the testing and selection of recruitment distributions for north atlantic fish stocks

ELSEVIER

The testing

Fisheries Research 25 (1996) 77-95

and selection of recruitment distributions for North Atlantic fish stocks

M. Power Dept. of Agricultural Economics. University of Manitoba, Winnipeg, Man. R3T 2N2, Canada

Accepted 15 February 1995

Abstract

The lognormal recruitment model has been widely applied as a description of the recruitment phenomenon. Despite its widespread use, recent work has suggested that a variety of distributional shapes might be expected to be descriptive of the available recruitment data. This paper applies randomization and goodness-of-fit tests to a hundred previously published North Atlantic fish stock recruitment data series as a means of establishing the suitability of the exponential, lognormal and Weibull distributions as descriptions of recruitment data. Randomization tests are required to establish the independence of data observations and act as a screening device for goodness-of-fit tests. Results of the testing procedure confirm that a variety of distribution models are often statistically adequate descriptions of the available recruitment data series. The Weibull model, however, best describes the largest number of data sets. The lognormal model best describes the remaining data sets and the exponential model is a poor description of the recruitment data. There were no patterns in the statistical suitability of any of the recruitment models on either a stock or species basis. No broad geographic patterns within the serially correlated data sets were found, however, the proportion of stocks dis- playing serial correlation was influenced by freshwaterdischarges or ocean current mixing. The results imply that recruitment should be viewed as a stock-specific attribute linked to life-history and environmental influences. Furthermore, managers should be made aware of the errors resulting from the inappropriate use of the lognormal recruitment assumption and the possible implications it might have on the development and implementation of fisheries management and exploitation policies.

Keywords: Recruitment models; North Atlantic fish stocks

1. Introduction

Empirical studies of recruitment have tended to use, or substantiate, lognormality as an appropriate model to describe the distribution of recruitment estimates and the distribution of values obtained from ratios such as the number of recruits to the number of spawners

0165-7836/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDIOl65-7836(95)00395-9

78 hf. Power/Fisheries Research 25 (19%) 77-95

(Shelton, 1992). For example, Allen ( 1973) using probability plotting demonstrated that the ratio of returns to escapement for Skeena River sockeye salmon (Onchorhynchus nerka) were lognormally distributed. Hennemuth et al. (1980) examined the null hypothesis that log-transformed recruitment data for 18 fish stocks were normally distributed and rejected

the hypothesis in only one instance. Myers et al. (1990) plotted the quantiles of log- transformed recruitment data against quantiles for the normal distribution, and assessed the fit using a Pearson correlation coefficient as a means of examining the viability of the null

hypothesis that the recruitment data for 100 North Atlantic fish stocks were lognormally distributed. At the 0.05 significance level the null hypothesis was rejected in only 19 cases. Finally, Lapointe et al. (1992) used the lognormal distribution to model recruitment in investigations into the effect of variable mortality on recruitment in virtual population analysis.

Shelton ( 1992), however, concludes that because of the potential effect of estimation error, empirical analyses cannot be interpreted as unequivocal evidence that the true recruitment distributions are generally lognormal. If recruitment is treated as the outcome of the probability of individual survival, then as Shelton (1992) shows, the recruitment distributional shape will depend on the degree of survival independence and its true shape may be obscured by estimation error. Based on the theory developed by Shelton, a variety of distributional shapes can be expected to be descriptive of recruitment data for different stocks, and the most appropriate function to describe the shape should be selected on a stock by stock basis.

Applications of the lognormal hypothesis in empirical work are largely based on the use of the Ricker recruitment model (Ricker, 1954) and the pioneering work of Walters and

Hilborn ( 1976). The model is defined as:

R,=S,_,+-@I-‘+Y” (1)

where R, defines recruits at the end of period t, S,_ 1 defines the number of spawners at the start of t - 1, a is a stock production parameter, p is an equilibrium stock parameter and u, a random error term assumed to be normally and independently distributed with mean 0 and variance u*. Walters and Hilborn (1976) developed an argument for lognormality based on the assumption about normality in the error term. If exp( u,) is viewed as a random survival factor resulting from M independent, multiplicatively acting environmental factors, si, that sequentially affect the survival of an initial number of individuals, Z, in a population such that:

R=Zfp i=l

(2)

In R=ln Z+ln s, fin s,+ . . . + In sM (3)

Furthermore, if s, through s,,,, are independent and identically distributed random variates, the central limit theorem implies that 1nR will be approximately normally distributed and that R will be lognormally distributed (Hogg and Craig, 1978). This proposition, however, will be true if, and only if, the survival rates are independent ( DeGroot, 1986), M is large and Z is constant, conditions which are unlikely to occur.

M. Power/ Fisheries Research 25 (1996) 77-95 79

Empirical evidence in support of the theoretical argument was provided by the log-

probability plot of Allen ( 1973). Re-examination of the data used by Allen (Shepard and Withler, 1958; Shepard et al., 1964) with empirical distribution function goodness-of-fit

tests (see below), however, indicates that while the lognormal model (p-value = 0.09 1) adequately describes the ratio of returns to escapement data in the Skeena River, the Weibull model (p-value=0.498) is an overall better description of the data. When coupled with Shelton’s assertion that a variety of distributional shapes can be expected and the strong assumptions required to establish the theoretical arguments for lognormality, the results suggest there is value in a comprehensive statistical testing of available data sets to determine the general applicability of the lognormal model versus other available distributional models. Accordingly, this paper discusses and applies appropriate testing procedures to the 100 data

sets contained in Myers et al. ( 1990) as a means of determining the suitability of several distributions as descriptions of the recruitment phenomenon in general, and the legitimacy of the widespread use of the lognormal distribution in particular.

