Towards built-in capture–recapture mixed models in program E-SURGE

Abstract

We consider the first steps towards implementing capture–recapture mixed models (CR2Ms) in program E-SURGE. The main issue when estimating the parameters of mixed models is that integrals associated with the random effects distributions need to be dealt with. Rather than using a Bayesian approach with Markov chain Monte Carlo and in line with Gimenez and Choquet (Ecology 91:951–957, 2010), we show that a frequentist approach using numerical integration can be tractable when independent clusters of individuals can be identified. In this case, the maximum likelihood approach is time-efficient because the dimension of the integral for the likelihood is small. This allows us to integrate the likelihood by an efficient and appropriate quadrature method with a procedure for error control. Building on program E-SURGE (Choquet et al. in Modeling demographic processes in marked populations, volume 3 of Springer series: environmental and ecological statistics. Springer, Dunedin, 2009b), we extend the GEMACO language (Choquet in Can J Stat 36:43–57, 2008) to incorporate random effects in a large set of capture–recapture models, including multievent models (Pradel in Biometrics 61:442–447, 2005). To illustrate the flexible implementation of CR2Ms in E-SURGE, we consider two real examples, one with an individual random effect and one with group random effects. Future developments and limitations are also discussed.

Appendices

Appendix 1: Implementation of the structure of $\user2{X}_i$ and $\user2{Z}_{l,i}$

$\user2{X}_i=\user2{A}\times \user2{B}_{i}$ where $\user2{B}_{i}$ is set of square matrices dependent on individuals. This structure allows us to consider model like: i+t.xind. In this case, $\user2{A}$ is a time dependent matrix and the matrix $\user2{B}_i$ is a (K − 1) × (K − 1) diagonal matrix which entries are the value of the covariates for individual i.

$\user2{Z}_{l,i}=\user2{C}_l\times \user2{D}_{l,i}$ where $\user2{D}_{1,i}$ is a square matrix depending of the individual and of the effect. This structure allows us to consider model like: i+xind+rand(xind). In this case $L=0, P=1,\, C_1$ is a constant vector of 1 and $\user2{D}_{1,i}$ is the value of the covariates for individual i.

Appendix 2: Influence of the finite difference scheme

We check that the tolerance (Tol = 10⁻⁶) used for the quasi-Newton algorithm on the gradient and the error (Err) made by the approximation of the gradient using a finite difference scheme and a numerical integration are consistant, i.e. of the same order. Let ɛ be the computer precision, we will demonstrate that the global error Err defined by:

$$ Err = K'(y_0) - \sum_{n=1}^{N} \omega_n \times \frac{k(y_0+\sqrt{\varepsilon}, \user2{x}^{(n)})-k(y_0,\user2{x}^{(n)})} {\sqrt{\varepsilon}} $$

can be decomposed as a sum of two sources of error; the error of the finite difference scheme applied to the gradient of k and the error made by the quadrature formulae applied to the gradient of k. By the Fubini theorem,

$$ K'(y_0)= \int\limits_{{\mathbb{R}}^d} \frac{\partial k(y,\user2{x}) } {\partial y}|_{y_0} \exp^{-\user2{x}^t\user2{x}} \mathbf{d}\user2{x} $$

The error made by approximating a derivatives by a first order finite-difference scheme is $O(\sqrt{\varepsilon})$ (see Dennis and Schnabel 1983) so

$$ \begin{aligned} Err &= K'(y_0) - \sum_{n=1}^{N} \omega_n \times \left( \frac{\partial k(y,\user2{x}^{(n)})}{\partial y}|_{y_0} +O(\sqrt{\varepsilon}) \right) \\ Err &= \int\limits_{{\mathbb{R}}^d} \frac{\partial k(y,\user2{x})} {\partial y}|_{y_0} \exp^{-\user2{x}^t\user2{x}} \mathbf{d}\user2{x} - \sum_{n=1}^{N} \omega_n \times \frac{\partial k(y,\user2{x}^{(n)})}{\partial y}|_{y_0} +O(\sqrt{\varepsilon}) \end{aligned} $$

For Matlab on a 32-bit Windows system, ɛ = 2.2204 × 10⁻¹⁶ so as soon as the error made by the quadrature formulae is lower than Tol, the global error is lower than Tol. For a lower tolerance (Tol = 10⁻⁶) used for the quasi-Newton algorithm on the gradient then a second-order finite-difference scheme should be used.

