Linking the CT hazard function with the DT detection function

The estimators from CT and DT models are compared later on in several simulation studies. The data for these studies are simulated in CT using a CT detection hazard but this hazard needs to be linked with the DT detection function to ensure that the detection process is not misspecified for the DT models. In other words we need the CT data to be in sync with a DT detection function that has particular parameter values. IfTj =tj−tj−1 wheretj−1 is the start of occasionj andtj the end, then this

is accomplished by specifying the detection probability for detector k in occasion j

of duration Tj in terms of the hazard function as follows.

pkj(s;θ) = 1−exp − Z tj tj−1 hk(u,s;θ)du ! (2.37)

Under the assumption of a constant hazard Rtj

tj−1hk(u,s;θ) du = hk(s;θ)×Tj

and hence solving Equation 2.37 for hk(s;θ) leads to a hazard with the following

form:

hk(s;θ) =

−log(1−pkj(s;θ))

Tj

(2.38) As explained in Section 2.5.1, the paramaterisation used when the hazard function is not constant through time has a component that depends on distanceh(s;θ)) and a component that depends on time h(t;ψ). Since h(s;θ)) does not depend on time,

R

h(t,s;θ,ψ)dt =h(s;θ)R

h(t;ψ)dt and as beforeh(s;θ) can be specified so that the CT detection hazard links with the DT detection function as follows:

hk(s;θ) =

−log(1−pkj(s;θ))

R

Tjh(t;ψ)dt

(2.39)

Note that for the cosine hazard, if the integrationR h(t;ψ)dt is carried out over a period that is equal to a multiple of the cycle length, the first cosine component of h(t;ψ) (cos(α×(t+ψ1))) integrates to zero, and in such cases the integration in

the denominator will become Tj ×ψ2.

It may be helpful to think about this linkage for a given distance. In other words, for a given distance from a trap, integrating the associated hazard over the duration of the jth occasion will lead to the probability of capture during that occasion for an individual with an activity centre that given distance away from the trap. If the hazard is specified according to Equation 2.37, and probabilities are calculated for a

range of distances, then the probabilities will follow the same shape as the detection function used in the specification (pkj(s;θ)).

In subsequent analyses, I use hazard functions in whichpkj(s;θ) has a half-normal

form: pkj(s;θ) =g0,Tjexp −dk(s)2 (2σ2₎

where as before dk(s) is the distance from s to detector k and g0,Tj and σ are

parameters to be estimated. The parameterg0,Tj is the probability that an individual

with activity centre located at a detector is detected in a time period of length Tj.

Specifying a model forpkj(s;θ) implies a model forhk(t,s;θ,ψ), although not a

unique one. The mean value of the detection hazard in intervaljmust be ¯hkj(s;θ,ψ) =

−log{1−pkj(s;θ)}/

R

h(t;ψ)dt and anyhk(t,s;θ,ψ) with this mean is consistent

with pkj(s;θ). When the hazard is constant in the interval there is a one-to-one

relationship between the detection hazard and the detection probability.

As shown above, the hazard function specified in this way will produce a detection function with a half-normal shape when it is integrated over a specific period of time (for example Ht1

t0 would be the integrated or cumulative hazard for the length of the

first interval from t0 to t1). When the hazard varies across time, integrating the

hazard over some other interval of time will give a different integrated hazard toHt1

t0

and consequently the detection function will have a shape that is different to the half-normal, i.e. the effect of distance will change over time. The discrete-occasion model can only achieve a similar degree of flexibility if occasion-specific parameters are estimated (although the shape of the detection function will still be constrained to be half-normal in each interval).

This can be seen mathematically for occasionj that runs fromt0 tot1 as follows:

pkj(s, t0, t1;θ) = 1−exp − Z t1 t0 h(u,s;θ,ψ)du = 1−exp −h(s;θ) Z t1 t0 h(u;ψ)du = 1−exp −h(s;θ)Ht1 t0 = 1−exp (−h(s;θ))H t₁ t0

Hence, when the cumulative hazard is equal to Ht1

t0 the detection function will

not hold for different values of Htj+1

tj . If a time-varying cyclic hazard has the same

cycle length as the occasions used then the integrated hazard for each occasion will be the same.

2.5.3

Models with varying effort

CT models readily accommodate detectors that are operational for different periods of time by setting the detection hazard at a detector to zero while the detector is out of operation (i.e.hk(t,s;θ,ψ) = 0 if detector k was not operating at timet). This is

the basis of the varying effort model for binary detectors and counts in Efford et al. (2013).

However it should be noted that in that case it was assumed that the effect on the hazard is proportional to a standard unit of effort, i.e. if the effort for a given occasion is twice that of a standard unit of effort then the detection hazard gets doubled. The varying effort model can therefore accommodate occasions that are of different lengths but in order to do this it implicitly assumes that the hazard is constant through time.

The CT framework generalises this idea and can for example recognise that a detector failing for a period of time during the day will have little effect on the detectability of a nocturnal species. The ability of the CT model to properly handle varying trap exposure is what leads to the construction of the single-catch trap likelihood. It is more difficult to accommodate detectors when their time of failing is unknown, and that problem is not addressed here.

Chapter 3

Models with a constant detection

hazard

This chapter presents models under the assumption of a constant detection haz- ard through time, firstly in the context of proximity detectors and secondly for single-catch traps. The first section illustrates how the general continuous-time (CT) proximity likelihood reduces to a discrete-time (DT) Poisson count model when the detection hazard is constant through time. Both a DT binary proximity detector model and a CT proximity detector model with a constant hazard are applied to the Belize jaguar dataset followed by two simulation studies that compare these estima- tors when captures are firstly independent of each other, and secondly when there is spatio-temporal correlation in captures.

The second section again uses two simulation studies to explore the robustness of the DT multi-catch estimator when it is incorrectly applied to single-catch trap data, and to assess the performance of the correct CT single-catch trap likelihood (Equation 2.33) presented in Chapter 2. The first simulation study examines and compares the performance of the estimators when the density component of the model is incorrectly specified. The second set of simulations is similar but has density correctly specified in the models.

3.1

Proximity detectors