2. Methods

The data sets in Myers et al. (1990) are time series of the annual recruitment to a 100 North Atlantic fish stocks. Recruitment was estimated from research surveys or from sequen-

tial population analysis. The boundaries used to define each of the stocks generally follow those of either the Northwest Atlantic Fisheries Organization (NAFO) or the International

Council for the Exploration of the Sea (ICES). Further details of the boundary descriptions and data assembly can be found in Myers et al. ( 1990).

As distributional testing procedures typically require that the data are uncorrelated (independent), the data sets contained in Myers et al. ( 1990) must first be tested for serial correlation before being used in goodness-of-fit testing. This required the adoption of a two stage testing procedure. In the first stage, each data series was tested for serial correlation using randomization tests. In the second stage, those series found to contain no significant serial correlation were further tested for compatibility with a variety of distributional assumptions to determine which, if any, distributional model best described the available recruitment data.

2. I. Randomization tests

The null hypothesis for goodness-of-fit tests require that the data being tested represent a random sample from an hypothesized distribution (Stephens, 1986). Accordingly, before goodness-of-fit tests may be appropriately used as a means of establishing whether the available data represent a random sample from a given distribution, the randomness assumption must be confirmed. In time series data this can be accomplished by assessing the significance of any observed serial correlation within the data.

Serial correlation among a time-ordered sequence of random variables indicates that the random variables at different time periods are correlated. In a non-random time series, observations that are a distance k apart will show a relationship, the strength of which can

80 M. Power/Fisheries Research 25 (1996) 77-95

be measured using sample autocorrelation coefficients. When the observations are equally spaced in time the kth sample autocorrelation coefficient is estimated by:

r,= CyLF (Xi-X)(Xi+,-X)/(?l-k)

C;=‘=(xi-X)*/n (4)

wherex,, . . . . X,, are the data values in the series with mean 8. For a random series r, will

be approximately normal with mean - 1 / (it - 1) and variance 1 /n when n is large (Madan- sky, 1988). Tests for significance can thus be constructed by comparing the statistic:

r,+ ll(n- 1)

zk= Jlln (3

with percentage points of the standard normal distribution for each k (Manly, 1992). Such tests, however, while useful, will not be exact.

A second approach, followed here, involves the use of randomization tests. The tests are

applied by calculating the observed rk (k = 1, . . . , 10 years) values from the recruitment

data using Eq. (4). The data are then randomly permuted and the resulting rk (k = 1, . . ., 10) values for the new data series are computed. The permutation and re-calculation of the rk values are completed a large number of times and the empirical distributions of the rk values at lag k constructed. The resulting empirical distributions provide convenient descriptions of the probabilities associated with observing given rk values and may be used for drawing inferences about the acceptability of the null hypothesis. This is done by calculating the proportion of all values of rk in the empirical distribution greater than or equal to the value of rk at lag k computed for the original data series. Since the proportion of such values is an estimate of the probability of observing the value of rk, it is also the significance attached to the computed r, in the original data series. If the proportion is less than or equal to a chosen level of significance, (Y, then the computed r, at lag k for the original data series is significant and the hypothesis of no significant serial correlation is rejected.

A test that involves sampling from a randomization distribution is exact in the sense that using a lOOa% level of significance has a probability of (Y, or less, of giving a significant result when the null hypothesis is true (Manly, 1992). While an infinitely large number of randomizations is not necessary, there is a minimum number which must be completed. The number completed should be sufficient to insure that the significance level estimated from the number of randomizations completed is close to the level that would be obtained from considering all possible permutations of the data.

The correspondence between the significance level yielded by the randomization testing procedure, and that obtained by considering all possible permutations of the data, can be computed by calculating the limits within which the significance level estimated from the randomization procedure will lie 99% of the time for a given significance level p from the full distribution using the approach given in Edgington ( 1987) as follows:

p=2.58 p(l-P)

d n

where IZ is the number of randomizations completed. Setting p = 0.01 and n = 5000 yields sampling limits of 0.006 and 0.014 within which the estimated significance level of the


randomization procedure will fall. As this is close to the desired 0.01 significance level that

would be obtained from the complete enumeration procedure, n was set at 5000 for the completion of the randomization tests reported below.

When a number of sample autocorrelation coefficients are tested for significance at the same time there is a probability of declaring at least one of them to be significant by chance alone. If K significance tests are conducted simultaneously, each with significance level aj, then the probability, PB, of all K tests being correct simultaneously will be given by the

Bonferroni inequality:

PB>l- 5 aj j=l

(7)

.whether or not the test statistics used are independent. Eq. (7) thus states that if K serial autocorrelations are tested using the (aj = a/K)% significance level for each test, then there is a probability of only cy or less of declaring any of them significant by chance. Accordingly, in selecting the appropriate level of significance for the completion of each of the significance tests on the r, values, due consideration must be given to the effect of the Bonferroni inequality on the implied level of significance for the randomization test as a

whole.

2.2. Goodness-of-j2 testing

Empirical distribution function (EDF) statistics are based on the empirical distribution function which is calculated as a step function from the sample data. EDF statistics are measures of the differences between the EDF and an hypothesized distribution function F(x) and are used in goodness-of-fit testing to assess the fit of the sample data to the hypothesized distribution.

