Interpretation of results and they may be due to data processing or transcription

7.4.4 Interpretation of results and they may be due to data processing or transcription

errors (see commentary on measurement error in

the calculation of year-class strength Chapter 13, this volume).

Once values for terminal F and terminal S have An example is given in Tables 7.10 and 7.11, for also been selected, the essential results of sepa- the same North Sea cod data set, using F s (t) = 1.0

rable VPA are a set of overall fishing mortalities and S(g) = 1.0, and these results are illustrated

F s (y), and a set of selection factors S(a), normalized in Figs 7.8 and 7.9. The exploitation patterns of to S(a*) = 1. These may immediately be used to gen- Fig. 7.9 may be compared with those of Figs 7.2 and erate a full matrix of fitted separable fishing mor-

7.3: they are clearly similar to those from the JAM talities using the underlying model equation

method, which is to be expected since they are, in this instance, based on similar assumptions.

Fya ¢( , )=()() FySa s .

7.4.5 Discussion

The result of separable VPA is in fact an inter-

pretation of the data which is as far as possible con- Separable VPA is a very convenient way of generat- sistent with this separable model, but not forced to ing exploratory VPAs, and very effective in forcing fit it exactly. The closeness with which the solu- the analyst to come to terms with ignorance con- tion conforms to the model depends on a number cerning estimates of terminal F and terminal S. It is of factors, and in some part of the matrix, such as in also a useful objective way of estimating exploita-

Chapter 7

tion patterns over limited periods of time, for ex- minimum periods in practice. Finally, the usual ample before and after an expected change: three (e.g. CEFAS, Lowestoft and ICES) implementa- years in the absolute minimum period of time for tions include the estimation of weighted average this, but four or five would be more preferable as terminal populations, which is a most useful fea-

ture, particularly with data of poor quality. Against this, the method does not permit exter-

Table 7.10 North Sea cod: fishing mortality estimates nal data such as CPUE from commercial fleets or from separable VPA.

surveys to be incorporated directly, and it really ex- poses the analyst’s ignorance, rather than helping

Age

him to overcome it. Furthermore, the method is 1990

Year

based on the assumption of a constant exploitation pattern, for the total international fishery, for

some period of time, and some range of ages. The 2 0.977

results are not, of course, forced to fit this model 3 1.016

exactly, but it does nevertheless underpin the solu- 4 0.916

tions obtained. In the case of changes of exploita- 5 0.886

tion pattern on the youngest ages, there is little Table 7.11 North Sea cod: population estimates from separable VPA.

Fishing mortality at ages 1 to 4 0.2

Fig. 7.8 Fishing mortality esti-

Year

mates from separable VPA.

Dynamic Pool Models I: Interpreting the Past

Population numbers at ages 1 to 3 20 000

1993 1994 1995 1996 from separable VPA.

Fig. 7.9 Population estimates

doubt that such changes are a priori quite likely, dent information, such as indices from a series of because of differing spatial distributions of juve- recruit surveys, is vitally important for this pur- nile fish from year to year. They are, however, pose. In the absence of such information the esti- also confounded with higher sampling errors on mates from separable VPA may, however, give younger ages. It is, indeed, not at all clear how some rough indication of recruiting year-class much larger residuals on the younger ages are due strength, but these should be handled with ex- to process errors through changes of exploitation treme caution, and alternative explanations pattern and how much to sampling errors (Sparre sought, especially if they are outside the historic and Hart, Chapter 13, this Volume). In any case, range of the data. any unexpectedly high or low catches of young

On balance, we consider that separable VPA is a fish due to either process in recent years will be useful tool, if carefully applied, and preferable to interpreted by separable VPA as being due to large more heuristic methods such as those discussed or small year classes, and since no long run of above, including the JAM method. When total data exists for these recent year classes there catch-at-age data is the only thing available, it may is no prospect of deciding which is the correct indeed be the method of choice, but in this situa- interpretation.

tion the risk of an incorrect interpretation is high, The estimates of newly recruiting (recent) year- and independent CPUE and/or survey data are class strengths obtained by separable VPA should greatly to be desired. Their analysis is discussed therefore be treated with appropriate scepticism. below. They are based on the application of ‘normal’ lev- els of F to the few catch data available, but may be due either to a real large or small year class, a shift

of exploitation pattern, or simple sampling error. 7.5 TUNING OF VPA USING

An analysis of the variability (log standard error) of

CPUE AND SURVEY DATA

historic fishing mortality on the age in question gives a rough idea of the probable imprecision of

7.5.1 The analysis of catchability

the estimates. The true interpretation cannot be The need for additional information to resolve the deduced from catch-at-age data alone: indepen- indeterminacy of VPA has been stressed above.

