Two-age analysis: using the PIM to provide age structure
The example is described on pg. 392 (Tables 16.24-16.26) of Chapter 16 of Williams et al. (2002). The data are for adult and young male mallards in Colorado banded during 1963-1971 (9 encounter occasions). Open these data using program MARK. Specify 2 groups, and label the first group 'adult' and the second group 'juvenile' (this is the order of the band recovery matrices in the input file). Open the PIMs for the adult males and young males. You should see a matrix
for "adults"
and
for "young", the usual time-and group specific survival model. However, the above group model does not take into account the fact that juveniles age into adults. That is, in the second calendar year juveniles have adult (not juvenile) survival. The adults are assumed to remain adults (i.e., there is no further age stratification) and the PIM for adults will work fine. However the juvenile PIM needs to be modified to
which captures the fact that the 'juvenile' survival trait persists only through the first calendar year following banding (as juveniles); thereafter these are treated as adults. You can do this manually (and painfully) by entering all the off-diagonal numbers in the PIM by hand, or easily by 1) resetting all the juvenile indices to match the adult values, and 2) using the 'initial|diagonal' toolbar feature to construct the diagonal of the matrix. Experiment until you get a matrix that looks like the one above.
You will need to make a similar adjustment for the transition of juvenile to adult recovery rate, directly analogous to the survival rates. You should wind up with an adult matrix similar to the adult survival matrix, with parameters 17-25 in the columns, and a juvenile matrix with parameters 26-34 on the diagonal but otherwise like the adult recovery matrix.
Finally, just as with the sex-specific grouping factor earlier, the above global model specifies that age interacts with time for both survival and recovery rates. You could describe this model as
S(a*t) f(a*t) where now a stands for the age grouping factor, with transition over time between age classes defined by the PIMs. Use the design matrix to build models in with age and time are additive (on the logit scale); this is accomplished exactly the same way it was when sex was the grouping factor. Try building models S(a+t) f(a*t) , S(a*t) f(a+t) , and S(a+t) f(a+t).