nest success as multi-state, fixed values are singular

[markmiller]

I analyzed the example mallard nest success data using the multi-state model, attempting to follow the approach described in Devineau et al (2014). I obtain the same estimates for Psi 1 to 2 as they report (shown below).

However, Psi 2 to 1, which I constrain to 0, is identified as a singular value. Why is that? Using the mlogit link does not resolve this issue.

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  SIN Link Function Parameters of {mallard nest success as multi-state b}
                                                                   95% Confidence Interval
 Parameter            Beta         Standard Error      Lower           Upper
 ----------------  --------------  --------------  --------------  --------------
    1:S 1:State 1  0.0000000       0.0000000       0.0000000       0.0000000     
    2:Psi 1 to 2   -1.1329283      0.0122286       -1.1568964      -1.1089602   
    3:Psi 2 to 1   -0.0995092      426.95820       -836.93760      836.73858     


        Real Function Parameters of {mallard nest success as multi-state model b}
                                                         95% Confidence Interval
  Parameter      Estimate    Standard Error    Lower          Upper
 ---------------  --------------  --------------  --------------  --------------
 1:S 1:State 1  1.0000000    0.0000000     1.0000000    1.0000000  Fixed               
 2:Psi 1 to 2     0.0471712   0.0025925      0.0423422    0.0525207                           
 3:Psi 2 to 1     0.0000000   0.0000000      0.0000000    0.0000000  Fixed               

 Attempted ordering of parameters by estimatibility:
  1 2
 Beta number 3 is a singular value.

Perhaps the issue is in the details. I constrain detection to 1 for all days and all nests, even for days when a nest was not checked, but I suspect that is not affecting estimates. Also, I did not remove a nest once failure was detected. Since nests are only included in the capture history on days they are checked, and once failed are always failed, I suspect removing failed nests does not affect parameter estimates, but perhaps I am mistaken.

Thank you for any thoughts on why a fixed value is identified as singular.

[jhines]

If you've constrained psi(2-1)=0, then the beta for psi(2-1) is never used in the likelihood and shouldn't be included in the design matrix. By default, MARK will use an identity design matrix and include a column for that beta. You can get rid of the singular error message by specifying a design matrix where there are 3 rows (1 for each real parameter), and 1 column (1 for each estimated beta). Since you've fixed S=0 as well, the design matrix should be:

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         beta1   
S         0
Psi 1-2   1
Psi 2-1   0

Usually, MARK handles the situation pretty well when you use the identity design matrix and gives correct variances for the other parameters.

Jim

[markmiller]

Thanks, Jim. That was very helpful.

Out of curiosity I wondered what would happen if I filled in the capture histories between visits to live nests, so that a capture history of:

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.....1.....1..2...

became

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.....1111111..2...

The resulting parameter estimate and SE were identical, as I was expecting, since to my knowledge, I was not providing any additional information by filling them in.

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          SIN Link Function Parameters of { S(~1)p(~1)Psi(~-1 + stratum:tostratum) }
                                                              95% Confidence Interval
 Parameter                   Beta         Standard Error      Lower           Upper
 ---------------------  --------------  --------------  --------------  --------------
1:Psi:stratum1:tostrat  -1.1329283      0.0122286       -1.1568964      -1.1089602   


          Real Function Parameters of { S(~1)p(~1)Psi(~-1 + stratum:tostratum) }
                                                             95% Confidence Interval
Parameter              Estimate    Standard Error      Lower           Upper
 -------------------  --------------  --------------  --------------  --------------
1:Psi s1 to2 g1 c1 c1   0.0471712    0.0025925    0.0423422    0.0525207                           
2:S s1 g1 c1 c1 a0 o1   1.0000000    0.0000000    1.0000000    1.0000000       Fixed               
3:Psi s2 to1 g1 c1 c1   0.0000000    0.0000000    0.0000000    0.0000000       Fixed               

     CPU Time in seconds for last procedure was 0.17