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Parameter Index Matrices (PIM)

The Parameter Matrices, or Parameter Index Matrices (PIM), define the set of real parameters, and allow constraints to be placed on the real parameter estimates.  There is a parameter matrix for each parameter in each group, with each parameter matrix shown in its own window.  As an example suppose that 2 groups of animals are marked.  Then, for live recaptures, 2 Apparent Survival (Phi) matrices (Windows) would be displayed, and 2 Recapture Probability (p) matrices (Windows) would be shown.  Likewise, for dead recovery data for 2 groups, 2 Survival S matrices and 2 reporting probability r matrices would be used, for 4 windows.  When both live and dead recoveries are modeled, each group would have 4 parameters: S, r, p, and F (Fidelity). Thus, 8 windows would be cascaded on the screen.

The parameter matrices determine the number of parameters that will be estimated.  Commands are available to set all the parameter matrices to a particular type, or to set the current window to a particular type.  See Constant Matrix for how parameters are specified constant for each occasion, Time Matrix for a parameter matrix that is time-specific, Age Matrix for a parameter matrix that is age-specific, Time and Age Matrix for an example where parameters are both time and age specific, and All Different Matrix for cases where you want every parameter value in the PIM to be different.

Included on the PIM window are push buttons to Close the window (but the values are not lost -- just not displayed), Help to display this help screen, PIM Chart to graphically display the relationship among the PIM values, and + and - in increment or decrement, respectively, all the values in the PIM Window by 1.

Although the most general method of setting up the PIM's is via the PIM Windows, the quickest and most efficient for most models is to use the right-click and block dragging features of the PIM Chart.

Parameter matrices can be manipulated to specify various models.  The following are the parameter matrices for live recapture data to specify a {Phi(g*t) p(g*t)} model for a data set with 5 encounter occasions and 2 groups.

Apparent Survival Group 1

1  2  3  4
2  3  4
3  4
4

Apparent Survival Group 2

5  6  7  8
6  7  8
7  8
8

Recapture Probabilities Group 1

9  10  11  12
10  11  12
11  12
12

Recapture Probabilities Group 2

13  14  15  16
14  15  16
15  16
16

To reduce this model to {Phi(t) p(t)}, the following parameter matrices would work.

Apparent Survival Group 1

1  2  3  4
2  3  4
3  4
4

Apparent Survival Group 2

1  2  3  4
2  3  4
3  4
4

Recapture Probabilities Group 1

5  6  7  8
6  7  8
7  8
8

Recapture Probabilities Group 2

5  6  7  8
6  7  8
7  8
8

In the above example, the parameters are equal across groups, i.e., the parameter estimates are forced to be the same for each group.

The following parameter matrices have no time effect, but do have a group effect.  Thus, the model is {Phi(g) p(g)}.

Apparent Survival Group 1

1  1  1  1
1  1  1
1  1
1

Apparent Survival Group 2

2  2  2  2
2  2  2
2  2
2

Recapture Probabilities Group 1

3  3  3  3
3  3  3
3  3
3

Recapture Probabilities Group 2

4  4  4  4
4  4  4
4  4
4

Additional constraints can be placed on the real parameters through the beta parameters defined with the Design Matrix.  An example of model {Phi(g+t) p(g+t)} is shown in the Design Matrix Advance 2 section.  The Pre-Defined Models option provides you with a set of pre-defined models, including both PIM and Design Matrix coding.  Thus, this option is a good way to learn how to code models in MARK.

Other options for building PIMs are to Copy the current PIM to the clipboard, and then Paste the clipboard into another PIM.

All of the examples above show upper-triangular PIMs.  However, many data types use single rows of parameters, such as the closed capture modelsMixture models will use a square PIM because each mixture is a separate row.  Even more complex PIMs are used with the Cormack-Jolly-Seber model that has mixtures on phi.