Simulation of tail weight distributions in biological
year 19862006 landings of brown shrimp, Farfantepenaeus aztecus, from
the Northern Gulf of Mexico fishery.




Abstract: 
Size distribution within reported landings is an important aspect
of northern Gulf of Mexico penaeid shrimp stock assessments. It reflects
shrimp population characteristics such as numerical abundance of various
sizes, age structure, and vital rates (e.g. recruitment, growth, and
mortality), as well as effects of fishing, fishing power, fishing
practices, sampling, sizegrading, etc. The usual measure of shrimp size in archived landings data is count (C) the number of shrimp tails (abdomen or edible portion) per pound (0.4536 kg). Shrimp are marketed and landings reported in pounds within tail count categories. Statistically, these count categories are count class intervals or bins with upper and lower limits expressed in C. Count categories vary in width, overlap, and frequency of occurrence within the landings. The upper and lower limits of most count class intervals can be transformed to lower and upper limits (respectively) of class intervals expressed in pounds per shrimp tail, w, the reciprocal of C (i.e. w = 1/C). Age based stock assessments have relied on various algorithms to estimate numbers of shrimp from pounds landed within count categories. These algorithms required underlying explicit or implicit assumptions about the distribution of C or w. However, no attempts were made to assess the actual distribution of C or w. Therefore, validity of the algorithms and assumptions could not be determined. When different algorithms were applied to landings within the same size categories, they produced different estimates of numbers of shrimp. This paper demonstrates a method of simulating the distribution of w in reported biological year landings of shrimp. We used, as examples, landings of brown shrimp, Farfantepenaeus aztecus, from the northern Gulf of Mexico fishery in biological years 19862006. Brown shrimp biological year, Ti, is defined as beginning on 1 May of the same calendar year as [T.sub.i] and ending on 30 April of the next calendar year, where subscript i is the place marker for biological year. Biological year landings encompass most if not all of the brown shrimp life cycle and life span. Simulated distributions of w reflect all factors influencing sizes of brown shrimp in the landings within a given biological year. Our method does not require a priori assumptions about the parent distributions of w or C, and it takes into account the variability in width, overlap, and frequency of occurrence of count categories within the landings. Simulated biological year distributions of w can be transformed to equivalent distributions of C. Our method may be useful in future testing of previously applied algorithms and development of new estimators based on statistical estimation theory and the underlying distribution of w or C. We also examine some applications of biological year distributions of w, and additional variables derived from them. 


Subject: 
Fish industry
(Influence) Fisheries (Influence) Shrimps (Research) 
Authors: 
Caillouet, Charles W., Jr. Hart, Rick A. Nance, James M. 
Pub Date:  03/22/2011 
Publication:  Name: Marine Fisheries Review Publisher: Superintendent of Documents Audience: Academic Format: Magazine/Journal Subject: Agricultural industry; Business Copyright: COPYRIGHT 2011 U.S. Department of Commerce ISSN: 00901830 
Issue:  Date: Spring, 2011 Source Volume: 73 Source Issue: 2 
Topic:  Event Code: 310 Science & research 
Product:  Product Code: 0900000 Fishing, Hunting & Trapping; 0913080 Shrimp NAICS Code: 114 Fishing, Hunting and Trapping; 114112 Shellfish Fishing SIC Code: 0912 Finfish; 0913 Shellfish; 0919 Miscellaneous marine products; 0921 Fish hatcheries and preserves 
Geographic:  Geographic Scope: United States Geographic Code: 1USA United States 
Accession Number:  272740311 
Full Text: 
Introduction Size distribution within reported landings is an important aspect of northern Gulf of Mexico penaeid shrimp stock assessments. It reflects population characteristics such as numerical abundance of various sizes, age structure, and vital rates (e.g. recruitment, growth, and mortality), as well as effects of fishing, fishing power, fishing practices, sampling, sizegrading, etc. (Kutkuhn, 1962; Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Caillouet et al., 2008; Nance et al., 2010; Parrack (1); Nichols (2)). Age of shrimp cannot be determined directly (Parrack, 1979; Rothschild and Brunenmeister, 1984; Neal and Maris, 1985). Therefore, age structure of shrimp in reported landings has been determined indirectly by estimating numbers of shrimp from pounds allocated to marketing size categories, and transforming size into age using growth curves (Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Parrack (1); Nichols (2)). Most but not all reported landings from northern Gulf of Mexico shrimp fisheries are sizegraded. The usual measure of shrimp size in archived landings data is count (C), the number of shrimp tails (abdomen or edible portion) per pound (0.4536 kg). Shrimp are marketed and landings reported in pounds within tail count categories. Statistically, these count categories are count class intervals or bins with upper and lower limits expressed in C. The upper and lower limits of most count class intervals can be transformed to lower and upper limits (respectively) of class intervals expressed in pounds per shrimp tail, w, the reciprocal of C (i.e. w = 1/C) Age based stock assessments have relied on various algorithms to estimate numbers of shrimp from pounds landed within count categories (e.g. Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Parrack (1); Nichols (2)). These algorithms required underlying explicit or implicit assumptions about the distribution of C or w. However, no attempts were made to assess the actual distributions of C and w. Therefore, validity of the algorithms and assumptions could not be determined. When different algorithms were applied to landings within the same size categories (e.g. Parrack (1) vs. Nichols (2)), they produced different estimates of numbers of shrimp (Cail louet, 2003). Estimating numbers of shrimp from pounds landed within size categories is statistically challenging for additional reasons. Some count categories representing the largest shrimp have an implied lower limit of zero (e.g. < 15 count), and some representing the smallest shrimp have an implied upper limit of [infinity] (e.g. > 67 count). Neither zero nor [infinity] can be transformed to real values of w. Count categories also exhibit considerable variability in width, overlap, and frequency of occurrence within the landings. Certain count categories dominate the landings, reflecting what are referred to as standard count categories: <15, 1520, 2125, 2630, 3110, 4150, 5167, and > 67 count (Caillouet et al., 2008). Our approach does not require a priori assumptions about the parent distributions of w or C, and it takes into account the variability in width, overlap, and frequency