Alaska red king crab: A relatively intractable target in a multispecies trawl survey of the eastern Bering Sea

Fisheries Research 85 (2007) 165–173

Alaska red king crab: A relatively intractable target in a multispecies trawl survey of the eastern Bering Sea C. Braxton Dew ∗ , Roberta G. Austring 1 National Marine Fisheries Service, Alaska Fisheries Science Center, 7600 Sand Point Way NE, Seattle, WA 98115-6349, United States Received 7 July 2006; received in revised form 19 January 2007; accepted 1 February 2007

Abstract The success with which Alaska’s groundfish stocks have been managed is not matched by management of its crab stocks, several of which have been declared overfished. It is unlikely that all species will be sampled equally well in a generic, multispecies trawl survey with a uniform distribution of sampling effort. We evaluated the relative sampling tractability of six species targeted by the eastern Bering Sea stock assessment survey conducted annually by the U.S. National Marine Fisheries Service to see if there were differences that might explain the dichotomy in management success observed for groundfish and crabs. Using relative niche breadth, Green’s Index of spatial patchiness, heterogeneity of variance and abundance-estimate precision, we analyzed 22 years of survey data to conclude that three major crab species, red king crab (Paralithodes camtschaticus), snow crab (Chionoecetes opilio) and Tanner crab (Chionoecetes bairdi), were less tractable than three major fish species, walleye pollock (Theragra chalcogramma), Pacific cod (Gadus macrocephalus) and yellowfin sole (Pleuronectes asper). In particular, our analysis established red king crab as the species for which abundance-estimate precision, niche breadth and sampling efficiency were lowest, patchiness was highest and heterogeneity of variance was the most severe. As a consequence, Bering Sea multispecies sampling is not likely to achieve the same degree of abundance-estimation success for red king crab as it might for other crab or fish species. © 2007 Elsevier B.V. All rights reserved. Keywords: Paralithodes camtschaticus; Chionoecetes opilio; Snow crab; Chionoecetes bairdi; Tanner crab; Spatial patchiness; Stock assessment; Overfished stocks; North Pacific

1. Introduction Advocates for Alaska as a model of sustainable fishing for the rest of the nation frequently remind us that there are no overfished groundfish stocks in the North Pacific. Lost in this message is the fact that there are four overfished stocks in the North Pacific (Rosenberg et al., 2006), all of which are Bering Sea crab stocks resuscitating under rebuilding plans. A substantial proportion of Alaska’s total crab fishery is represented by these overfished stocks, which include Bering Sea snow crab (Chionoecetes opilio) and Tanner crab (Chionoecetes bairdi) (NPFMC, 2006). Some would include as overfished the Bristol Bay red king crab (Paralithodes camtschaticus) stock, given the population’s sudden and as-yet unexplained collapse in 1981, after years of historically high levels of exploitation and unmon-

∗ 1

Corresponding author. Tel.: +1 206 526 4132; fax: +1 206 526 6723. E-mail address: [email protected] (C.B. Dew). Present address: 3233 Bay View Drive, Kodiak, AK 99615, United States.

0165-7836/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fishres.2007.02.001

itored bycatch (Dew and McConnaughey, 2005). Our objective is to examine 22 years of Bering Sea trawl-survey data to see if there are behavior-related, species-specific differences in sampling tractability and spatial distribution that might explain the dichotomy in management success observed for groundfish and crabs. The assessment of crab and fish stocks in the eastern Bering Sea is conducted annually by the U.S. National Marine Fisheries Service (NMFS) as a multispecies trawl survey with effort distributed approximately uniformly over a systematic grid of geographically fixed stations. Except for additional sampling within areas near the Pribilof and St. Matthew Islands (Otto, 1986), the sampling of the NMFS survey is unstratified. That is, the spatial distribution of the sampling, by design, does not vary with respect to the species-specific spatial distributions of the target organisms. Uniform sampling-effort distributions tend to be most efficient for wide-ranging species found throughout the entire survey area (e.g., Pacific cod, Gadus macrocephalus). Species whose geographic range occupies only a small proportion of the total survey area (e.g., red king

