Design Matrix — Advanced Applications 2

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Design Matrix — Advanced Applications 2

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 

Consider the additive effects model that is only possible by using the design matrix: {Phi(g+t) p(g+t)}.  In this model, the  time effect is the same for each group, with the group effect additive to this time effect.  In the following matrix, column 1 is the group effect for apparent survival, columns 2-5 are the time effects for apparent survival, column 6 is the group effect for recapture probabilities, and columns 7-10 are the time effects for the recapture probabilities.  Note that the group effect is zero for the second group, and 1 for the first group.  

1  0 0 0 1 0 0 0 0 0 
0 1 0 0 1 0 0 0 0 0 
0 0 1 0 1 0 0 0 0 0 
0 0 0 1 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 1 
0 0 0 0 0 0 1 0 0 1 
0 0 0 0 0 0 0 1 0 1 
0 0 0 0 0 0 0 0 1 1 
0 0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 0 

An alternative way of coding the above design matrix is shown next.  This alternative uses an intercept term, followed by the time effects, followed by the group effects.  The first column corresponds to the estimate of survival for the second group on the fourth interval.  The resulting estimates of the real parameters will be identical to the example above, but the interpretation of the beta estimates will be different.  In general, an infinite number of design matrices can produce identical estimates of the real parameters.

1  1 0 0 1 0 0 0 0 0 
1 0 1 0 1 0 0 0 0 0 
1 0 0 1 1 0 0 0 0 0 
1 0 0 0 1 0 0 0 0 0 
1 1 0 0 0 0 0 0 0 0 
1 0 1 0 0 0 0 0 0 0 
1 0 0 1 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 1 0 0 1 
0 0 0 0 0 1 0 1 0 1 
0 0 0 0 0 1 0 0 1 1 
0 0 0 0 0 1 0 0 0 1 
0 0 0 0 0 1 1 0 0 0 
0 0 0 0 0 1 0 1 0 0 
0 0 0 0 0 1 0 0 1 0 
0 0 0 0 0 1 0 0 0 0 

The following example is also equivalent to the 2 preceding examples.  However, in this case, the design matrix is constructed to compute the overall mean survival and recapture rates as the “intercept” parameter.  This approach requires the use of 1 and -1 coding, instead of 0 and 1 coding.  The group effect is now half of the difference between the 2 groups, with the “half” effect added onto group 1, but subtracted from group 2.

1  1 0 0 1 0 0 0 0 0 
1 0 1 0 1 0 0 0 0 0 
1 0 0 1 1 0 0 0 0 0 
1 -1 -1 -1 1 0 0 0 0 0 
1 1 0 0 -1 0 0 0 0 0 
1 0 1 0 -1 0 0 0 0 0 
1 0 0 1 -1 0 0 0 0 0 
1 -1 -1 -1 -1 0 0 0 0 0 
0 0 0 0 0 1 1 0 0 1 
0 0 0 0 0 1 0 1 0 1 
0 0 0 0 0 1 0 0 1 1 
0 0 0 0 0 1 -1 -1 -1 1 
0 0 0 0 0 1 1 0 0 -1 
0 0 0 0 0 1 0 1 0 -1 
0 0 0 0 0 1 0 0 1 -1 
0 0 0 0 0 1 -1 -1 -1 -1 

See Design Matrix Scaling Covariates for why “reasonable” values should be used in the design matrix.