‘Tis the Season for Modelling Mortalities

Post provided by ELIE GURARIE

A warning: Halloween is nigh, and the following post contains graphic real-life imagery of maggot-eaten eye-sockets and deadly pianos. Read on… if you dare!

A Death in the Woods 

In the vast and often frozen boreal forest of northern Canada there is a slow-burning forensic investigation into a death. The victim: a woodland caribou, an iconic species that is threatened or endangered throughout its range.

Kyle Joly

The scene is very much made for TV neo-Scandinavian neo-noir. From a not-too-luxurious regional office in the town of Fort Smith, just north of the Alberta border, over a steaming cup of coffee, world-weary biologist Allicia Kelly – who’s seen it all and then some – is monitoring the movements of collared animals on her computer screen. It’s the middle of May. The females, nearly all pregnant, are scattering to higher ground to find suitably cozy and secluded sites to calve. All is as peaceful and idyllic as a bunch of blips on a computer screen can be.

But then (cue slightly unsettling dissonance in the soundtrack) one of the little blips seems to have stopped moving. Kelly raises her eyebrow, tells herself to keep an eye out. A moment later she makes the call: “Team, we’ve got another ringer … let’s roll!The music intensifies as Kelly and a crack team of wildlife techs sprint to a helicopter, swing up under the deafening rotors, head out over the dark matchstick spikes of the black spruce into the vast bog-flecked boreal forest. The pilot settles gingerly on a slightly elevated sedge-covered clearing, the team leaps out in rubber boots handheld GPS devices drawn, to find the source of the signal. Once at the spot, they separate, bushwhacking through the undergrowth, searching for evidence.

Crime victim. ©Danny Allaire

There she is!” yells one of the search team. Pushing through a blueberry thicket, Kelly is greeted by the dark stench of death, beneath which lies the body. Arranged as if for ritual sacrifice on a great bed of whitish fur, itself arranged on a bed of yellow needles. The legs neatly folded under the belly, ribs glistening in the summer sun. The GPS collar – hanging loosely over nothing but vertebrae – dutifully transmits its location. The rumen sac – filled with indigestible plant-stuff of no interest to the carnivorous killer – lies a few meters away, as if surgically severed.

Kelly leans forward to investigate the damage, flips the skull over, barely flinches as a thick stream of maggots oozes out of the eye-socket.

There’s no time to be sentimental. A data point, after all, is a data point.

A Silent Demon of Mortality 

Kelly will be quick to point out that some liberties have been taken in this portrayal. It might take a few days, rather than a few moments, to make the call. The helicopter might be a three and a half hour drive – rather than a quick sprint – away, but boy would she kill (not literally!) to have a helicopter in the regional office parking lot. And the crack team might consist of just one (crack) tech.

Maggotty eye-socket. ©Allicia Kelly

But the carcasses are real, and the mystery is real, and the maggots and the smell are very much real. Like many biologists deep in the field Kelly has deep appreciation for the insights provided by any collared animal, and the animal’s sacrifice is never taken for granted. An awful lot, after all, can be learned from a forensic examination of even a single carcass.

But in 14 years of study, the million square kilometres of the Northwest Territories have been slowly sprinkled with 136 recovered carcasses of collared caribou. And – for a species that is as threatened throughout its range – perhaps the bigger question is: What can be learned from a lot of deaths in the woods?

This brings us to a concept that is no less spooky, if much more abstract, than a maggot-filled eye socket: the hazard function. The hazard is a number that chases you at every moment in your life and coolly determines the probability that you’ll die. Think of it as a silent bookkeeping demon of death continuously rolling (hopefully) very asymmetric dice. No matter how small your hazard might be at any given moment, all that exposure to hazard cumulatively and inexorably drives your probability of mortality up and up and up and only up, until that fateful moment when the piano comes crashing down upon your head while walking down the street.

Black Friday is coming up. Beware that deadly killer… the piano! ©Jean and Fred

For all its subtle, incremental labour, the hazard function can be very fragmented, complex and structured. Your getting-crushed-by-a-falling-piano-hazard (a very specific hazard that we all carry with us at all times) has plenty of spatial structure. It’s highest for the residents of the notorious Piano-Packing district with its teetering tall buildings and ziplines connecting factory floors to showrooms. It also has seasonal structure: peaking right around Black Friday, when every piano dealer is offering Incredible-One-Time-Only-Gotta-See-It-To-Believe-It-Deals!!!, and product is moving fast along those dodgy ziplines.

