Ecological Archives E091-067-A4 O. Gimenez and R. Choquet. 2010. Individual heterogeneity in studies on marked animals using numerical integration: capture–recapture mixed models. Ecology 91:951–957. Appendix D: Incorporating individual random effects using program E–SURGE In this appendix, we provide a step-by-step procedure to fit model {φ(x + x2 + x3 + h), p} that was used in the Sociable weaver case study using numerical integration as implemented in program E–SURGE. E–SURGE is available from http://www.cefe.cnrs.fr/BIOM/logiciels.htm, and is described at length in Choquet et al. (2009) and in the E–SURGE user manual. The implementation of capture–recapture mixed models in E–SURGE is also described in Choquet and Gimenez (2010). 1 Read in data E–SURGE accepts capture–recapture (CR) 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. Below are the CR histories of the 6 first individuals of the data set we considered in the Sociable weaver case study (MARK format), the last three columns being standardized body mass, squared body mass and body mass raised to the power three. 10000000 1 -0.8305 -0.2190 0.0969; 11100000 1 0.7385 -0.2086 -0.0545; 11100000 1 -1.8007 1.1382 0.8611; 2 11100000 1 -0.2436 -0.5342 0.0241; 11000000 1 0.6307 -0.2965 -0.0269; 11000000 1 0.6426 -0.2874 -0.0296; From E–SURGE, start a new session named ”result.mod” from pull-down menu ”Start”. Read in the data file (from ”Data” menu) and specify in E-SURGE that there are three individual covariates when the question ”Is there external variables” pops up. Amend the number of capture occasions, groups and age classes if needed: here we have 8 capture occasions, 1 age class and 1 group (use ”Modify” button in the ”Data Status” panel). Now because we work in a multi–event framework (Pradel, 2005) we need to define the events and states: we consider a standard Cormack-Jolly-Seber-type (CJS) model, which is easily obtained by using 2 events – non detection or detection – and 2 states – alive or dead. By default, E–SURGE gets it right here. 2 Fitting the model Fitting a model in E–SURGE involves four steps: • 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 and which ones correspond to impossible events or transitions and are fixed to zero; • The GEMACO step: specifying effects (time, age) acting on the parameters; 3 • The IVFV step: specifying initial values for the optimization procedure and / or fixed values for the active parameters; • The RUN step: launching the optimization procedure. 2.1 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): • The initial state probabilities Π; • The transition probabilities Φ make the link between states at time t given states at time t − 1; • The event probabilities B make the link between events at time t and states at time t. To enter the GEPAT interface, click the GEPAT button at the bottom of the main window. By default, E–SURGE leaves all live states available as initial steps. The state dead, last in the list, is impossible and does not 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 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; 4 • 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; 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. Note that the order of the states is chosen by the user except for the dead state that is always positioned last. Here, by default, the pattern proposed by E–SURGE is correct for the model we want to fit, therefore press the EXIT button to leave the GEPAT interface. The vector of initial state is Π= 1 0 which gives in the GEPAT syntax ∗ . In the Sociable weaver case study, because we have only two states, the transition matrix is simple with only 1 parameter for survival to be estimated φ 1−φ Φ= 0 1 5 which gives in the GEPAT syntax ψ ∗ − ∗ (the first column corresponds to the probability of being alive conditional on being alive or dead at the previous occasion). For the event, we have also a single parameter to estimate for detection p 1−p B= 0 1 which gives in the GEPAT syntax ∗ β ∗ − (the second column corresponds to the probability of being detected conditional on being alive or dead). 2.2 Specifying the model using the GEMACO interface The GEMACO interface uses keywords to create a modelling sentence that specifies 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 6 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. • Modelling the initial states For the CJS model, the set of initial state probabilities is empty. Click on ”Initial State” button on top of the panel to proceed to the ”Transition” window. • Modelling the transition probabilities To specify the model on transition probabilities, we used the keyword ”i” for the intercept, the keyword ”xind” for slopes corresponding to individual covariates and the keyword ”ind” for an individual random effect. To combine them all, the GEMACO sentence should be ”i + xind(1,2,3) + ind”. On top of the panel, click on the ”Transition” button to proceed to the ”Event” window. • Modelling the event probabilities 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 state 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 7 only one active event which is the capture on the relevant state, and the other possible event taken as the complement is ”not encountered” (first column). At the time of the first encounter, the capture is certain and the capture probabilities will all be set to 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”. 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. Then, press the EXIT button to return to the main window. 