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crab) receive a concomitantly small proportion of the sampling effort. Notwithstanding the generic multispecies approach, it is possible for uniform sampling to provide an adequate sample of all species of interest; but because all species will not be sampled equally well, a hierarchy of abundance-estimation reliability will exist among data sets collected from multiple-species distributions. For example, we might expect the abundance-estimate precision of an unstratified collection to be relatively high for samples drawn from a spatial distribution in which the organisms are spread evenly across the entire sampling area. Precision would be relatively low for the same number of samples drawn from a contagious, patchy distribution (e.g., Gunderson, 1993). If podding behavior and the associated contagious spatial distributions reported for red king crab juveniles (e.g., Dew, 1990) continue into adulthood, we might expect red king crab population estimates from the NMFS multispecies trawl survey to be imprecise relative to estimates for other major Bering Sea target species. Here we use relative niche breadth, spatial patchiness, heteroscedasticity (inequality of variance) and precision of abundance estimates to rank the relative statistical tractability of several major species that are targets of the Bering Sea trawl survey. 2. Methods The NMFS annual multispecies stock-assessment survey in the eastern Bering Sea collects bottom-trawl samples (tows) at more than 360 fixed stations, each representing a grid square of 1372 km2 (400 nm2 ). The sampling is distributed systematically or evenly over a uniform grid of stations. In this paper we report results based on the analysis of more than 8000 tows conducted during the 22-year period from 1983 to 2004. The data are species-specific catch weights from the haul and catch tables in the trawl-survey data base (RACEBASE) maintained at the Alaska Fisheries Science Center, NMFS, Seattle, Washington, USA. For each tow we calculated a catch rate (kg/h for males, females and juveniles combined) for each of six species, namely: walleye pollock (Theragra chalcogramma), Pacific cod, yellowfin sole (Pleuronectes asper), red king crab, Tanner crab and snow crab. The six species analyzed represent, 60–80 wt.% of all roundfish, flatfish and commercial crab caught in the NMFS Bering Sea survey during 1983–2004. The eastern Bering Sea bottom-trawl standardization protocol states that the duration of a standard tow is 30 min and that acceptable tows may be no less than 10 min. Accordingly, we dropped from analysis a 2003, 4 min tow with a pollock catch rate of 134,849 kg/h, which was both the shortest tow and the highest catch rate for pollock recorded during 1983–2004.

Niche breadth = −

k 

pi log10 pi

(1)

i=1

for each of k sampling stations where the species of interest was collected. There appears to be no common agreement as to logarithmic base, and we used base-10 logarithms for convenience. For each station, pi is the proportion of the survey’s total number (or weight) of the species collected at that station. In the extreme example where all individuals of a species were found at a single station, and none at the other stations sampled, niche breadth for the species would be 0 because pi is 1 and the logarithm of 1 is 0. Niche breadth approaches a maximum as pi approaches equality at all stations. For a given survey, maximum niche breadth is equal among species and takes the value log10 (n), where n is the total number of stations sampled. Maximum niche breadth also describes the distribution of sampling in a survey where 1/nth of the effort is expended at each of n stations (pi = 1/n). Relative niche breadth (RNB) is species niche breadth standardized to a proportion of maximum niche breadth:  − ki=1 (pi log10 pi ) RNB = (2) log10 (n) The denominator of Eq. (2) is the same for all species in a given survey, and therefore RNB is a valid comparison of niche breadth among species. We used arcsin-transformed (Zar, 1999) values of RNB for comparisons between species and between years. Also, inasmuch as RNB is a ratio of a species’ distribution to its sampling distribution, we used RNB as an index of sampling efficiency; e.g., for a species with a uniform distribution (pi = 1/n), sampling efficiency would be 100%. 2.1.2. Spatial pattern We used Green’s Index to compare the degree of nonrandomness, patchiness or contagiousness of each species’ spatial distribution within its geographical range. Green’s is the only index whose value for maximum contagiousness is independent of the number of stations sampled or the total number (weight) of organisms collected. It is therefore a suitable index for comparing the spatial distributions of species whose geographical ranges differ (Elliott, 1977; Ludwig and Reynolds, 1988). Green’s Index (GI) is calculated as: ¯ −1 (s2 /X) GI =  , x−1

