Closed Population Models
Recall that closed-capture models assume that no individuals enter (through births
or immigration) or leave (through deaths or emigration) the population during the study.
Our ultimate goal is to estimate population size but we must estimate other parameters to
achieve this goal. The numbers of parameters we must estimate depends on how
complex our model of population structure is.
Model M0 assumes that there is no heterogeneity in capture probability and that
capture probability is constant across capture occasions. Likelihood for this model is:
P{x } | N , p  
N!
KN  n.
p n. 1  p 
 x ! N  M K 1 !


Model Mt allows capture probabilities to vary among capture occasions but
assumes no heterogeneity in capture probability.

 

P {x } | N , p 
K
N!
n
N n
  p j j 1  p j  j
x ! N  M K 1 ! j 1
 

Model Mh allows for heterogeneity in capture probabilities among individuals.
Program Mark implements mixing models (Norris and Pollock 1995, Pledger 1998,
1999) to address heterogeneity, in which a fixed (and usually small) number of groups
are specified for which capture probabilities are estimated. These models require
estimation of the proportion of the total population in each group.
Likelihood for heterogeneity models replaces p with the following expression:
p  pL  (1   ) pH
Model Mb allows for the capture probabilities of individuals to be a function of
their previous capture history.
GETTING STARTED
1.
2.
Move the data set closedcaptures.inp from the Lab 5 subdirectory of the
Public folder into your personal folders.
Open Program Mark.
a. Click on the File button in the upper left hand corner, then select New
File.
b. In the Window that opens select “Click to select file” and open the file
you just moved to your personal directory.
c. Click “View File” and count the number of capture occasions.
d. Increment the number of encounter occasions until you are at the
correct number. This tells Mark how many encounter occasions to
expect.
e. Then, on the left side of the window, under data type, find closed
captures and select it. A new window will open and select “closed
captures”. When you are finished click OK.
f. We have just set up Mark to begin analysing our data.
ANALYSIS
Behavioural response but no heterogeneity
We are going to begin with analysis of M0, Mt, and Mb. You should be looking at
the Parameter Index Matrix (PIM) for the ps, capture probabilities for each
capture occasion. We need to open two other PIMs to get started. Select the PIM
button on the top toolbar and Select all, then click OK. Now you should have
three PIMs open. If you want to see them all at the same time select the Tile
option from the Window button.
Each PIM tells us how many different values each parameter type can have. For
example the PIM for p, has 10 values, one for each capture occasion. The PIM
for c (recapture probability) has 9 values, one for each potential recapture
occasion. There is only one possible estimate of N.
We test hypotheses about the structure of the data by modifying the PIMs. For
example, if we set all of the PIM values equal in the PIM for p, we are stating the
hypothesis that p is constant across capture occasions (p only can take on one
value). If we set the cs = ps, we are saying that recapture probability equals initial
capture probabililty. (Note that this is the same as saying there is no effect of
capture on subsequent capture probability.)
Models M0, Mt, Mb, Mbt
If we run the model specified by the PIMs that automatically popped up we are
running a model equivalent to Mbt. This is because we allow capture and
recapture probabilities to differ from each other (representing a response to being
captured in subsequent capture probabilities) and we allow ps and cs to be
different on every occasion.
1. Go ahead and run this model by clicking the button with the green arrow (3rd
from the left) on the top of any of the PIMs. A new window will pop up
asking you for a model name, what link function you want to use and some
other information. We’ll help you with a standard protocol for naming
models (see below). Use the logit link and don’t worry about other details
now. After clicking OK, you will get a window with the question do you
want to use the Identity matrix since none was specified. (More on Design
matrices later). Click Yes. When Mark is finished estimating parameters, it
will ask if you want to append results to the Results Browser. Say yes.
2. Click on the Model Output button on the top menu bar then on specific model
results, parameter estimates, real estimates and view estimates, etc., in
notepad. You will see estimates of all of the ps and cs for each capture
occasion. Note that the confidence interval for the estimate of N was zero,
indicating that this parameter was not effectively estimated. This suggests we
had insufficient data to estimate all 20 parameters in this model.
