I'm interested in measuring the diversity of each sample out of many independent samples. I know a priori all possible species that could appear in a sample, and expect the counts of species to be from 0 to about 10, although often times there will be none of a particular species within a sample. For example, some data might look like:
Sample # | Species A | Species B | Species C
--------------------------------------------------
1 0 4 0
2 1 1 1
3 0 1 1
4 2 2 2
5 0 1 0
The most diverse sample here is #4. The least is #5. Theoretically, #5 would also be the least diverse ever encountered (assuming there is at least one of some species in each sample).
I encountered trouble when using Simpson's Index of Diversity since that formula would make the diversities of Sample 2 and Sample 3 both 1.0. Intuitively, with this data, I am confident that #2 is more diverse. This is because I know all possible species that may show up.
I'm relying on this definition: $$ D = \frac{\sum n(n-1)}{N(N-1)} $$
Are there measures of diversity which account for a priori knowledge of possible species (and perhaps their bounds) and expect near-zero counts?