Death Pulses and Unknown Fates

Our goal, essentially, was to figure out when Black Friday happens for the caribou. The challenge is to infer the shape of that invisible hazard function. Note that in the far north everything changes with the seasons. But unlike the daylight (reliably longer in the summer) or temperature (only slightly less predictable), not nearly as much is known about seasonal survival patterns of animals in the wild. Considering the ubiquity of seasonal processes and the importance of survival to ecological studies, we were surprised to find that no handy tools existed to readily characterise seasonal patterns of mortality.

As real as mortality may feel at that moment when the piano’s meeting your head, the hazard function itself is very abstract. It is, for one, impossible to directly measure or observe, it can only be inferred. And to infer a hazard function you need (a) data, and (b) statistical tools.

It is unfortunately not enough to simply look at a histogram of mortality times. Animals are captured and collared at a certain time of the year and the number of mortalities always depends on the number of animals that are still wandering about the landscape. Adding to the bias, collars are programmed to drop off an animal’s neck via timed release, or they stop working, or are lost, or simply run out of battery. There’s still information there – the animal survived at least as long as the collar was transmitting, but the animal’s fate is unknown (or – as these kinds of data are referred to – right-censored).

Sir David Cox, eagerly anticipating more parametric survival modelling.

It turns out, analysis of right-censored survival data in the wildlife sciences is absolutely dominated by the Cox proportional hazard model. This model takes advantage of a tidy mathematical trick to sweep the hazard completely under the rug to focus on comparing the different factors that relatively influence mortality with no regard to the shape – or even magnitude – of the baseline hazard. Unfortunately for us, in all its non-parametric glory, it has nothing to say about seasonal patterns because it truly doesn’t care about it. The Cox model’s progenitor, Sir David Cox himself (whose contributions to survival modelling were apparently significant enough to get knighted), seems to have had rather ambivalent feelings to the “cottage industry” surrounding the Cox proportional hazards model. His understated declaration to “normally [prefer to] tackle problems parametrically” we interpret as a full-throated endorsement of our approach.

Other excellent and well-developed tools in survival analysis do focus on parametric descriptions of hazard functions and how they change in time. A typical application is in disease ecology, where the virulence (or hazard) of a disease might increase rapidly after infection, and then slowly decrease. But for the apparently simple question of identifying an (annual) season of increased mortality, there was nothing to do but go back to the drawing board.

Voldemort’s Little Cousin: Cyclomort

In ‘For Everything There is a Season: Analysing periodic mortality patterns with the cyclomort R package’, our solution was to develop a periodic hazard model and fitting tools. We bundled them up in a (we feel) easy-to-use R package called cyclomort.

It works like this: you provide data (i.e. start times, end times, and the fate of each individual: whether Dead or Fate-Unknown), and a number of seasons to estimate. The function then tells you when the peaks of mortality occurred, how long those mortality ‘seasons’ last, how much of the total mortality each ’season’ accounts for, and what the total average hazard is.

That’s it! Try it! It’s free!

Left plot: NWT mortality data, with grey lines indicating “unknown fate” and black lines represent death (i.e. likely maggotty eye-sockets). Right plot: Cyclomort estimates of seasonal hazard pattern overlain on a histogram of mortalities.

Merry Mortalities!

So, what did we learn about the woodland caribou of the Northwest Territories? Well, there are not one, not two, but three mortality seasons! We found two concentrated peaks on April 21 (Easter Mortality Week!) and July 13 (Bastille Day Mortality Week!), and one more extended one centred on November 8 (Halloween Hangover Mortality Month!) That dramatic dip in late May and early June? Most likely the sudden pulse of calves in the woods absorbing the brunt of the predation? (The Bad Mother Hypothesis.) Or, perhaps, cows are even more focused and skilled at remaining cryptic and hidden as they nurse their newborn calves? (That would be the more generous Good Mother Hypothesis.)

©peupleloup

Given the severity of winter conditions and difficulty of accessing food, it’s striking that the deep winter months (January – mid-March) have the lowest mortality of the year. Ultimately, that’s a testament to the incredible adaptations caribou have evolved for surviving in the North. Perhaps – also – confounded with relatively lower activity from predators and human hunters.

As is so typical in ecological investigations, there are always multiple, interacting explanations for even the most basic observations – a reminder that that the mysteries of the Why’s and How’s of caribou mortality are still out there for biologists like Kelly to unravel. But – just as the GPS blips tell us Where to look – at least with cyclomort, we now know When.

To find out more read our full Methods in Ecology and Evolution article ‘For everything there is a season: Analyzing periodic mortality patterns with the cyclomort R package‘ (No Subscription Required)

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