High Med Low Survey

1000 Fig. 7.10 North Sea cod: the ‘tuning’ problem. The method changes the terminal F value in an

Population numbers and survey index attempt to make the trend line for population numbers parallel the

100 trend line found in the survey data. 0 1 2 3 4 5 6 High, medium and low F values are

Age

used.

Common forms of information suitable for this The use of CPUE data on its own to provide in- purpose are indices of abundance such as catch per dices of abundance, and hence estimates of total unit effort (CPUE) from commercial fleets or mortality, using the log:catch ratio method, i.e. research vessel surveys, or indices derived from Z (y,a) = ln[u(y,a)/u(y + l,a + l)], has a long history acoustic surveys. It will be assumed here that age (see e.g. Gulland 1983). Its use for VPA tuning is, composition information is available to go with however, relatively recent. Many methods have these data, although this is not always the case, been invented, particularly by assessment work- and that a reasonable time series of say five or more ing groups during the 1970s, when VPA became years of data are available.

widespread, and as its indeterminacy became un- Such data therefore provide estimates of the rel- derstood (Ulltang 1977). The reason for the multi- ative abundance of each year class through time, plicity of methods is that there are many possible and may therefore be used in principle to choose choices to be made, even assuming that one uses the VPA interpretation in terms of absolute abun- some sort of regression method. Should one regress

dance most closely in accordance with them. This F on effort or CPUE on population size? What type process is usually referred to as ‘tuning’ the VPA, of regression is appropriate – predictive, functional by analogy with the process of tuning a musical or calibration? Should the regression be forced instrument to the correct absolute pitch. The through the origin? Should a logarithmic transfor- process is essentially a synthesis of two different mation be applied? What criterion of best fit sorts of information, the estimates of relative should be used? Different answers to these ques- abundance from abundance indices, and the esti- tions all generate different methods of analysis mates of absolute abundance from VPA applied to which, given the quite substantial sampling errors catch data. In the interpretation of the former we of catch-at-age and CPUE data, can easily generate have an unknown multiplicative calibration con- results which differ from one another by amounts stant, which is the catchability, and in the latter an which are of considerable practical significance, unknown additive constant, the terminal popula- even if the differences are not statistically signifi- tion. By putting the two data sets together we can cant! This is a classic example of the need to exer- in principle determine both of these unknowns, cise more care in choosing methods for the and thus the full sequence of absolute population analysis of poor data than is required if one has size, as illustrated in Fig. 7.10.

good (precise) data. However, Laurec and Shepherd

143 (1983) showed that many of the doubts and confu- for example, by fitting a regression of C/P against

Dynamic Pool Models I: Interpreting the Past

sions can be clarified if one takes care to treat the E .

problem as an exercise in statistical modelling, Similarly, using model (7.9) one would natural- and if one regards the whole process as an exercise ly apply the same procedure to in the analysis of catchability. The reasons for this are as follows.

(7.11b) The use of fishing effort or CPUE data relies

CE ª qP ,

on the assumption that fishing mortality should However, the results of these analyses would

be strongly related to fishing effort, preferably not generally be the same. Firstly, use of standard proportional to it, and that CPUE should be predictive regression formulae to fit these equa- similarly related to abundance. It is perhaps tions would imply different assumptions about the not quite obvious that the constants of propor- nature of the errors, and therefore lead to different tionality are the same. If E is fishing effort, u is results. In case (7.11a), effort (E) would implicitly CPUE and q is the catchability coefficient, we may

be treated as exact, and uniform additive errors write

would implicitly be assumed for the ratio C/P. In case (7.11b), P would be treated as exact, with uni-

F = qE (7.8) form additive errors in C/E. The effects of these different assumptions would usually be greatly in- or alternatively,

creased if one permitted non-zero intercepts for the regressions, which might seem to be a normal

u = q¢ , P

(7.9) and natural thing to do.

