### Michael Schaub, Olivier Gimenez, Benedikt R. Schmidt, and Roger Pradel. 2004. Estimating survival and temporary emigration in the multistate capture–recapture framework. Ecology 85:2107–2113.

Appendix A. MAPLE session for assessing the intrinsic identifiability of the temporary emigration models.

A pdf-version of Appendix A is available for viewing.

This appendix provides the MAPLE session for assessing the intrinsic identifiablity of the temporary emigration models (Appendix B).

Catchpole and Morgan (1997) developed the mathematical principles for testing the intrinsic identifiablity. While applying to multistate models, one has to do the following steps (Gimenez et al. 2003):

1. Write down the elementary probabilities of the m-array Omega, and store the relevant components in a vector Mu;
2. Differentiate Mu with respect to any parameter in the model (gathered in a vector th); one thus obtains a derivative matrix denoted A;
3. Determine the rank of A. The rank of A is the number of the separately identifiable quantities. If it is equal to the number of the parameters in the model, all parameters are separately identifiable and the model is said to be full rank. Otherwise, the model is parameter redundant; in this case, go to the following step;
4. Determine the linearly independent solutions of the equation alpha × A = 0 where alpha forms a base of the left kernel of A. The vector alpha indicates which parameters are identifiable: at each place where a component of alpha is 0, the corresponding component of th indicates an identifiable parameter (see below).

Gimenez et al. (2003) developed a maple code to calculate the elementary probabilities of the m-array freely available at the URL: ftp://ftp.cefe.cnrs-mop.fr/biom/PRM (see text files Maple_procedures.txt and readme.txt). In what follows, we used the function CreateMarray which called the two external functions Merge and ProdM. These three procedures have to be loaded by copying and pasting at the beginning of the maple session, after having loaded the linear algebra and partial differential equations toolboxes (see below).

To save space, we present the MAPLE sessions for only two models – the syntax is easily adapted for other models. The MAPLE syntax is in the courier font and comments are flaged by a #.

# Testing the identifiability of model {S., ,, p.} (model #64 in Appendix B)

# This model has 4 parameters: S, , and p. We consider five capture occasions.

# Load the linear algebra and partial differential equations toolboxes.

with(linalg):with(PDEtools):

# Load the function CreateMarray (maple_procedures.txt)

# Form the matrix of probabilities

Phi1 := matrix([[S*(1-psiOU),S*psiOU],[S*psiUO,S*(1-psiUO)]]):

Phi2 := matrix([[S*(1-psiOU),S*psiOU],[S*psiUO,S*(1-psiUO)]]):

Phi3 := matrix([[S*(1-psiOU),S*psiOU],[S*psiUO,S*(1-psiUO)]]):

Phi4 := matrix([[S*(1-psiOU),S*psiOU],[S*psiUO,S*(1-psiUO)]]):

P2 := diag(p,0):

P3 := diag(p,0):

P4 := diag(p,0):

P5 := diag(p,0):

Phi := augment(Phi1,Phi2,Phi3,Phi4):P := augment(P2,P3,P4,P5):

Omega := CreateMarray(Phi,P);

# Omega is a square matrix with dimension 2(t – 1) (where t is the number of capture occasions) and contains the probabilities of the multistate model in the m-array format. For t occasions of capture and the two states O (observable) and U (unobservable), Omega has the following form:

 Time of first recapture (State of recapture) Time of release State of release 2 (O) 2 (U) 3 (O) 3 (U) … t (O) t (U) 1 O 1,1 0 1,3 0 … 1,(2t–3) 0 1 U 2,1 0 2,3 0 … 2,(2t–3) 0 2 O 0 0 3,3 0 ... 3,(2t–3) 0 2 U 0 0 4,3 0 ... 4,(2t–3) 0 …. ... ... ... ... ... ... ... ... t–1 O 0 0 0 0 ... (2t–3),(2t–3) 0 t–1 U 0 0 0 0 ... 2(t–1),(2t–3) 0

where, e.g., 1,3 is the probability to recapture an animal at the third occasion that was captured at the first occasion and was not captured at the second occasion. Omega in the present form is, however, not yet the correct m-array for the model. Because state U is unobservable, there are no releases from this state, and thus 2,1, 2,3, 2,(2t-3), 4,3, 4,(2t–3) and 2(t–1),(2t–3) must be set to zero. This is done in the next step – the vector Mu contains only the non-null probabilities of the m-array of the model.

Mu:=vector(10):

Mu[1]:=Omega[1,1];Mu[2]:=Omega[1,3];Mu[3]:=Omega[1,5];

Mu[4]:=Omega[1,7];Mu[5]:=Omega[3,3];Mu[6]:=Omega[3,5];Mu[7]:=Omega[3,7];Mu[8]:=Omega[5,5];Mu[9]:=Omega[5,7];Mu[10]:=Omega[7,7];

# define vector th containing all model parameters

th:=vector(4,[S,psiOU,psiUO,p]);

# create the matrix A

A:=array(1..4,1..10):i:='i': j:='j':

for i to 4 do

for j to 10 do

A[i,j]:=diff(log(Mu[j]),th[i]):

od;od;evalm(A);

# calculation of the rank of A

rank(A);

# result: rank = 4

# Interpretation: the rank of A is equal to the number of parameters in the model. All parameters in this model are therefore separately identifiable; the model is full rank.