Consider a random sample of size n, X,, X,, . . . X,. Let Xc,, < X,,, < . . . < Xc”) be the order statistics for the sample. The EDF is denoted as F,,(x) and is defined as:

F,(.x) = 0 x<X,,,

Fn(x)= ; X(i) Gxcx(i+l)

F,(x) = 1 &I, GX (8)

For any X, F,,(x) records the proportion of observations in the sample data less than or equal to X. On the otherhand, F(x) is the probability of an observation being less than or equal to X for the hypothesized distribution. If F,(x) can be expected to approximate F(x), then any statistic measuring the difference between F,(x) and F(x) will provide a measure of the goodness-of-fit of F,,(x) to F(x).

The EDF statistic employed here for goodness-of-fit testing is the Anderson-Darling statistic, A2. It concentrates on the vertical differences between F,,(x) and F(x) over the entire range of the considered data and is drawn from the Cramer-von Mises family of

statistics defined by Q:

82 M. Power/Fisheries Research 25 (1996) 77-95

co

Q=n j- [F,(x) -Yx)l*W) Wx) --m

where q(x) is a function that weights the squared differences between the empirical and hypothesized distribution functions using the expression [F(x) / ( 1 - F(x) ) 1.

Stephens ( 1986) reports that power studies for normality and exponentiality tests indicate that A2 is the recommended omnibus test statistic for EDF tests in which the required distributional parameters for F(n) are estimated from the sample data. In the cases discussed below, the parameters of the hypothesized recruitment distributions were estimated directly from the data with appropriate maximum likelihood (MLE) techniques. Furthermore, the data were tested directly for normality or exponentiality, or transformed such that they could be tested for normality or exponentiality. Hence, the choice of A2 as the test statistic is appropriate. Finally, following Stephens ( 1986) the A2 values were used only to indicate the appropriateness of the hypothesized distribution function as a model of the true recruitment distribution. Since the distribution of A2 will vary with F(x), the value of A2 itself will not indicate which of a series of hypothesized distribution functions best fits the data. Instead, the p-value attached to A2 must be used as an indicator of the true population from which the data were sampled, with a larger p-value (measured from the upper-tail) indi-

cating a better fit. This suggests a two phase testing procedure. In the first phase tests are used to establish

the candidacy of a particular distributional model as the best description of the data by determining the statistical adequacy of a hypothesized distribution as a description of the data. Since the distribution of,A* will vary with F(x), a second phase of testing is required in which the p-values are computed and used to arbitrate between candidate models as a means of selecting which, if any, of the considered distributional models best describes the data.

2.3. Recruitment models

Treating recruitment as the outcome of the probability of individual survival makes it analogous to statistical survival analysis. An important aspect of survival analysis is the selection of a distributional model to fit the available data. Biologically there are many causes which can lead to the death of an individual organism at a particular point in time. The causes may be as diverse as genetic, predation, starvation or toxic stress effects and it is difficult, if not impossible, to isolate each cause and mathematically account for its action. This suggests that the choice of an appropriate recruitment distribution may be as much an art as a science. Furthermore, it suggests that no single distribution will adequately account for the large number of possible combinations of factors acting on individuals within a population that ultimately determine recruitment. Accordingly, it seems appropriate to consider a number of distributional models which have been widely applied in statistical survival analysis. A number of the more common distributions used include: the exponential, Weibull, lognormal and gamma distributions. Of these, the exponential, Weibull and lognormal distributions can be fitted and directly compared using the testing procedure described above and are, therefore, used in the testing reported below.

M. Power/ Fisheries Research 25 (1996) 77-95

2.4. The exponential model

83

The exponential model has been widely applied to descriptions of survival and life-

expectancy in both the engineering and biological sciences. Specifically, it has been suc- cessfully applied in animal and human studies of both chronic and infectious diseases (Lee, 1992). The distribution is characterized by a single scale parameter, 0, denoting a constant rate of decline in the probability of successively larger observations. The distribution is appropriately descriptive of recruitment phenomenon dominated by a large number of smaller values and a few larger values. As such, it is probably most aptly descriptive of recruitment in relatively stable environments for long lived, high fecundity species with high juvenile mortality.

2.5. The Weibull model

The Weibull distribution is a generalization of the exponential distribution described by both a shape, A, and a scale, y, parameter. The two parameters allow the distribution to take on a wide variety of shapes from the extremes of the J-shaped exponential distribution to the skewed, bell-shaped curve more characteristic of the lognormal distribution. Testing for the suitability of the Weibull distribution can be completed by transforming the sample data. Data X,, X,, . . . , X, are distributed following a Weibull distribution if, and only if, Xt, Xi, . . . . Xi are exponentially distributed with scale parameter 0 = 9 (Law and Kelton, 199 1) . Thus estimating the shape (A) and scale ( 7) parameters for the Weibull distribution from the data and completing the transformation allows the data set Xt, Xi, . . ., Xi to be directly tested for exponentiality using the Anderson-Darling statistic.

2.6, The lognormal model

The lognormal distribution has enjoyed a long history of use in the biological sciences. It is typically used to describe phenomena whose distributions are skewed to the right or processes representing the product of many independent, multiplicatively acting factors (Bulmer, 1979). The distribution is defined by shape, u, and scale, CL, parameters that can

be directly estimated from the data. Data are lognormally distributed if, and only if, the logarithms of the data are normally distributed. Thus, if the data Xi, X,,. . ., X, are thought to be lognormal, the logarithms of the data points can be treated as normally distributed for purposes of hypothesizing a distribution, parameter estimation and goodness-of-fit testing (Law and Kelton, 1991). The relationship is particularly important both because it makes lognormally distributed data readily useable in EDF goodness-of-fit tests, and because of the number of easily accessible statistical techniques that exist for the study of normally distributed data.