of occurrence of count categories within the landings. Simulated biological year distributions of w can easily be transformed to equivalent distributions of C. Our method may be useful in future testing of previously applied algorithms and development of new estimators based on statistical estimation theory and the underlying distribution of w or C. We also examine some applications of biological year distributions of w and additional variables derived from them. Materials and Methods Fishery The brown shrimp fishery of the northern Gulf of Mexico is bounded by statistical subareas 1021, and comprises inshore (estuarine) and offshore (Gulf of Mexico) territorial waters of Texas, Louisiana, Mississippi, Alabama, and a portion of Northwestern Florida, as well as adjoining Federal waters landward of the 50 fm depth contour within the U.S. Exclusive Economic Zone (EEZ) (Fig. 1). Brown shrimp produce annual crops (Neal and Maris, 1985), with recruitment to the fishery occurring in MayJuly (Rothschild and Brunenmeister, 1984). Although life span is 2027 mo (Baxter, 1971), most brown shrimp are harvested within 6 mo of age. (3) Neal (1967) conducted virtual population analyses of brown shrimp in statistical subareas 18 and 19 (Fig. 1), and found that estimated numbers of brown shrimp in reported landings during biological year 1964 represented 97.7% of the total virtual population over a 17mo period. This finding indicated that only 2.3% (by number) of the shrimp recruited as new cohorts in biological year 1964 contributed to the landings in biological year 1965. If shrimp landed in a given biological year within our time series (19862006) included survivors from cohorts recruited in preceding biological years, this could have affected our biological year simulations of w and other variables derived from them. However, such a carryover would be small, because it would involve only the larger sizes of shrimp which are lowest in pounds and fewest in numbers within the landings. [FIGURE 1 OMITTED] Landings Data Brown shrimp landings data are archived by the National Marine Fisheries Service (NMFS) Galveston Laboratory, Texas. Statistically, reported landings are fisherydependent samples taken without replacement from the brown shrimp population. They are multitudinous but have limitations (Kutkuhn, 1962; Snow, 1969; Prytherch, 1980; Parrack (1); Nichols (2); Poffenberger (4)) which may bias not only our simulated distributions of w and additional variables derived from them, but also may have biased previous estimates of numbers of shrimp from pounds landed within count categories. Not all brown shrimp that are caught are landed, and not all that are landed are reported (Kutkuhn, 1962; Berry and Benton, 1969; Baxter, 1973; Snow, 1969; Prytherch, 1980; Nance et al., 1991; Caillouet et al., 2008; Poffenberger (4)). Nonreported catch includes shrimp marketed directly to consumers, marketed as fishing bait (not all, but some), discarded for various reasons, kept for personal use by shrimpers, or otherwise not reported. Thus, reported landings are less than the actual catches, and also represent incomplete samples of the actual landings (Caillouet et al., 2008). Reported shrimp landings data are recorded by calendar year, month, statistical subarea (Fig. 1), depth zone, shrimping trip, and count category or unknown size category, along with other information (Kutkuhn, 1962; Snow, 1969; Prytherch, 1980; Poffenberger4). We treated the unknown size category as a catchall category. In selecting records for a working file of sizegraded landings data for our simulations, we excluded all landings originally reported in the unknown size category, as well as landings added to the unknown size category after we judged their count categories to be outliers (see Data Selection and Preparation below). The resultant unknown size category contained landings that were: 1) not sizegraded, 2) sizegraded incorrectly or size limits not recorded, 3) not assigned to a count category for other reasons (e.g. pieces of shrimp tails), or 4) sizegraded but reported in count categories we judged to be outliers. Previous investigators (e.g. Rothschild and Brunenmeister, 1984; Parrack (1); Nichols (2)) also excluded certain landings from their analyses for various reasons. Two methods of grading shrimp, boxgrading and machine grading, were described by Kutkuhn (1962), Snow (1969), Prytherch (1980), and Poffenberger (4). Differences between these grading methods and variations in their relative contributions to sizegraded landings over time may have biased our simulated distributions of w and variables derived from them, but they may also have biased previous estimates of numbers of shrimp within count categories. Data Selection and Preparation Our final working file contained archived landings records selected from biological years 19862006, but only those we considered to have legitimate count class limits. We initially consulted NMFS port agents (who collect landings data) to obtain their opinions about the true range in size of brown shrimp tails in the landings. It was agreed that the maximum C (smallest shrimp) for brown shrimp in the landings was around 250 tails per pound (equivalent to w = 0.004 lb, or 1.8 g), and minimum C (largest shrimp) around 9 tails per pound (equivalent to w [approximately equal to] 0.111 lb, or [approximately equal to] 50.3 g). Preparation of the working file involved filtering and editing a copy of archived data from biological years 19862006 as follows: 1) If a record was originally coded as belonging to the unknown category, it was excluded. 2) If an upper or lower limit of a count category was not recorded (i.e. left blank), the record was excluded. 3) If a recorded lower limit exceeded the recorded upper limit of a count category, the limits were assumed to have been inadvertently transposed at data entry, and the record was retained in the working file after being recoded by interchanging its count category limits. 4) If recorded upper and lower limits of a count category were both C = 0, the record was excluded. 5) If the recorded upper limit of a count category was 0 < C < 9, both the lower and upper limits were recoded as C = 9, and the record was retained in the working file. 6) If only the recorded lower limit of a count category fell within C < 9, but the recorded upper limit was [greater than or equal to] 9, the lower limit was recoded as C = 9, and the record was retained in the working file. 7) If the recorded lower and upper limits of a count category were C > 250, the record was excluded. 