# Testing the identifiability of model {St, ,, pt} (model #18 in Appendix B)

# We consider 5 capture occasions. The model has 24 parameters: S1, S2, S3, S4, ,,,,,,,,,,,,,,,, p1, p2, p3, and p4.

# Load the linear algebra and partial differential equations toolboxes.

with(linalg):with(PDEtools):

# Load the function CreateMarray (maple_procedures.txt)

# Form the matrix of probabilities

# group A

PhiA1 := matrix([[S1*(1-psiOU1A),S1*psiOU1A],[S1*psiUO1A,S1*(1-psiUO1A)]]):

PhiA2 := matrix([[S2*(1-psiOU2A),S2*psiOU2A],[S2*psiUO2A,S2*(1-psiUO2A)]]):

PhiA3 := matrix([[S3*(1-psiOU3A),S3*psiOU3A],[S3*psiUO3A,S3*(1-psiUO3A)]]):

PhiA4 := matrix([[S4*(1-psiOU4A),S4*psiOU4A],[S4*psiUO4A,S4*(1-psiUO4A)]]):

PA2 := diag(p1,0):

PA3 := diag(p2,0):

PA4 := diag(p3,0):

PA5 := diag(p4,0):

PhiA := augment(PhiA1,PhiA2,PhiA3,PhiA4):

PA := augment(PA2,PA3,PA4,PA5):

OmegaA := CreateMarray(PhiA,PA);

# group B

PhiB1 := matrix([[S1*(1-psiOU1B),S1*psiOU1B],[S1*psiUO1B,S1*(1-psiUO1B)]]):

PhiB2 := matrix([[S2*(1-psiOU2B),S2*psiOU2B],[S2*psiUO2B,S2*(1-psiUO2B)]]):

PhiB3 := matrix([[S3*(1-psiOU3B),S3*psiOU3B],[S3*psiUO3B,S3*(1-psiUO3B)]]):

PhiB4 := matrix([[S4*(1-psiOU4B),S4*psiOU4B],[S4*psiUO4B,S4*(1-psiUO4B)]]):

PB2 := diag(p1,0):

PB3 := diag(p2,0):

PB4 := diag(p3,0):

PB5 := diag(p4,0):

PhiB := augment(PhiB1,PhiB2,PhiB3,PhiB4):

PB := augment(PB2,PB3,PB4,PB5):

OmegaB := CreateMarray(PhiB,PB);

Omega:=augment(OmegaA,OmegaB);

# select the non-null probabilities of the m-array

Mu:=vector(20):Mu[1]:=Omega[1,1]:Mu[2]:=Omega[1,3]:Mu[3]:=Omega[1,5]:Mu[4]:=Omega[1,7]:Mu[5]:=Omega[3,3]:Mu[6]:=Omega[3,5]:Mu[7]:=Omega[3,7]:Mu[8]:=Omega[5,5]:Mu[9]:=Omega[5,7]:Mu[10]:=Omega[7,7]:Mu[11]:=Omega[1,9]:Mu[12]:=Omega[1,11]:Mu[13]:=Omega[1,13]:Mu[14]:=Omega[1,15]:Mu[15]:=Omega[3,11]:Mu[16]:=Omega[3,13]:Mu[17]:=Omega[3,15]:Mu[18]:=Omega[5,13]:Mu[19]:=Omega[5,15]:Mu[20]:=Omega[7,15]:

# define vector th containing all model parameters

th:=vector(24,[S1,S2,S3,S4,psiOU1A,psiOU2A,psiOU3A,psiOU4A,psiOU1B,psiOU2B,psiOU3B,psiOU4B,psiUO1A,psiUO2A,psiUO3A,psiUO4A,psiUO1b,psiUO2B,psiUO3B,psiUO4B,p1,p2,p3,p4]);

# create the matrix A

A:=array(1..24,1..20):i:='i': j:='j':for i to 24 do

for j to 20 do

A[i,j]:=diff(log(Mu[j]),th[i]):

od;od;evalm(A);

# calculation of the rank of Arank(A);

# result: rank=19; the rank is less than the number of the parameters in the model (24)

# inidicating that the model is parameter redundant; there are only 19 estimable quantities.

# determination of alpha

zero:=vector(coldim(A),0);alpha:=linsolve(transpose(A),zero,'r',t);

# result:

# By comparing with the vector th components, one will conclude that 11 parameters (the number of null components in alpha) are separately identifiable:

S1,S2,S3,psiOU2A,psiOU3A,psiOU2B,psiOU3B,psiUO3A,psiUO4A,psiUO3B,p2,p3

The other parameters

S4, psiOU1A, psiOU4A, psiOU1B, psiOU4B, psiUO1A, psiUO2A, psiUO4A, psiUO1B, psiUO2B, psiUO4B, p1 and p4

are not separately identifiable. Indeed, the parameter psiOU1A (e.g.) cannot be estimated separately, because at the fifth place in vector alpha there is an expression and not a 0. If model {St, ,, pt} would be used with real data, one would nevertheless get an apparent estimate of psiOU1A. However, this estimate cannot be interpreted. Actually, these 13 not separately identifiable parameters are functions of 19 – 11 = 8 (rank of A minus number of zeros in alpha) complex quantities. These complex quantities are also estimable (see Gimenez et al. 2003 for further details).

Literature cited

Brownie, C., J. E. Hines, J. D. Nichols, K. H. Pollock, and J. B. Hestbeck. 1993. Capture–recapture studies for multiple strate including non-Markovian transitions. Biometrics 49:1173–1187.

Catchpole, E. A., and B. J. T. Morgan. 1997. Detecting parameter redundancy. Biometrika 84:187–196.

Catchpole, E. A., B. J. T. Morgan, and A. Viallefont. 2002. Solving problems in parameter redundancy using computer algebra. Journal of Applied Statistics 29:625–636.

Gimenez, O., R. Choquet, and J.-D. Lebreton. 2003. Parameter redundancy in multistate capture-recapture models. Biometrical Journal 45:704.

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