Here’s an interesting problem involving the time interval between cougar “kills”…meaning cougars killing prey, not cougars being killed. (By the way, “cougar” is synonymous with “mountain lion”, “catamount”, and “puma”. Same animal.) The data I’ll discuss below were collected by Polly Buotte and other researchers guided by Toni Ruth of the Selway Institute, funded by the Hornocker Wildlife Institute and Wildlife Conservation Society.

Cougars in and around Yellowstone National Park are monitored in two ways. Researchers try to put a radio collar on every adult cougar; there are typically about a dozen adult cougars in the park.

Most of the collars used, now and historically, are old-style radiotelemetry collars. These emit a periodic signal that can be used, through triangulation, to determine the approximate location of the animal (spatial error less than 100m). More recently, some of the collars are GPS collars that report the exact location of the animal every three hours. The GPS collars, a new technology, are expensive, relatively short- lived, and somewhat failure-prone.

One of the issues of interest to researchers is the statistical distribution of intervals between kills, called the “inter-kill interval” or IKI. A specific question of interest is the extent to which the IKI distribution has changed due to the reintroduction of wolves to Yellowstone. Some change might be expected because (1) wolves sometimes steal a cougar’s kill before the cougar is done with it, so the cougar might have to kill more frequently to make up for the lost meat, and (2) prey availability might change, as prey change their behavior to try to avoid areas favored by wolves, thus possibly changing the types of prey available to cougars or their density in cougar habitat.

In addition to the statistical distribution of IKI overall and its change since the reintroduction of wolves, a related question of interest is how the IKI differs for different “social classes” of cougars, where “social class” distinguishes adult female, adult male, or maternal female (i.e. female with cubs).

Based on the radio collar data, 121 IKIs were determined for 11 cougars over 8 years. The following figure shows the IKI data for the three social classes, as determined by the two different methods (GPS and “ground”).

With the help of the radio collars, researchers have tried to characterize every cougar kill made by certain cougars during certain time periods. “Characterizing” the kill means determining the date, time, and location of the kill and the type of animal killed: a large bighorn sheep, a young elk, and so on. For the standard telemetry collars, this involves using the collar to track the cougar’s movements; a researcher essentially tracks the cougar every day (without disturbing its behavior) searches locations the day after the cat leaves, and locates the carcass from each kill. This method, which we refer to below as the “ground” method, is very labor- intensive. By contrast, with the GPS collars, the researcher compiles a list of the locations where the cougar spent a substantial amount of time, and visits each of those locations to characterize the kill. (Cougar usually stay on or near a kill for at least 3 days, unless driven off, and are rarely stationary for that long unless they have made a kill). This method (the “GPS” method) is much less time-intensive because the researcher can proceed from kill location to kill location rather than following the cougar.

All of the above is important background to the statistical issue, which is this: both the GPS and ground methods of determining IKI lead to errors. In the case of the ground method, kills can be missed if, for instance, they are made at night when the researcher is not monitoring the cougar’s location. If a kill is missed, then a long IKI will be recorded where two short IKIs ought to be recorded. (For example, suppose a cougar makes a kill on days 1, 5, and 10, but the kill on day 5 is missed. Instead of two IKIs, of 4 and 5 days, the researcher will record one IKI of 9 days).

In the case of the GPS method, a kill can be mistakenly attributed to a cougar. For example, if a cougar comes upon a carcass killed by another predator, it may chase off the other predator and feed. Using the “ground” method, the researcher would be on the spot fairly soon after and might be able to tell that the carcass is too old to have been the prey of this cougar; also, if the predator was another collared cougar, the researcher would already know that the other cougar was already at this spot and will realize or at least suspect that the other cougar made the kill. “Crediting” a cougar with a kill that it didn’t make will lead to two small IKIs where there should be one larger one, the reverse of the problem with the ground data.

The IKI data thus have an unusual type of error. Most of the data have no error at all (to within the desired 1-day resolution, anyway), but some of the data are seriously wrong: some single IKI values determined from the ground method should be replaced by two IKI values (that sum to the original single value), while some pairs of small IKI values determined by the GPS method should be replaced by a single value equal to their sum.

It is possible to get a rough estimate of what error rates might be expected. From the GPS data, about 10% of cougar kills occur at night, and would likely be missed by the ground method. From ground data, about 2.2 percent of kills that would be attributed to a cougar by the GPS method are known not to have been made by the cougar; since these are just the known cases of error, the actual error rate may be higher. These estimated error rates, though not large, are large enough to cause concern when looking for changes in IKI distributions, and particularly for comparing data collected with the GPS method with data collected with the ground method.

I would like to use the “usual methods” for investigating various effects…perhaps linear regression of log(IKI) on indicator variables for social class, detection method, and presence of wolves, perhaps including individual cougar random effects if appropriate. But what to do (if anything) about the peculiar error distribution? If you have any suggestions or you’ve heard of any related work, please mention them here!