## Observation Models

There are two general types of observation models that are commonly used in fisheries stock assessment: (1) models for comparing trend information, and (2) models for comparing composition information. Examples of trend information include the CPUE index described in the data section for trends in relative abundance, trends in absolute abundance, and trends in mean age or mean body weight. Relative abundance indices are not always informative about the underlying population parameters in the population dynamics models. CPUE indices that do not vary over time (e.g., catch rates are relatively constant) provide no contrast in which to inform model parameters. This is nearly equivalent to performing a regression analysis in which there is no variation in the independent variable. Furthermore, CPUE indices that display a one way trip (either a continuous decline or continuous increase over time) do not allow for separation of variables in the population model. That is, in one-way trip data, we cannot distinguish between a small productive population or a very large unproductive population.

The basic observation model for trend information is defined by eqn , and it is common to assume that observation errors are multiplicative and thus log-normally distributed. Given an estimate of abundance from the population model (e.g., Bt conditioned on the observed catch history or effort and initial guesses for the population parameters) we can reexpress eqn  in terms of the residuals:

where q is a scalar term, or an observation model parameter. Equation  forms the statistical criterion (e.g., as least-squares criterion) in which to evaluate alternative parameter combinations (both population model parameters and observation model parameters). It is also possible to eliminate the nuisance parameters (e.g., scalar parameters such as q) by computing the maximum-likelihood estimate of the parameter conditional on other model parameters (see 'Further reading'). In cases where the index is an absolute measure of biomass or mean body weight, etc., it is not necessary to include the scaling term (q) into the parameter set, and estimation of model parameter proceeds by making direct comparisons.

Compositional information such as catch-at-age or size-frequency data are assumed to come from a multinomial distribution. To utilize these types of data to estimate model parameters an age-structured or stage-structured population model is necessary to compute the predicted proportions. Thus, the observation model is simply the population model that generates the predicted proportions-at-age or proportions-at-length and the assumed sample size. It is common to use the actual number of fish measured or aged as the effective sample size for the multinomial distribution; however, many studies have demonstrated that this approach tends to overweight the compositional information and has the potential to produce strong biases or confounding in the parameter estimates. The effective sample size for the multinomial distribution is nearly always much less than the total number of fish aged (or measured), and it is often thought that the number of vessels sampled or the number of sampling trips conducted to collect the information is a better approximation of the effective sample size. 