2.7. Estimation and computation

The randomization tests were completed with the aid of the testing subroutines given in Manly ( 1992). The test statistic A2 was computed using the Davis and Stephens ( 1989) algorithm AS 248. The algorithm computes A2 and appropriately modifies it such that only

84 M. Power/Fisheries Research 25 (19%) 77-95

one line of significance points is required to judge the acceptability of the null hypothesis. At the 5% level of significance non-serially correlated data sets were accepted as adequate descriptions of the exponential, lognotmal or Weibull distributions if the computed A2 statistics were less than 1.321, 0.752 and 1.321 respectively. Approximate p-values were

also calculated using the formulae from Stephens ( 1986). The values are most accurate in the upper-tail and provide a convenient means of distinguishing between candidate models in terms of which provides the best fit to the data. Higher p-values indicate a better fit and

the distributional model having the highest p-value was selected as the best description of the recruitment data.

For the exponential distribution the scale parameter, 8, was estimated by its MLE esti- mator as follows:

(10)

The scale (A) and the shape (7) parameters of the Weibull distribution were estimated directly from the data following the MLE procedure of Thoman et al. ( 1969). Finally, the shape ((T) and scale ( p) parameters of the lognormal distribution were estimated by their

respective MLE estimators as follows:

(11)

s=

i

Cy=* (In Xi-/Z)*

n (12)

3. Results and discussion

The number of series for which the null hypothesis of no serial correlation can be accepted depends critically on the choice of the level of significance required of the testing procedure. When the first ten sample autocorrelation values are tested simultaneously, the Bonferroni inequality implies that for the level of significance associated with the randomization test

to be at least 1 - cr, each of the ten tests on the significance of the r, values must be conducted at the ( 1 - (u/ 10) level of significance. When the level of significance for the randomization test is set at 5%, each of the tests on r, is completed at the 0.005 level of significance. When the level of significance is set at lo%, each of the tests on r, is completed at the 0.01 level of significance. While it is tempting to increase the level of significance associated with the testing procedure, the gain in significance would come at the cost of an increase in the probability associated with the mistaken acceptance of the null hypothesis when it was false (type 2 error) and a consequent loss of power in the testing procedure (Hogg and Craig, 1978). To avoid unnecessary loss of power, and to keep the level of significance associated with each of the individual r, tests within the range of values commonly used for hypothesis testing procedures, the level of significance for the randomization procedure was set at 10%. This resulted in 65 of the 100 data series being accepted as consistent with the null hypothesis


Table 1

Results of the randomization and goodness-of-fit testing on a species basis

Species NR R Adequate Best

EXP LOG WE1 EXP LOG WE1

American Plaice

Argentine

Blue Whiting

Butterhsh

Cod

Greenland Halibut a

Haddock ’

Herring a

Mackerel

Menhaden

Plaice a

Pollock

Pout a

Redfish a

Scallops

Silver Hake

Sole a

White Hake

Whiting

Yellowtail Flounder

1

9

5

9

1

1

1 _

2 _

2

1

1

2

2 1

1 - 1 _

1 1

14 6

1 -

8 3

15 2

1 1 _

2 - 6 _

1 -

4 - 1 _

1 _

3 - _

2 1

1 -

2

1

1

_ 1

_ 2

1 5 _ 8

6

3

1

2

_ 1

1 _ _

_ I

1 _ 1

2

I

1 I _

12

3

6

I

_ 5

2

1

Totals 35 65 15 55 58 1 24 35

The species name given is the common North American name from Leim and Scott ( 1966). NR defines the

number of recruitment data sets that showed evidence of significant serial correlation. R defines the number of

recruitment data sets that showed no evidence of serial correlation. For each species, the number of stocks

adequately and best described by the exponential (EXP), lognormal (LOG) and Weibull (WEI) distributional

models are given in columns 4 through 6 and 7 through 9, respectively. For species marked with the superscript a,

R is greater than the sum of the best columns because one of the random recruitment data series could not be

adequately explained by any of the considered distributional models. For species marked with the superscript “,

the sum of the best columns exceeds R by one because no distinction could be made between the Weibull and

exponential models on the basis of completed testing.

of no serial correlation rather than 74, as would have been the case if the significance level had been set at 5%.

Table 1 reports the summary results of the randomization and distributional testing procedures on a species basis. The species name given is the common North American name (Leim and Scott, 1966). For multi-stock species, such as Cod, Haddock and Herring, there is evidence of both significant and insignificant serial correlation in the recruitment

data series. While serially correlated recruitment data do not dominate for any single multi- stock species, their existence does suggest local environmental influences may be important in determining the year-to-year recruitment for some stocks. This has been suggested elsewhere in the literature. Myers and Drinkwater ( 1989), for example, concluded there was evidence to suggest that reductions in the recruitment of groundfish were associated with local environmental influences. Species such as Argentine, Blue Whiting, Butterfish,

86 M. Power/ Fisheries Research 25 (1996) 77-95

Greenland Halibut, Pollock, Pout, Scallops and Whiting provided no substantive evidence of serial correlation in year-to-year recruitment processes. On the otherhand, species such as Menhaden and White Hake provided strong evidence of serially correlated recruitment processes. For the remainder of the species, tests on recruitment data provided evidence of both serial and non-serially correlated recruitment data. A fact corroborating the conclusion of Myers and Drinkwater ( 1989) concerning the localized nature of many of the environmental influences operating on the recruitment process.

Significant serial correlation is evidence of the persistence of anomalous recruitment for a given stock arising from the specifics of environmental influences. It suggests that information about the underlying processes generating the data may be contained within the data itself (Box and Jenkins, 1976). For example, Thompson and Page (1989) attempted to determine the degree of synchronous behaviour exhibited both within and between autocorrelated species stock recruitment data as a means of establishing the degree to which large-scale environmental forcing influenced recruitment.