Appendix 3: Application 1 with E-SURGE

E-SURGE can accept capture–recapture data either in MARK or BIOMECO format. The two types of data file are not very different: each row corresponds to a particular capture history followed by the number of individuals with that history. In MARK format, this count is followed by a semi-colon and in BIOMECO format the data are preceded by the number of different capture histories and the number. In this study, there is only one site. We will fit here the model ϕ (flood) p(partial(m) + ind) described in Application 1.

Starting E-SURGE

From E-SURGE, start a new session named ’result.mod’. Read in the data file and tell E-SURGE that there are no individuals covariates. Check that the numbers of capture occasions, groups and events are correct (in this case, 7 capture occasions, 1 group and 2 events): E-SURGE makes assumptions about the number of states, but these need to be modified depending on the problem you have to treat. In this case, we change the number of states to three. We will also have to set the number of age classes to one as for the present, we will not consider any age effect.

Fitting the model

Press the Modify button, and change the settings to specify that there a single age class, a single group and 3 states (see Fig. 1).

Fig. 1

E-SURGE: the number of states is set to 3 and the number of age classes is set to one

The main menu should show the changes. Fitting the models in E-SURGE involves four steps:

• 1. The Gepat step: specifying which ones of all the potential parameters have to be estimated, which ones will be calculated as the complement to 1 (there is one such parameter per multinomial) and which ones correspond to impossible events or transitions and are fixed to zero;

• 2. The Gemaco step: specifying the effects (time, age…) acting on the active parameters;

• 3. The IVFV step: specifying initial values for the optimization procedure and/or fixed values for the active parameters;

• 4. The RUN step: launching the optimization procedure.

Specifying the pattern matrices using the GEPAT interface

There are three types of parameters used in the definition of a multi-event model (Pradel 2005):

• 1. the initial state probabilities;

• 2. the transition probabilities;

• 3. the event probabilities.

Each type of parameter is gathered into a row–stochastic matrix, i.e. each row corresponds to a multinomial. (Each matrix can be further decomposed into a product of several stochastic matrices allowing for example to estimate separately survival and transition parameters. However, for the current model, only 1 step is required for each type of matrix.)

To enter the GEPAT interface, click the GEPAT button at the bottom of the main window. The GEPAT interface screen for the initial state pattern matrix appears.

By default, E-SURGE lets all live states be available as initial steps. The state dead, last in the list, is impossible and does not even appear. The last live step is taken by default to be the one whose probability will be calculated indirectly, as the complement of those of the other live states. This is specified in the above window using the following general conventions for Gepat:

• a minus sign (−) indicates that a potential parameter is unavailable in the current model (impossible transition, for instance). This is equivalent to fixing it to zero but more explicit;

• any Greek letter (strike a Latin letter and E-SURGE will show its Greek counterpart) indicates a free parameter, one to be estimated directly. Note that the particular Greek letter entered is totally irrelevant to E-SURGE. In particular, the same Greek letter is used repeatedly by default within a pattern matrix (by default for initial states); this does NOT mean that the parameters are being forced to be equal;

• the asterisk (*) indicates the parameter that is calculated indirectly, as the complement to 1 of all the other parameters on the same row. There MUST always be one and only one asterisk per row because each row corresponds to a multinomial.

Note that the order of the states is chosen by the user except for the dead state that is always positioned last. Here, the default pattern is not correct for transition and event. We need to change them.

For initial states, we implement the pattern given in Fig. 2. For transitions, we need two steps, one for survival and one for capture. So we set the number of steps to 2. For survival, we implement the pattern given in Fig. 3. For capture, we implement the pattern given in Fig. 4. For event, we implement the pattern given in Fig. 5. Press the “EXIT” button to return to the main window.

Fig. 2

E-SURGE: pattern for the initial states vector

Fig. 3

E-SURGE: pattern for the survival matrix

Fig. 4

E-SURGE: pattern for the capture matrix for the trap-dependent model

Fig. 5

E-SURGE: pattern for the event matrix

Specifying the model using the GEMACO interface

The GEMACO interface uses keywords to create a modeling sentence that indicates how parameters vary by time, over groups, over age classes, etc. At the end of the GEMACO procedure, a design matrix is created for each type of parameters. Each row of the design matrix will correspond to a parameter of the full model (all potential variability: time, age, group…) and each column corresponds to a parameter of the actual model.