(3)

 ¯ the data mean and x is the where s2 is the data variance, X  sum of the variates. The range of GI is from −1/( x − 1) for a maximally uniform distribution to 1 for a maximally contagious distribution where all of the individuals were collected at a single station. A value of GI = 0 is obtained for a random distribution ¯ (s2 = X).

2.1. Relative measures of sampling tractability 2.1.1. Niche breadth Niche breadth is a measure of the distribution of species individuals (or their weight) among habitats (sampling stations), calculated as (Levins, 1968)

2.1.3. Variance inequality Heteroscedasticity is the inequality (heterogeneity) of variances among samples and is often caused by a functional dependence of the variance on the mean; e.g., the variance increases as the mean increases. To test for variance inequality

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among years we used a robust version of the Levene test, in which the test statistic was calculated using the median rather than the mean (Brown and Forsythe, 1974). For each of the species studied here, the functional relationship between the variance and mean, based on data from 22 annual trawl surveys, conformed to the exponential growth model VX¯ = V0 e

¯ GX

.

(h/(P − Pc)) − (h/(P + Pc)) uhi − ulo = u u (h/P)((1/(1 − c)) − (1/(1 + c))) = h/P =

(4)

1 1 1+c 1−c 2c − = − = . 1−c 1+c 1 − c2 1 − c2 1 − c2

Relative heteroscedasticity for each species was evaluated as the slope of the line

3. Results

¯ loge (VX ) = loge (V0 ) + GX,

3.1. Niche breadth

(5)

¯ is the sample mean (kg/h) for each survey year, V the where X X variance for a given mean, loge (V0 ) the regression intercept and G is the rate of increase of loge variance with the mean. For G to be directly comparable among species, we standardized the data by expressing each species’ annual mean (and variance) as a proportion of the overall mean (or variance) of the entire 22-year data set for that species. 2.1.4. Precision We used the non-parametric bootstrap percentile method (Efron and Tibshirani, 1993; Mooney and Duval, 1993; Manly, 1997) to estimate a distribution of population means for each year and to obtain empirical 95% confidence intervals for each year’s mean abundance. For each of 22 years we drew 3000 resamples from the sample data and calculated a mean for each resample. The frequency distribution of the resample means was a bootstrap estimate of the distribution of means from the population. Using a sorted vector of the 3000 bootstrapped means for each year, a count up 2.5% to the 75th lowest value and down 2.5% to the 75th highest value gave the endpoints of a 95% bootstrap confidence interval for that year’s mean catch rate (kg/h). 2.1.5. Uncertainty To demonstrate that the uncertainty surrounding a utilization or harvest rate is greater than the uncertainty of the corresponding population estimate, the following definitions apply: c is a proportion of the population mean (P) representing the halfwidth of a symmetric confidence interval, where ±c is identical to 2c, P the population estimate (mean), h the known harvest, Plo the population estimate’s lower confidence limit, Phi the population estimate’s upper confidence limit, u the utilization rate, ulo the lower confidence limit for u, uhi the upper confidence limit for u and (uhi − ulo )/u is the total width of the confidence interval derived for the utilization rate, expressed as a proportion of u. These definitions lead to the following relationships: u=

