The statistical properties of recreational catch rate data for some fish stocks off the northeast U.S. coast.
Major recreational fisheries in the waters off the northeast U.S. coast target a wide variety of pelagic and demersal fish species (NMFS, 1995, 1996). Fishery data collected in the National Marine Fishery Service (NMFS) Marine Recreational Fishery Statistics Survey (MRFSS) are the basis of fishery catch and effort estimates for most of these recreational fisheries and for indices of population abundance used in stock assessments (USDOC, 1992, 2001). For some stocks, reliable fishery-independent data such as research trawl survey indices are not available, and therefore the recreational fishery data are essential for tracking stock abundance. The intercept (creel sampling) portion of the MRFSS is an interview-type survey of recreational fishing trips and is conducted at public fishing sites such as marinas, launching ramps, fishing piers, and beaches. MRFSS catch estimates are made by expanding intercept survey sample catch rates in numbers, calculated on a per trip basis, by the estimated total number of recreational fishing trips. The estimated total number of fishing trips is calculated from data collected in a MRFSS telephone survey of households located in coastal counties. The U.S. Department of Commerce (USDOC, 1992, 2001) has provided overviews of the MRFSS intercept and telephone survey methods and catch estimation procedures.
In many cases recreational and commercial catch rates used as abundance indices are standardized by using general linear models that assume a lognormal error distribution (Gulland, 1956; Robson, 1966; Gavaris, 1980; Kimura, 1981). Commercial fishery catch-rate data generally meet tests of normality when log-transformed (Gulland, 1956; O'Brien and Mayo, 1988). Because of the efficiency and "integrating" property of commercial fishing gear (including trawls, fixed nets, and longlines), even catch rates on a per tow or per set basis are usually lognormally distributed (Taylor, 1953). An important characteristic of commercial data is that catch rates of zero (tows or sets with no catch of the target species) are rare.
With the assumption that there is an underlying lognormal error distribution, general linear models have of Len been used to standardize recreational fishery catch rates and compute indices of abundance. This approach has been used in the assessments of bluefin tuna (Brown and Browder, 1994), summer flounder (Terceiro (1)), black sea bass (NEFSC (2)), tautog (NEFSC (2)), winter flounder (NEFSC (3)), and bluefish (NEFSC (4); Gibson and Lazar (5)). However, Bannerot and Austin (1983) noted that the sampling distribution of recreational catch data is often highly skewed with a longer right-hand tail than might be expected even from a lognormal distribution. Furthermore, depending on the way the catch rate is defined (i.e. catch per trip, day, or hour), recreational fishery catch-rate distributions may contain a high proportion of zero catches.
Hilborn (1985) presented a frequency distribution of numbers of salmon caught per trip in the British Columbia sport fishery that appears to be best characterized by the negative binomial distribution, with a catch per hour frequency best characterized by the Poisson distribution. Jones et al. (1995) investigated the statistical properties of recreational fishery sampling data collected in angler surveys in Virginia and noted that the non-normality of recreational fishery data may violate assumptions of lognormality in methods used to develop indices of abundance, and especially the validity of confidence intervals. Power and Moser (1999) expressed similar concerns about sampled distributions of fish and plankton collected by research trawl nets, noting that the assumption of an underlying normal or lognormal distribution for these types of data is commonplace, and perhaps in error, and that distributions such as the Poisson or negative binomial may be more appropriate. Smith (1990, 1996) recommended various nonparametric resampling methods (e.g. bootstrap confidence intervals) for characterizing the dispersion of highly skewed research trawl survey catch distributions having a large proportion of zero catches. Smith (1999) modeled angling success for salmon, expressed as the catch after the first hour of angling, using a negative binomial distribution model.
In addition to the Poisson and negative binomial, alternatives to the lognormal error model for recreational fishery catch rates also include the delta-lognormal and delta-Poisson error models. These models are combinations of the delta distribution (Pennington, 1983) and lognormal or Poisson model approaches. The delta distribution has been used in modeling fish and plankton abundance indices from research trawl survey data, which are characterized by highly skewed distributions with a relatively high proportion of zero catches (Pennington, 1983). In the combined delta-lognormal and delta-Poisson approaches, indices of abundance are modeled as a product of binomially distributed probabilities of a positive catch and lognormal or Poisson distributed positive catch rates. The delta-lognormal model has been used in modeling fish-spotter data (Lo et al., 1992) and in the standardization of recreational fishery catch rates for bluefin tuna (Brown and Porch, 1997; Turner et al., 1997; Brown, 1999; Ortiz et al., 1999), both characterized by a highly contagious spatial distribution and a large proportion of zeroes. Bluefin and yellowfin tuna catch rates in the commercial and recreational fisheries have also been standardized by using Poisson (Brown and Porch, 1997), negative binomial (Turner et al., 1997), and delta-Poisson error distributions (Brown, 2001; Brown and Turner, 2001) to address these distributional characteristics.
In this study I first examine the statistical properties of recreational fishery catch-rate data as sampled by the MRFSS. Next, I examine the goodness of fit to different statistical distributions of empirical MRFSS catch rates, on both per trip and per hour bases. I then explore the effects of five different assumptions about the error structure of the catch-rate frequency distributions (lognormal, delta-lognormal, Poisson, delta-Poisson, and negative binomial) in deriving standardized indices of abundance with general linear models, using simulated recreational fishery and empirical MRFSS catch per trip (zero catches included) data.
Materials and methods
Overview of statistical methods
This work focuses on catch number per trip sampled in the MRFSS as the index of abundance. The distributional properties of MRFSS catch-per-hour rates are also examined, in order to explore whether the general conclusions reached for catch-per-trip rates are likely to be similar to catch-per-hour rates. Directed trips are defined as those for which interviewed anglers indicated that they were intending to catch a particular species as a primary or secondary target, whether successful or not (zero catches included). In analyses of trips for all species, all trips were used regardless of target or success (zero catches included). Catch rates were expressed as integer (natural) numbers offish per trip or per hour.
A value of 1 was added to all observations when applying a lognormal transformation to allow inclusion of the zero catch rate observations (this constant was subtracted upon retransformation to the original scale). Expected sample values for the lognormal distribution were calculated by using the normal distribution and log-transformed catch rates (Sokol and Rohlf, 1981). Previous work on MRFSS catch-per-trip data has shown that the value of 1 is the appropriate constant to be added (Terceiro (1)); NEFSC (3)) because it tends to minimize the sum of the absolute value of skew and kurtosis for these distributions (Berry, 1987). The standard logarithmic transform bias correction was applied to express results in the original arithmetic scale (Finney, 1951; Bradu and Mundlak, 1970). No constant was added when data were analyzed under the assumption of binomial, Poisson, or negative binomial error distributions.
