The dataset, stownet,
contains length-frequency data from a size selectivity experiment on
haddock from a fishing trawl. The dataset contains the length classes
and counts of fish retained in the codend and the cover. The codend is
the aft part of a trawl and the cover is a fine mesh placed around the
codend to determine the number and length of fish able to swim through
the codend and represent those released from the trawl. The data are for
19 replicate hauls, but we will ignore haul ID for this exercise. This
exercise is described in detail in Millar, R.B. (2011), data are from
Clark (1957).
Fit a logistic regression model to the data using a binomial
distribution in TMB:
\[\begin{align}
\eta_{i} &= \beta_{0} + \beta_{1}L_{i}\\
&\\
p_{i} &= \frac{1}{1+exp(-\eta)}\\
&\\
N &= Codend + Cover\\
&\\
Codend &\sim Binomial(N, p)
\end{align}\]
Plot the observed probability of catch in the
codend by length along with the expected probability vector from the
model.
Of particular interest to management is the length of 50% retention, or \[l_{50} = -\beta_{0}/\beta_{1}\] Find the estimate and standard error for this value.
References:
- Clark, J.R. 1957. Effect of length
of haul on cod end escapement. ICNAF/ICES/FAO workshop on selectivity,
Lisbon. Paper S25.
- Millar, R.B. 2011. Maximum likelihood
estimation and inference: With examples in R, SAS, and ADMB. Wiley,
Auckland, New Zealand. pg. 155-160.