alternative closed capture design matrices: parsimony?

[achatinella]

I am writing with a question about alternative parameterizations of design matrices for closed population models with individual covariates and groups. One type of parameterization is generally favored in the Mark Book over other options, and I am wondering why this is.

Bear with me as I explain this problem in detail.

Consider a Huggins closed population model for two trapping occasions and two groups with one individual covariate, which we will call ‘mass.’ The ‘standard’ design matrix for the fully parameterized version of this model would be:

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11
1 m 1 1 m m 1 0 0 0 0
1 m 0 1 0 m 0 0 0 0 0
1 m 1 0 m 0 0 0 0 0 0
1 m 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 m 1 m
0 0 0 0 0 0 0 1 m 0 0

Here, B1 is the intercept for the P parameters. B2 is a continuous covariate (mass). B3 is a dummy variable for group. B4 is a dummy variable for time (1 if first occasion, 0 otherwise). B5 is a mass*group interaction. B6 is a mass*time interaction. B7 is a group*time interaction. B8 is the intercept for the C (recapture) parameters. B9 is a continuous covariate (mass) for the C parameters. B10 is a dummy variable for group for the C parameters. B11 is a mass*group interaction for the C parameters.

The first four rows of the design matrix are for P parameters, and the last two rows are for C parameters. The particular parameter estimated by each row is Pij, where i indicates group membership and j indicates time. Rows 1 - 6 therefore estimate P11, P21, P12, P22, C12, and C22 respectively. This correspondence between rows and P/C parameters is not the default in Mark, but I rearranged the PIMS (swapped parameter 2 and 4) to make it this way. You will see why in a moment. Note that the P parameters and C parameters are estimated with separate slopes and intercepts. The equations for each real parameter are therefore:

P11 = (B1 + B3 + B4 + B7) + (B2 + B5 + B6)*mass

P21 = (B1 + B4) + (B2 + B6)*mass

P12 = (B1 + B3) + (B2 + B5)*mass

P22 = B1 + B2*mass

C12 = (B8 + B10) + (B9 + B11)*mass

C22 = B8 + B9*mass

An alternative matrix is possible, which, in my opinion, is more easily interpretable, and it has one fewer beta parameter.

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10
1 m 1 1 m m 1 0 0 0
1 m 0 1 0 m 0 0 0 0
1 m 1 0 m 0 0 0 0 0
1 m 0 0 0 0 0 0 0 0
1 m 1 0 m 0 0 1 m 1
1 m 0 0 0 0 0 1 m 0

B8 is now a dummy variable that codes for a 'behavioral effect' i.e. whether a C parameter or a P parameter is being estimated. B9 is a mass*behavioral effect interaction. B10 is a behavioral effect*group interaction. Note that B11 is absent, we are estimating the same real parameters as before, but with one less beta parameter.

The equations for each of the P parameters remain unchanged. But the equations for the C parameters (and the interpretations of C parameters) differ. Notably,

C12 = (B1 + B3 + B8 + B10) + (B2 + B5 + B9)*mass

C22 = (B1 + B8) + (B2 + B9)*mass

Using the second design matrix, the equation for C12 (the recapture rate for group 1, which is necessarily for time 2) can be directly compared to that for P12 (the probability of initial capture for group 1 in time 2). Here, (B8 + B10) describes the change in intercept for C12 versus P12, and B9 describes the change in slope for C12 versus P12. No similar easy comparisons are possible for the first design matrix, because the P and C parameters do not share common beta parameters. A similar comparison can be constructed for C22 and P22.

So my question for the list is: given the emphasis on matrices like the first one in the Mark Book, is there some reason why I should avoid matrices like the second design matrix presented here, which seems more parsimonious and straightforward?

Thank you in advance for your responses. Feel free to respond off list: jkellner@plantbio.uga.edu. I will summarize responses and post them to the list.

Jim

[cooch]

...
Note that B11 is absent, we are estimating the same real parameters as before, but with one less beta parameter.

Meaning, its a different model. The number of reconstituted 'real' parameters is determined by the PIM structure - the number of columns in the design matrix represents the number of potentially estimable parameters. If your DM has one fewer beta, then in fact its a reduced parameter model, relative to the first DM you describe. So, its hard to compare relative degrees of 'interpretability', since the models are not equivalent.

Your general point about 'structure of the DM' and 'interpretability' is a good one - and one that is perhaps not emphasized enough in Chapter 7 - basically, you have a large number of DM which are equivalent in terms of model fit (i.e., the model deviances and values of reconstituted parameters will be identical), but where the interpretation of the beta terms differs. The DM approach discussed most often in 'the book' is standard reference cell coding - where betas are interpreted as deviations from a control (intercept). There are, of course, other ways to code the DM (there is a reason that most of the textbooks on linear models are many hundreds of pages long, after all).