The binomial distribution is a discrete frequency (probability) distribution of the number of times an event occurs in a sample in which some proportion of the members possess some variable attribute (Snedecor and Cochran, 1967). Each event is assumed independent of other prior 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 distribution 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 maximum 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 distributions (Bliss and Fisher, 1953).
The negative binomial is a discrete frequency distribution 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 converge 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 of k decreases and the "over-dispersion" of the distribution in relation to a Poisson distribution increases. The most efficient estimate of the sample parameter, k', is estimated 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). Evaluation of the most appropriate distributional fit to the data was based on inspection of the frequency distribution plots, the parametric chi-square [chi square] and G-statistic goodness-of-fit tests, and the nonparametric Kolmogorov-Smirnov (D-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 intervals 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. Unrealistic (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.
Standardized annual indices of abundance derived from the simulated recreational and empirical MRFSS data were calculated by using maximum likelihood estimation to fit generalized linear models with the SAS GENMOD procedure (SAS, 2000). The SAS (2000) defaults for model specification were generally followed. An identity link function was used under the lognormal distribution assumption (catch rates were In-transformed prior to analysis). A logistic link function was used under the binomial distribution assumption applied for the probability of positive catch component in the delta-lognormal and delta-Poisson model approaches. A logarithmic link function was used under the Poisson and negative binomial assumptions (SAS, 2000). Type-3 general linear models were fitted in all cases because the results of this type of analysis do not depend on the order in which the terms of the model are specified. The significance of the individual classification effects (factors) in the models was judged by the chi-square statistic (Searle, 1987; SAS, 2000).
The overall goodness of fit of the standardization models was evaluated by using the deviance and log-likelihood statistics. The deviance is defined to be twice the difference between the maximum achievable log likelihood and the log likelihood at the maximum likelihood estimates of the model parameters (McCullagh and Nelder, 1989). The deviance has a limiting chi-square distribution, and so significance is judged by comparison to critical values of the chis-quare distribution. The scale parameter (i.e. for normal distributions) was held fixed at 1 for all models to facilitate evaluation of goodness of fit and the degree of overdispersion for models with different error distribution assumptions. Holding the scale parameter fixed has no effect on the estimated intercept or model regression coefficients (e.g. in the study, the year coefficients that serve as the annual indices of abundance), but allows equivalent calculation among models of a "dispersion estimate" (SAS, 2000). This "dispersion estimate," measured after model fitting as the deviance divided by the degrees of freedom (deviance/df), is used to judge whether the data are overdispersed or underdispersed with respect to the error distribution used in model fitting and is therefore useful in evaluating whether the correct error distribution assumption has been used in the model (McCullagh and Nelder, 1989; SAS, 2000).
Descriptive statistics for MRFSS catch rates
The descriptive statistics (mean, median, variance, skewness, and Kolmolgorov-Smirnov (D) normality test statistic) and frequency distributions of MRFSS sample catch rates for 1981, 1988, and 1996 were examined for four species from U.S. Atlantic coast waters (Maine to the east coast of Florida), and in aggregate for all species sampled along the U.S. Atlantic coast. The following individual species were considered: bluefish (Pomatomus saltatrix, an example of a Atlantic coast predatory "gamefish"); summer flounder (Paralichthys dentatus, a Mid Atlantic Bight demersal flatfish); Atlantic cod (Gadus morhua, a New England demersal roundfish); and scup (Stenotomus chrysops, a Mid-Atlantic demersal schooling roundfish, likely to yield a relatively high catch per trip). These species were selected as examples because they occur over a broad range along the northeast U.S. coast, are among the most frequently caught by recreational fishermen, and their catch-rate distributions are representative of most species caught by recreational fishermen in the northeast U.S (USDOC, 1992). Four configurations of catch rate distributions were examined: 1) catch per trip distributions including zero catches, 2) catch per trip distributions with positive catches only, 3) catch per hour distributions including zero catches, and 4) catch per hour distributions with positive catches only.
Goodness-of-fit statistics for the lognormal, Poisson, and negative binomial distributions were calculated for the four individual species and for all species to help judge which error structure best characterized the MRFSS catch-rate data. A single year (1996) is presented because of the similarity of the catch distributions across species and time. Given the results of the Kolmogorov-Smirnov D tests from the descriptive statistics work, which indicated that none of the catch rates were normally distributed (see "Results" section), that error structure was not examined further. As with the descriptive statistics analysis, both catch-per-trip and catch-per-hour rates were examined in the goodness-of-fit exercise, both for all directed trips including zero catches and for positive catches only.
Simulated recreational fishery catch rates
To isolate the consequences of possible model misspecification in deriving standardized indices of abundance, negative binomial distributions with characteristics like those of MRFSS recreational catch-per-trip distributions were simulated by using the SAS RANTBL function (SAS, 2000). The simulated distributions were arranged to provide continuously decreasing, continuously increasing, and peaked (increasing to a peak and then decreasing) trends in an 11-year time series of catch per trip. For the decreasing trend, the simulation procedure began with year 1 set at a mean catch per trip = 3.0, maximum catch per trip of 50 fish per trip, and variance = 81.0, which are characteristic of the MRFSS catch-per-trip distributions for all species (Table 1). For year 1, this combination of mean and variance provided a maximum likelihood estimate of the negative binomial dispersion parameter, k, of 0.23.
The vector of expected probabilities of catch per trip for these initial moments, assuming a negative binomial distribution, was then used to randomly generate 1000 observations of catch per trip (including zeroes) for year 1 (n=1000). The initial mean for year 2 was then set at 10 percent less than year 1 (i.e. 2.7) and the year 2 set of 1000 observations generated under the negative binomial assumption. The dispersion parameter, k, was held constant at the year 1 maximum likelihood estimate of 0.23, resulting in a decrease in variance, a relatively stable coefficient of variation (CV), and less frequent occurrence of large catch-per-trip values, as the mean decreased. These conditions were felt to best reflect the true changes in angler catch per trip as stock abundance declines. The exercise was repeated for years 3 to 11, providing a time series of decreasing simulated recreational fishery catch per trip. The simulated annual distributions, scaled (normalized) to the 11-year time series mean of 1.75, were re-ordered to create the increasing and peaked time series. Standardized indices of abundance were then calculated from the simulated, trended series by using lognormal, Poisson, negative binomial, delta-lognormal, and delta-Poisson models, with year serving as the single classification variable and index of abundance. Modeled in this way, the negative binomial model is expected to provide year-effect coefficients very close in absolute value to the unstandardized, mean simulated catch per trip of the true underlying negative binomial distribution because no other classification effects are present to account for variance from the unstandardized mean. The deviance of the year coefficients provided by the models, assuming the other error distributions, then provides an indication of the degree of model misspecification because virtually all the estimated variance in this particular exercise is due to model (process) error, except for the small amount generated by the random draw from the starting probability distributions.
