions.
In order to obtain a binomial distribution, the scoring of the response categories is as follows.
strongly disagree = 0, disagree = 1, neutral = 2, agree = 3, strongly agree = 4
These two assumptions lead to a binomial distribution for the response frequencies provided that there are no systematic differences between the respondents.
I hope you can agree. I would appriciate very much if you could improve my english in the above text.