The GEMACO syntax is fairly intuitive but the “sentences” you enter in the GEMACO interface must respect some priority rules that we will not develop here. We encourage the user to read the E-SURGE user manual and Choquet (2008) in which the GEMACO syntax is fully explained.

In this example, we only want to show how to use E-SURGE to fit our model.

For the trap-dependent model, the set of initial state probabilities and the set of event probabilites are empty. Click on the top “Initial State” button to go to the “Transition” screen.

To specify the model on survival, we use the phrase t(1 4 5 6,2 3) for the flood effect, and the new keyword ind for the individual random effect. As we combine all, the GEMACO sentence becomes t(1 4 5 6,2 3)+ind. Select the next step for transitions which corresponds to the modeling of trap dependence. We use the phrase t(1:4)+t(5 6).f+ind.

At this stage, to create the design matrices, we click on the Gemaco item in the top menu and select the “call GEMACO (all phrases)” submenu. At this stage, all the model structures are specified and the design matrices appear in the left window of each screen of the GEMACO interface; press the “EXIT” button to return to the main window.

Specifying the initial and fixed values using the IVFV interface

In E-SURGE, the user can choose the way to generate the initial values of the optimization procedure. They can be either “constant”, “randomly generated” or “equal to the estimates of a previously fitted model”. Once the type of initial values is chosen, the user can also fix the values for some parameters using the IVFV interface. Press the IVFV button to enter the interface (Fig. 6). As there is no need to specify either fixed values or initial values, click on the “EXIT” button to return to the main window.

Fig. 6

E-SURGE: the initial values fixed values interface. The two last parameters are those associated to the two random effects

Running the model

Before running the model, we have to specify the method of integration; here, we choose the classical Gauss-Hermite method (set by default) described in the paper with 29 quadrature nodes (r = 15, set by default). We also tick the “compute C-I(Hessian)” option to get confidence intervals. The model is now ready to be fitted to the data. Press the RUN button. We observe in Fig. 7 that at the end of the fit, the estimate error made by the GH scheme to get the likelihood is lower than 10⁻⁶ as $-15.64\le-6$.

Fig. 7

E-SURGE: the Output during the RUN step shows that 20 iterations are needed for convergence and that the error due to numerical integration is small (lower than 10⁻⁶)

In Fig. 8, we get the estimates for the model ϕ(flood + ind) p(partial(m) + ind) in the mathematical scale.

Fig. 8

E-SURGE: estimates and standard errors for the model ϕ(flood + ind) p(partial(m) + ind)

Appendix 4: Application 2 with E-SURGE

We will fit here the model ϕ (flood) p(partial(m) + ind) described in part 2 with E-SURGE.

Starting E-SURGE

From E-SURGE, start a new session and load the dataset. For this application, we only need to change the number of age classes to one as, for the present, we will not consider any age effect.

Fitting the model

Press the Modify button, and change the settings to specify that there is a single age class, nine groups and 2 states (see Fig. 9).

Fig. 9

E-SURGE: the number of age classes is set to one

The main menu should show the changes. Fitting the models in E-SURGE involves four steps:

Specifying the pattern matrices using the GEPAT interface

Note that the order of the states is chosen by the user except for the dead state always positioned last. Here, the default pattern is one of the CJS model. We do not need to change them.

Specifying the model using the GEMACO interface

In this example, we only want to show how to use E-SURGE to fit our model.

• The initial states modeling

For the CJS model, the set of initial state probabilities is empty. Click on the top “Initial State” button to go to the “Transition” screen.

To specify the model on survival, we use the phrase i (see Fig. 10).

• Click on the top “Transition” button to go to the “Event” screen.

Fig. 10

E-SURGE: the sentence ’i’ builds the model ϕ (.)

Because the model conditions on the first capture occasion of each individual, the only event to model at the time of the first encounter is the site of capture. It is only later on that the event ’not encountered’ becomes possible. Thus, the event probabilities at the time of the first encounter must be treated separately. This is achieved through the use of the keywords “firste”, which stands for ’first encounter’, and “nexte”, which stands for ’next encounters’, respectively.

