Multi-age Open model exercise

This exercise is designed to show how to run program MARK to compute survival and capture-probabilities from 'capture-recapture' data.

Background

Data for this example came from the trapping of meadow voles, Microtus pennsylvanicus, at Patuxent Wildlife Research Center, Laurel, MD (Nichols et al., 1984). Data were collected on a 10 x 10 grid of trapping stations spaced at 7.6-m intervals in old field habitat. A single, modified Fitch live trap (Rose, 1973) was placed at each station. Hay and dried grass were placed in the traps and whole corn was used as bait. Sampling occurred for five consecutive days each month, from June 1981 through December 1981. During each 5-day trapping session, traps were opened in the evening of the first day, checked the following morning, locked open during the day, and reset in the evening, with the sequence repeated each day until 5 days had elapsed. A racoon, Procyon lotor (later captured), visited the traps on the final two nights of the second trapping session, essentially leaving only 3 days of trapping for this session. At each capture, animals were examined for a tag, sexed, weighed, and examined for external reproductive characteristics. Tagged animals were ear-tagged with numbered fingerling tags, and tag numbers of marked animals were recorded at each capture.

We used 'adult' (>22g) and 'young' (<=22g) animals and collapsed the 5 days of sampling each month into a single assessment of presence or absence, leaving six monthly sampling occasions.

Data input

Input data consists of summarized frequencies of capture-histories.

The example we'll use for this model contains 4 groups of animals: Males and females, initially captured as adults or young. So, each capture-history will have 4 frequencies following the capture-history. The input file has already been created and is named: 'mp2age.inp'.

Here are the steps to run MARK on this input file:

Start MARK by clicking 'Start' button (lower left corner usually), 'Programs', 'MARK 4.3', then 'MARK'. Or, double-click the MARK icon on the desktop.

When the form appears, go to the 'File' menu and select 'New'.

A dialog window will appear where you can specify the data type, title, filename, and occasion. For this example, the data type is 'Recaptures only' so you do not need to change it. Click in the textbox under 'Title for this set of data', and type:'multi-age open model exercise'. Next, click the button labeled 'Click to select file'.

The program will present a dialog box asking for the name of the input file. Find the folder containing the sample input file (mp2age.inp), then click on the filename (mp2age.inp), then click 'Open'.

Change the number of encounter occasions to 6 (click up-arrow). Change the number of groups to 4, and click the 'Enter group labels' button. When the dialog box appears, change the name of group 1 from 'Group 1' to 'aF'. Change the name of group 2 from 'Group 2' to 'aM', 'Group 3' to 'j F', and 'Group 4' to 'jM'. Then, click OK.

Click 'OK' and the program will create a database file containing the results for this data.

Model phi(.),p(.)

For the 'Recaptures only' models, there 2 sets of parameters: survival and capture probability. Once the database file is created, MARK will open one PIM window for you. Let's close this window and open the PIM chart (Pim menu, Parameter index chart). Notice that there are 8 parameter-sets on the y-axis this time. As in the single-age model, right-click each box and select 'Constant' in order to build the simple (phi(.),p(.)) model. Then, right-click anywhere in the chart and select 'Renumber with overlap'. You should see 8 boxes for 8 parameters. This would be the 'phi(g),p(g)' model as it is now. We want to create the 'phi(.),p(.)' model, so drag 3 of the boxes corresponding to Phi so they line up with parameter index number 1. Also, drag all 4 of the 'p' boxes so they all line up with parameter 2. Close the chart and run the model, naming it 'phi(.),p(.)'.

View the output by clicking the fourth button from the left in the Results Browser, then close the notepad window.

Model phi(a),p(.)

The next model to run with this example is: S(a),p(.) [ survival age-specific, but equal among sex, capture probability constant and equal among ages and sexes]. Here's where we need to use the Parameter Index matrices instead of the chart to see how to number the parameters.

Start by going to the 'PIM' menu and selecting 'Open parameter Index matrix'. Click '(Phi) j F' and '(Phi) j M', then 'OK'. Go to the 'Window' menu and select 'Tile'.

