Bayes and the Bees

Over on Mastodon, I was alerted to a paper about bee populations on the prairie.¹ The authors demonstrate that after the prairie is burned, ground-nesting bees move in in greater numbers than in similar patches of un-burned prairie. This is good stuff — it’s another contribution to our growing understanding that wildfires play a constructive role in the ecology of grasslands. (So say both the National Park Service and the Nature Conservancy, among others.)  This paper is very well done. The next time I think my job is tedious and nit-picky, I can think of these researchers examining bees under a microscope to see whether their mandibles are worn down from digging holes.

Figure 3 from Brokaw et al.

I’m writing this post because the graph of their primary result makes me sad. The results they’re showing are for matched sets of burned and un-burned plots at four sites. On the left, graph (a) shows their estimate of the number of active subterranean bee nests. On the right, (b) shows the “effective number of species” calculated by a tricky mathematical process² from the sample of bees they caught.³

The authors want to show that there’s a significant difference between the burned (orange) and unburned sites, but this graph doesn’t do that. The horizontal line indicating the mean of each case is inside the range of the other case.  Putting a couple of asterisks at the top of  plot (a) (which was statistically significant) doesn’t help the visual impact of the overlapping boxes. I guess we could read the text around the figure to find out what tests they used and why they think the difference is significant, but then what’s the figure for? It’s supposed to be worth a thousand words.

Plot of Table S3 from Brokaw et al.

Fortunately, this is an open-access paper and the authors have made their data available. Here are their data, as histograms of the number of nests they found in each of the sixteen samples at each of their four sites. The top row, burned, looks like they might have more nests than the bottom row, especially in the first and third sites (Devil’s Run and Miller), but the differences in sites 2 and 4 are harder to see. The main purpose of statistics is to make sure I’m not fooling yourself when I look at a graph, and this is an ideal place for it.

As it happens, my old computer joined the bleeding Choir Invisible a couple of weeks ago, and I’m just getting the new one properly furnished. I installed Richard McElreath’s “rethinking” package for R last night, so I’m all set to do some statistics.  Let’s see what I can do with Brokaw’s bees.

First, the number of nests they find is an integer. If we can make one more assumption, we can do a lot more with the data on hand. I’m going to say that the chance that a nest appears in a given sample site is independent of whether there’s already a nest there or not. I think this is pretty good. Their sample sites are squares 2-4 feet on a side. Every bee’s nest I ever dug up was a few inches across (the exact sizes were hard to determine while I was running away).⁴ The number of nests maxes out below 16, which means they’ve all got at least a foot between them. I think this is a good approximation.  [Edited to add: The Principal Investigator informs me that the things I dug up are wasp nests, which is indicated by the fact that I needed to run away. Bee nests are only a couple of millimeters across, the bees aren’t harmful, and the assumption is even better than I thought.]

That assumption lets us do a simple model: The number of nests they found is distributed according to a Poisson distribution. Poisson distributions have one parameter, the log of whose mean is a different number for each site, plus a constant for whether the site was burned or not. Prior distributions for all parameters were uninformative gaussians. The burning-constant is assumed to be the same for all sites. Here’s what comes out of the model.

Posterior distribution of the parameter for each site and treatment.

This graph makes it clearer how good Brokaw et al.’s research finding was. There’s no question here — using a Bayesian Poisson regression model eliminates almost all the overlap between burned and unburned results. The burning-constant is +82%, with a 95% confidence interval from +45% to +227%. (Some of those peaks are really broad.) That is, regardless of which site we start at, the number of nests we’d expect to see in a recently-burned site would be 82% higher.

How good is the Poisson assumption? Pretty good, based on just looking at how far apart the nests are. One other argument, though: this model would break down if there were lots and lots of bee nests, but in that case we wouldn’t be worried about whether we were harming bee populations by stamping out grass fires, and nobody would have done these measurements in the first place.

Notes

1. Brokaw, J., Portman, Z.M., Bruninga-Socolar, B. & Cariveau, D.P. (2023) Prescribed fire increases the number of ground-nesting bee nests in tallgrass prairie remnants. Insect Conservation and Diversity, 1– 13.
2. Dauby, G. and Hardy, O.J. (2012), Sampled-based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species. Ecography, 35: 661-672. https://doi.org/10.1111/j.1600-0587.2011.06860.x
3. Disclaimer: I do not completely understand this.
4. ‘He that flies counts every foeman twice’ LR 3.07.029