The output of Gepat is shown in the ’transitions pattern’ subwindow at the bottom left. For each state (in row), there is only one active event which is the capture on the relevant site (where the stands), and the other possible event taken as the complement is ’not encountered’ (first column). For instance, for state 11, the first row, the individual can be encountered on site 1 (second column, probability to be estimated) or ’not encountered’ (first column, probability calculated as 1 − the other probability). The active parameters are thus just the capture probabilities. At the time of the first encounter, the capture is certain and the capture probabilities will all be 1. At this stage, we cannot specify a fixed value, but we can specify that we need just one parameter common to all states by keeping “firste” by itself. Later, capture probability will be constant. Thus, the complete sentence is “firste+nexte+random(nexte.g)”.

At this stage, to create the design matrices, we click on the Gemaco item in the top menu and select the “call GEMACO (all phrases)” submenu (see Fig. 11). All the model structures are now specified and the design matrices for fixed effect appear in the left window of each screen of the GEMACO interface; press the “EXIT” button to return to the main window.

Fig. 11

E-SURGE: the sentence ’firste+nexte+r(nexte.g)’ builds the model ϕ (i + r(g))

Specifying the initial and fixed values using the IVFV interface

In E-SURGE, the user can choose the way to generate the initial values of the optimization procedure. They can be either “constant”, “randomly generated” or “equal to the estimates of a previously fitted model”. Once the type of initial values is chosen, the user can also fix the values for some parameters using the IVFV interface. Press the IVFV button to enter the interface.

The initial states probabilities In this case, there is no need to specify neither fixed values nor initial values. Click on the top “Initial State” button to arrive at the “Transition” screen.

The survival-transitions probabilities There is no need to fix values for the transition probabilities so this screen can be left in its default state. Click on the top “Transition” button to arrive at the “Event” screen.

The event probabilities We can see here the different capture rate appearing in the definition of the model. The series of number indicate successively:

• the line in the event matrix (corresponding to the state),

• the column in the event matrix (corresponding to the event),

• the capture occasion,

• the age class,

• the group,

• the step in the matrix decomposition of the event matrix (here 1).

Thus, the first parameter corresponds to the capture rate at the first capture occasion for the first age class, i.e. time of first encounter (A=1). This is the only parameter with A=1 because we have gathered all the capture rates relative to the first encounter into a single parameter. This parameter needs to be fixed to 1. We do this by entering the value 1 as “Initial Value” and ticking the box nearby (see Fig. 12).The other parameter corresponds to the following capture rates (A=2); there is no need to fix these parameters.

Fig. 12

E-SURGE: the first mathematical parameter corresponding to ’firste’ is fixed to 1 by ticking the box nearby. The third mathematical parameter is the starting value for the square root of the standard error; it must be strictly positive

After all the fixed values have been specified, press the EXIT button.

Running the model

Before running the model, we have to specify the method of integration. Click on the button ’Advanced Numerical Options > Modify’ in the main window of E-SURGE. Here we choose the Adaptive Gauss-Hermite method described in the paper (the fourth value is set to 1) with 29 quadrature nodes (the fifth value is set to 15) (see Fig. 13).

Fig. 13

E-SURGE: we choose the Adaptive Gauss-Hermite method (the fourth value is set to 1) with 15 quadrature nodes (the fifth value is set to 15)

We also tick the submenu ’Random Effects for Independent Group Only’ in the menu ’Models’ (Fig. 14).

Fig. 14

E-SURGE: to speed-up calculations and improve the precision of the integration, we set that the random group effect is i.i.d

We also tick the ’compute C-I(Hessian)’ option to get confidence intervals and an estimated of the model rank. The model is now ready to be fitted to the data. Press the RUN button. We observe in Fig. 15 that at the end of the fit, the estimate error made by the GH scheme to get the likelihood is lower than 10⁻⁶ as $-6.45\le-6$.

Fig. 15

E-SURGE: the Output during the RUN step shows that only 10 iterations are needed for convergence and that the error due to numerical integration is small (lower than 10⁻⁶)

In Fig. 16, we get the estimates for the model ϕ(.) p(i + r(g)) in the mathematical scale.

Fig. 16

E-SURGE: estimates and standard errors for the model ϕ(.) p(i + r(g))