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The NMFS annual multispecies trawl survey provided more information for wide-ranging, randomly distributed species than it did for clumped species with relatively small habitat ranges. Of the six species considered, only pollock and cod had geographical ranges that extended throughout the 371-station area sampled (on average) each year (Fig. 1). About 97% of the survey tows conducted during 1983–2004 caught pollock and 95% caught cod, indicating that both species were distributed relatively evenly throughout their range. At the other extreme were red king crab, whose range included less than half (49%) of the survey range. Moreover, within this relatively small range only 44% of survey tows caught red king crab, suggesting the crab were distributed non-randomly in clumps. Relative niche breadth (RNB) is a measure that includes consideration of a species’ geographical range as well as the non-randomness of the species’ distribution within that range. From 1983 through 2004 the RNB of Bering Sea red king crab was consistently lower than the RNB for the other species considered (Fig. 2). The 22-year average RNB for red king crab was 54% of the maximum, whereas the average RNB for the other five species ranged from 72% for C. bairdi to 88% for Pacific cod. Single-station encounters between the trawl and podded crab (e.g., Dew, 1990) resulted in especially low RNB values for red king crab in 1984, 1986 and 1991. Multiple-comparisons testing (Scheff´e’s test) confirmed that the mean RNB was signif-

h p

uhi =

h h = Plo P − Pc

ulo =

h h = Phi P + Pc

Fig. 1. The average number of survey tows within a species’ range and the average number of within-range tows that were positive (non-zero) for that species.

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Fig. 3. Box-and-whisker plot (Tukey, 1977) showing the distribution of 22 annual Green’s Index values for six species. The boxes, bounded by the 25th and 75th percentiles, include 50% of the values. The horizontal line within the box is the median. The black circles represent outliers beyond the 10th and 90th percentiles marked by the “whiskers”.

Fig. 2. Relative niche breadth (RNB) for: (A) three fish species and (B) three crab species that are primary targets of the annual trawl survey of the eastern Bering Sea. Densely aggregated species, whose behavior results in a large proportion of the annual total being caught in a single tow, are characterized by low RNB.

crab weight (standardized to kg/h) in each year. In 20 of the 22 years studied, patchiness, as measured by Green’s Index, was more pronounced for red king crab than for any of the other five species considered. Although the other species were distributed more evenly than red king crab, C. opilio exhibited some clumping in 2000 when a single tow collected 23% of the total survey weight for that species. However, even at its most extreme, the

icantly (p < 0.001) lower for red king crab than for other major species during 1983–2004. Using RNB as an index of sampling efficiency, the efficiency for Bering Sea fish species (pollock, Pacific cod, yellowfin sole) was, on average, 84%; and for crab species (red king crab, C. bairdi, C. opilio), efficiency was 68%. The 54% sampling efficiency for red king crab was the lowest of the species studied. Recognizing Eq. (1) as a measure of information content (e.g., Krebs, 1978), it is apparent that the species-specific information content provided each year by the NMFS annual Bering Sea multispecies survey was lower for red king crab than for any of the other species considered. 3.2. Spatial pattern Analysis of Bering Sea trawl-survey data demonstrates that the spatial distributions of the three crab species, particularly those of red king crab, were more patchy than the spatial distributions of the three fish species (Figs. 3 and 4). The 22-year average GI values for cod (GI = 0.008) and yellowfin sole (GI = 0.007) indicate that these species’ spatial distributions were essentially random. At the other end of the spectrum were red king crab, with an average GI of 0.081, consistent with the fact that red king crab were often found in extremely non-random clumps. In 1984, for example, a single red king crab catch of 3 metric tonnes accounted for 59% of the total red king crab weight collected in the 1984 survey. Similarly, in 1986 and 1991, singletow red king crab catches accounted for 39% of the total red king

Fig. 4. Trends in spatial patchiness during 1983–2004, as measured by Green’s Index (GI) for three fish and three crab species. Numbers in parentheses are 22-year average GI values.

