## It’s All Relative: Measuring Abundance In The Face of Detection Bias

### There are many papers out there discussing estimates of abundance and occurrence of a variety of plants and animals. Sometimes you’ll also see references to relative abundance and relative occurrence. What makes researchers go for one estimate over the other? When might you face a similar choice? The goal of this post is to try to shed some light on when you might want to keep things relative.

If your goal is to measure the total abundance of your favorite species, you probably know the drill. You would go out and count every one you saw. If you are looking for something large and not subtle, like say a hippo, then you’d probably be able to spot every one that you walked by, and so you’d feel pretty good about your abundance estimate. But while hippos or elephants might be easy enough to count, if you’re tallying up most other species, what we call “imperfect detection” bias is probably going to come into effect. That is, there is a non-negligible probability that you don’t see every individual you cross paths with. They could be small, they could be hiding, you might have blinked when they passed by.

You could try to use a model to account for imperfect detection by using covariates to help describe your ability to spot the species. For example, if you are roughing it through dense tree cover in the rain close to nightfall, you suspect that you might not be at the peak of your detection game. However, we know that no model is perfect, and we still might be missing something. Like we might know that the size of the individual species impacts whether or not we spot it, but we have no way of incorporating that information into the model. After all, if we happened to have the individual measurements, we would also get the abundance by default.

So what else can we do? We can give up on having an absolute abundance measure and go for a measurement that gives us a general sense of abundance without having a concrete number. A strategy we can use if we are focusing on one species is to come up with some proxy for “search effort” and then judge abundance *relative* to that effort.

Effort could be approximated by something like the time spent at each site looking for the species since with more time spent at a site we’d expect to have a better chance of spotting the species if it was actually there. Effort could also be approximated by the time we’ve been out searching overall since we’re maybe more likely to see a species earlier in the day when we aren’t as fatigued from a day of traipsing through the woods. But let’s be real, imperfect proxies are still a bummer. Are we out of options?

If we happened to be searching for many species at the same time, we could borrow information *across* species to give us another way to estimate relative abundance. If we assume our detection problems are consistent across species, then our relative measure of abundance becomes the proportion of Species A out of all the species that we saw. Or in fact, any other sampling biases that we assume are the same across species we can sweep under the rug by just considering relative abundance. Why is that?

By looking at proportions, and under the right conditions, we get to cancel a lot of the problematic terms in our estimate. Think of any sampling bias as a thickening or thinning of true abundance. In the case of oversampling in places that are easier to get to, we may inflate the true abundance. In the case of imperfect detection; we dampen the true abundance of species k, denoted ak, by a detection probability, denoted p, less than one.

When we collect data about, say, three species, the relative abundance of the first species is then equal to (p * a1 ) / (p * a1 + p * a2 + p * a3) = a1 / (a1 + a2 + a3). We no longer have to worry about estimating the detection probability.

However, we know that not everything goes according to plan in the wild. Can relative abundance still lead us astray? If we look closely at that formula, we reveal a big assumption that we made: the thinning or bias (here we’re considering it to be the detection probability) is assumed to be the same for every species (one p, no pk) so that we can cancel it out in the numerator and denominator. This may be plausible if all of our species are similar. But it would be much harder to learn about how many small lizards we may be missing from how many bobcats we are missing. Their detection probabilities are likely very different, even if we account for differences in size.

Despite its caveats, this multi-species version of relative abundance is the version that is also relevant to biodiversity. If we care about answering questions about the numbers of one species *in comparison to another*, then we are in luck. There are lots of questions we can answer by prioritizing our understanding of the ranking of species with respect to one another rather than worrying about their absolute numbers. For example, is Species A consistently spotted more than Species B?

Can we detect changes over time using abundance and/or relative abundance? We have to make some more assumptions, but yes, it’s possible. In the absolute case we need to model the detection probability (and any other sampling biases) and make sure we can account for any changes in those quantities over time as well. In the relative case, we can still cancel out those thinning parameters, but we need to further assume that as they change over time, they change consistently across species. Whether or not that assumption is defensible is context dependent. For some species, it may well be defensible. Yet other species may become more visible at certain times of the year, for instance if they become more brightly coloured for mating season, or need to venture out into the open more as food becomes scarce.

There are some questions that only absolute abundance can really answer though. For example to make particular conservation decisions, absolute values are needed to make sure species aren’t dropping below abundance thresholds where they change protection status. If we make too optimistic a a guess of our detection probability that turns out to overestimate a species population, the resulting conservation measures might not do enough, and the outcome for that species could be severe.

While we are at it, all of this logic follows for absolute occurrence v. relative occurrence. Estimating absolute occurrence would result in having a calibrated sense of the probability we will encounter a particular species. Estimating relative occurrence would help us understand whether we are more or less likely to observe one species over another.

At the end of the survey day it really depends on what questions you are trying to answer and what assumptions you are willing to make about your survey approach. That’s nothing new. However, by understanding how that one word “relative” impacts the interpretation of your results, we can still make progress under the constraints of the data we have.

Have a quantitative term or concept that mystifies you? Want it explained simply? Suggest a topic for next month →  @sastoudt. You can also see more of Sara’s work at Ecology for the Masses at her profile here.