3. Let’s constrain parameters to see if we can produce a better estimate. Set all
of the numbers in the PIMs for pand c equal (within each matrix). You should
have a three-parameter model now. This model corresponds to the model Mb,
a model with behavioural but no temporal variation in capture probability.
Note: you can change all of the indices in the p matrix to 1 by hitting the
minus key until all of the indices are 1. Repeat this process for the c matrix
then hit the plus key once to change all indices to 2. (If you right click the
mouse you will see a number of additional options for making changes to the
PIM.). Now change the index in the N PIM to 3. Run this model and append
the results to the Results browser. Examine the parameter estimates as you
did above. You should see a single estimate for p and c and you got a
reasonable estimate for N.
4. We might also check a model in which we set p = c. What assumption are we
making about capture probability with this model? Go ahead and make the
necessary changes to the PIMs and run the model. Add the results to the
Results browser. Check the parameter estimates. What do you notice about
the estimate of N? Also note that the models are being ranked by the value of
their AIC score. We’ll talk more about this later but for now it suffices to
know that lower AIC scores indicate a more parsimonious model.
Heterogeneity in capture probability
Remember that we address heterogeneity in capture probability (variation in capture
probability among individuals in the population) using mixing models. In these models
we replace a single capture probability (assuming no behavioural response and no
temporal variation) with a parameter describing the proportion of the population in each
of a predefined set of groups, each with its own capture probability. If we assume two
groups the mixing formula would look like:
p  pL  1    pH ,
where  is the proportion of the population in the group with the lowest capture
probability.
Models Mbth, Mth, Mh
1. 1. We can shift the models for closed capture data by selecting the “Change data
type” command from the PIM menu on the top toolbar. Click the PIM button
then the change data type button. Select “Full closed captures with
heterogeneity”. Now open the PIMs (Parameter Index Matrices). Select all
matrices then tile the display so you can see all of the PIMs. You should now
have a model with 40 parameters (the highest PIM number is 40). Note that there
are now two rows of parameter indices for both the ps and the cs. These represent
capture (and recapture probabilities) for the groups with low and high capture
probability, respectively. There is also a PIM for the  parameter which is the
mixture proportion. Run this model and append the results to the Results
Browser. Check the parameter estimates. What do you see?
2. Constrain the capture and recapture parameters to be constant with time but allow
for heterogeneity in both capture and recapture probabilities. This should produce
a model with six parameters (pi, two ps, two cs and N). Run this model and
append the results to the Results Browser.
3. We can produce models without heterogeneity by constraining  = 1, and setting
all of the parameter indices within the p and c matrices constant. That is, we
estimate a single p and a single c. Run this model and append the results to the
Results Browser. The results should resemble those for Mb. Do they? These last
model runs should convince you that there are multiple ways to construct the
same models to describe a particular set of data.
4. Overall, what do you conclude about the pattern of capture probabilities within
this particular population? Which model did the best job of estimating population
size?
Model naming nomenclature:
We name models by identifying the variables that could influence each parameter and the
nature of the allowed variation in the parameter. For example, a model of survival and
capture probability that allowed fully interactive variation between groups and across
time intervals for both survival and capture probability would be coded as follows:
{sg*t,pg*t}.
The g*t subscript indicates that each parameter is allowed to vary independently among
groups and across time periods. A model in which we allowed survival to differ only
between groups but was constrained to be constant across time would be coded:
{sg,pg*t}.
A model in which survival was constant both across time and groups is called the dot
model:
{s.,pg*t}.
We could use the same notation to indicate models of capture probability. A class of
models we will consider later is called additive models. These models constrain (in our
example) variation to be parallel between groups across time and are designated (for
survival):
{sg+t,pg*t}.
We could think of many other variables over which we might allow model parameters to
vary. Nomenclature described above can be adapted to any of these other scenarios.