MRFSS standardized indices of abundance, 1981-98
The potential effect of the assumed error structure on the calculation of standardized indices of abundance was further explored with empirical examples using the 1981-98 MRFSS time series of catch-per-trip rates (zero catches included) for bluefish, summer flounder, Atlantic cod, scup, and for all species. Annual indices of stock abundance were developed from these MRFSS catch rate data following procedures in previous Atlantic coast bluefish and summer flounder stock assessments (Terceiro (1)); NEFSC (3); Gibson and Lazar (5)). Standardized indices were calculated by applying lognormal, Poisson, negative binomial, delta-lognormal, and delta-Poisson models, using the main effects classification variables determined in these stock assessments to be statistically significant factors: year, fishing mode (shore, private or rental boat, party or charter boat), state of landing (Maine to Florida), fishing wave (two-month sampling period, e.g. Jan-Feb), fishing area (>3 miles from shore, [less than or equal to] 3 miles from shore), and days 12, the angler-reported days of saltwater fishing during the previous 12 months (a proxy for angler avidity, experience, or skill, or a proxy for all three characteristics). The retransformed, bias-corrected (when necessary) year coefficients serve as the annual indices of stock abundance. Calculation and evaluation of the MRFSS standardized indices followed the general procedures described in the "Overview of statistical methods" in the "Materials and methods" section.
Descriptive statistics for MRFSS catch rates
Descriptive statistics of MRFSS catch rates for the four catch rate configurations, four individual species, and for all species are presented for the years 1981, 1988, and 1996 (Tables 1-4). These three years are characteristic of the 1981-2002 time series of MRFSS data. Given the similarity among these years, frequency distributions are plotted only for 1996 (Figs. 1-4). Catch rate means, both with and without zero catches, are generally much higher than medians, variances are much larger than the means, skewness is always much larger than zero, and there is a high proportion of zero catch and one-fish catch-rate observations. In all cases, the Kolmogorov-Smirnov D test statistics were significant at the 1% level. All of these factors indicate that MRFSS catch-rate distributions are highly contagious and overdispersed in relation to the normal distribution (Sokol and Rohlf, 1981). Scup has highest frequency of high catch rates (Figs. 1-4). The scup and Atlantic cod samples exhibit modes at regular intervals of high catch-per trip rates (e.g. 10, 15, 20, 25, and 30 fish per trip) that may indicate some degree of digit bias in the sampling.
[FIGURES 1-4 OMITTED]
For the catch-per-trip configurations, catch rates were best characterized by the negative binomial distribution (Tables 5-6, Figs. 5-6). Note that the calculated chi-square, G-, and D-test statistics were generally significant at the 1% level, so that based on strict interpretation of these results, the null hypothesis that the observed distributions come from one of the theoretical distributions was rejected in all cases. However, the calculated test statistics for the negative binomial distributions were at least an order of magnitude smaller than those for the Poisson and lognormal distributions, suggesting that an underlying negative binomial distribution was much more likely. The distributions of the catch-per-hour rates generally had a truncated range compared to the catch-per-trip rate configurations (Figs. 1-4). For most of the catch-per-hour distributions, the maximum likelihood solution for the negative binomial k parameter occurred at very large values (>1000). The expected frequencies for the negative binomial distribution therefore converged to those expected for a Poisson distribution, resulting in identical test statistic values and indicating that the catch-per-hour rates are best characterized by the Poisson distribution (Tables 7-8, Figs. 7-8).
[FIGURES 5-8 OMITTED]
Simulated recreational fishery catch rates
The eleven simulated distributions of catch per trip had means ranging from 2.80 to 0.98 fish per trip, variances ranging from 31.81 to 4.39, and CVs of about 200%. Simulated variance decreased as the simulated mean decreased because the negative binomial dispersion parameter, k, was held constant at 0.23. The resulting unstandardized, simulated index of abundance declined by 65% over the 11 year series (Table 9).
All standardization model fits were highly significant (P<0.001), as characterized by the chi-square statistics for the year effect (Table 10). The three different time series trends had no effect on the results, and therefore only the results for the decreasing series are reported. The Poisson and negative binomial models generated year coefficients as standardized indices of abundance that were very similar to each other and, as expected, virtually identical to the unstandardized annual means, indicating a 65% decline over the time series (Fig. 9). Interestingly, the diagnostic statistics indicated a better determined year effect (more precise year coefficients) for the Poisson than for the negative binomial. However, the dispersion estimate (deviance/ df) for the Poisson model was much greater than 1.0, indicating that the input data were overdispersed with respect to the Poisson distribution (Table 10). The latter was the expected result, given that the variance of the annual simulated data sets was much larger than the mean. The results indicated that the negative binomial was a more appropriate model, with a dispersion estimate closer to 1.0, which was also the expected result given the true negative binomial distribution of the simulated data (SAS, 2000).
[FIGURE 9 OMITTED]
The consequence of assuming a lognormal model for the true underlying negative binomial distribution was a more extreme smoothing of the true time series trends than with the other model assumptions, with a decline of only 28% over the time series (Fig. 9). The diagnostic statistics for the lognormal model indicated a significant model fit, but the dispersion estimate was much less than 1.0, indicating that the input data were underdispersed with respect to the lognormal distribution (Table 10). This finding is reflective of the large number of 0 and 1 catch-per-trip observations, and a lack of observations near the mean of the input probability distribution (SAS, 2000). In this simulation exercise, therefore, the lognormal model dispersion estimate of much less than 1.0 is indicative of model misspecification.
As noted in the "Materials and methods" section, the indices of abundance from the delta models are calculated as the product of the year-effect coefficients from the two component models. The interaction of the year coefficients from the binomial proportion positive catches and lognormal or Poisson positive catches components of the delta models provided some interesting results in this simulation exercise. The binomial model component, common to both delta models, provided a highly significant year effect and indicated a 41% decline in abundance over the time series. The dispersion estimate indicated some overdispersion of the data with respect to the binomial distribution (Table 10).