C.B. Dew, R.G. Austring / Fisheries Research 85 (2007) 165–173

Fig. 5. The ratio (ρ) of the largest to the smallest of 22 (1983–2004) annual sample variances, by species. For a data set to qualify as moderately heteroscedastic, the ratio should not exceed 20 (Green, 1979).

patchiness of C. opilio in 2000 (GI = 0.061) was less than the 22-year average patchiness of red king crab. 3.3. Variance inequality A robust Levene’s test that minimized the probability of falsely detecting unequal variances (Brown and Forsythe, 1974), showed that variance heterogeneity was significant (p < 0.0001) for each of the six species examined, both before and after logtransformation of the variates. To rank the degree of variance heterogeneity for each species, we used three methods: first we used the ratio of the annual maximum to the annual minimum variance, which showed that variance inequality among years was greatest for red king crab (Fig. 5). Secondly, inasmuch as each species’ potential for heteroscedasticity increased with the growth rate of variance with respect to the mean, we used standardized data to obtain Eq. (5) parameter values for each species, demonstrating that the variance–mean curve of red king crab accelerated more rapidly than that of other species (Fig. 6). ¯ Thirdly, we relied on the fact that the model VX¯ = V0 eGX ¯ by an amount 1/G causes implies that a change in the mean (X) a change in the variance (VX¯ ) by a factor of e (2.718). So, as G increases, a smaller change in annual mean abundance is required to cause a 2.7-fold change in the variance. For red king crab, a G-value of 0.0978 meant that an increase (or decrease) in average abundance of 10.2 kg/h (1/0.0978) increased (or decreased) variance by a factor of e. For pollock, a change in

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Fig. 6. Relationship between the (standardized) mean and variance for six Bering Sea species. The rate of increase (G) of the variance with the mean is greatest for red king crab (RKC).

mean abundance of 526 kg/h (1/0.0019) was necessary to change the variance by a factor of e. The probability of a change in mean abundance by an amount 1/G is greater if 1/G is small relative to the expected range of mean abundance. For red king crab and C. bairdi, 1/G was 24% as large as their 22-year ranges of mean abundance. For pollock and yellowfin sole, 1/G was 67 and 63% of their respective 22-year ranges of mean abundance (Table 1). Therefore, variance stability, or the probability of equal variances among years, was highest for pollock and yellowfin sole and lowest for red king crab and C. bairdi, reflecting the fact that annual shifts in variance by a factor of 2.7 were relatively common for the two crab species. Although the annual abundance of red king crab varied by a factor of 4.7 during 1983–2004, parametric analysis of variance (ANOVA) detected no significant difference (at α = 0.05) among years using either untransformed (p = 0.277) or log-transformed (p = 0.073) data. Given that ANOVA detected significant (p < 0.0001) differences among years for all of the other species tested (yellowfin sole, pollock, cod, C. bairdi and C. opilio), using either untransformed or log-transformed data, it is likely that severe heterogeneity of variance invalidated the assumptions of parametric testing with regard to red king crab. In support of this conclusion, a non-parametric Kruskal–Wallis test found significant differences among years for all species (p < 0.0001), including red king crab (p = 0.032). According to Harris (1975) and Green (1979), moderate heteroscedasticity, typified by a maximum–minimum variance ratio (ρ) that does not exceed 20, may not debilitate parametric significance testing. Of the species compared in this study, only

