Concept
Counting or trying to observe the presence/absence of unmarked animals during surveys can often seem like it’s a unique study design but it’s maybe more helpful and correct to think about it as an extension of capture-recapture models. Historically, counts or the presence/absence of animals has been treated as if animals were perfectly observed. This is incorrect because you do not detect all individuals, just like you do not capture every individual in a capture-recapture study. When making the jump from physically capturing an animal to observing or counting them, (re)capture probability is no longer a relevant term or parameter. Instead, the new parameter is called detection probability and it attempts to account for imperfect detection. It does function in a manner similar to capture probability but it starts to encompass more “things” beyond what capture probability does. Detection probability is really a composite of several processes. They are classified into one or two (or three) groups that are recognized as combining to affect detection probability: availability and sightability. You may then see notation in some articles that present detection probability, p_., as a product of the two processes: p_.=p_{\ availability} \times p_{\ sightability}
Availability
Temporary Emigration
Availability is not unique to detection probability or unmarked sampling. For capture-recapture of marked or unmarked individuals, it can refer to two separate processes. The first way is being physically present. Just like capture probability only applies to or describes the probability of capture for animals that are available to be captured, detection probability only applies to individuals that are available to be detected. An exaggerated example in capture-recapture could be if you are trying to capture bears using culvert traps, during the winter. Most bears are hibernating and would be unavailable to be captured. When using something like a trap grid, this typically means that only animals with home ranges inside your trapping grid are considered available.
An animal that is not present, because they’re not inside the study area, hibernating in a den, etc., has a detection/capture probability of 0. More commonly, whether or not an animal is available varies. If an animal is moving in and out of the study area during surveys or captures, their capture/detection probability changes between p and 0. They are sometimes available and sometimes not. This process is often called temporary emigration. This is most often associated with physical presence in a study area, but need not always be. The best way to differentiate this form of availability from the next is recognizing that availability has two discrete states: \left\{ \begin{aligned} unavailable: p=0 \\ available: p>0 \end{aligned}\right\} This form of availability applies both to traditional or physical capture-recapture methods as well as surveys of unmarked individuals.
Behavior
The second form of availability (and possible third group) is related to animal behavior, not their physical presence; is the animal behaving in a way that makes them observable? Unlike temporary emigration, these differences don’t always result in a bifurcation of detection probabilty (i.e., 0 or p > 0). For example, calling by frogs is sensitive to precipitation and temperature. Trying to perform counts of frogs during drought or cold weather will likely have fewer individuals available to be observed than during warmer and wetter weather. Availability in this second form is more of a continuous scale, and not a discrete yes/no like with temporary emigration, but an animal must still be behaving in a way that makes them available in order for an observer to detect them (Farnsworth 2002)1.
Sightability/ perceptability
The last component of detection probability is sightability or perceptability (not all detections are visual). Sightability is often affected by attributes and characteristics of the observer or factors that affect the ability of an observer to detect an animal, given the animal is present and behaving in a way that makes the animal observable (singing, strutting, not hibernating, etc.) Differences in hearing, eyesight, or awareness among observers, distance between an observer and an animal, or ambient road noise during a bird survey or shadows during an aerial survey are all part of sightability process, but not availability. Generally, this is unique to studies where animals are not physically being captured, as the “observer” in traditional capture recapture is usually a net or trap, not a person.
Study Design
Given the above issues, how do you deal availability and sightability when accounting for imperfect detection in your surveys? You do so by carefully selecting and implementing a specific study design that allows you to at least try to account for the various sources of imperfect detection. In general, this means recording specific information at the visit, or increasing the number observers, sampling visits to the same site, or both. It’s rare and very difficult to model all the above components separately. Most focus on only one or maybe two sources, and may not uniquely model each process separately. Most often, behavioral availability and sightability are estimated jointly as…..detection probability.
Temporary immigration
If temporary immigration occurs in either traditional capture-recapture or unmarked methods, the only certain method of dealing with it is via the robust design (Pollock 1982)2. Under this framework, you perform a simple capture-recapture framework with >1 capture opportunity where closure is maintained (secondary occasions) multiple times (primary occasions). In typical robust design methods,you might repeat 3 consecutive nights of surveys (secondary) on three consecutive weeks (primary), assuming no births or deaths across weeks. Alternatively, you can treat each survey visit to a site is a primary occasion and apply double observer methods in place of secondary occasions, or break the survey into smaller time intervals, treating repeated counts as independent samples (Royle 2004)3 or removal samples (Chandler et al. 2011)4.
