Questions about the adjusted lognormal error distribution used in
calculating goodness of fit for survey and commercial age composition
data in preliminary ASPM models

(*Contribution to Agenda item 2*)

February 28, 2008

Larry Jacobson

Northeast Fisheries Science Center

The likelihood of age composition data and details of calculation are
important in SCCA modeling. For example, in preliminary SS2 models for
Georges Bank yellowtail flounder, results for recent years depend on
whether the model solution favors (tends to fit) survey and commercial
age composition data at the expense of survey trend data, or vice-versa.
All approaches are approximate because the statistical distribution of
age composition data is hard to specify and model misspecifications
complicate the matter further. Still, it is important to understand and
weight the advantages of the approach taken, particularly if it is new.

According to equation A21.9 in the working paper 2.2-a by Butterworth
and Rademeyer, the negative log likelihood of survey and commercial age
composition data in the preliminary ASPM for Gulf of Maine cod is:

$- \text{ln}\left( L^{\text{CAA}} \right) = \sum_{y}^{}{\sum_{a}^{}\left\{ \text{ln}\left( \sigma_{\text{com}}/\sqrt{p_{y,a}} \right) + p_{y,a}\left\lbrack \text{ln}\left( p_{y,a} \right) - \text{ln}\left( {\hat{p}}_{y,a} \right) \right\rbrack^{2}/2\sigma_{\text{comm}}^{2} \right\}}$
A2.19

Where $p_{y,a}$and${\hat{p}}_{y,a}$are observed and predicted
proportions of the total catch age *a* during year *y*, and the standard
deviation$\sigma_{\text{comm}}$associated with the catch at age, which
is estimated in the fitting procedure using:

${\hat{\sigma}}_{\text{comm}} = \frac{\sqrt{\sum_{y}^{}{\sum_{a}^{}\left\{ p_{y,a}\left\lbrack \text{ln}\left( p_{y,a} \right) - \text{ln}\left( {\hat{p}}_{y,a} \right) \right\rbrack^{2} \right\}}}}{\sum_{y}^{}{\sum_{a}^{}1}}$
A2.21

Thus, the log likelihood term is calculated assuming that measurement
errors in catch at age arise from a weighed log-normal distribution.
Lognormal distributions are often used for catch at age data and not
cause for concern. However, the weighting factor (i.e. the observed
proportion$p_{y,a}$) for each age is unusual and has uncertain
statistical properties that should be evaluated and explained. The text
following equation A2.2.1 explains "*Punt (pers. comm.) advised
weighting by the observed proportions (as in equation A2.19) so that
undue importance is not attached to data based upon a few samples
only.*"

The following are three concerns that occur to this reviewer
immediately. First, the magnitude of the proportion$p_{y,a}$has no
necessary relationship to the number of samples. A very low observed
proportion would naturally occur for a rare event in any number (small
or large) of samples. Secondly, this approach gives zero weight to data
in age composition bins with zero observations, even though the data
likely contain valid and reliable information concerning the rarity of
fish of that particular age. Finally, use of the observed, rather than
the predicted, proportion implies that the observed proportions are more
precise as estimates of the population proportion than the predicted
ones. If the observed proportions are more precise, then why bother
calculating the predicted values? In any case, the observed values
probably include measurement and process errors that contribute variance
that is not desirable in weights.