Table 1 Parameter values used to calculate variance stability V0

Pollock YFS Cod C. opilio C. bairdi RKC

617451.2 82693.9 6900.6 3628.8 209.3 506.0

G

0.0019 0.0037 0.0135 0.0209 0.1083 0.0978

r2

0.58 0.76 0.65 0.70 0.85 0.78

1/G

526.0 268.4 74.0 47.9 9.2 10.2

Range of means Low

High

547.2 263.1 105.8 25.4 4.4 11.8

1329.9 685.9 267.8 171.0 43.2 55.2

Range width

Ranked variance stability, (1/G)/width

782.8 422.7 162.0 145.6 38.8 43.4

0.672 0.635 0.457 0.329 0.237 0.235

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yellowfin sole (ρ = 5) and pollock (ρ = 8) qualified as moderately heteroscedastic (Fig. 5). Red king crab, with a 1983–2004 maximum–minimum variance ratio of ρ = 278, were severely heteroscedastic. Calculation of a standard error by √ Var (6) Standard error = √ n shows why it is expected that an estimate of a population mean based on a large sample size (n) should have a lower standard error than an estimate of the same parameter based on a small n (Zar, 1999). However, this is true only for populations where the variance is reasonably independent of the mean, i.e., heteroscedasticity is low. For populations where the variance is functionally dependent on the mean, an approximation to the standard error can be calculated as  V0 eGX √ Standard error ≈ (7) n Consistent with Eq. (4), an increase of 1/G√in mean abun¯ means that the numerator of Eq. (7) ( variance) will dance (X) √ increase by a factor of e or 65%. For the standard error to √ remain stable, the denominator ( n) also must increase by 65%; but n itself must increase by a factor of e (172%). Therefore, for species with high G-values, slight increases in abundance require large increases in sample size merely to maintain a given level of estimate error. Because of its relatively high G-value (Fig. 6), red king crab was the only species for which a doubling in sample size from 30 to 60, accompanied by a 25% increase in mean abundance, resulted in an increase in the standard error of the estimate rather than a decrease (Table 2). The proportional standard error (PSE), which is the standard error expressed as a proportion of the mean, indicates that the increase in sample size improved the precision of the mean for all six species. However, the PSE values for red king crab, higher than for any other species in Table 2, were substantially greater than the generally accepted limit (PSE < 0.30) for the reliability of an estimate (e.g., NOAA, 2006). 3.4. Precision We used the 22-year average width of bootstrap confidence intervals to rank the precision of abundance estimates for each

Fig. 7. Bootstrap 95% confidence intervals for mean catch rates (kg/h) expressed as a ±% of the mean: (A) 22-year (1983–2004) average interval for each species and (B) most extreme 22-year confidence interval width and the year in which it occurred.

species (Fig. 7A). The 95% confidence interval width for red king crab (−45% to +63% of the mean) was substantially greater than that of any other species. Estimates for yellowfin sole (−16% to +17%) and Pacific cod (−15% to +18%) were the most precise, with interval widths approximately one-third that of red king crab. Comparing each species for the single survey-year in which confidence intervals were widest did little to change the ranking except that C. opilio, because of its clumped distribution in the 2000 survey, moved to second place (Fig. 7B). It was necessary to use asymmetrical, bootstrap confidence intervals to estimate precision because symmetrical, normalbased intervals (mean ± 1.96 × standard error) were relatively

Table 2 A hypothetical example using parameters from Table 1, showing red king crab to be the only species for which the standard error increased with a 100% increase in sample size (n1 = 30, n2 = 60) and a 25% increase in mean abundance 25% increase

YFS C. bairdi Pollock C. opilio Cod RKC

Mean 1

Mean 2

460.0 16.0 980.0 90.0 160.0 30 .0

575.0 20.0 1225.0 112.5 200.0 37.5

n1 = 30, standard error 1

n2 = 60, standard error 2

Standard error (%) change

PSE 1

PSE 2

123.0 6.3 364.0 28.2 44.7 17.8

107.6 5.5 324.8 25.2 41.4 18.2

−12.5 −12.2 −10.8 −10.7 −7.4 2.0

0.27 0.39 0.37 0.31 0.28 0.59

0.19 0.28 0.27 0.22 0.21 0.49

The proportional standard error (PSE) shows that precision was lowest for red king crab but improved for all species with increased sample size.

