I have a population of animals that were categorised into 3 groups (beginning, continuous and later). The assignment to each of the groups results from intrinsic characteristics and not by chance (e.g. -sorting a population by female, male and other - a certain individual does not end up into the group female by chance, it will end up there if it is indeed a female and not on the others). This is also not the result of a sampling method, it is more like a census. The number of individuals that ended up assigned to each of these groups is different. How do I test that these differences among them are significant? Do I use a test of proportions? I guess, assuming that without any difference among the groups, each would have a 3rd of the total (population). Or do I even have to make a statistical test? Descriptive statistic suficies?
*Example
Beginning: 27
Continuous: 13
Later: 4
Thank you
A) You can do a hypothesis test against the null hypothesis that the counts would be equally distributed (1/3, 1/3, 1/3). As Blaine Tomkins suggests, this can be done by a chi-square goodness-of-fit test. There are also alternatives to this test that would be similar in purpose, like a chi-square goodness-of-fit test by Monte Carlo simulation, or an exact multinomial test.
B) You can also do a hypothesis test against different theoretical proportions. For example, if in theory the distribution would be (50%, 35%, and 15%). You would use the same sort of tests for this scenario.
C) A plot of the counts or proportions is pretty clear way to represent the results. Here is a plot where I compare observed proportions of participants by racial categories against theoretical proportions.
https://rcompanion.org/handbook/images/image300.png
D) A different approach is to calculate the multinomial confidence intervals for the observed proportions. There are specific methods for the multinomial case (for example, the Sison-Glaz method, or see the options for "method" and References at the following link.
https://www.rdocumentation.org/packages/DescTools/versions/0.99.45/topics/MultinomCI
This allows you to put confidence interval bars on your plot.
Here, I have a plot of some proportions of the color of nail polish on women’s toes from an observational study. (Not my data !!! ) With bars indicating the multinomial confidence intervals.
https://rcompanion.org/handbook/images/image093.png
If you are interested, these approaches are shown in more detail at the following link, with the caveat that I wrote it.
https://rcompanion.org/handbook/H_03.html
Thank you very much for all the detailed suggestions Salvatore!
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