# Creating a probability distribution that is truncated skewed

I have a dataset I want to use to generate a probability distribution. The distribution is skewed and can only include positive integers. I've tried normal (both skewed and truncated, although I can't figure out how to try both at once using R), poisson, and gamma. I've also plotted it on a Cullen and Frey graph (below). Gamma is by far the best fit of these. Is there an option with a better fit than a gamma distribution or way to find a better option? I'm concerned because the gamma distribution doesn't take into account the requirement that the data be positive integers. Thanks in advance.

These are my data (number of species per genus in Catarrhini, if that helps at all): c(16,5,2,1,4,2,1,1,1,1,19,5,5,8,1,10,2,4,1,1,7,3,1,2,2,2,1)

Edited for tags.

You have count data. The gamma distribution is continuous. As your question anticipates, it would not normally be the case that one would use a continuous distribution to describe discrete data. You can of course discretize continuous distributions on $(0,\infty)$ in various ways to achieve a distribution on the positive integers (such as round up, or round off and then truncate, for example).

It would be more typical to first attempt to describe discrete data with a discrete distribution. However, it's not clear to me that one should expect "number of species per genus" to follow any of the standard discrete distributions; it seems more like some form of birth-and-death process (new species arise -- "birth", and go extinct -- "death"), and surely something like that kind of understanding would inform the choice of distribution, if a specific distribution were actually needed.

It may be that some other process (perhaps something such as a Chinese restaurant process) might happen to give suitable approximations in some situations, but I'd be inclined to start with a natural model like a birth-death process. I don't know a stationary distribution for that (though I think recursion formulas exist for the pmf), but it at least gives a starting point for simulations from which more suitable distributional choices might arise.

Failing that, I'd have started with a heavy-tailed discrete distribution on the positive integers, like a logarithmic series distribution (which has been used for a host of biological problems such as relative species abundance and even species per genus**) or perhaps a zero-truncated negative binomial$^\dagger$.

$\dagger$ [Note that the logarithmic series arises as a special case of the truncated negative binomial, so these aren't really separate suggestions]

Is there a reason you need to identify a distribution at all?

** e.g. see

Williams, C.B. (1964),
Patterns in the balance of nature,

which discusses its use for modelling species per genus and genera per sub-family among other uses.

In response to the revelation of the existence of explanatory variables (rather than just building a univariate model as the question seems to suggest):

If you have several explanatory variables, that's a really important thing to mention; it does give some reason to look at distributional models. But if you do have such IVs for your observations, it means you can't look across observations that might have different values for the IVs in order to try to figure out what the distribution is -- instead of the conditional distribution which is what matters for the model, you'd be looking at the marginal distribution (essentially a mixture of those conditionals). The two may look nothing alike. This means that an exercise akin to the one you carry out in your question is misguided -- if the IVs in your model have any value, consideration of the marginal distribution in order to try to choose a model for the conditional distribution may be pretty much useless.

Incidentally, if you still conclude a logarithmic series distribution is suitable, it appears to be exponential family, so it should be possible to do a generalized linear model to include your covariates. (Actually the R package VGAM looks like it can fit a glm with this distribution, via vglm using the logff family. The default link there is logit, but other links are possible.)

• Thanks a lot for the reply. I want the distribution because I want to compare them for different extant and fossil taxa. Catarrhini is the first one I'm trying. Birth-death process makes sense, thank you. I'll see if I can make it work. – Jeff Aug 7 '15 at 11:17
• In the end you'll be comparing estimates of distributions (or more likely estimates of parameters) -- and those estimates based on the samples. Which means you'll ultimately be comparing sample statistics. Which in turn suggests that rather than trying to guess at some distribution and be stuck with whatever sample statistics that leads to, why not simply choose to compare a sample statistic (or several) that will be meaningful to your actual research question? That is ... deliberately compare things that make sense, rather than being led by accident of choice to compare things that may not. – Glen_b Aug 7 '15 at 20:37
• Part of the question I'm interested in involves making slightly more educated guesses at where holes exist in the fossil record. I know it will be rough but the point is to be able to update the model as new data comes in. I assumed I'd need a distribution for that. Is that wrong? Also, the log series distribution seems to work very well, thank you. – Jeff Aug 7 '15 at 22:31
• Also so far I'm having trouble doing what I want to do with the birth-death process, but I think that's because I'm not familiar with them. I do think there is potential there so thank you for that suggestion as well. – Jeff Aug 7 '15 at 22:46
• It's not clear what you're doing to "identify holes" but I expect you don't necessarily need a distribution to do it. When you say "update the model" ... what model is that? On the other hand if the logarithmic series distribution works well for you, who am I to argue? – Glen_b Aug 7 '15 at 23:25