The lognormal positive catches component of the delta-lognormal model also provided a highly significant year effect and indicated a 39% decline in abundance over the time series, producing a smoothing effect similar to that observed for the lognormal model of catch per trip including zeroes. The dispersion estimate indicated some underdispersion of the data with respect to the lognormal distribution (Table 10). The product of the annual year coefficients from the two delta-lognormal model components, which individually indicated less decline than the unstandardized indices, provided final indices of abundance that declined 64% over the time series (due to the product of two positive fractional values <1 providing a even smaller value <1)--nearly identical to the unstandardized, Poisson, and negative binomial series (Fig. 9).
The Poisson positive catches component of the delta-Poisson model provided a highly significant year effect and indicated a 51% decline in abundance over the time series. The dispersion estimate was much greater than 1.0, indicating overdispersion of the data with respect to the Poisson model (Table 10). The product of the annual year coefficients from the two delta-Poisson model components provided indices of abundance that declined 71% over the time series, a slightly greater decrease than for the other models (Fig. 9). Note again that the delta-lognormal and delta-Poisson models share the same binomial proportion positive catch model components, and therefore Annual year coefficients for this component. The decrease estimated by the delta-Poisson model was greater than that for the delta-lognormal because the year coefficients from the Poisson positive catch model were all smaller, and more closely matching the unstandardized positive catch series, than the comparable lognormal positive catch year coefficients over the course of the time series. For example, the year-11 coefficient from the binomial proportion positive catches model was 0.59; the year-11 lognormal positive catches coefficient was 0.61, providing a product for the year-11 index of 0.36. In contrast, the year-11 Poisson positive catches coefficient was 0.49, providing a product for the year-11 index of 0.29. When these and the other annual coefficients were scaled to the respective series means, the delta-Poisson model indicated a slightly greater decline over the time series.
MRFSS standardized indices of abundance, 1981-98
All standardization models of the MRFSS catch per trip (including zero catches), for the four individual species and for all species, fitted well. In part because of the large number of observations, the overall model fits and the individual classification effects (year, mode, state, wave, and days 12) were all highly significant. Only the year effect chi-square statistics are tabulated because the year effect coefficients serve as the annual indices of abundance (Tables 11-15). The year effect was generally the second or third most important effect in the models, after mode and state. The dispersion estimates (deviance/df) for the lognormal models indicated the data were generally underdispersed with respect to the lognormal; the dispersion estimates for the Poisson models indicated overdispersion with respect to that distribution. The dispersion estimates for the negative binomial models and binomial components of the delta models were generally close to 1.0, indicating appropriate model specification (Tables 11-15).
As in the simulated catch-rate exercise, the lognormal standardized abundance indices generally show lower rates of change in abundance than do the unstandardized, Poisson, or negative binomial indices, with the CV of the lognormal series about 25-50% of the CV of the unstandardized indices (Figs. 10-14). In effect, the lognormal standardization of MRFSS per trip catch rates had an unintended (and undesirable) smoothing effect on the independent annual indices abundance. The Poisson and negative binomial models generally provided interpretations of the trend and annual changes in abundance very similar to those of the unstandardized indices.
[FIGURES 10-14 OMITTED]
For bluefish, the delta-lognormal, and delta-Poisson models provided time series of indices with about the same variability and trend, but slightly different annual changes, as those from the unstandardized, Poisson, and negative binomial models. For summer flounder, Atlantic cod, scup, and all species, the delta-lognormal and delta-Poisson models provided time series of abundance indices that were more variable, with slightly different trends and annual changes, than the unstandardized, Poisson, and negative binomial series. This last result is comparable to that observed for the delta models used with the simulated data and is therefore likely due in part to model misspecification of the positive catch-per-trip component (recall that catch per trip for these examples is best characterized by the negative binomial distribution) and a comparable interaction of the binomial, lognormal, and Poisson model year coefficients.
The frequency distributions of recreational fishery catch-rate data as sampled by the MRFSS are highly skewed, often with a significant proportion of zero catch observations. The present study indicates that MRFSS catch rates generally are not normally or lognormally distributed but usually best characterized by the Poisson or negative binomial distribution, depending on the manner in which the catch rate is configured. This finding suggests that standardization methods for MRFSS catch-rate data where Poisson (in the case of per hour rates) or negative binomial (for per trip rates) error structures are assumed would usually be more appropriate than methods where normal or lognormal error structures are assumed.
The modeling of both the simulated and empirical MRFSS catch rates indicates that one may draw erroneous conclusions about stock trends by assuming the wrong error distribution in procedures used to developed standardized indices of abundance. The results demonstrate the importance of considering not only the overall model fit and significance of classification effects, but also the possible effects of model misspecification, when determining the most appropriate model construction. In particular, the simulation exercise indicates that assuming a lognormal model in the calculation of indices of abundance from recreational fishery catch-per-trip data with a true underlying negative binomial distribution will provide indices that will strongly underemphasize the true trends in the indices, and therefore in stock abundance. This underestimation applies equally to populations that may be declining or increasing faster than the lognormally standardized indices might indicate.
The MRFSS catch-per-trip indices standardized with the negative binomial model, which the descriptive statistics and goodness-of-fit results suggest should be the appropriate model, differ relatively little from the unstandardized indices, indicating that the model effects accounted for a low percentage of the variation in mean catch rate. The classification categories recorded in the general MRFSS sampling are broad, and even measures of angling avidity such as "angler-reported days of saltwater fishing during the previous 12 months" may not be adequate proxies for the real factors (besides stock abundance) that account for variation in recreational fishery mean catch rates. To make standardization analysis of MRFSS catch rate data potentially more useful, by accounting for a significantly larger part of the unexplained variance and thus providing more accurate indices of abundance, more information on the characteristics of individual fishing trips may be needed. Such information might include details on the type of equipment used, the skills, experience, avidity, and identity of the individual fishermen, and detailed temporal and spatial information about fishing trips. In the future, collection of detailed trip data for general recreational fisheries may be best accomplished by the identification and sampling of "test fleets" of known, individual fishermen.
