Descripción textual de los Casos de Usos del Sistema (CU)

We start by considering the basic Cormack–Jolly–Seber (CJS) model, where both the survival probabilities and recapture probabilities vary with time.

Let φ_j denote the probability of survival between times j and j + 1, and let p_j denote the probability of recapture at time j. For a capture history with K = 7, we would have the following setup:

1−→ 2^φ1 _p

2

φ2

−→ 3_p

3

φ3

−→ 4_p

4

φ4

−→ 5_p

5

φ5

−→ 6_p

6

φ6

−→ 7_p

7

.

Numbers 1 to 7 denote the sampling occasions. In this example, we can think of them as being j = 1, 2, . . . , 7. The survival probabilities {φj}⁶1 are labelled above the arrow as each φ_j is defined to be the survival probability between times j and j + 1. Beneath each j, we have pj, which is defined as the proba-bility of recapture at time j.

Using the capture history from Equation (3.1), we have

0−→ 1−→ 0^φ2

1−p3

φ3

−→ 1p4

φ4

−→ 1p5

φ5

−→ 1p6 −→ 0.

We replace j with the capture history at time j and change the labels of φ_j and

3.2. CAPTURE–RECAPTURE 31

pj accordingly. For example, the individual was not captured at time j = 3, and so we change the label beneath position 3 from p₃to 1−p³. The individual was last seen at time j = 6, since at position 7 we have a record of 0. So we no longer have φ₆ between position 6 and 7. It follows that the probability of this capture history Pr(CH_i) is given by

φ2(1− p³)φ3p4φ4p5φ5p6(1− φ⁶) + φ2(1− p³)φ3p4φ4p5φ5p6φ6(1− p⁷). (3.2)

As the fate of the animal is not known after occasion 6, the first term of the above probability represents the probability that the animal did not survive after time 6, and the second term represents the probability that the animal had survived after time 6, but it was not recaptured at time 7.

The probability of a capture history from the CJS model can be written in the following form (Pledger et al.,2003),

Pr (CH_i) =

K

X

d=`i

( _d−1 Y

j=fi

φ_j

!

(1− φ^d)

d

Y

j=fi+1

p^x_j^ij(1− p^j)^1−xij

!)

. (3.3)

Note. We define the empty productsQd−1

j=fiandQd

j=fi+1to be 1 when f_i > d−1 and f_i + 1 > d respectively. We also set φ_K = 0 since we do not have any information on survival after the study between times K and K +1, and p1 = 1 as recapture can only start from time 2. There are K− 1 survival probabilities and K − 1 recapture probabilities. The CJS model has a total of 2K − 2 parameters.

Example 3.2.2 (CJS Model). Consider a study with length K = 3. For completeness, we consider all 2³ − 1 = 7 observable capture histories. Using

3.2. CAPTURE–RECAPTURE 32

Equation (3.3), we then obtain the following probabilities,



We do not consider the probability Pr(000) in the CJS model as we do not observe it. The CJS model is constructed to be conditional on the first capture.

Among the 7 probabilities above, we observe that

Pr(001) = 1 (3.5a)

Pr(010) + Pr(011) = 1 (3.5b) Pr(100) + Pr(101) + Pr(110) + Pr(111) = 1. (3.5c)

It follows that there are only four independent capture histories (with associ-ated probabilities). This means that we only have four pieces of usable infor-mation from K = 3 years of data. In practice, if there are missing capture histories, we could have less usable information.

Figure3.1shows a tree diagram of the CJS capture histories. At each sam-pling occasion, there are 2^j−1 terms that correspond to animals which were first caught in the same year j, j = 1, . . . , K and the probabilities of those capture history terms sum to 1. In general for a K–year study, we have 2^K− 1 observable capture histories. But only 2^K− K − 1 independent capture histo-ries can be used to estimate the parameters. This can be interpreted as there

3.2. CAPTURE–RECAPTURE 33

1 10 100 1000 . . .

. . .

1001 . . .

. . .

101 1010 . . .

. . .

1011 . . .

. . .

11 110 1100 . . .

. . .

1101 . . .

. . .

111 1110 . . .

. . .

1111 . . .

. . .

K=4

K=1 K=2 K=3

01 010 0100 . . .

. . .

0101 . . .

. . .

011 0110 . . .

. . .

0111 . . .

. . .

001 0010 . . .

. . .

0011 . . .

. . .

0001 . . .

. . .

Figure3.1:IllustrationofcapturehistoriesforCJSmodelsofcapture–recapturedata

3.2. CAPTURE–RECAPTURE 34

being 2^K ways of arranging 0s and 1s less K constraints and less 1 invalid way of arrange them (the one correspond to a string of 0s). In other words, among the 2^K− 1 observable histories, there are K constraints, one for each year of first capture. Hence, there are only 2^K− K − 1 independent capture histories.