8) If the recorded upper limit of a count category was C > 250, but the recorded lower limit was C [less than or equal to] 250, the upper limit was recoded as C = 250, and the record was retained in the working file. 9) All other archived records were retained in the working file. We then performed statistical analyses of the working file to identify and remove records having count class limits we judged to be outliers. For each biological year, we used SYSTAT (5) to fit preliminary weighted linear regressions of upper limits on lower limits of the count categories, where the weighting factor was the number of observations (i.e. shrimping trips) associated with each unique count category (i.e. unique combination of upper and lower limits). Figure 2A is an example of a preliminary regression and data plot for biological year 2006. Statistical weighting by number of shrimping trips was our way of dealing with variability in frequency of occurrence of count categories in the working file. Records removed from the working file by filtering, editing, and identification of residual outlier count categories represented a higher percentage of observations than percentage of pounds landed (Table 2); i.e. they contained relatively low pounds per observation. [FIGURE 2 OMITTED] We fitted final weighted linear regressions of upper limits on lower limits within the final working file for each biological year (Table 3). The weighting factor for these regressions was the number of observations (i.e. shrimping trips) associated with each unique count category remaining in the final working file. These final regressions characterized the relationship between legitimate count category upper and lower limits for each biological year. Figure 2B is an example final regression and data plot for biological year 2006. Slopes and intercepts of the final linear regressions (Table 3) for each biological year were examined for trends, using polynomial regression. Coded biological year ([T.sub.i] 1996) was substituted for [T.sub.i] in these polynomial regressions, to avoid problems that otherwise might have been caused by correlations among powers of [T.sub.i] (Sokal and Rohlf, 2000). Aggregation and Cumulation of Landings Landings from the final working file were aggregated (summed) by biological year and count category lower limits, [C.sub.ij], where j is the place marker for the [C.sub.ij] within a biological year; j = 0, ..., [m.sub.i], where [m.sub.i] is the total number of [C.sub.ij] in each biological year (Table 1). Upper limits of count categories (equivalent to lower limits of class intervals of w) were ignored. Because the number of unique count categories in the final working file varied among biological years, the number of [C.sub.ij] also varied among biological years, as did [m.sub.i]. Summing the landings by biological year and [C.sub.ij] produced a subset of data with much lower spatialtemporal resolution than that of more detailed data sets used in previous, bottomup approaches to estimating numbers of shrimp within count categories (Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Parrack (1); Nichols (2)). Biological year summations of landings combined all spatialtemporal influences (statistical subarea, depth zone, and month) on size of brown shrimp in the landings. These influences included sex ratio, recruitment, growth, mortality, fishing effort, fishing power of shrimp trawlers, experience of captains and crews, gear selectivity, discarding, data collection procedures, grading methods, and possibly other factors that affect count category landings within a biological year. Spatial influences were collapsed to the level of the entire fishery, and temporal influences to the level of biological years. Summation of landings by [C.sub.ij] combined landings within count categories having [C.sub.ij] as their lower limit. The simple hypothetical example below depicts this process: The sum of observations over all count categories having [C.sub.ij] as their lower limit became the weighting factor, [q.sub.ij], for each [C.sub.ij] and the sum of pounds associated with it. In the hypothetical example above, [C.sub.ij] = 9, [q.sub.ij] = 6, and both are associated with 1,740 lb landed. Examples of variation in [C.sub.ij] and [q.sub.ij] for biological years 1986, 1996, and 2006 are shown in Figure 3. Dominant [C.sub.ij] were conspicuous as indicated by their [q.sub.ij], and many were identical or close to the [C.sub.ij] of standard count categories, as expected. Within each biological year, the pounds associated with [C.sub.ij] were cumulated over the observed range of [C.sub.ij], from the highest to the lowest [C.sub.ij] (i.e. from the smallest to largest shrimp tails). These cumulative pounds were then converted to proportions of cumulative pounds landed, [P.sub.ij] (Table 1), from the highest to the lowest [C.sub.ij]. Figure 4A is an example of the stairstepped relationship between [P.sub.ij] and [C.sub.ij] for biological year 2006, and Figure 4B is the equivalent stairstepped relationship between [P.sub.ij] and [w.sub.ij], where [w.sub.ij] = 1/[C.sub.ij]. Modified Richards Function We searched for an asymptotic, asymmetrical sigmoid regression model to convert the stairstepped relationship between [P.sub.ij] and [w.sub.ij] to a smooth curve for each biological year. The regression model we chose was a simplified form of the Richards function (Richards, 1959): [FIGURE 3 OMITTED] P = Pmax [(l  [e.sup.abw]).sup.c] (1) where, P is the cumulative proportion of pounds landed at w, w is shrimp tail weight in pounds, over the observed range from minimum to maximum w, Pmax is the upper asymptote, a is the parameter which allows w at which P = Pmax/2 to vary, b is the parameter which represents the maximum intrinsic rate of increase in P per unit w, which occurs at the inflection point on the curve, c is the parameter that allows the sigmoid shape of the curve to vary (symmetrical or asymmetrical), and e is the base of natural logarithms. Because we constrained Pmax to equal 1 in fitting all the regressions, Eq. (1) was simplified into the following regression model: P = [(l  [e.sup.abw]).sup.c]. (2) For each biological year, we used GraphPad Prism (version 5.02) to fit Eq. (2) to [P.sub.ij] on [w.sub.ij] by weighted nonlinear regression, where the weighting factor was [q.sub.ij]. In this way, parameters [a.sub.i], [b.sub.i], and [c.sub.i] (Table 1) were estimated for each biological year (Table 4). The lower case parameter [c.sub.i] should not be confused with the upper case count [C.sub.ij]. We tried fitting a number of other asymmetrical sigmoid functions available in GraphPad Prism, but Eq. (2) was the best fitting of those we examined. While we recognize that additional curve fitting methods and models could have been tested, Eq. (2) was adequate for purposes of demonstrating our simulation approach. By fitting Eq. (2), we smoothed the relationship between [P.sub.ij] on [w.sub.ij], and obtained an equation representing this relationship for each biological year (Table 4). We also calculated the adjusted [r.sup.2] as an approximation of how well Eq. (2) fit the data points for each biological year (Table 4), but recognize it is not strictly applicable to nonlinear regression. [FIGURE 4 OMITTED] Simulating Biological Year Distribution of Tail Weight The next step toward simulating the distribution of w was to generate a new set of data pairs for each biological year, using the fitted equations in Table 4. First, we generated equally spaced values of [w'.sub.k] (Table 1), from a minimum, [w'.sub.0] (= 0.005155 lb), to a maximum, [w'.sub.999] (=0.111111 lb), where the kth place marker for the [w'.sub.k] was k = 0, ..., 999 (Table 1). The increment, g, between the [w'.sub.k] was then calculated as g = ([w'.sub.999]  [w'.sub.0])/999 = 0.000106 lb. The [w'.sub.k] were generated by [w'.sub.k] = k(g) + [w'.sub.0]. We then generated values of [P'.sub.ik] for each [w'.sub.k] for each biological year, using the following equation and estimates of parameters [a.sub.i], [b.sub.i], and [c.sub.i] from Table 4: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3) Three reasons for applying [w'.sub.0] = 0.005155 lb (derived from 1/194) as the minimum shrimp tail weight for all biological year simulations were: 1) The lowest maximum [C.sub.ij] observed (in the working file) among all biological years was 194 count, the reciprocal of the highest minimum [w.sub.ij]. 