Design Matrix — Advanced Applications
Advanced applications of the Design Matrix involve constraining parameters across groups. Suppose that a set of recapture data have 2 groups, say males and females. Then, the 4 parameter matrices for a time model for each parameter for each group might look like the following.
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
An identity design matrix, or Full Design Matrix, results in the model {Phi(g*t) p(g*t)}. The identity matrix for this example would look like the following.
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Each row corresponds to a parameter, and in this case, each column corresponds to a parameter.
Suppose that the model {Phi(t) p(t)} is desired, but with no change in the parameter matrices, shown above. Thus, 16 parameters need to be constrained to 8 parameters. The following design matrix will make these constraints. There are still 16 rows in the matrix, because there is still 16 parameters in the model. However, the rows of the matrix are identical for the first parameter and the fifth parameter, because these 2 parameters are now the apparent survival rate for the survival across the first interval. Likewise, row 2 and row 6 are identical, to model the separate apparent survival rates for interval 2 for each group as the same parameter. This process is continued for all 8 parameters with the following matrix.
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
This model could also be developed by manipulating the parameter matrices. To demonstrate this, the following 4 parameter matrices would result in the same estimates at the above design matrix would generate with the original 4 parameter matrices with 16 parameters. Note that the parameter matrix is identical for each group, hence, the parameter estimates are identical for each group, and thus model {Phi(t) p(t)}results.
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
See Design Matrix Advanced 2 for examples of building additive effects models.
See Design Matrix Scaling Covariates for why “reasonable” values should be used in the design matrix.