[R-sig-ME] Non linear mixed models on binomial distributed data

[Juan Santos]

Dear list members,

I frecuently use this model:

p(l)=\frac{s\times r(l)}{(1-s)+s\times r(l)} (1)

To estimate the expected catch rates in a selective fishing gear (the 
test gear) , which has been fishing in paralel with a non selective gear 
(the control gear) . p(l) depends on:

s=the split parameter: defining the probability of a fish to enter in 
the test gear (s) or the control gear (1-s)
r(l)= the likelihood of fish retention in the test trawl. This 
likelihood is conditioned to fish body length, and describes the test 
gear size selection.

the relationship between the retention likelihood and the fish length 
r(l) is used to be defined as a logit link function:

r(l)=\frac{exp(\beta_1+\beta_2\times l )}{1+exp(\beta_1+\beta_2\times l 
)} (2)


In overall, there is a total of three parameters to be estimated: s, 
\beta_1, \beta_2. I use a non linear approach to estimate such 
parameters, by maximizing the binomial log-likelihood function:

\sum_l{N_{l}^T\times \log{p(l)}+N_{l}^C\times \log{1-p(l)}} (3)

where N_{l}^T is the number of fishes per length-class caught in the 
test gear, and N_{l}^C is the numbers caught in the control codend.


for a given Test gear, we use to perform several hauls, showing in many 
cases a high variation of the estimated p(l) between hauls. To account 
for such variation I use the bootstrap (with resampling scheme based on 
resampling between hauls and fishes within hauls) to estimate the error 
of the mean curve (estimated using the data after pooling the different 
hauls).

I ´m wondering if I can shift to a non linear mixed modeling approach, 
where the hauls are considered as a random component. At the moment I 
only could find such approach but using leats squares as minimization 
criteria. But I could not find a way to keep using the equation (3) as 
target criteria.


All the best,

Juan Santos