Terceiro: The statistical properties of recreational catch data off the northeastern U.S. coast 



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events in the same sample (Sokal and Rohlf, 1981). In the 

 present study, the binomial distribution was used only to 

 model the probabilities of a positive catch (as opposed to a 

 zero catch; thus the variable attribute of the observation is 

 either catch or no catch ) in the combined delta-lognormal 

 and delta-Poisson models. 



The Poisson distribution is also a discrete frequency dis- 

 tribution of the number of times an event (such as catching 

 a fish during a trip ) occurs in a sample and is characterized 

 by a small mean value in relation to the observed maxi- 

 mum number of events within the sample (Sokal and Rohlf, 

 1981). For a Poisson distribution, the expected variance is 

 equal to its mean, and Poisson frequency distributions are 

 more highly skewed than normal or lognormal distribu- 

 tions (Bliss and Fisher, 1953). 



The negative binomial is a discrete frequency distribu- 

 tion with a higher degree of dispersion than the Poisson 

 distribution, such that the variance is significantly larger 

 than the mean. A negative binomial distribution will con- 

 verge to a Poisson as the variance approaches the mean 

 (Bliss and Fisher, 1953). Although not as widely applied 

 as the Poisson in the analysis of count data, there is a 

 growing literature describing the properties of negative 

 binomial regression methods to be used when analyzing 

 "over-dispersed Poisson" frequency distributions (Manton 

 et al., 1981; Lawless, 1987). The dispersion parameter of 

 the negative binomial distribution, k, is a positive exponent 

 relating the mean and variance of the distribution such 

 that as the variance of a distribution exceeds the mean, the 

 value ofk decreases and the "over-dispersion" of the distri- 



bution in relation to a Poisson distribution increases. The 

 most efficient estimate of the sample parameter, k', is esti- 

 mated by maximum likelihood (Bliss and Fisher, 1953). 



Descriptive statistics and frequency distributions of 

 MRFSS catch per trip and catch per hour observations 

 were compiled by using the SAS FREQ and UNIVARIATE 

 procedures (SAS, 2000 ). Tests of normality were made with 

 the Kolmolgorov-Smirnov D-statistic for normality (test 

 significance expressed as probability < D; SAS, 2000). Eval- 

 uation of the most appropriate distributional fit to the data 

 was based on inspection of the frequency distribution plots, 

 the parametric chi-square (x^) and G-statistic goodness- 

 of-fit tests, and the nonparametric Kolmogorov-Smimov 

 (Z3-statistic) goodness-of-fit test for an intrinsic hypothesis 

 (because the expected distributions were calculated from 

 the observed sample moments; Sokol and Rohlf, 1981). 

 For the chi-square and G-tests, when intervals (classes) 

 of catch per trip with fewer than 3 expected instances 

 occurred, expected and observed frequencies for these in- 

 tervals were pooled with the adjacent intervals to obtain a 

 joint class with an expected frequency of occurrence of 3 or 

 more (Sokol and Rohlf, 1981). Because of the large sample 

 sizes involved (»100), the G-test correction suggested 

 by Williams (1976) proved to be very small in a few test 

 calculations and therefore was not routinely applied. Un- 

 realistic (for recreational fishery catch-rate data) negative 

 expected values computed for the lognormal distributions 

 were excluded, and the remaining positive distribution was 

 raised to the observed sample total, so that the expected 

 proportions at each interval summed to 1.0. 