In the previous examples, when we looked at the parameter index matrices, there was 1 cell for each occasion. In the open-model capture-recapture models, there is matrix of cells for each parameter. The columns of the matrices correspond to the year of recapture, and the rows correspond to the year of release. So, the first cell (row 1, col 1) is the survival parameter for animals released in time 1, and surviving to time 2. The cell just to the right of it is the survival parameter for animals released in time 1 and surviving from time 2 to time 3.

The first cell in the 2nd row is the survival parameter for animals released in time 2, surviving from time 2 to time 3. So, if this matrix is for young animals, the first cell in the 2nd row would be the young survival rate parameter for time 2 (phi-y(2)). The 2nd cell in the 1st row would be the survival rate parameter for animals released as young in year 1, surviving from time 2 to time 3. Since these particular animals mature in a month, they would be considered adults in time 2. So, the 2nd cell in the 1st row would be phi-a(2), not phi-y(2). Here's a table showing a 2-age, time-specific matrix:

phi-y(1)  phi-a(2)  phi-a(3)  phi-a(4)  phi-a(5)     1  6  7  8  9
          phi-y(2)  phi-a(3)  phi-a(4)  phi-a(5)        2  7  8  9
                    phi-y(3)  phi-a(4)  phi-a(5)  or       3  8  9
                              phi-y(4)  phi-a(5)              4  9
                                        phi-y(5)                 5

If there was only 1 age-class, it would look like this:

phi(1)    phi(2)    phi(3)    phi(4)    phi(5)      1  2   3   4   5
          phi(2)    phi(3)    phi(4)    phi(5)         2   3   4   5
                    phi(3)    phi(4)    phi(5)  or         3   4   5
                              phi(4)    phi(5)                 4   5
                                        phi(5)                     5

If there were 3 age-classes (y,s,a), it would look like this:

phi-y(1)  phi-s(2)  phi-a(3)  phi-a(4)  phi-a(5)      1  6  10  11  12
          phi-y(2)  phi-s(3)  phi-a(4)  phi-a(5)         2   7  11  12
                    phi-y(3)  phi-s(4)  phi-a(5)  or         3   8  12
                              phi-y(4)  phi-s(5)                 4   9
                                        phi-y(5)                     5

Since we're interested in a phi(a),p(.) model, we want the matrix to look like this:

phi-y  phi-a  phi-a  phi-a  phi-a        1  2  2  2  2
       phi-y  phi-a  phi-a  phi-a           1  2  2  2
              phi-y  phi-a  phi-a  or          1  2  2
                     phi-y  phi-a                 1  2
                            phi-y                    1

The first step will be to open the PIM windows for young males and young females. Click 'PIM' menu, select 'open parameter index matrix', and select the 'yF survival' and 'yM survival' PIM's. From the last model, the matrices should contain 1's in all cells. Instead of entering 2's in all off-diagonal cells, go to the 'Initial' menu, and select 'age'. Enter '2' for the maximum number of age-classes and click 'OK'. Repeat this for the other sex and close these two windows. Now, we can use the PIM chart to set the indices for the adult Phi's and p's. Click the PIM chart button (camera), and drag the boxes for the Phi-aM and Phi-aF to the right until they are over parameter number 2. (Notice that adult survival in the PIM matrix for young is parameter number 2.) Also, drag the boxes for the p's to the right until they all line up over parameter index number 3.

Close this window and run the model, naming it 'phi(a),p(.)'. View the output file to see if there is a difference between young and adult survival rates.

Model Phi(a*s),p(.)

This model is the same as the previous one except that survival will now be different for the 2 sexes. This change can be done using the PIM chart by dragging the box for Phi-yM to the right until it is over index numbers 4 and 5. Drag the box for Phi-aM to the right until it is over index number 5. Close the window and run the model, naming it 'phi(a*s),p(.)'.

Model Phi(a*s),p(s)

This model differs from the previous model in that the recapture probability (p) for males is different from recapture for females. This modification can easily be done with the PIM chart by dragging the p-M box to the right until it is over parameter index number 6. After running this model you should get AICc=871.6779

Model Phi(ast),p(s)

For this model, the survival parameter index matrix for young Females should look like the first of the 4 sample matrices shown above. To do this, we'll need to open the parameter index matrices (PIM menu, 'open parameter index matrix') for all 4 survival PIM's. After opening them, spread them out on the screen using 'Window', 'Tile'. Click on the Phi-yF PIM window, then click 'Initial', 'Time'. Next, click 'Initial', 'Diagonal' and enter 1 for the 'diagonal to modify', and 6 for the 'starting parameter value' (and click OK). So, adult Female survival will be parameters 1-5, and young female survival will be parameters 6-10.