The upshot is, you need to have a design that has primary and secondary sampling in some manner to be able to deal with temporary immigration.
Behavior
Removal sampling is considered the primary way to account for availability in surveys. It assumes that the only reason an individual was not detected during a preceding time interval was they were not behaving in a way that made them detectable. If you assume that is the only reason why an individual wasn’t detected (i.e. ignore sightability), then removal sampling does account for availability.
Behavior-related availability can also be addressed by making repeat visits to a site while closure is maintained. In doing so, and documenting the conditions that might influence availability, you can model detection as a function of these covariates. In frog surveys, this might mean recording the amount of rainfall in the previous 24 hours, the ambient humidity, or temperature.
Alternatively, you might restrict your surveys to meet conditions where availability will be high and consistent. For example, state biologists will try to conduct roadside counts of pheasants only on mornings with heavy dew, because pheasants are believed to try to “dry out” along roads, making them more available for detection. You still should use some method to account for imperfect detection, but this is one way to reduce variation due to animal behavior.
Sightability
Sightability itself is comprised of individual observer differences as well as environmental factors. An animal must be physically present, and behaving in a way that makes them available to be sighted by an observer. Depending on the study design, some components of sightability can be modeled explicitly (i.e. distance sampling) while others may be included as covariates on a model for detection probability, similar to behavioral availability. From a design perspective, to deal with sightability, you need multiple observers and/ or visits, recorded covariates that might affect an observers ability to detect an animal (road noise, visual obstructions, etc.), and/or collecting distance data.
Double-observer methods, by virtue of the sampling design, implicitly account for differences in sightability between two observers. Sightability can be modeled explicitly as well. Distance sampling explicitly accounts for a decrease in sightability as the distance between the observer and animal increases. Some hierarchical approaches explicitly model behavioral availability and sightability separately. Some studies may restrict the distance at which they will count animals (e.g., 50m radius point count instead of unlimited distance) to minimize or eliminate the need to explicitly model the effect of distance.
Putting it all together
While these processes are, conceptually at least, unique, when trying to account for them they are not always treated as such. There are certainly approaches where only one component of detection is addressed. When dealing with behavior-based availability and sightability, it’s most common to model detection probability as a composite process, rather than separate processes. In these cases, a linear model is constructed for detection that includes covariates related to sightability along with availability. In the frog example, you might include covariates for rain as well as the observer and road noise: p. = \mu+\beta_1X_{rain}+\beta_2X_{road\ noise}+e_{observer} In this approach, you do not model availability and sightability separately. It’s not “correct” despite frequently being done this way, but it’s still a step in the right direction when dealing with imperfect detection and well accepted. More complex methods that tackle temporary immigration, or explicitly model sightability and availability separately do exist. For example, using a robust design with removal, double observer, or repeated counts, one can account for temporary immigration and availability (Chandler et al. 2011)5. Using distance sampling methods and removal methods, you can separately (and “more correctly”) account for availability and sightability (Amundson et al. 2014)6. Given the advancement of methods for fitting hierarchical models in the past 20 years or so, the sky is the conceptual limit to how many processes you can stack when trying to account for all the different components of detection probability. There are real limits though, as you will generally require more and more data to fit these complex models. And while conceptually, they are plausible, in practice, you might not be able to actually treat these processes separately.
Last thoughts
Accounting for imperfect detection is an important part of any survey or study. It’s also important not to obsess over it, but to make a good faith effort to account for it. It is unlikely that you will be able to adequately account for all aspects that contribute to imperfect detection. Some components you may be able to address by adjusting when or how you conduct surveys (optimal weather conditions, like with frogs; 50m vs unlimited distance point counts) or by implementing specific methodologies (removal or double-observer sampling). The important thing is to have a sound understanding the the ecology of your organism so you can identify what sources of imperfect detection are the most important, and address those.