The CJS model assumes that all animals have the same{φj} and {pj}. We can relax this assumption by introducing different groups of animals. Within each group, the animals share the same survival and recapture probabilities.

Pledger (2000) noted that it is not necessary to assume animals have actual different groups. Often the groups are essentially an artefact to detect hetero-geneity in the data if it is present.

To describe heterogeneity of both capture and survival, it is assumed that there are C classes with probabilities {wc} of an animal being in class c. Each animal in class c has survival probability {φ^jc} and capture probability {p^jc} for c = 1, . . . , C and j = 1, . . . , K. The mixture version of Equation (3.3) was derived in Pledger et al. (2003) to give the following probability of a capture history

Pr(CH_i) =

K

X

d=`i

C

X

c=1

( w_c

d−1

Y

j=fi

φ_jc

!

(1− φ^dc)

d

Y

j=fi+1

p^x_jc^ij(1− p^jc)^1−xij

!) , (3.6) where PC

c=1w_c = 1 is a proper constraint. For example, we choose to set w_C = 1−PC−1

c=1 w_c.

When heterogeneity is present, we model{φ^jc} and {p^jc} through different link functions g(φ_jc) (and/or g(p_jc)), using the notations defined in Nota-tion 3.2.1.

Notation 3.2.1. For the survival probabilities {φjc}, we use µφ as baseline from group 1, τ_φj as a time component and η_φc as a heterogeneous component.

Since we have µφ describing the baseline from group 1, we set up constraints

3.2. CAPTURE–RECAPTURE 35

τφ1 = 0 and ηφ1 = 0. So that τφj are the differences in the time component between the baseline and time component at j; η_φc are the differences in the heterogeneous component between the baseline and group c.

Similarly for the capture probabilities {p^jc}, we use µ^p as baseline from group 1, τ_pj as a time component and η_pc as a heterogeneous component, with constraints τ_p2 = 0 (since the recapture probability starts from time j = 2) and η_p1 = 0. By convention, we set φ_Kc = 0 and p_1c = 1 for all c. That is, the survival probability after the length of study K is 0 and the capture probability at time 1 is 1 for all groups.

For K = 4, C = 3, for example, we can model φ_jc using a linear link function,

φjc = µφ+ τφj+ ηφc (3.7a)

[φ_jc] =



















µ_φ µ_φ+ η_φ2 µ_φ+ η_φ3 µ_φ+ τ_φ2 µ_φ+ τ_φ2+ η_φ2 µ_φ+ τ_φ2+ η_φ3 µ_φ+ τ_φ3 µ_φ+ τ_φ3+ η_φ2 µ_φ+ τ_φ3+ η_φ3

0 0 0



















, (3.7b)

3.2. CAPTURE–RECAPTURE 36

or using a logistic link function,

log

In the matrix [φ_jc], each row describes the time effect and each column de-scribes the heterogeneous effect.

It is also possible to include interactions between the time component and the heterogeneous component in the model by setting

φ_jc = µ_φ+ τ_φj+ η_φc+ (τ η)_φjc, (3.9)

where the matrix (τ η)_φ takes the following form

(τ η)_φ =

Extra constraints are needed on (τ η)_φ, such as requiring each row and column to sum to zero. For simplicity, we set the first column and last row of (τ η)_φ to

3.2. CAPTURE–RECAPTURE 37

zero as constraints. Note that using different constraints can result in different point estimates (for example each entry in the matrix φ_jc). Models with in-teractions are often more complicated than they should be and have too many parameters for successful model fitting (Pledger et al., 2010). We will discuss this feature at more length in Section6.4. A similar setup can be used for p_jc.

Often we model probabilities using the logistic link function, so that the point estimates are between 0 and 1. When different link functions are used, we have different point estimates of the parameters. In particular, when using the method of maximum–likelihood estimation, we will get different results from using different link functions. The effect of using different link functions will be discussed in more detail in Section 6.3.

Note. Depending on the values taken by K and C, models can have different numbers of parameters. Table 3.2 summarises the numbers of model parame-ters of the 25 models given in Pledger et al. (2003). We use ‘·’ for a constant parameter, ‘t’ for a time varying parameter, ‘h_C’ for a heterogeneous parameter with C classes and ‘×’ for a parameter with interactions.

The square brackets indicate that both parameters are heterogeneous but they share the same {w^c}s; see Pledger et al.(2003).

The authors give the total number of estimable parameters in the last column of Table 1 in Pledger et al. (2003) as the column ‘Total’. In our table, we present the column ‘Total’ as the total number of parameters in the model. We will discuss the problem of ‘estimable parameters’ in Chapter 4 when considering parameter redundancy.

In document Analisis y Diseno del Sistema de Deteccion y Control de Vectores : version 2.0 (página 61-69)