Table 1 Descriptive statistics for MRFSS (Marine Recreational Fishery Statistics Survey), 1981, 1988 northeast U.S. coast catch per trip, including zero catches. Catch is given in numbers of fish. CV is the coefficient of variation (%). D is the Kolmogorov test statistic for normality. Test statistics significant at the 1% level (P<0.01) are shown by **, indicating rejection of the null hypothesis that catch rates follow a normal distribution. Species No. of trips Mean Median Variance 1981 Bluefish 4615 3.80 0.00 155.09 Summer Flounder 3135 1.88 0.00 14.69 Atlantic cod 509 2.55 1.00 13.49 Scup 269 8.44 2.00 275.10 All species 20,280 3.45 0.00 355.94 1988 Bluefish 7294 1.60 0.00 18.65 Summer Flounder 4779 2.26 0.00 18.48 Atlantic cod 1558 4.56 2.00 21.55 Scup 960 9.28 3.00 312.65 All species 48,423 2.29 0.00 43.66 1996 Bluefish 5457 1.20 0.00 13.12 Summer Flounder 7047 2.33 1.00 13.49 Atlantic cod 1099 3.97 1.00 43.34 Scup 643 13.83 4.00 524.60 All species 81,057 2.57 0.00 47.32 Species CV Skew D 1981 Bluefish 328 27.78 0.380 ** Summer Flounder 204 4.71 0.312 ** Atlantic cod 144 2.48 0.244 ** Scup 196 4.08 0.305 ** All species 547 65.48 0.427 ** 1988 Bluefish 270 6.42 0.355 ** Summer Flounder 190 3.70 0.300 ** Atlantic cod 154 3.68 0.258 ** Scup 190 4.48 0.300 ** All species 289 8.94 0.365 ** 1996 Bluefish 301 8.40 0.370 ** Summer Flounder 157 3.40 0.263 ** Atlantic cod 166 3.29 0.273 ** Scup 165 3.44 0.273 ** All species 268 10.45 0.354 ** Table 2 Descriptive statistics for MRFSS (Marine Recreational Fishery Statistics Survey) 1981, 1988 and 1996 northeast U.S. coast catch per trip, including zero catches. Catch is given in numbers of fish. CV is the coefficient of variation (%). D is the Kolmogorov test statistic for normality. Test statistics significant at the 1% level (P<0.01) are shown by **, indicating rejection of the null hypothesis that catch rates follow a normal distribution. No. of Species trips Mean Median Variance 1981 Bluefish 2288 7.66 4.00 283.32 Summer Flounder 1380 4.26 2.67 23.21 Atlantic cod 298 4.36 3.00 15.19 Scup 165 13.76 7.33 375.88 All species 9484 7.33 3.00 732.27 1988 Bluefish 2445 4.75 2.33 40.35 Summer Flounder 2326 4.64 3.00 26.92 Atlantic cod 1065 6.67 4.00 58.02 Scup 614 14.52 7.67 413.03 All species 19,094 5.76 3.00 90.39 1996 Bluefish 1666 3.93 2.00 32.26 Summer Flounder 4196 3.91 2.66 16.46 Atlantic cod 679 6.43 4.00 54.39 Scup 438 20.31 12.50 638.90 All species 39,094 5.30 2.67 83.43 Species CV Skew D 1981 Bluefish 220 22.02 0.340 ** Summer Flounder 113 3.87 0.222 ** Atlantic cod 89 2.26 0.188 ** Scup 141 3.39 0.262 ** All species 368 47.02 0.395 ** 1988 Bluefish 133 4.40 0.254 ** Summer Flounder 112 2.94 0.209 ** Atlantic cod 114 3.46 0.219 ** Scup 140 3.88 0.245 ** All species 165 6.49 0.278 ** 1996 Bluefish 144 5.61 0.258 ** Summer Flounder 104 3.13 0.203 ** Atlantic cod 115 2.85 0.210 ** Scup 124 3.05 0.220 ** All species 172 8.35 0.286 ** Table 3 Descriptive statistics for MRFSS (Marine Recreational Fishery Statistics Survey) 1981, 1988, and 1996 northeast U.S. coast catch per hour, including zero catches. Catch is given in numbers of fish. CV is the coefficient v of variation (%). D is the Kolmogorov test statistic for normality. Test statistics significant at the 1% level (P<0.01) shown by (**), indicating rejection of the null hypothesis that catch rates follow a normal distribution. No. of Species trips Mean Median Variance 1981 Bluefish 4615 0.77 0.00 5.57 Summer Flounder 3135 0.36 0.00 0.64 Atlantic cod 509 0.38 0.17 0.43 Scup 269 1.59 0.44 7.96 All species 20,280 0.74 0.00 13.23 1988 Bluefish 7294 0.39 0.00 1.26 Summer Flounder 4779 0.45 0.00 0.82 Atlantic cod 1558 0.96 0.50 1.99 Scup 960 2.12 0.67 14.03 All species 48,423 0.54 0.00 3.07 1996 Bluefish 5457 0.34 0.00 1.23 Summer Flounder 7047 0.52 0.22 0.729 Atlantic cod 1099 0.86 0.28 1.99 Scup 643 3.06 1.17 27.78 All species 81,057 0.62 0.00 4.51 Species CV Skew D 1981 Bluefish 305 12.21 0.371 ** Summer Flounder 221 6.43 0.325 ** Atlantic cod 172 4.04 0.280 ** Scup 178 2.79 0.287 ** All species 491 40.11 0.419 ** 1988 Bluefish 287 7.26 0.363 ** Summer Flounder 200 6.11 0.309 ** Atlantic cod 147 3.32 0.249 ** Scup 177 3.60 0.286 ** All species 325 15.18 0.379 ** 1996 Bluefish 322 7.73 0.378 ** Summer Flounder 164 4.17 0.271 ** Atlantic cod 165 3.12 0.272 ** Scup 172 3.79 0.281 ** All species 341 35.84 0.385 ** Table 4 Descriptive statistics for MRFSS (Marine Recreational Fishery Statistics Survey), 1981, 1988 northeast U.S. coast catch per hour, positive catches only. Catch is given in numbers of fish. CV is the coefficient of variation (%). D is the Kolmogorov test statistic for normality. Test statistics significant at the 1% level (P<0.01) are shown by (**), indicating rejection of the null hypothesis that catch rates follow a normal distribution. No. of Species trips Mean Median Variance 1981 Bluefish 2288 1.56 0.75 10.00 Summer Flounder 1380 0.82 0.50 1.07 Atlantic cod 298 0.65 0.45 0.56 Scup 165 2.56 1.67 10.41 All species 9484 1.58 0.67 26.96 1988 Bluefish 2445 1.16 0.63 2.85 Summer Flounder 2326 0.93 0.60 1.24 Atlantic cod 1065 1.40 1.00 2.29 Scup 614 3.31 1.71 17.98 All species 19,094 1.37 0.67 6.67 1996 Bluefish 1666 1.13 0.55 3.15 Summer Flounder 4196 0.87 0.58 0.90 Atlantic cod 679 1.39 0.80 2.49 Scup 438 4.49 2.73 34.38 All species 39,094 1.29 0.63 8.49 Species CV Skew D 1981 Bluefish 203 9.48 0.317 ** Summer Flounder 