2) Imaginary numbers were generated by Eq. (3) for the minimum [P'.sub.ik] in some biological years when the actual minimum [w.sub.ij] observed in those years was applied (this probably was due in part to the fact that Eq. (3) did not fit the data points representing very small shrimp tails closely in those years). 3) It was consistent to constrain [w'.sub.k] to be the same for all biological years. The first derivative of Eq. (3), [delta][P'.sub.ik]/[[delta]'.sub.wk], was [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4) For each biological year, we used Eq. (4) to generate first derivatives for each [w'.sub.k]. To transform these first derivatives (Eq. (4)) into [p'.sub.ik] (Table 1), which was the proportion of pounds landed at [w'.sub.k] for each biological year, we divided them by the sum of all first derivatives over the range in [w'.sub.k], for each biological year. This sum was calculated as [999.summation over (k=0)]([delta][P'.sub.ik]/[delta][w'.sub.k]). In other words, for each biological year, [p'.sub.ik] at each [w'.sub.k] was calculated as [P'.sub.ik] = ([delta][P'.sub.ik]/[delta][w'.sub.k])/[999.summation over (k=0)]([delta][P'.sub.ik]/[delta][w'.sub.k]). Biological year yield, [Y.sub.i], encompassed all landings within a biological year, including those retained in our final working file as well as those that had been excluded from it. For each biological year, number of shrimp tails, [f'.sub.ik] (Table 1), at each [w'.sub.k] was calculated by [f'.sub.ik] = [Y.sub.i] ([p'.sub.ik])/[w'.sub.k]. (5) Equation 5, describing the relationship between [f'.sub.ik] and [w'.sub.k], is the simulated distribution of w for the ith biological year. We would have been able to exclude some steps in our simulation sequence had the final working file represented total reported landings from each biological year (i.e. [Y.sub.i]). However, the final working file was a subset of sizegraded landings selected from the archived landings, and it did not contain landings we excluded (i.e. those relegated to the unknown category), whereas [Y.sub.i] contained all landings for each biological year. Therefore, Eq. (5) applied the subset of proportions [p'.sub.ik] to the total yield Yi to estimate [f'.sub.ik] for each biological year. We recognize that relative distributions of w for each biological year, and their corresponding cumulative relative distributions, also could have been derived from our simulated distributions of w. They might be of interest in some applications of our approach, but they were not essential to the purpose of our paper. They can easily be calculated from the information provided in this paper. However, the concept of cumulative relative distribution of w in biological year landings of brown shrimp is important in that it would estimate the probability of occurrence of tail weight [less than or equal to] w; i.e. it would be an approximation of the cumulative distribution function (CDF) for w. This is the major part of the explanation of why we chose lower limits, [C.sub.ij] (equivalent to upper limits of [w.sub.ij]), for aggregating and cumulating landings, and then transformed [C.sub.ij] to [w.sub.ij] in preparation for fitting Eq. (2). Because a simulated distribution of w can be used to calculate the relative distribution of w and cumulative relative distribution of w, it is relevant to future testing of past algorithms and development of new ones to estimate numbers of shrimp from pounds landed within class intervals of w or C in the landings. Although we excluded certain landings (unknown size category) and ignored upper limits of legitimate count categories in simulating biological year [p'.sub.ik], our simulations of [f'.sub.ik] included all biological year landings ([Y.sub.i]); i.e. all biological year landings contributed to simulation of biological year distributions of w. Biological Year Total Number of Shrimp Tails ([N.sub.i]), Mean [C.sub.i], and Mean [w.sub.i] The total number of shrimp tails, [N.sub.i], in the landings from a biological year [T.sub.i] was simulated by [N.sub.i] = [999.summation over (k=0)][f'.sub.ik]. (6) Crude estimates of biological year mean count ([N.sub.i]/[Y.sub.i]) and its equivalent mean tail weight ([Y.sub.i]/[N.sub.i]) were calculated. We examined trends in both of these means via polynomial regression, where coded years ([T.sub.i]  1996) were substituted for [T.sub.i]. Tail Weight at Half of [Y.sub.i] Given that a fitted equation representing the relationship between [P.sub.ij] and [w.sub.ij] was available for each biological year (Table 4), we estimated tail weight, [w50.sub.i], at which half of the annual yield, [Y.sub.i]/2, was harvested in each biological year (note that when Pmax is constrained to equal 1, [w50.sub.i] = Pmax/2 = 0.5). Each equation (Table 4) was solved for [w50.sub.i] as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This statistic is similar in concept to LD50, the estimated lethal dose (concentration) of a toxic substance at which 50% mortality occurs in exposed subjects. In our application, it is a potentially useful index of the relationship between brown shrimp size and yield (see Caillouet et al., 2008). We examined w50i via polynomial regression, where coded years ([T.sub.i]  1996) were substituted for [T.sub.i]. Results Polynomial Regressions We recognize that polynomial regression is an empirical approach to fitting a curve to a time series of data, and that the resulting polynomial terms have no structural meaning (Sokal and Rohlf, 2000). We applied it only to detect possible trends in the variables we simulated, and to demonstrate possible applications of our simulated distributions of w. Obviously, many other curve fitting approaches could have been used to examine the time series for each variable. Causes and effects within this brown shrimp fishery could have influenced the detected polynomial trends, despite variability (deviations from regression) caused by fluctuations in annual recruitment and other factors which are typical in shrimp populations (Caillouet et al., 2008). Weighted Linear Regressions of Upper vs. Lower Limits of Count Categories Data plots and preliminary weighted linear regressions of upper on lower limits of unique count categories in each biological year (see the example for the year 