Next, go to the adult female PIM window, click 'Initial', then 'Time'. Then, repeat these steps for males. Adult male survival parameters should be numbers 11-15, and young males should be numbers 16-20.

Note: The 'Initial' menu fills in a matrix by looking at the first cell of the matrix. So, you might have to make sure the first cell contains the starting parameter number before filling in the rest of the matrix.

We're done with the survival PIM's, so close each of the 4 windows, and open the PIM chart. Notice that the boxes for the p's now overlap the boxes for the Phi's. We definitely don't want to force any p value to be equal to a phi value, so we need to move the p boxes to the right. Drag the boxes for p-aF and p-yF to index number 21, and the boxes for p-yM and p-yF to index number 22.

Model phi(a*t),p(s)

To build this model, we'll base it on the previous model and use the design matrix to force survival for males to be the same as survival for females. Start by clicking on the previous model name in the results window, then go to the 'Retrieve' menu and select 'Current model'. This sets all of the PIM's to that model. Next, open the design matrix window (Design menu, select 'Identity'). Remember that parameters 1-10 were survival rates for females, and 11-20 were for males. To make them equal in the design matrix, we need to duplicate the pattern of 1's that appear in rows 1-10 in rows 11-20. So, click on the first cell in row 11 (under B1), then go to the 'FillCol' menu, select 'Partial identity' and enter 10 as the number of columns to fill. Since the male survival rates will be estimated from B1-B10 (same as females above), we don't need B11-B20. So, click on any cell in column B11 and go to the 'DelCol' menu, select 'multiple columns' and enter 11 for the first column and 20 for the last column to delete. Now, male and female survival rates will be the same (since they're computed from the same B's), and p-M will be different from p-F. Run this model.

Model phi(a+t),p(s)

Any model with the '+' relationship will require the design matrix. We could have built the last model without using the design matrix, but sometimes it's easier to work through a progression of models by keeping the same PIM structure and modifying the design matrix.

To build this model, we can modify the design matrix from the last model. If you haven't exited MARK, the design matrix window should still be open. If it isn't, just retrieve the last model and MARK will open it. Let's start with female survival rates (parameter indices 1-10). We want them to be different for each of the 5 intervals, but we want the young survival to be different from adult survival by a constant amount. So, we need to estimate 6 things (survival for each of the 5 intervals and the difference between adult and young survival). This means we'll need 6 columns in the design matrix for the 1st 10 rows. First, lets make young survival equal to adult survival by putting a diagonal of 1's starting at row 6, column 1. Click on this cell, then go to the 'FillCol' menu, select 'partial identity' and enter 5 for the number of columns.

So, B1-B5 will be the time-specific survival estimates for adults, and we'll make B6 the difference between young and adult survival. Click on the cell in row 6, column 6, go to 'FillCol', select 'partial intercept' and enter 5 for the number of rows.

Since survival for the sexes are the same for this model, rows 11-20 should be exactly the same as rows 1-10. So, we'll need to put a partial identity at row 16, column 1 (for 5 columns), and a partial intercept at row 16, column 6 (for 5 rows).

By making young survival equal to adult survival (plus an offset), we don't need the columns which made young survival time-specific and different from adults. These were columns 7-10, so delete them. (DelCol menu, 'multiple columns', enter 7 and 10.)

Notice that rows 1-10 are the same as rows 11-20 (male survival equal to female survival), and rows 6-10 differ from rows 1-5 by an additional column of 1's (young survival equal to adult survival + constant). Run this model (click 'run button).

Discussion

This is an example of the model selection procedure, where you start with a simple model, building up to more complicated models based on results of previous models. You've proabaly noticed that there can be an overwhelming number of models and you might be tempted to try all (or as many as possible) of them. The 'proper' way is to stick to a set of pre-defined pheasable models and try the ones that make sence. In this analysis, it makes sence to think that sex, age or time might affect survival or recapture, so we've tried combinations of them. We couldn't estimate recapture rates for young since we never recapture young animals, so that limits what we can try with the structure of recapture rates.