Examination of the r, values for the serially correlated series indicated that most, 77%, had the largest correlations at a lag of 1 year. This suggests that local environmental influences, whose effects act immediately on the recruitment process, contribute more heavily to the incidence of anomalous recruitment. Thompson and Page ( 1989) found similar results in a study of recruitment synchrony. They concluded that the variability seen in recruitment data was due to small spatial scale variability and noise in the data rather than large-scale environmental forcing. For example, for Cod their study found that only 40% of the recruitment variance could be explained by large-scale environmental factors. However, large correlations at lags 2, 3,7 and 8 in the remaining 23% of the series suggest that life-history adaptations cannot be completely ruled out as the cause of some of the observed serial correlations in the data.

The distributional testing results reported in Table 1 indicate the statistical adequacy of

both the lognormal and Weibull distributional models as descriptions of much of the recruitment data. The lognormal and Weibull models usefully describe 55 and 58 of the 65

non-serially correlated data sets, respectively. The exponential model, which is a statistically adequate description of only 15 data sets, is a less useful explanation of the recruitment phenomenon. Six of the data sets (Greenland Halibut V & XIV, Herring Via (south) and VII bc, Plaice North Sea, Pout Norway, Redfish 5YZ and Sole North Sea) could not be adequately described by any of the considered models. For these stocks the uniform, normal, logistic and gamma distributional models were tested for adequacy with probability plotting techniques. As none of the alternative models were found to describe the data, they were omitted from further analysis.

While the results given in Table 1 substantiate the acceptability of the lognormal distributional model as a description of recruitment data, they do not suggest that it is universally the ‘best’ available description of the data. When the p-values are calculated and used to arbitrate between the candidate models, the lognormal model is considered to be the ‘best’ description of the considered data sets in only 24 of 59 (4 1%) cases. The Weibull model, on the otherhand, is ‘best’ in 35 of 59 (59%) cases, including one case in which the Weibull and the exponential models provide equally good descriptions of the available data. The last result undoubtedly arises because of the close theoretical connection between the two distributions as the shape parameter of the Weibull distribution approaches one.

M. Power/Fisheries Research 25 (I 996) 77-95 87

Table 2 reports a more detailed set of goodness-of-fit testing results for the 59 data sets found to be adequately described by at least one of the considered distributional models. As in Table 1, the common North American species name is given. Stock boundaries for each species follow those of NAFO or ICES. In some instances a region is referred to by name if it commonly applies to the stock in practice, or if the NAFO or ICES boundaries do not properly describe currently used stock boundaries. N defines the number of years of recruitment data contained in the data set, max r, defines the largest of the first ten sample

autocorrelation coefficients computed for each data set using Eq. (4). p defines the smallest probability associated with any of the estimated sample autocorrelation coefficients for a given data set. For each data set, the upper-tail p-values associated with the A2 statistic are reported in columns 6 through 8 for those distributions determined to be statistically adequate descriptions of the data at the 0.05 level of significance. The p-value for the distribution giving the ‘best’ fit is underlined. The percent error measure of column 7 reports the error that would result in predicting mean recruitment using the lognormal model for those stocks whose recruitment data were more appropriately described by the Weibull model. The error was calculated as the difference between the lognormal and Weibull model means expressed as a percentage of the Weibull model mean. Positive errors, accordingly, indicate an over- prediction of mean recruitment by the lognormal model. Summaries of the average predictive errors, which omit consideration of the statistical outlier for Redfish 4VWX, are given at the bottom of the table.

The errors introduced by the inappropriate use of the lognormal recruitment model vary within and between species. In 20% of the cases in which the lognormal model of recruitment could be mistakenly used in place of the Weibull model, substantive over-prediction of

mean recruitment in the order of greater than 10% occurs. In 54.3% of the cases marginal over-prediction of mean recruitment of less than 10% occurs and in 25.7% of the cases marginal under-prediction occurs. Averaged over all stocks, predictive errors resulting from the improper use of the lognormal model equalled 6.80%. The tendency toward over- prediction of mean recruitment at 9.93%, however, was considerably larger than that toward under-prediction, - 1.9 1%. The variability in the magnitude and direction of the predictive errors suggests that managers will need to complete sensitivity analysis on the effect any

assumed recruitment distribution assumption has on recruitment predictions, before conclusions about the design and implementation of optimal management policies may be confi- dently stated.

Combined, the results of Tables 1 and 2 indicate that no one distributional model appro-

priately describes the specifics of recruitment for a given species. For multi-stock species serial correlation predominates in some data sets but not in others. Amongst those data sets that are free of serial correlation, the lognormal model ‘best’ describes some of the data sets and the Weibull model others. The fact that recruitment data can take on a variety of distributional shapes at the stock level requires that the most appropriate function to describe

the shape of the recruitment distribution be selected on a stock by stock basis. As Shelton (1992) has shown, under heavy exploitation there may be a considerable reduction in population size leading directly to less independence in survival among individuals and a more positively skewed distributional shape. This may, in part, help to explain the prevalence of the Weibull distribution as the most appropriate description of much of the available recruitment data. When coupled with concerns about scientific uncertainty and parameter

88

Table 2

M. Power/Fisheries Research 25 (1996) 77-95

Detailed results of the goodness-of-fit testing on a species and stock basis

Species and Stock

N Max P Percent p-values

rk error

Lognomlal Weibull Exponential

American Plaice 3Ps 4v

Argentine 4vwx

Blue Whiting Northern Area

Butte@sh 58~6

Cod 1 3M 3PnRS 4x 5Y 5ze Iceland North Sea Celtic Sea VII f&g VIIde Kattegat Skageraak Baltic Areas 22&24 Baltic Areas 25-32