2006 in Fig. 2A) showed that count category outliers remained in the data after filtering and editing. In the year 2006 example, the outliers were concentrated near the minimum lower limit count of 9 (largest shrimp), which elevated the intercept of the fitted line in Figure 2A as compared to the intercept of the fitted line in Figure 2B, in which residual outliers had been removed. The unusually wide class intervals of outlier count categories could lead to serious biases in estimating numbers of shrimp within such count categories. Also, we emphasize that each data pair (upper and lower limits) was weighted, so the actual numbers of residual outliers are much higher than the number of data points representing outliers in Figure 2A (Table 2). As expected, final weighted linear regressions of upper limits on lower limits of count categories were close fitting in all biological years as shown by high adjusted [r.sup.2] (Table 3, Fig. 2B). These final regressions characterized the relationship between upper and lower limits of what we considered to be legitimate count categories in each biological year. All slopes of these final regressions were slightly greater than 1 (Table 3), indicating that count class intervals in the working file widened as their lower limits increased. Trends in slopes and intercepts of these regressions are shown in Figures 5A and 5B, respectively. Biological Year [m.sub.i] and Weighted Regressions of [P.sub.ij] on [w.sub.ij] The biological year total number, [m.sub.i], of [C.sub.ij] exhibited a concave quadratic (parabolic) trend (Fig. 6); [m.sub.i] dropped from 77 in 1986 to 35 in 1995, then increased but not to its earlier highest level. This trend in [m.sub.i] reflected changes in the total number of legitimate count categories over the biological years. However, the total number of count categories in biological year [T.sub.i] exceeds [m.sub.i], because upper limits of count categories were ignored in our simulations; i.e. landings at the count category level were combined at the count category lower limit level, [C.sub.ij] (see Aggregation and Cumulation of Landings). Wide variation in biological year numbers of count categories and the consequential quadratic trend in mi (Fig. 6) are interesting and worthy of further investigation. They could reflect changes in sizerelated marketing strategies, recruitment, and perhaps other influences on choices of count categories in the landings. [FIGURE 5 OMITTED] Weighted nonlinear regressions of [P.sub.ij] on [w.sub.ij] for all biological years were close fitting, as indicated by very high adjusted [r.sup.2.sub.i] (Table 4). Over all biological years, adjusted [r.sup.2.sub.i] equaled or exceeded 0.981. Examples of plotted data points [P.sub.ij] vs. [w.sub.ij] and fitted curves for 1986, 1996, and 2006 are shown in Figures 7AC, respectively. Inflection points of the regressions were far to the lower left in such plots (Fig. 7AC), suggesting that brown shrimp were fully recruited to the landings at very small sizes, which is a very important finding. [FIGURE 6 OMITTED] The total sample size, [[m.sub.i].summation over (j=0)][q.sub.ij] (Fig. 8), for each biological year regression (Table 4), and the adjusted [r.sub.i.sup.2] (Fig. 9) for these regressions, declined over biological years. In other words, adjusted [r.sub.i.sup.2] and total sample size were dependent, as expected (Fig. 10); i.e. the larger the sample size the higher the adjusted [r.sub.i.sup.2]. We emphasize that the total sample size (Fig. 8) used in fitting the regressions of [P.sub.ij] on [w.sub.ij] for each biological year was less than the actual number of shrimping trips in the archived data for each biological year, because landings from some trips were initially in the unknown category or later placed there by filtering, editing, and outlier removal from the working file. Therefore, the data points and trend in Figure 8 should not be taken to represent total shrimping trips in the biological years. As is common in fitting models containing more than one parameter, the parameter estimates often are not independent (i.e. orthogonal). GraphPad Prism provided estimates of dependency of estimated parameters [a.sub.i], [b.sub.i], and [c.sub.i] within each biological year regression (dependency = 1 represents complete dependency, and dependency = 0 indicates orthogonality). Over biological years, dependency was 0.8650.985 for parameter [a.sub.i], 0.9680.981 for parameter [b.sub.i], and 0.9780.994 for parameter [c.sub.i]. Not only did all these parameters show strong dependency within each biological year regression, but they also appeared related to each other over biological years (Fig. 11AC). [FIGURE 7 OMITTED] [FIGURE 8 OMITTED] Simulated Distributions of w Example distributions of [w'.sub.k] for biological years 1986, 1996, and 2006 are shown in Figures 12AC. All were strongly skewed to the right. Their most striking feature was their likeness to negative exponential curves. Therefore, we plotted them in the form of ln([f'.sub.ik]) vs. [w'.sub.k] for all biological years (Fig. 13). Straight lines for ln([f'.sub.ik]) vs. [w'.sub.k] would have indicated that these simulated distributions of w followed a negative exponential pattern, once full recruitment to the landings was reached at very small sizes (Fig. 13). Only slight concavity was evident in all the curves. [FIGURE 9 OMITTED] [FIGURE 10 OMITTED] Biological Year Total Number of Shrimp Tails and Yield Interestingly, although the biological year total number of shrimp tails, [N.sub.i] (Fig. 14), and yield, [Y.sub.i] (Fig. 15), showed hints of declines, they exhibited no significant trends over biological years, because of wide year to year variation. A close linear relationship between [N.sub.i] and [Y.sub.i] (Fig. 16) was expected; i.e. the more pounds landed the greater the number of shrimp tails in the landings, and vice versa. However, biological year mean count, [N.sub.i]/[Y.sub.i] (Fig. 17A) was not constant, because simulated distributions of w and [Y.sub.i] were not constant over biological years (Fig. 12AC, Fig. 13). Biological year mean tail weight ([Y.sub.i]/[N.sub.i]) also was not constant (Fig. 17B). We emphasize that [N.sub.i]/[Y.sub.i] and [Y.sub.i]/[N.sub.i] are crude estimates of mean count and mean tail weight, respectively, and do not represent biological year central tendency of C and w in the landings very well. Trends in [N.sub.i]/ [Y.sub.i] (Fig. 17A) and [Y.sub.i]/[N.sub.i] (Fig. 17B) were cubic (sigmoid), mirroring each other as expected. [FIGURE 11 OMITTED] Tail Weight at Which Half of the Biological Year Yield was Harvested The cubic trend in [w50.sub.i] is shown in Figure 18. As expected, it is similar in shape to that of [Y.sub.i]/[N.sub.i] (Fig. 17B). However, the two trends (Fig. 17B, Fig. 18) were not parallel, because the slope of the regression of [w50.sub.i] on [Y.sub.i]/[N.sub.i] did not equal 1 (Fig. 19). Instead, [w50.sub.i] was 1.459 times [Y.sub.i]/[N.sub.i]. Although significantly different from zero, the intercept of the regression of [w50.sub.i] on [Y.sub.i]/[N.sub.i] was very small (i.e. near the origin). Discussion It is clear that brown shrimp landings data should be filtered, edited, and residual records representing outlier count categories removed before distributions of shrimp tail weight are simulated. The same should be (and in most cases have been) done before numbers of shrimp are estimated from landings within count categories, regardless of the algorithm used to estimate numbers of shrimp within count categories, unless the algorithms are based on actual sampling of size distributions within count categories (Ehrhardt and Legault, 1996). The problem of unreported landings and other limitations of reported landings data affect not only our simulations, but all other uses of reported landings to estimate numbers of shrimp within count categories. These data problems cannot be rectified retroactively, but should be addressed in the future. Our simulated biological year distributions of brown shrimp tail weight could be biased to unknown degrees by many factors. This is true of all estimates of numbers of shrimp derived from landings within count categories, whether at the highest possible level of data resolution (i.e. an individual shrimping trip within a statistical subarea, depth zone, and month), or at lower levels of data resolution represented by various spatialtemporal aggregations of landings data, including ours. Our simulated distributions of shrimp tail weight should not be taken as equivalent to distributions of brown shrimp tail weight in the population of the northern Gulf of Mexico. However, our simulated distributions of w in biological year landings no doubt have some yet undetermined relationship to actual distributions of shrimp tail weight in the brown shrimp population in biological years. This relationship cannot be determined retroactively due to lack of or paucity of required data. Unreported landings are much less than reported landings, but our simulated distributions of shrimp tail weight only represent landings that were reported and archived. Despite landings data deficiencies, our simulated distributions of w, and other fisherydependent statistics derived from them, can be useful in examining changes in the brown shrimp fishery over biological years. Their relationships to other important fisherydependent and fisheryindependent variables could be examined in attempts to explain causes and effects. Our method could be applicable to fisheries of other penaeid shrimp species for which landings are recorded within size categories expressed in C or w. It might also be applicable to finfish fisheries in which landings are reported within size categories expressed in number of fish per unit weight or in weight per fish. The method may also be applicable to shrimp landings aggregated at spatialtemporal levels lower (i.e. higher resolution) than that of an entire fishery and biological year. Our results suggest that brown shrimp were fully recruited to the fishery at small sizes in each biological year, then declined in number with w in a pattern similar but not identical to that of a negative exponential curve. In a study of distributions of growth rates of shrimp in captivity, Banks et al. (2009) examined effects of bin width, sample size, and sampling frequency on distributions of weight per shrimp. Interestingly, the shapes of their distributions of weight per shrimp were similar to those of our simulated distributions of w. Although we did not simulate relative distributions of w or corresponding cumulative relative distributions of w, we noted that they could be simulated from our approach, and they too might be of interest and use in shrimp stock assessments. [FIGURE 12 OMITTED] [FIGURE 13 OMITTED] [FIGURE 14 OMITTED] [FIGURE 15 OMITTED] [FIGURE 16 OMITTED] Simulated biological year distributions of w could be used to estimate numbers of parents and recruits, for purposes of determining parentrecruit relationships (Rothschild and Brunenmeister, 1984; Gracia, 1991; Ehrhardt and Legault, 1996; Parrack (1); Nichols (2)). Numbers of parents or recruits could be extracted from curves representing distributions of w by integrating them over the size ranges of parents and recruits. However, estimates or assumptions about size at maturity and growth patterns of males and females would be required, as well as estimates of sizespecific sex ratios in the landings (Gracia, 1991; Ehrhardt and Legault, 1996 Parrack (1), Nichols (2)). [FIGURE 17 OMITTED] It may be possible to estimate instantaneous total mortality rate (Z) from simulated biological year distributions of w by transforming them to bounded length distributions and applying lengthbased models similar to those of Ehrhardt and Ault (1992) (Ehrhardt (6)). Alternatively, the lengthbased models used by Ehrhardt and Ault (1992) might be reformulated for direct application to biological year distributions of w for purposes of estimating Z (Ehrhardt (6)). Biological year distributions of w could also be transformed to agefrequencies for agestructured stock assessments. This would require conversion of tail weight to age using sexspecific growth curves and knowledge of sizespecific sex ratios in the landings (Parrack, 1979; Rothschild and Brunenmeister, 1984; Gracia, 1991; Ehrhardt and Legault, 1996; Parrack (1); Nichols (2)). [FIGURE 18 OMITTED] [FIGURE 19 OMITTED] Simulated distributions of w of brown shrimp in biological year reported landings are linked, by definition and calculations, to biological year yield. Fishing effort influences sizecomposition of the landings and therefore influences yield, although environmental variables affecting recruitment also affect yield (Caillouet et al., 2008; Nance et al., 2010). Numbers of shrimp estimated from landings within count categories have been used in evaluating the influence of environmental factors on abundance, growth, and survival (Diop et al., 2007). Our method provides an alternate way to estimate abundance of shrimp in reported annual landings, as compared to algorithms used by previous investigators. However, the relationship between abundance of shrimp in the landings and in the population remains undetermined. Our simulated distributions of w provide examples for comparison with explicit or implicit assumptions made by previous investigators about the distributions of C and w. They also provide information of potential use in developing new estimators of number of shrimp from landings data, based on statistical estimation theory and the underlying distribution of w or C. Finally, there may be other useful applications of our approach and results that we have not realized or anticipated. Acknowledgments Special recognition goes to Charles H. Lyles, Jr., for his pioneering development of the system used to collect and report shrimp fishery statistics, and to all shrimp industry participants whose cooperation made it possible over the years. We are especially grateful to those who perpetuated and improved this system, and to those who collected, processed, and archived shrimp fishery statistics, making them available for analyses such as ours. Joseph H. Kutkuhn's