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ineffective at capturing the right-shifted means of positively skewed catch–frequency distributions. Using the 2004 red king crab data (skewness = 7.2) as an example, the confidence level (Abell et al., 1999) of the normal-based interval was 87%, meaning that the ±56% interval failed to include the mean 13% of the time rather than the 5% expected of a 95% confidence interval. For the 1984 red king crab data (skewness = 9.7) the confidence level of the normal-based interval (±101%) was only 67%, for an effective error rate of α = 0.33, or six to seven times the nominal rate (α = 0.05). 3.5. Uncertainty If a stock-size estimate is highly uncertain, as indicated by wide confidence intervals, even less will be known about the fishing pressure on the stock. Fishing pressure from the directed fishery can be estimated as a utilization (harvest) rate, which is the known harvest expressed as a proportion of the preharvest population, ignoring all bycatch mortality. Using the bootstrap to calculate 95% bounds for the population mean, confidence intervals for the utilization rate can be estimated by expressing the harvest as a fraction of the upper and lower 95% population-estimate bounds. The bounds for this utilizationrate estimate are always wider (relative to the mean) than the bounds for the corresponding population estimate. Consistent with the algebra shown in Section 2, the relationship between the width of a population-estimate confidence interval (left side of the inequality) and the width of a corresponding utilizationrate interval (right side of the inequality) can be represented as: 2c 2c < , 1 − c2 where c, a proportion of the population mean, is the half-width of a symmetric confidence interval around the population mean. For example, given a population estimate with a confidence interval width of 0.30 + 0.39 = 0.69 or 69% (c = 0.345), which was the 95% confidence interval width for the mean catch rate of pollock in 1995 (Fig. 7B), the corresponding interval width for a utilization rate would be 78%. 4. Discussion A recurring question during the 25 years since the collapse of the Bristol Bay red king crab population in the early 1980s is: “Why have federally managed groundfish stocks in Alaska fared better than crab stocks?” Part of the answer may lie in the fact that the major crab species, all of which are characterized by relatively patchy spatial distributions, are more difficult than fish as survey targets. In particular, red king crab occupied an extreme position within the spectrum of spatial distribution, and statistical intractability. Our analysis of 22 years of Bering Sea survey data establishes red king crab as the species for which abundance-estimate precision, niche breadth and sampling efficiency were lowest, patchiness was highest and heterogeneity of variance was the most severe. As a consequence, Bering Sea multispecies sampling is not likely to achieve the same degree

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of abundance-estimation success for red king crab as it might for other crab or fish species. Our findings differ from the management position that the Bristol Bay red king crab stock “is extremely amenable to trawl surveys because almost all of its habitat consists of smooth bottom and the habitat area is large enough to ensure adequate sample sizes during a large-scale, multipurpose survey.” (Otto, 1986, p. 98). While it is true that red king crab reside largely in habitat consisting of smooth bottom, the same is true for the other five species compared here. Also, because of their limited geographical range and their clumped distribution within that range (factors which, when combined, result in a low niche breadth), the information content for red king crab from each year’s survey was comparatively low. Effective sample sizes (the number of tows in which the target species was caught) were relatively small for red king crab (Fig. 1), and in order to approach the sample-size levels typical of other major species, the sampling density within red king crab habitat would have to increase substantially. The practical implications of the relationship between red king crab abundance, sample size, and estimate error became apparent in 1995. In an effort to obtain a more precise estimate of the number of egg-bearing females in Bristol Bay, NMFS trawl sampling in a 27,440 km2 area was increased from the standard 20 tows to 80 tows. However, because the 60 extra tows, by chance, happened to result in a 26% increase in the mean abundance of female crab over that obtained in the earlier standard 20-tow survey, the standard error also increased, consistent with the relationship shown for red king crab in Table 2. The standard error is commonly used as an indicator of precision (e.g., Zar, 1999), so the increase in the red king crab standard error, despite a quadrupling of the sample size, was of some concern. Not until the standard error was recast as a proportional standard error (PSE), expressing the standard error relative to the mean, did it become evident that the mean estimated from the larger sampling effort was indeed more precise. However, it was now clear that the absolute value of a standard error should not be used directly to evaluate precision when dealing with species such as red king crab, whose catch data are characterized by extreme heterogeneity of variance. Typical violations of the assumptions of parametric statistical procedures (e.g., ANOVA) include non-normality and heteroscedasticity (non-homogeneity of variances), of which the most serious violation is heteroscedasticity (Green, 1979). Heteroscedasticity causes an increase in the type II or β-error rate so that there is a loss of power (1 − β) in significance tests (Cochran, 1977; Green, 1979). However, it is well known that inequality of variance among sample sets is endemic to most real data sets collected from the aquatic environment, and moderate heteroscedasticity may not debilitate significance testing. But as heteroscedasticity increases, the loss of power becomes more serious and the probability increases that investigators will accept null hypotheses that are false and should be rejected. A maximum–minimum variance ratio (ρ) among years that does not exceed ρ = 20 is symptomatic of moderate heteroscedasticity (Harris, 1975; Green, 1979). The fish species in this study, with a three-species average ratio of ρ = 15, qualified as moderately