Haddock 3NOl a 3N02 = 3Psl a 3Ps2 a 4x Iceland Via North Sea

Herring 4R (spring) 4R (spring) 4T (fall) AB CD EF I Baltic Areas 22.24 South Central Baltic Gulf of Riga Baltic Areas 28&29s Bothnian Sea Bothnian Bay Gulf of Finland

13 0.517 4.80 0.894 0.869 20 0.276 8.02 57.55 0.278

17 -0.314 6.60 69.14 0.319 0.626

16 0.505 2.94 0.79 0.512 0.972

19 0.473 2.92 0.913 0.802

34 0.393 1.74 13.17 0.511 0.517 27 0.255 10.16 14.15 0.780 0.925 13 0.405 6.46 0.28 0.054 0.873 40 0.284 5.32 0.87 0.117 0.394 24 0.464 1.20 7.41 0.391 0.902 24 - 0.257 17.82 0.872 0.522 32 -0.351 3.52 - 0.23 0.107 26 - 0.220 17.42 0.80 0.713 0.862 15 0.333 8.42 3.64 0.286 0.818 13 -0.291 4.16 0.199 0.159 15 - 0.396 2.62 -0.19 0.268 0.580 9 -0.187 1.34 0.05 0.863 0.906

17 0.331 8.12 1.36 0.550 0.992 20 0.521 1.34 -0.56 0.204 0.326

25 0.372 2.60 0.860 0.505 9 - 0.587 2.60 0.151 0.061

23 - 0.399 2.52 21.27 0.079 0.090 10 - 0.504 2.50 -6.11 0.101 0.182 25 0.209 3.02 0.123 0.118 25 - 0.325 5.46 1.93 0.309 0.709 19 -0.313 10.10 0.382 0.305 27 - 0.457 1.12 0.505 0.331

21 -0.311 4.60 0.149 0.117 22 - 0.325 2.14 - 6.98 0.739 0.909 18 0.547 1.08 3.52 0.277 0.579 14 0.598 1.60 0.517 0.331 14 0.566 2.58 30.23 0.629 0.768 14 0.444 1.28 0.515 0.496 14 0.295 2.84 0.507 0.497 18 0.470 3.24 -0.00 0.387 0.775 26 0.331 9.18 0.22 0.317 0.921 18 - 0.215 17.38 0.798 0.523 16 - 0.276 7.02 0.879 0.777 13 - 0.342 11.42 0.873 0.798 14 - 0.522 2.04 - 2.07 0.117 0.196 18 -0.193 35.90 0.885 0.866

0.124

0.066

0.110 0.791

0.804 0.577

0.117 0.061

0.090

0.328 0.316

0.145 0.218

M. Power /Fisheries Research 25 (1996) 77-95 89

Species and

Stock

N Max P Percent p-values

rk error

Lognormal Weibull Exponential

Mackerel Western

Plaice Irish Sea VIIa

Pollock 4vwx5 Northeast Arctic b

Iceland

Farm

Via

North Sea

Redjish 3NOb 3Ps 4vwx

Scallops 5z.e

Silver Hake 4vwx

Sole Celtic Sea

IIIa

Whiting Via

North Sea

Yellowtail Flounder 3N0

15 0.274 17.76 6.60 0.080

22 -0.313 10.36

16 -0.371 7.20 20 0.257 25.26 19 0.513 1.74 24 0.470 1.50 26 0.351 3.18 26 0.449 1.50

0.38

0.00

- 0.85

0.42

0.25

9 - 0.464 4.44

14 - 0.526 1.76

17 - 0.308 7.04

8.71

443.33

15 0.358 7.32 0.64

14 - 0.337 9.86 0.359 0.246

15 - 0.345 2.24

18 0.484 2.18

4.93

26 0.394 1.94

27 - 0.270 8.68

18 0.514 1.84 - 0.23

Average percent error ’ 34 6.80

Average over-prediction ’ 25 9.93

Average under-prediction 9 - 1.91

0.714

0.566 0.078

0.438

0.515

0.091

0.511

0.084

0.622

0.569 0.144 0.206 0.643 0.784 0.436

0.983

0.069

0.112

0.489

0.779

0.541 0.668

0.162 0.557 0.911 0.481

0.571

0.053

0.445

0.287 0.052

0.492

a Non-overlapping time series for Haddock stocks 3N0 and 3Ps.

’ Stocks for which the upper-tailp-value for the lognormal distribution < 0.05.

’ Excludes the Redfish 4VWX stock from the reported average.

The species name given is the common North American name from Leim and Scott ( 1966). The boundaries used

to define stocks follow those defined by NAFO and ICES. N defines the number of data in the series, max r,

defines the largest of the first ten sample autocorrelation coefficients calculated using Eq. (4). p defines the smallest

probability associated with any of the estimated r, values for a data set. Percent error defines the amount by which

mean recruitment is over or under-estimated by the lognormal model in terms of the mean recruitment predicted

by the Weibull model. The upper-tailp-values for each of the tested distributions are reported in columns 6 to 8.

with the value for the distribution giving the ‘best’ fit being underlined.

estimation error, errors unnecessarily introduced through the inappropriate use of the lognormal recruitment assumption could hold significant implications for our ability to properly manage and conserve critical marine fish stocks.