comprehensive statistical examination and evaluation provided an early and important understanding of the usefulness and limitations of landings data in shrimp stock assessment. Succeeding investigators expanded and improved this understanding, for which we are grateful. We greatly appreciated reviews of our manuscript by Nelson M. Ehrhardt, and three anonymous reviewers. We thank Jo Anne Williams for assistance in drafting the figures; James Primrose and John Cole for assistance in landings data compilation; and Brian Linton, James A. Bailey, and Stephen A. Bailey for assistance with derivatives of the modified Richards function, and in solving this function for w50. This paper is dedicated to the memory of the senior author's parents, Charles W. Caillouet, Sr. (19081971) and Elida P. Millet Caillouet (19062004). Literature Cited Banks, H. T., J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, and A. K. Dhar. 2009. Experimental design and estimation of growth rate distributions in sizestructured shrimp populations. Inverse Problems 25(9):9500395030. Baxter, K. N. 1971. Brown shrimp live longer than many biologists believe. Commer. Fish. Rev. 33(3):2. . 1973. Shrimp discarding by the commercial fishery in the western Gulf of Mexico. Mar. Fish. Rev. 35(9):26. Berry, R. J., and R. C. Benton. 1969. Discarding practices in the Gulf of Mexico shrimp fishery. FAO Fish. Rep. 57:983999. Caillouet, Jr., C. W. 2003. Improved assessments and management of shrimp stocks could benefit sea turtle populations, shrimp stocks and shrimp fisheries. Mar. Turtle Newsl. 100:2227. , R. A. Hart, and J. M. Nance. 2008. Growth overfishing in the brown shrimp fishery of Texas, Louisiana, and adjoining Gulf of Mexico EEZ. Fish. Res. 92:289302. Diop, H., W. R. Keithly, Jr., R. F. Kazmierczak, Jr., and R. F. Shaw. 2007. Predicting the abundance of white shrimp (Litopenaeus setiferus) from environmental parameters and previous life stages. Fish. Res. 86:3141. Ehrhardt, N. M., and C. M. Legault. 1996. Tuned lengthbased cohort analysis (TLCA) computer program. Crustacean stock assessment techniques incorporating uncertainty. Report of the CFRAMP/FAO/DANIDA Stock Assessment Workshop on the Shrimp and Groundfish Fisheries of the GuianaBrazil Shelf, Port of Spain, Trinidad. FAO Fish. Rep. 544, Suppl. 111131. and J. S. Ault. 1992. Analysis of two lengthbased mortality models applied to bounded catch length frequencies. Trans. Am. Fish. Soc. 121:115122. Gracia, A. 1991. Spawning stockrecruitment relationships of white shrimp in the southwestern Gulf of Mexico. Trans. Am. Fish. Soc. 120:519527. Kutkuhn, J. H. 1962. Gulf of Mexico commercial shrimp populationstrends and characteristics, 195659. Fish. Bull. 62:343402. Nance, J. M., N. Garfield, and J. A. Paredes. 1991. A demographic profile of participants in two Gulf of Mexico inshore shrimp fisheries and their response to the Texas Closure. Mar. Fish. Rev. 53(1):1018. , E. X. Martinez, and E. F. Klima. 1994. Feasibility of improving the economic return from the Gulf of Mexico brown shrimp fishery. N. Am. J. Fish. Manage. 14:522536. , C. W. Caillouet, Jr., and R. A. Hart. 2010. Sizecomposition of annual landings in the white shrimp fishery of the northern Gulf of Mexico, 19602006: its trend and relationships with other fisherydependent variables. Mar. Fish. Rev. 72(2):113. Neal, R. A. 1967. An application of the virtual population technique to penaeid shrimp. Proc. 21st Annu. Conf. Southeast Assoc. Game Fish Comm. 21:264271. and R. C. Maris. 1985. Fisheries biology of shrimps and shrimplike animals. Chapter 1: Fisheries biology of shrimps and shrimplike animals. In A. J. Provenzano, Jr. (Editor), Economic aspects: fisheries and culture, vol. 10, The biology of crustacea, p. 1110. Acad. Press, Inc., N.Y. Parrack, M. L. 1979. Aspects of brown shrimp, Penaeus aztecus, growth in the northern Gulf of Mexico. Fish. Bull. 76:827836. Prytherch, H. F. 1980. A directory of fishery data collection activities conducted by the statistical surveys division in the southeast region of the United States. U.S. Dep. Commer., NOAA Tech. Memo. NMFSSEFC16, 91 p. Richards, F. J. 1959. A flexible growth function for empirical use. J. Exper. Bot. 10:290300. Rothschild, B. J., and S. L. Brunenmeister. 1984. The dynamics and management of shrimp in the northern Gulf of Mexico. In J. A. Gulland and B. J. Rothschild (Editors), Penaeid shrimpstheir biology and management, p. 145172. Fish. News Books Ltd., Farnham, Surrey, Engl. Snow, G. W. 1969. E/54 Detailed shrimp statistical program in the Gulf states. In M. N. Mistakidis (Editor), Proceeding of the World Science Conference on the Biology and Culture of Shrimps and Prawns, FAO Fish. Rep. R57, Vol. 3, p. 947956. (online at http://www. fao.org/docrep/005/ac741t/AC741T25.htm). Sokal, R. R., and F. J. Rohlf. 2000. Biometry, the principles and practice of statistics in biological research, 3rd edition, W. H. Freeman and Co., N.Y., 887 p. (1) Parrack, M. L. 1981. Some aspects of brown shrimp exploitation in the northern Gulf of Mexico. Presented at the Workshop on the Scientific Basis for the Management of Penaeid Shrimp, Key West, Fla., Southeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Miami, Fla. Unpubl. rep., 50 p. (2) Nichols, S. 1984. Updated assessments of brown, white and pink shrimp in the U.S. Gulf of Mexico. Presented at the Workshop on Stock Assessment, Miami, Fla., Southeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Miami, Fla. Upubl. rep., 54 p. (3) Fishery Management Plan for the Shrimp Fishery of the Gulf of Mexico, United States Waters. Gulf of Mexico Fishery Management Council, Tampa, Fla., Nov. 1981 (online at http://www. gulfcouncil.org). (4) Poffenberger, J. R. 1991. An overview of the data collection procedures for the shrimp fisheries in the Gulf of Mexico, National Marine Fisheries Service, Southeast Fisheries Science Center, Miami, Fla. (online at http://www.sefsc. noaa.gov/gssprogram.jsp). (5) Mention of trade names or commercial firms does not imply endorsement by the National Marine Fisheries Service, NOAA. (6) Ehrhardt, N. M. Rosentiel School of Marine and Atmospheric Science, Miami, FL. Personal commun., August 2010. Charles W. Caillouet, Jr., conservation volunteer, retired in 1998 from the Galveston Laboratory, National Marine Fisheries Service, NOAA, 4700 Avenue U, Galveston, TX 77551. Rick A. Hart and James M. Nance are with the Galveston Laboratory, National Marine Fisheries Service, NOAA, 4700 Avenue U, Galveston, TX 77551 (corresponding author: rick.hart@noaa.gov). Table 1.Symbols and descriptions of variables used in analyses of biological year reported landings of brown shrimp from the northern Gulf of Mexico fishery. These apply only to sizegraded landings in legitimate count categories; i.e. data selected by filtering, editing, and removing residual outliers from archived landings data. Symbols Descriptions of variables [T.sub.i] biological year, from 1 May of a given calendar year through 30 April of the next calendar year, where i = 0, ..., 20 is the place marker for biological years 19862006 [C.sub.ij] the jth lower