I acknowledge so much the help that very proficient people are giving in this forum, I want to remark how useful it is.
I have some questions about parameter redundancy and identifiability that is something E-SURGE handles very well and, at the same time, something very important for these models.
Choquet wrote:In the same way, E-SURGE gives you a reliable rank (not based on the hessian)
and the list of parameters which are redundant when the model is not full rank.
This is really crucial for this kind of model.
I have read carefully the E-Surge manual as well as a very useful paper on this (Gimenez et al 2004).
Unfortunately my reduced mathematical background makes my learning curve somehow slow and let me with some doubts I would like someone could clear by answering to the following queries.
1. I am not sure to catch the difference between the identifiability and redundancy concepts. To me (i) a parameter or a function of parameters are identifiable if the model reach a global minimum for one specific value of that parameter (or function of parameters) and in the above paper is said that (ii) some parameters are redundant when the model can be expressed as a function of fewer than the original number of parameters. So for example, in the classical CJS example, the last phi and p are unidentifiable (not estimable separately) but their product (as said above, function of parameters) is identifiable. Are they also considered redundant? any further explication on these concepts?
2. As told in the above cited paper, the best method to deal with this given it provides answers to intrinsic (due to model structure) and extrinsic (due to data characteristics) redundancy and allows to know which parameters are redundant, would be the formal derivative matrix, also called CMF (Catchpole, Morgan and Freeman).
In E-SURGE, the numerical version of the CMF approach is implemented (same features but, unlike the formal derivative matrix, no estimable functions of the redundant parameters are explicitly identified).
Even though, I don't understand very well the output of parameter identifiability for my models that is explained in the Figure 32 of the manual where it is said:
E-Surge manual wrote: ...The above temporary window displays for each of the 5 points near the MLE the number of singular values of the derivative matrix, the number of additional singular values below a less selective threshold and indices of the potentially redundant mathematical parameters.
Here there are two real data examples of the (1) parameter identifiability excel sheet and the (2) reduced set of parameters excel sheet of a over-parameterized model I have run {IS(t) S(g.f.t) T(g.f) C(a(1)+a(2).f.t) SA(a(1).g.f.t(1)+a(2).g.f.t)} (2 groups, 3 states, 4 events, 9 occasions and 1 age class).
1)
2)
2.1 What is going on with parameters 38, 39, 40, 41, 60 and 61? and to which parameters do they correspond (there is a sheet of reduced parameters and another one with parameters)?
I know there is said that "5 quantities solutions of 1 partial derivative equations, made of redundant parameters (indices below) are estimables" but I don't catch what does it means (shame on me!).
2.2 Finally it is said that 13 quantities solutions of 1 partial derivative equations, made of redundant parameters (indices below) are estimables (the final sentence is referred to the MLE itself, the other prior four are points near the MLE). Again, what does it means?
2.3 Once I have found the redundant parameters I could fix them (must I fix them to one or whatever value?) and repeat the analysis, isn't it?
If you could explain me these results in some simple words, it would make things easier and my learning curve would improve a lot for sure.
Simone