Table 3 reports the details of the serial correlation testing for the 35 stocks whose recruitment data were found to be significantly autocorrelated. As elsewhere, the common North American species name, stock designations and the number of years for which data


Table 3 Detailed results of the serial correlation testing for stocks exhibiting anomalous recruitment

Species and stock

American Plaice

3LN0 Cod 2J3KL 3N0 3Ps 4TVn 4vsw Northeast Arctic Fame Plateau Via Irish Sea VIIa Haddock 4TVW 5Y 5z Northeast Arctic Faroe Plateau Herring 4wx 4R (fall) GH Iceland (summer) Iceland (spring) Norway (spring) Via (north) North Sea Northern Irish Sea Mackerel Areas 2-6 Menhaden US Atlantic Plaice Kattegat Redfish Iceland I & II Silver Hake 5Ze 5Zw6 Sole

Irish Sea Whire Hake 4T Yellowtail Flouruier 5z S.N.E.

N

21

27 29 29 37 28 27 23 19 16

39 25 29 26 23

21 21 14 40 23 37 18 41 18

25

27

19

14 21

31 31

16

16

25 25

Max

rk

0.743

0.785 0.749 0.457 0.676 0.425 0.585 0.479

- 0.378 - 0.549

0.702 0.472 0.360 0.506 0.565

- 0.461 0.591 0.464 0.541 0.717 0.430

- 0.424 0.666 0.686

0.525

0.601

0.623

0.757 0.638

0.905 0.899

- 0.526

0.837

0.565 0.692

k P

1 0.02

1 0.02 1 0.02 1 0.92 1 0.02 1 0.10 1 0.14 8 0.16

10 0.56 3 0.96

1 0.02 1 0.70 2 0.18 1 0.64 1 0.22

8 0.28 1 0.28 2 0.62 1 0.04 1 0.02 1 0.58 7 0.90 1 0.02 1 0.04

1 0.42

1 0.06

1 0.18

1 0.08 1 0.02

1 0.02 1 0.02

3 1.00

1 0.02

1 0.16 1 0.04

The species name given is the common North American name from Leim and Scott ( 1966). The boundaries used to define stocks follow those defined by NAFO and ICES. N defines the number of data in the series, max r, defines the largest of the first ten sample autocorrelation coefficients calculated using Eq. (4). k defines the lag in years at which the largest r, value occurs andp defines the smallest probability associated with any of the estimated r, values for a data set.

M. Power/Fisheries Research 25 (I 9%) 77-95 91

Table 4

Geographic distribution of random and non-random recruitment stocks a

Geographic area R NR

European Continental Shelf Baltic Sea

Barents Sea, Spitsbergen Bank Celtic Shelf

English Channel

Faroe Plateau Greenland Sea, Denmark Strait

Hebridean Shelf

Iceland

Irish Sea North Sea

Norwegian Sea

Rockall, Porcupine Banks

Skageraak, Kattegat

North American Continental Shelf

Davis Strait

Flemish Cap

George’s Bank

Grand Banks

Gulf of Maine

Gulf of St. Lawrence

Labrador Shelf

N.E. Coastal US

Saint Pierre, Green Banks

Scotian Shelf

US Atlantic

31 16

9 _

2 3 2 _

1

1 2

1 _

4 2

4 3

1 3

6 I 2

3

30 23

a The sum of R and NR exceed 100 as the figures above include the Butter Fish 5 & 6, Greenland Halibut V &

XIV and Mackerel Areas 2-6 stocks that are defined by NAFO and ICES with respect to multiple geographic

areas.

The geographic areas are defined with respect to common water body and continental shelf area names. R gives

the number of stocks whose recruitment data showed no evidence of significant serial correlation as determined

by the randomization testing. NR gives the number of stocks whose recruitment datashowed evidence of significant

serial correlation as determined by the randomization testing. For these stocks, statistical evidence of anomalous

recruitment exists.

were available are reported. The largest sample autocorrelation coefficient, max r,, and the year of the lag, k, at which it occurred are also given. p defines the probability associated with observing a value for rk at least as large as that reported in the table as determined by the randomization testing. The tendency for the largest correlation coefficient to be found at a lag of 1 year, and the wide geographic distribution of the stocks in which such significant lags occur, suggests species-specific sensitivity to environmental influences critically affects the incidence of anomalous recruitment in marine fish stocks.

Table 4 presents the results of the serial correlation testing on a geographic basis. Common water body and continental shelf area names have been used in the table. Stocks were assigned to a geographical area on the basis of the stock boundaries defined by NAFO and ICES. Accordingly, the 5Y stock boundaries correspond to the Gulf of Maine and those of

92 M. Power/ Fisheries Research 25 (19%) 77-95

3LN0 correspond to the Grand Banks. Comparisons of the number of serially and non- serially correlated recruitment data sets within each geographical area yield an approximate index of the importance of local environmental influences for the incidence of anomalous

recruitment in each area. The proportion of non-serially correlated recruitment data sets among European and North American stocks was 0.30 and 0.43, respectively. A significance test on the difference in the proportion of serially correlated recruitment series among European and North American stocks was completed by transforming the observed differences to a standard normal variate and determining significance with reference to the standard normal table (Bulmer, 1979). As the observed difference was significant at the 0.0793 level, there is insufficient evidence to conclude that the observed difference is real. The existence of both serially and non-serially correlated recruitment series, however, does

indicate that local environmental influences play some part in determining year-to-year recruitment.