126 5.35 0.229 ** Atlantic cod 115 3.64 0.220 ** Scup 125 2.16 0.242 ** All species 328 29.00 0.381 ** 1988 Bluefish 145 4.91 0.253 ** Summer Flounder 120 5.45 0.212 ** Atlantic cod 108 3.14 0.196 ** Scup 128 3.09 0.223 ** All species 188 11.08 0.301 ** 1996 Bluefish 157 4.80 0.270 ** Summer Flounder 109 3.93 0.198 ** Atlantic cod 114 2.68 0.205 ** Scup 131 3.38 0.228 ** All species 226 28.09 0.331 ** Table 5 Summary of goodness of fit tests for 1996 MRFSS (Marine Recreational Fishery Statistics Survey) catch per trip distributions, including zero catches, for bluefish, summer flounder, Atlantic cod, scup, and all species. [chi Expected number Degrees of square] Species of intervals freedom statistic Bluefish Mean = 1.20 Variance = 13.12 n = 5457 Lognormal 9 6 7314 Poisson 7 5 5315 Negative binomial 23 20 68 Summer flounder Mean = 2.33 Variance = 13.49 n = 7047 Lognormal 11 8 8654 Poisson 10 8 10,902 Negative binomial 22 19 139 Atlantic cod Mean = 3.97 Variance = 43.34 n = 1099 Lognormal 12 9 2138 Poisson 12 10 8284 Negative binomial 25 22 48 Scup Mean = 13.83 Variance = 524.60 n = 643 Lognormal 28 25 389,173 Poisson 25 23 6.67e+07 Negative bionomial 51 48 305 All species Mean = 2.57 Variance = 47.32 n = 81,057 Lognormal 14 11 180,754 Poisson 13 11 306,000 Negative binomial 51 48 1577 [[chi G square]. D [D.sub. Species statistic sub.0.01] statistic 0.01] Bluefish Mean = 1.20 Variance = 13.12 n = 5457 Lognormal 5722 17 0.462 0.014 Poisson 4251 15 0.394 0.014 Negative binomial 27 38 0.007 0.014 Summer flounder Mean = 2.33 Variance = 13.49 n = 7047 Lognormal 4714 20 0.281 0.012 Poisson 5772 20 0.307 0.012 Negative binomial 101 36 0.011 0.012 Atlantic cod Mean = 3.97 Variance = 43.34 n = 1099 Lognormal 1068 22 0.360 0.031 Poisson 2212 23 0.425 0.031 Negative binomial 22 40 0.015 0.031 Scup Mean = 13.83 Variance = 524.60 n = 643 Lognormal 3850 44 0.541 0.041 Poisson 6391 42 0.544 0.041 Negative bionomial 235 74 0.053 0.041 All species Mean = 2.57 Variance = 47.32 n = 81,057 Lognormal 83,230 25 0.382 0.004 Poisson 129,928 25 0.440 0.004 Negative binomial 1146 74 0.020 0.004 Table 6 Summary of goodness-of-fit tests for 1996 MRFSS (Marine Recreational Fishery Statistics Survey) catch per trip distributions, positive catches only, for bluefish, summer flounder, Atlantic cod, scup, and all species. Expected number Degrees of [chi square] Species of intervals freedom statistic Bluefish Mean = 3.93 Variance = 32.26 n = 1666 Lognormal 9 6 1803 Poisson 11 9 2091 Negative binomial 21 18 425 Summer flounder Mean = 3.91 Variance = 16.46 n = 4196 Lognormal 10 7 6068 Poisson 12 10 3821 Negative binomial 20 17 699 Atlantic cod Mean = 6.43 Variance = 54.39 n = 679 Lognormal 12 9 3376 Poisson 14 12 3419 Negative binomial 27 24 121 Scup Mean = 20.31 Variance = 638.90 n = 438 Lognormal 30 27 374e+11 Poisson 32 30 8.09e+7 Negative binomial 50 47 204 All species Mean = 5.30 Variance = 83.43 n = 39,094 Lognormal 11 8 70,234 Poisson 16 14 169,662 Negative binomial 50 47 12,293 [[chi square]. G sub. D [D.sub. statistic 0.01] statistic 0.01] Species Bluefish Mean = 3.93 Variance = 32.26 n = 1666 Lognormal 1026 17 0.312 0.025 Poisson 1211 22 0.312 0.025 Negative binomial 347 35 0.196 0.025 Summer flounder Mean = 3.91 Variance = 16.46 n = 4196 Lognormal 2863 18 0.270 0.016 Poisson 2234 23 0.240 0.016 Negative binomial 587 33 0.143 0.016 Atlantic cod Mean = 6.43 Variance = 54.39 n = 679 Lognormal 962 22 0.379 0.040 Poisson 925 26 0.365 0.040 Negative binomial 88 43 0.147 0.040 Scup Mean = 20.31 Variance = 638.90 n = 438 Lognormal 6565 47 0.543 0.049 Poisson 3477 51 0.475 0.049 Negative binomial 147 72 0.089 0.049 All species Mean = 5.30 Variance = 83.43 n = 39,094 Lognormal 24,957 20 0.254 0.001 Poisson 59,516 29 0.391 0.001 Negative binomial 10,217 72 0.201 0.001 Table 7 Summary of goodness-of-fit tests for 1996 MRFSS (Marine Recreational Fishery Statistics Survey) catch per hour distributions, including zero catches, for bluefish, summer flounder, Atlantic cod, scup, and all species. Expected Degrees [chi number of of square] Species intervals freedom statistic Bluefish Mean = 0.35 Variance = 1.23 n = 5457 Lognormal 6 3 2806 Poisson 5 3 514 Negative binomial 5 2 514 Summer flounder Mean = 0.52 Variance = 0.72 n = 7047 Lognormal 7 4 2022 Poisson 5 3 1408 Negative binomial 5 2 1408 Atlantic cod Mean = 0.86 Variance = 2.00 n = 1099 Lognormal 7 4 289 Poisson 5 3 51 Negative binomial 5 2 51 Scup Mean = 3.06 Variance = 27.78 n = 643 Lognormal 11 8 546 Poisson 10 8 1209 Negative binomial 19 16 54 All species Mean = 0.62 Variance = 4.51 n = 81,057 Lognormal 13 10 144,556 Poisson 7 5 54,675 Negative binomial 7 4 54,675 [[chi G square]. D Species statistic sub.0.01] statistic Bluefish Mean = 0.35 Variance = 1.23 n = 5457 Lognormal 3047 11 0.323 Poisson 41 11 0.028 Negative binomial 41 9 0.028 Summer flounder Mean = 0.52 Variance = 0.72 n = 7047 Lognormal 2430 13 0.245 Poisson 1209 11 0.193 Negative binomial 1209 9 0.193 Atlantic cod Mean = 0.86 Variance = 2.00 n = 1099 Lognormal 300 13 0.243 Poisson 11 11 0.051 Negative binomial 11 9 0.051 Scup Mean = 3.06 Variance = 27.78 n = 643 Lognormal 346 20 0.312 Poisson 583 20 0.347 Negative binomial 39 32 0.032 All species Mean = 0.62 Variance = 4.51 n = 81,057 Lognormal 126,529 23 0.575 Poisson 72,657 15 0.036 Negative binomial 72,657 13 0.036 [D.sub. Species 0.01] Bluefish Mean = 0.35 Variance = 1.23 n = 5457 Lognormal 0.014 Poisson 