2.3 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 (although it might be worth trying several different initial values for the standard deviation of the random effect). On top panel, click on the ”Initial State” button to proceed to the ”Transition” screen. 8 • The transition 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 ”Transition” button to proceed to the ”Event” screen. • The event probabilities We can see here the different capture parameters 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. The other parameter correspond to the following capture rates (A=2); there is no need to fix these parameters. After all the fixed values have been specified, press the EXIT button. 2.4 Running the model Before running the model, we tick the ”compute C-I (Hessian)” option in the ”Advanced Numerical Options” panel to obtain confidence intervals of the parameters. We also need to 9 specify the method of numerical integration. Here, we go for the standard Gauss–Hermite quadrature method. Click ”Modify” in the panel ”Advanced Numerical Options” and set to 0 for non-adaptive Gauss-Hermite in the 4th line, and specify 15 quadrature nodes in the 5th value for the order of integration. Note that the adaptive Gauss–Hermite quadrature is also available by specifying 1 in the fourth line. The model is now ready to be fitted to the data, press the RUN button. 3 Interpreting the outputs The output file (see next page) contains several pieces of information. The first paragraph reminds you the characteristics of the data set, the second paragraph the GEMACO syntax and the initial and fixed values, while the third and fourth paragraphs give numerical details. The next paragraph tells us the number of estimable parameters: here we have 4 regression parameters, 1 standard deviation for the distribution of the individual random effect, plus 1 detection parameter, therefore 6 parameters. We are also given the values of AIC and deviance. Finally, the parameter estimates are provided (on the scale of the link function), together with standard errors and confidence intervals: beta1 is the intercept c0 , beta2, beta3 and beta4 are the regression parameters c1 , c2 and c3 , beta5 is the standard deviation of the individual random effect, given on the scale of the square root. Using program R, we back–transform to get the standard deviation by squaring the value E–SURGE, which gives us 10 > 0.907975363*0.907975363 [1] 0.8244193 and we use the delta–method (resorting to the package ”msm”) to get its standard error > library(msm) > ?deltamethod > deltamethod (~ x1*x1, 0.907975363 , 0.089961882*0.089961882) [1] 0.1633663 Finally beta6 is the detection probability on the logit scale, which needs to be back-transformed to obtain the detection probability > 1/(1+exp(0.292820001)) [1] 0.4273136 and again use the delta–method to obtain its standard error > library(msm) > deltamethod (~ 1 / (1+exp(-x1)),-0.292820001 , 0.095215528*0.095215528) [1] 0.02330083 11 OUTPUT FILE : C:\OG\...\phi(x1+x2+x3+RE) avec CI.out Data File :C:\OG\...\weavers.inp Number of occasions : 8 Number of states : 2 Number of events : 2 Number of groups : 1 Number of age classes : 1 ---------------------------------------------Model Name : phi(x1+x2+x3+RE) avec CI Model formula (or file) : For Initial State:IS - Step 1 - (0): For Transition:T - Step 1 - (5): i+xind(1,2,3)+ind For Event:E - Step 1 - (2): firste+nexte Init(values) 0.135152,0.250624,0.826779,0.917867,0.485031,0.913677, Init(indices) 1, 3, 4, 5, 6, 7, Fix(values) 1.000000, Fix(indices) 2, ---------------------------------------------Link function : logitgen Explicit gradient : off TOLF : 0.000001 12 TOLX : 0.000010 ---------------------------------------------Time for optimisation : 2.912564e+002 seconds Time for Hessian : 3.266891e+002 seconds Number of iterations for optimisation : 17 ---------------------------------------------- dev(PI)+dev(PHI,B) = 2376.181 QAIC = 2388.181 Number of mathematical parameters : 6.000000 Estimated model rank applicable to the data : 6.000000 Estimated number of boundary parameters : 0.000000 WARNING : there might be 0 further non identifiable parameters c hat used for the QAIC : 1.000000 -----------------------------------------------------------------------------------------------------Beta (Mathematical parameters) 13 --------------------------------------------------------Index Beta | Lower & Upper 95 percent CI | S.E. --------------------------------------------------------Beta# 1# | +0.509340913 +0.280820455 +0.737861371 +0.116592070 Beta# 2# | +0.015738742 -0.241075099 +0.272552582 +0.131027470 Beta# 3# | -0.268172515 -0.441169542 -0.095175489 +0.088263789 Beta# 4# | +0.081036919 -0.005817684 +0.167891522 +0.044313573 Beta# 5# | +0.907975363 +0.731650075 +1.084300651 +0.089961882 ######################################################################### Beta# 6# | -0.292820001 -0.479442436 -0.106197567 +0.095215528 14 References Choquet, R. 2008. Automatic generation of multistate capture recapture models. The Canadian Journal of Statistics, 36:4357. Choquet, R. and O. Gimenez. 2010. Towards built-in capture-recapture mixed models in program E-SURGE. Journal of Ornithology. Choquet, R., L. Rouan, and R. Pradel. 2009. Program E-SURGE: a software application for fitting multievent models. In D. L. Thomson, E. G. Cooch, and M. J. Conroy, editors, Modeling Demographic Processes in Marked Populations, pages 845865. Series: Environmental and Ecological Statistics, Vol. 3. Pradel, R. 2005. Multievent : an extension of capture-recapture models to uncertain states. Biometrics, 61:442447. 15

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