The results of the distributional testing, and the geographic pattern of anomalous recruitment obtained here, reflect the debate in the literature about the specificity of recruitment

processes. Sutcliffe et al. (1976) showed that outflows from the Gulf of St. Lawrence affected water temperatures throughout the Scotian Shelf and Gulf of Maine regions to the extent that commercial catch could be correlated with freshwater discharge. Koslow ( 1984) found large-scale patterns of biological production in the ocean were linked to processes operating over distinct oceanographic or climatic regions. The conclusion that such large- scale environmental influences were important in explaining the widespread year-class synchrony among Northwest Atlantic gadoid stocks would suggest a similarity in the pattern

of distributional testing results should exist. The results, however, are hard to interpret with any degree of unanimity. For example, while 12 of 14 Cod stocks tested were best described by the Weibull model, only 14 of 23 data sets could be regarded as non-serially correlated

in the first place. This suggests that some 40% of the tested stocks are influenced by specific environmental factors which distinguish them from the majority (52%) of the stocks whose recruitment processes are governed by the Weibull model. Thompson and Page ( 1989) in re-examining the synchrony of recruitment processes found that for Cod the variance in recruitment was due largely to small spatial scale variability rather than large-scale environmental forcing. A fact which would explain the pattern of both distributional and serial correlation testing results found here.

Myers and Drinkwater ( 1989) provide evidence of very specific environmental influences on the recruitment process in the form of warm core rings, or large eddies, generated

by unstable meanders from the Gulf Stream current along the Northwest Atlantic continental shelf area. The eddies, which entrain large volumes of water from the shelf area, were shown to transport enough fish eggs and larvae to significantly reduce groundfish recruitment. The local action of such effects is consistent with the pattern of serial correlation observed within and between species in Table 1.

The combined results of Sutcliffe et al. ( 1976) and Myers and Drinkwater ( 1989) suggest the pattern of serial correlation may relate to the influence of physical processes such as the conjunction of ocean currents, warm core rings or freshwater discharges that act to promote environmental heterogeneity over wide geographic areas. The Barents Sea, Iceland, the Fame Plateau, Grand Banks, Scotian Shelf, Gulfs of Maine and St. Lawrence and George’s Bank are areas that have identifiable water mixing processes. In the Barents Sea mixing


occurs as the northward extension of the North Atlantic current, the Norway current,

discharges into the Barents basin. Mixing also occurs to the north and east of Iceland at the confluence of the clockwise circulation of the minor arm of the Irminger current and the eastward flowing branch of the East Greenland current. On the Faroe Plateau, mixing occurs at the intersection of the minor arm of the Irminger current and the northeast flowing North Atlantic current. The Grand Banks is dominated by the meeting of the Labrador Current and the Gulf Stream, as well as the complex set of minor rotational currents resulting from the meeting of the two currents. Meanders, warm water core rings and freshwater discharges respectively influence the off-shore and in-shore regions of both the Scotian Shelf and

George’s Bank. Finally, freshwater discharges affect the Gulfs of St. Lawrence and Maine. Within these regions some 46% of the stocks have serially correlated recruitment data, whereas outside these regions only some 29% of all stocks have serially correlated data. The difference is significant (p = 0.039) and suggests that the environmental heterogeneity resulting from oceanographic processes that affect the mixing of warm and cold and salt

and freshwater water may help to explain the pattern of anomalous recruitment. Perhaps the clearest summary, however, of the recruitment puzzle has been provided by

Koslow et al. (1987) who argue the evidence of large scale, periodic behaviour in the

meterological, physical and biological oceanographic variables of the Northwest Atlantic clearly points to the need for an improved understanding of the regional climatic processes and their interactions which may regulate recruitment to the region’s major groundfish

stocks. The evidence for both large- and small-scale environmental influences on recruitment, the incidence of serial correlation in some recruitment data series, the pattern of geographic results and the variety of distributional descriptions of the non-serially correlated data series all suggest that recruitment is the result of a complicated set of environmental and life-history interactions which are only likely to be adequately described on a stock- specific basis.

For management and policy design, specificity holds particularly important implications. As assumptions about the recruitment distribution hold direct implications for the variability presumed in the recruitment data and the degree of precision associated with predictions about management and exploitation policies, it is particularly important to validate recruitment assumptions whenever possible. Furthermore, as has been shown, the acceptance of several competing distributional models as adequate descriptions of a given set of recruitment data implies that single distributional hypothesis tests cannot be relied on to select the ‘best’ distributional model for the available data. This necessitates the use of a variety of distributional models and the implementation of more comprehensive testing regimes for the selection of recruitment models. Where no one model can clearly be accepted as ‘best’, managers must be prepared to complete sensitivity testing on the design and predicted efficacy of their management and exploitation policies. Such testing would entail measuring the responsiveness of policy recommendations to selected distributional assumptions and

help to clarify for all the implications of incomplete information on the nature of recruitment mechanisms for fisheries management practice.

4. Conclusion

The use of statistical testing has shown that a variety of distributional shapes can be expected to be descriptive of the recruitment data. The results of testing multi-stock species


also indicates that the ‘best’ distributional model varies as much on a stock by stock basis as it does on a species basis. Evidence from both Europe and North America indicates the incidence of anomalous recruitment is significantly higher in areas dominated by freshwater or ocean current mixing, but is not exclusively confined to those areas. Accordingly,

recruitment should be viewed as being a stock-specific attribute, closely linked to life- history and local environmental influence interactions. For fisheries managers this poses the challenge of completing appropriately designed risk analyses on any, and all, proposed management policies. The effect of assumptions underlying the selection of the recruitment model must be made explicit so that more fully informed management decisions can be made. In light of recent concern, particularly in Canada, about the sustainability of the fishery, the need for examining the implications of such assumptions on the development of fisheries policy is great. Only when fisheries managers are made aware of the possible

implications that different recruitment models may have on conclusions concerning the sustainability of fisheries practices can optimal policies be designed and implemented.

Acknowledgments

Comments by anonymous reviewers assisted in the revision of the manuscript. An NSERC operating grant to M. Power supported this work.

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the testing and selection of recruitment distributions for north atlantic fish stocks

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