Use of the delta method to evaluate the precision of assessments that fix parameter values

Many stock assessments fix values for influential, but poorly known, parameters such as the natural mortality rate or the steepness of the stock-recruitment relationship, which leads to published estimates of uncertainty being underestimates. The delta method, in which the partial derivatives of the model outputs with respect to all of the parameters of the model can be easily obtained numerically using likelihood profiling, can be used to provide quick, but approximate, estimates of precision for model outputs when some key parameters are fixed.

► Fixed parameters cause underestimated imprecision.
► The partial derivatives needed for the delta method can be obtained numerically rather than analytically.
► Likelihood profiling is the correct approach for numerical estimation.
► Delta method results are very close to the corresponding asymptotic estimates.
► Tail probabilities are unreliable, and confidence intervals are best expressed at the 50% level (inter-quartile range).