limit of a legitimate count (number per pound) category in landings data from the ith biological year, where y = 0, ..., [m.sub.j] [m.sub.i] the total number of [C.sub.ij] in landings data from the ith biological year [w.sub.ij] the jth upper limit of a pounds per shrimp tail category, where [w.sub.ij] = 1/[C.sub.ij], in landings data from the ith biological year [P.sub.ij] the jth cumulative proportion of pounds landed at [w.sub.ij] in ith biological year [q.sub.ij] the jth weighting factor for the [P.sub.ij] and [w.sub.ij] data pairs in the ith biological year. This weighting factor, [q.sub.ij] is the sum of observations over all count categories having [C.sub.ij] as their lower limit (or [w.sub.ij] as their upper limit), regardless of the recorded upper limits of these count categories [w'.sub.k] the kth simulated value of weight per shrimp tail, where 0.005155 lb [less than or equal to] [w'.sub.k] [less than or equal to] 0.111111 lb, k = 0, ..., 999, and the interval between the [w'.sub.k] is 0.000106 [P'.sub.ik] the kth cumulative proportion of pounds landed at [w'.sub.k] in the ith biological year, which is simulated from the modified Richards function fitted to [P.sub.ij] on [w.sub.ij] in the ith biological year [a.sub.i] the parameter, estimated from the modified Richard's function fitted to [P.sub.ij] on [w.sub.ij] in the ith biological year, which allows the [w'.sub.k] at which [P'.sub.ik] = [Pmax.sub.i]/2 to vary among biological years [b.sub.i] the parameter, estimated from the modified Richard's function fitted to [P.sub.ij] on [w.sub.ij] in the ith biological year, which represents the maximum intrinsic rate of increase in [P'.sub.ik] per unit [w'.sub.k] at the inflection point of the curve [c.sub.i] the parameter, estimated from the modified Richard's function fitted to [P.sub.ij] on [w.sub.ij] in the ith biological year, which allows the sigmoid shape of the curve to vary (symmetrical or asymmetrical) among biological years [p'.sub.ik] the kth simulated proportion of pounds landed at [w'.sub.k] in the ith biological year [Y.sub.i] the ith biological year yield, which includes pounds of brown shrimp tails landed in legitimate count categories and in the unknown size category combined [f'.sub.k] the kth simulated number of shrimp tails at [w'.sub.k], where 0.005155 lb [less than or equal to] [w'.sub.k] [less than or equal to] 0.111111 lb, in the ith biological year [N.sub.i] the simulated total number of shrimp tails landed in the ith biological year [w50.sub.i] the simulated pounds per shrimp tail at which half of [Y.sub.i] is harvested in the ith biological year [N.sub.i]/[Y.sub.i] the simulated mean count of brown shrimp in the landings from the ith biological year [Y.sub.i]/[N.sub.i] the simulated mean pounds per shrimp tail of brown shrimp in the landings from the ith biological year Table 2.Number of observations (shrimping trips) and pounds landed in the NMFSarchived records, compared to those remaining after filtering, editing, and removal of residual outlier count categories, for brown shrimp landings in the northern Gulf of Mexico fishery in biological years 19862006. Observations Records (shrimping trips) Pounds (tails) Archived 2,425,373 1,682,806,769 100.0% 100.0% After filtering and 2,319,554 1,668,305,100 editing 95.64% 99.14% After residual outlier 2,308,674 1,664,449,467 removal 95.19% 98.91% Count Pounds Category Observations landed 912 2 500 915 3 1,200 920 1 40 Total 6 1,740 Table 3.Final weighted linear regressions of upper (U on lower (L) limits of count categories in brown shrimp landings data selected by filtering, editing, and removal of residual outliers from the NMFSarchived landings data. The weighting factor was the number of shrimping trips associated with each unique count category (i.e. unique U and L data pair) in the landings data selected from each biological year. Sample size was the sum of these weighting factors for each bio logical year (see Fig. 2). Biological year, [Intercept Sample Adjusted [T.sub.i] .sub.i] [Slope.sub.i] size [r.sub.i.sup.2] 1986 0.0609420 1.172878 141,523 0.988 1987 0.7467180 1.189633 159,010 0.988 1988 0.4795405 1.160103 158,733 0.992 1989 0.0479237 1.171595 147,315 0.992 1990 0.6609263 1.157006 137,647 0.993 1991 0.1564869 1.180132 122,065 0.992 1992 0.1879885 1.169404 117,633 0.991 1993 0.5079871 1.187639 105,907 0.989 1994 0.0533226 1.173414 111,968 0.992 1995 0.1264397 1.179191 102,643 0.993 1996 0.7873293 1.195364 97,111 0.989 1997 1.4422680 1.210573 98,415 0.987 1998 1.0609130 1.199557 91,378 0.988 1999 1.2453040 1.204808 92,638 0.985 2000 0.3466840 1.179789 95,775 0.990 2001 0.1584804 1.169094 89,022 0.992 2002 0.2696291 1.187891 122,160 0.992 2003 1.0236740 1.206445 103,013 0.993 2004 1.1202140 1.208740 82,006 0.993 2005 0.3479283 1.192750 69,662 0.994 2006 0.9036610 1.208740 63,050 0.995 Table 4.Biological year yield ([Y.sub.i]), parameter estimates, and other statistics for weighted nonlinear regressions (modified Richards function, Eq. (2)) of cumulative proportions of pounds landed, [P.sub.ij], on pounds per shrimp tail, [w.sub.ij], in brown shrimp landings data selected by filtering, editing, and removal of residual outliers from the NMFSarchived landings data. The weighting factor was the number of shrimping trips, [q.sub.ij], associated with each data pair, [P.sub.ij] and [w.sub.ij], in the selected landings data. For a given biological year, the number of data points analyzed (total sample size) was the sum of these weighting factors, [[m.sub.i].summation over (j=0)] [q.sub.ij]. Biological Yield Estimated parameters year, [Y.sub.i] [T.sub.i] pounds [a.sub.i] [b.sub.i] [c.sub.i] 1986 94,738,424 0.2770842 53.75477 1.016937 1987 89,394,421 0.3177634 61.64658 1.074417 1988 79,859,436 0.2713897 57.15293 1.286208 1989 94,170,525 0.2802385 58.89570 1.243767 1990 105,121,282 0.2865627 55.59358 0.968642 1991 85,602,708 0.1627544 47.08183 1.095961 1992 68,425,417 0.2294646 55.76027 1.150789 1993 66,431,237 0.2427682 55.10823 0.865503 1994 67,049,354 0.2126820 51.46945 1.107677 1995 75,859,021 0.2123855 48.21137 0.829590 1996 73,500,416 0.2459783 55.83692 0.888528 1997 65,389,618 0.2837078 55.32308 0.761172 1998 80,514,861 0.2723718 61.82822 0.975136 1999 81,035,496 0.2308989 56.10879 0.836558 2000 94,463,851 0.3038908 59.25881 1.084649 2001 87,660,251 0.3287329 74.62214 1.352838 2002 73,180,653 0.3917993 81.88587 1.447248 2003 82,309,001 0.3194503 79.86258 1.376817 2004 74,233,767 0.2973424 57.98183 0.884165 2005 58,819,403 0.2349768 56.86499 1.169126 2006 85,047,627 0.0818478 51.68214 1.640370 Total sample size Biological [[m.sub.i] Adjusted year, .summation [r.sup.2 [T.sub.i] over (j=0)] .sub.i] 1986 141,523 0.994 1987 159,010 0.997 1988 158,733 0.997 1989 147,315 0.996 1990 137,647 0.992 1991 122,065 0.993 1992 117,633 0.995 1993 105,907 0.989 1994 111,968 0.996 1995 102,643 0.991 1996 97,111 0.991 1997 98,415 0.994 1998 91,378 0.992 1999 92,638 0.987 2000 95,775 0.995 2001 89,022 0.987 2002 122,160 0.988 2003 103,013 0.986 2004 82,006 0.981 2005 69,662 0.981 2006 63,050 0.991 
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