0.014 Negative binomial 0.014 Summer flounder Mean = 0.52 Variance = 0.72 n = 7047 Lognormal 0.012 Poisson 0.012 Negative binomial 0.012 Atlantic cod Mean = 0.86 Variance = 2.00 n = 1099 Lognormal 0.031 Poisson 0.031 Negative binomial 0.031 Scup Mean = 3.06 Variance = 27.78 n = 643 Lognormal 0.041 Poisson 0.041 Negative binomial 0.041 All species Mean = 0.62 Variance = 4.51 n = 81,057 Lognormal 0.004 Poisson 0.004 Negative binomial 0.004 Table 8 Summary of goodness of fit tests for 1996 MRFSS (Marine Recreational Fishery Statistics Survey) catch per hour distributions, positive catches only, for bluefish, summer flounder, Atlantic cod, scup, and all species. Expected [chi number of Degrees of square] Species intervals freedom statistic Bluefish Mean= 1.13 Variance = 3.15 n = 1666 Lognormal 6 3 1508 Poisson 5 3 590 Negative binomial 5 2 590 Summer flounder Mean = 0.87 Variance = 0.90 n = 4196 Lognormal 6 3 3005 Poisson 5 3 838 Negative binomial 5 2 838 Atlantic cod Mean = 1.39 Variance = 2.49 n = 679 Lognormal 6 3 414 Poisson 5 3 145 Negative binomial 5 2 145 Scup Mean = 4.49 Variance = 34.38 n = 438 Lognormal 10 7 601 Poisson 11 9 725 Negative binomial 17 14 127 All species Mean = 1.29 Variance = 8.49 n = 39,094 Lognormal 7 4 38,171 Poisson 8 6 31,475 Negative binomial 8 5 31,475 [[chi G square]. D Species statistic sub.0.01] statistic Bluefish Mean= 1.13 Variance = 3.15 n = 1666 Lognormal 1462 11 0.446 Poisson 590 11 0.269 Negative binomial 590 9 0.269 Summer flounder Mean = 0.87 Variance = 0.90 n = 4196 Lognormal 3146 11 0.416 Poisson 921 11 0.210 Negative binomial 921 9 0.210 Atlantic cod Mean = 1.39 Variance = 2.49 n = 679 Lognormal 368 11 0.356 Poisson 113 11 0.213 Negative binomial 113 9 0.213 Scup Mean = 4.49 Variance = 34.38 n = 438 Lognormal 292 18 0.270 Poisson 324 22 0.280 Negative binomial 99 29 0.166 All species Mean = 1.29 Variance = 8.49 n = 39,094 Lognormal 33,641 13 0.434 Poisson 16,429 17 0.270 Negative binomial 16,429 15 0.270 [D.sub. Species 0.01] Bluefish Mean= 1.13 Variance = 3.15 n = 1666 Lognormal 0.025 Poisson 0.025 Negative binomial 0.025 Summer flounder Mean = 0.87 Variance = 0.90 n = 4196 Lognormal 0.016 Poisson 0.016 Negative binomial 0.016 Atlantic cod Mean = 1.39 Variance = 2.49 n = 679 Lognormal 0.040 Poisson 0.040 Negative binomial 0.040 Scup Mean = 4.49 Variance = 34.38 n = 438 Lognormal 0.049 Poisson 0.049 Negative binomial 0.049 All species Mean = 1.29 Variance = 8.49 n = 39,094 Lognormal 0.001 Poisson 0.001 Negative binomial 0.001 Table 9 Summary statistics for the simulated recreational fishery catch per trip assuming a negative binomial distribution, configured to decline by 10% in successive time periods (years). For year 1, starting maximum catch per trip was 50 fish per trip, mean was 3.0, variance was 81.00, coefficient of variation (CV) of 300%, and the dispersion parameter of the negative binomial distribution, k, was 0.23. In years 2-11, k was held constant at the year-1 value of 0.23, allowing the variance to decrease as the mean catch declined. Annual simulated means were scaled to the 11 year time series mean (1.75) for comparability with standardized indices calculated for decreasing, increasing, and peaked time series trends. Simulated Simulated mean catch maximum catch Simulated Year per trip per trip variance 1 2.80 47 31.81 2 2.49 39 24.75 3 2.27 37 20.97 4 2.05 34 17.23 5 1.85 31 14.24 6 1.67 29 11.84 7 1.49 26 9.65 8 1.32 21 7.36 9 1.21 21 6.47 10 1.09 19 5.38 11 0.98 17 4.39 Scaled Simulated simulated catch Year CV (%) per trip 1 201 1.60 2 200 1.42 3 202 1.30 4 203 1.17 5 204 1.06 6 206 0.95 7 209 0.85 8 206 0.75 9 211 0.69 10 213 0.62 11 215 0.56 Table 10 Summary of model fits for simulated recreational fishery catch per trip (including zero catches) with a decreasing time series trend. Total model degrees of freedom were 10,989; for the positive catches component of the delta models, degrees of freedom were 4,184. Year-model-effect degrees of freedom were 10, and the year-model effect was highly significant (P<0.0001) in all five models. Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 7330 0.6670 Log-likelihood -13,773 Year chi-square 183 Poisson model Deviance 51,719 4.7064 Log-likelihood -7483 Year chi-square 2049 Negative binomial model Deviance 10,699 0.9736 Log-likelihood 7524 Year chi-square 239 Delta models: binomial proportion positive catch Deviance 14,546 1.3237 Log-likelihood -7273 Year chi-square 78 Delta-lognormal model: lognormal positive catches Deviance 3474 0.8303 Log-likelihood -5557 Year chi-square 119 Delta-Poisson model: Poisson positive catches Deviance 15,822 3.7815 Log-likelihood 10,466 Year chi-square 936 Table 11 Summary of model fits for estimating indices of abundance from empirical MRFSS (Marine Recreational Fishery Statistics Survey) bluefish catch per trip (including zero catches), 1981-98. Total model degrees of freedom (df) were 130,300; for the positive catches component of the delta models, degrees of freedom were 48,447. All model fits and classification effects were highly significant (P<0.001). Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 84,150 0.6458 Log-likelihood -156,444 Year chi-square 1835 Poisson model Deviance 675,791 5.1864 Log-likelihood -19,680 Year chi-square 20,604 Negative binomial model Deviance 99,393 0.7628 Log-likelihood 190,140 Year chi-square 2104 Delta models: binomial proportion positive catch Deviance 157,674 1.2101 Log-likelihood -78,837 Year chi-square 854 Delta-lognormal model: lognormal positive catches Deviance 39,963 0.8249 Log-likelihood -64,129 Yera chi-square 1240 Delta-Poisson model: Poisson positive catches Deviance 249,112 5.1419 Log-likelihood 193,660 Year chi-square 10,501 Table 12 Summary of model fits for estimating indices of abundance from empirical MRFSS (Marine Recreational Fishery Statistics Survey) summer flounderf catch per trip (including zero catches), 1981-98. Total model degrees of freedom(df) were 102,162, for the positive catches component ofthe delta models, degrees of freedom were 52,507. All model fits and classification effects were highly significant (P<0.001). Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 66,452 0.6505 Log-likelihood -122,989 Year chi-square 2663 Poisson model Deviance 444,657 4.3525 Log-likelihood -14,827 Year chi-square 14,053 Negative binomial model Deviance 96,698 0.9465 Log-likelihood 97,777 Year chi-square 2560 Delta models: binomial proportion positive catch Deviance 130,341 1.2758 Log-likelihood -65,171 Year chi-square 2498 Delta-lognormal model: lognormal positive catches Deviance 36,780 0.7005 Log-likelihood -65,202 Year chi-square 1203 Delta-Poisson model: Poisson positive catches Deviance 183,019 3.4856 Log-likelihood 115,991 Year chi-square 5675 Table 13 Summary of model fits for estimating indices of abundance from empirical MRFSS (Marine Recreational Fishery Statistics Survey) Atlantic cod catch per trip (including zero catches), 1981-98. Total model degrees of freedom(df) were 20,629, for the positive catches component ofthe delta models, degrees of freedom were 13,160. All model fits and classification effects were highly significant (P<0.001). Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 19,425 0.9416 Log-likelihood -28,697 Year chi-square 380 Poisson model Deviance 142,834 6.9239 Log-likelihood 54,501 Year chi-square 4090 Negative binomial model Deviance 21,824 1.0579 Log-likelihood 98,335 Year chi-square 323 Delta models: binomial proportion positive catch Deviance 24,997 1.2117 Log-likelihood -78,837 Year chi-square 191 Delta-lognormal model: lognormal positive catches Deviance 11,657 0.8858 Log-likelihood -17,920 Year chi-square 353 Delta-Poisson model: Poisson positive catches Deviance 75,359 5.7264 Log-likelihood 88,239 Year chi-square 2805 Table 14 Summary of model fits for estimating indices of abundance from empirical MRFSS (Marine Recreational Fishery Statistics Survey) scup catch per trip (including zero catches), 1981-98. Total model degrees of freedom (df) were 17,604; for the positive catches component of the delta models, degrees of freedom were 11,124. All model fits and classifi- cation effects were highly significant (P<0.001). Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 32,270 1.8331 Log-likelihood -30,346 Year chi-square 332 Poisson model Deviance 375,924 21.3545 Log-likelihood 309,490 Year chi-square 12,094 Negative binomial model Deviance 18,668 1.0604 Log-likelihood 466,529 Year chi-square 369 Delta models: binomial proportion positive catch Deviance 22,027 1.2512 Log-likelihood -11,013 Year chi-square 174 Delta-lognormal model: lognormal positive catches Deviance 14,340 1.2891 Log-likelihood -17,225 Year chi-square 350 Delta-Poisson model: Poisson positive catches Deviance 212,250 19.0804 Log-likelihood 391,327 Year chi-square 8793 Table 15 Summary of model fits for estimating indices of abundance from empirical MRFSS (Marine Recreational Fishery Statistics Survey) catch per trip (including zero catches), 1981-98. Total model degrees of freedom(df) were 1,033,367, for the positive catches component ofthe delta models, degrees of freedom were 457,598. All model fits and classification effects were highly significant (P<0.001). Dispersion estimate Criterion Value (value/df) Lognormal model Deviance 861,881 0.8341 Log-likelihood -1,372,576 Year chi-square 7246 Poisson model Deviance 8,048,246 7.7884 Log-likelihood 277,042 Year chi-square 28,734 Negative binomial model Deviance 870,357 0.8422 Log-likelihood 3,118,822 Year chi-square 2243 Delta models: binomial proportion positive catch Deviance 1,351,532 1.3079 Log-likelihood -675,766 Year chi-square 11,867 Delta-lognormal model: lognormal positive catches Deviance 466,644 1.0198 Log-likelihood 653,838 Year chi-square 655 Delta-Poisson model: Poisson positive catches Deviance 3,773,909 8.2472 Log-likelihood 2,414,210 Year chi-square 10,785
I thank Vic Crecco of the Connecticut Department of Environmental Protection, for raising questions about the best way to calculate indices of abundance from recreational fishery catch rate data during debates over the bluefish assessments; Paul Rago of the Northeast Fisheries Science Center, for numerous discussions about statistical distributions and tests; and two anonymous Fishery Bulletin referees, whose comments helped improve the quality of the analyses and therefore the usefulness of the results.
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Manuscript approved for publication 30 January 2003 by Scientific Editor.
Manuscript recived 4 April 2003 at NMFS Scientific Publications Office.
Fish Bull. 101:653-672 (2003)
Northeast Fisheries Science Center
National Marine Fisheries Service, NOAA
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|Date:||Jul 1, 2003|
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