A Parametric Bootstrap of Catch-Age Compositions Using the Dirichlet Distribution

Analyses of length and age distributions using continuation-ratio logits

“…Thirdly, overdispersion is easily incorporated in the analyses allowing the data to vary more than expected in a traditional multinomial distribution. If the variance in the collected data is greater than expected in the multinomial case, this may alternatively be addressed by analysing compound multinomial distributions in which the probabilities of the different outcomes are assumed to vary between samples (Smith and Maguire 1983) or by assuming the probabilities of the outcomes to be dirichlet distributed (Williams and Quinn 1998). However, neither of these methods incorporate the smoothness of the distributions, and they are both sensitive to the inadequacy of the 2 χ distribution in the small sample case.…”

“…However, the χ 2 -distribution has the disadvantage of being an inaccurate approximation when the expected number of outcomes in a category is less than five (Cramér 1946) and the variance in the collected data is often greater than can be described by the multinomial distribution Maguire 1983, Williams andQuinn 1998).…”

Review and evaluation of likelihood functions for composition data in stock-assessment models: Estimating the effective sample size

“…Williams and Quinn (1998) found that the effective sample size for an assessment of Pacific herring was substantially less than the actual sample size and even less than the maximum of 400 suggested by Fournier and Archibald (1982). Williams and Quinn (1998) suggest that, since the effective sample size can be much smaller than the actual sample size, a re-evaluation of the sample size requirements in catch-at-age models is needed. Unfortunately, there have been few studies dedicated to developing methods to determine the appropriate sample size to use when including catch-at-age data in stock assessment models (e.g.…”

“…Williams and Quinn (1998) showed how the Dirichlet distribution can be used to estimate the effective sample size from observed and predicted catch-at-age proportions. This distribution is considered a generalization of the multinomial distribution which takes account of over-dispersion (Kitada et al, 1994).…”

“…The multinomial distribution can represent only the observation (sampling) error under the assumption of random sampling (Williams and Quinn, 1998). Fournier and Archibald (1982) suggest that the multinomial based on the actual sample size is an inappropriate representation of the true estimation error because residuals from a catch-at-age analysis include both observation and process error.…”