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heteroscedastic (Fig. 5). The three crab species, with an average of ρ = 174, were severely heteroscedastic, a fact which should serve to warn investigators who rely on parametric statistical methods to test untransformed data, and who use symmetric, normal-based confidence intervals to estimate precision. This 12-fold difference in the statistical tractability of fish versus crab may help to explain why Alaskan groundfish stocks, none of which are overfished, have fared better than crab stocks, several of which have been designated by NMFS as overfished. Statistically sound estimates of population size are a mainstay of fisheries-management efforts. The reliability of an estimate (e.g., mean abundance) is often presented in the form of a confidence interval around the mean, and the wider the interval the less precise the estimate of the mean. Investigators commonly use the width of confidence intervals (±% of the mean) to make inferences as to the patchiness of a population’s spatial distribution and the species behavior that leads to such distributions. These inferences are only as good as the confidence intervals upon which they are based. Bakkala and Wilderbuer (1991) reported normal-based 95% confidence intervals of ±12 to 17% for population estimates of Bering Sea yellowfin sole, consistent with the average bootstrap interval of −16 to +17% we calculated for yellowfin sole (Fig. 7A). Based on findings from the present study, confidence intervals for red king crab should be considerably wider than those for yellowfin sole. Inconsistent with this expectation were confidence intervals of ±12 to 27% (average ±17%) published by Otto (1986) for large male red king crab in the Bering Sea during 1975–1983, similar to the level of precision reported for yellowfin sole. Although McConnaughey and Conquest (1992) and Gunderson (1993) believed that the Otto (1986) confidence intervals were overly precise due to a flawed statistical treatment (poststratification), others (e.g., Incze et al., 1986; Armstrong et al., 1993) used these narrow confidence intervals to conclude that adult red king crab do not aggregate in extremely dense clusters or pods. However, given the intractability noted for Bering Sea red king crab in the present study, and taking into account the many direct observations of podding by juvenile, sub-adult and adult red king crab at Kodiak, Alaska (Dew, 1990; Dew et al., 1992; Dew and McConnaughey, 2005) and in southeast Alaska (Stone et al., 1992, 1993), it is likely that the historical levels of precision reported for Bering Sea red king crab population estimates (e.g., Reeves, 19752 ; Pereyra et al., 1976; Reeves et al., 1976; Otto, 1981, 1986) are considerably overstated. With regard to the length-based model used to manage the Bristol Bay red king crab fishery since 1995, chronic overstatement of survey precision (e.g., Dew, in press) has led to the expectation that the model, to be a credible management tool, must produce estimates that closely fit those of the annual trawl survey. However, given that survey-measurement error is likely to be substantially greater than previously acknowledged, it is counterproductive for modelers (e.g., Zheng et al., 1995a,b) to fine-tune model parameters such as natural mortality (M), about which little is known (e.g., Beverton and Holt, 1956), so that pop-

2

Cited in Pereyra et al. (1976).

ulation estimates from the model more closely fit those of the survey.

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