Analysis method for count data with unequal sample sizes and categorical predictors?

Question

I have a problem I've been going over and over for months to find the right statistical analysis method. I'm planning to execute the analysis in R, so any mention of appropriate packages is also appreciated. I appreciate any advice anyone can offer. Here's an analogy to my data.

Consider that I have 5 colleges. My hypothesis is that one particular college has awarded more degrees than the other colleges. So I want to test whether the number of degrees awarded by the different colleges differs significantly. My data includes 3 data sets as follows:

1)The total number of degrees earned by students from each college, since the opening of each respective college. This is not a random sample, but the actual count. All colleges have different opening dates, so they differ in age. I do have the age of each college if it turns out to be relevant. This data includes second, third, and fourth degrees earned by individual students in the count. So the data includes two columns (college name, and number of degrees awarded), and 5 rows (one for each college).

2)The number of degrees per student per school. I have the list of about 5000 students in the total data set, and the number of degrees each student obtained at each school. Some students attended more than one of the schools, and many earned degrees from more than one school. Every degree ever awarded by these five colleges is represented in this data set as well. This data set does not include students who attended, but never earned a degree at these five colleges. This data set has six columns (one for student ID, and five for college name), and about 5000 rows (one for each student). The totals for each college column are equal to the values in the first data set.

3)The total number of students attending each school over the 30 year period, including those who did not earn a degree. The total number of students, of course, varies at each school. So five rows (one for each college) and two columns (college name, and total students attended in the history of the college).

So I have count data with unequal sample sizes. My categorical predictor needs to be the college, as that's what I'm interested in addressing.

I've considered different types of regression models, but I'm not sure if these would be most appropriate. For a regression model I guess my response would simply be the number of degrees for each school. Or maybe the proportion of degrees with respect to the total students per school. Would this be sufficient to correct for the unequal sample sizes?

Any thoughts on which regression model would be most appropriate? Binomial doesn't work, because I need to take into account students with more than one degree, so it's not a binomial response. I'm not sure what type of distribution should be assumed, as picking one to test against seems very arbitrary to me.

Would some type of contingency table be more appropriate?

Thanks in advance for any suggestions you can offer.

Answer 1

If you are interested in the proportion of students who earn at least one degree from school $y$, among those students who enroll at school $y$, then logistic regression is probably the best way to go. The only snag is that you would have to reduce the numerator for each school to exclude those people who earned more than one degree from that school. That is not difficult: the number of over-counts can be easily computed from your second table.

If the question of interest really does require you to count individuals earning multiple degrees from one college with corresponding multiplicity, then you are right that a binomial based method is inappropriate. In this case, I think that so long as none of the cells are underpopulated a $\chi^{2}$ goodness of fit test should work, using the $2\times 5$ contingency table generated directly from your first and third datasets.

(From the way that you have phrased your hypotheses, I don't think that the possibility of individuals moving between colleges is really relevant to the question of interest.)

However, if you want to know whether admission to college $y$ is associated with the completion of any advanced degree(s), whether at $y$ or not, then I think that you have a problem. You have no way of linking students who attend $y$ but do not graduate there to the schools at which they subsequently graduate from elsewhere. If such students are very rare - i.e., if the vast majority of students who enroll at each college do subsequently graduate from that college, then this might not matter much. But otherwise, I think that this version of the question cannot be answered from the data you have.