Secondly, the effects of variations of effort and where P is the population averaged over the year-class strength on these two possible relation- year in the same way as is the CPUE, and ships are quite different. Consider a typical situa- we allow the possibility that q¢ may possibly tion where effort does not change very much, but

be different from q. However, by definition, u = year-class strength does. The first model (F versus

C /E, and C = FP [1 - exp(-Z)]/Z, so C = FP ¯ , since [1 - effort) leads to a plot like Fig. 7.11(a), whilst the exp(-Z)]/Z is just the ubiquitous averaging factor second (CPUE versus population) leads to a plot relating the average of an exponentially declining like Fig. 7.11(b). quantity to its initial value. Thus, from equation

The first plot exhibits no convincing relation- (7.9)

ship between F and effort. One has a cloud of points some way away from the origin. The correlation

qP ¢== u CE = FP E (7.10) coefficient will be very small and probably not even statistically significant, and the usual predic- and thus q¢ = F/E = q. The models (7.8) and (7.9) are tive regression coefficient will likewise be very therefore precisely equivalent, and involve the small. With no convincing evidence of any rela- same catchability coefficient, so that in principle tionship, the faint-hearted analyst could easily it should not matter which of these is used as the give up and conclude that the CPUE data are basis of the analysis.

worthless. Contrast this with the situation shown However, in practice, different results may in Fig. 7.11(b). Here one sees a good spread of points arise because of fitting procedures. The basic ‘ob- tracing a fairly convincing linear relationship. The servable’ quantities may be regarded as C and E correlation coefficient will be high, with a highly (raw data), and P which are simply derived from cu- significant regression coefficient, and an intercept mulative catches by VPA. Using model (7.8), one near zero. Yet these are simply different presenta- would naturally fit the relationship

tions of the same data! And the position could in fact be reversed if one had a stock where year-class

CP ª qE , (7.11a) strength was rather invariable, but effort had

1.4 able to apply a logarithmic transformation to the

1.2 data, since both u and P are fundamentally non-

1.0 negative quantities, and both are probably mea-

0.8 sured with something approaching a constant

0.6 coefficient of variation, rather than with constant

0.4 additive errors.

Fishing mortality

0.2 Most of this confusion can, however, be avoided

0 if we remember that the whole analysis is founded

0.2 0.4 0.6 0.8 1.0 1.2 on the presumption of a more or less constant Fishing effort

catchability, which is of course given by (b) 120 000

q = uP = ( CEP ) = ( CPE ) = FE .

80 000 Thus q is just the slope of a straight line joining 60 000

each of the points to the origin, in both Figs 7.11(a) CPUE

40 000 and 7.11(b). Given the statistical properties of

C and P, mentioned above, it would be reasonable 20 000

and preferable to work with the logarithm of this

0 10 000 20 000 30 000 40 000 50 000 60 000 equation, i.e. Population size

ln ()=()-()-() q ln C ln P ln E . (7.13) Fig. 7.11 (a) Relationship between fishing mortality and

effort; (b) relationship between CPUE and population This has the incidental benefit of circumvent- size. The graphs illustrate the relationship implied by

ing any argument as to whether one should in fact (a) equation 7.8 and (b) equation 7.11(a).

try to estimate the reciprocal catchability (l/q), since after log transformation this becomes an equivalent procedure. If one is prepared to treat

changed considerably during the period covered catchability as a constant, for each fleet and age, by the data.

then one needs to do no more than estimate the This confusion is reduced considerably if one mean of ln(q) from applying equation (7.13) to the realizes that allowing a non-zero intercept in these data. This is equivalent to estimating and using regressions is in principle inconsistent with the the geometric mean slope of the lines joining postulated models (equations 7.11(a) and 7.11(b)). the points to the origin in either of Figs 7.11(a) and If one forces the regressions through the origins the 7.11(b). This commonsense approach was suggest- discrepancy between the results using these two

ed by Laurec and Shepherd (1983), and leads alternative models is much reduced: the correla- to a VPA tuning method often known as the tion coefficients based on deviations from the Laurec–Shepherd method. origin will be similar, as will the regression coeffi-

The approach implied by this procedure can, cients, which are the estimates of catchability, however, be extended. One can treat the individual though they will still not be quite the same.

estimates of log catchability from equation (7.13) Even then, however, there is still scope for con- as a derived quantity, whose variation with time, siderable confusion and argument. One could population size, effort and weather can potentially make an excellent case for doing the regressions

be examined, and tested against the underlying de- the other way round, which would mean using a fault null hypothesis that it should be a constant. calibration rather than a predictive regression (see This constant will be different for each fleet, al- Shepherd 1997). Furthermore, when working with though preferably not varying with age for older model equation (7.11b), it would be very reason- fish. We are therefore led to consider the statistical be examined, and tested against the underlying de- the other way round, which would mean using a fault null hypothesis that it should be a constant. calibration rather than a predictive regression (see This constant will be different for each fleet, al- Shepherd 1997). Furthermore, when working with though preferably not varying with age for older model equation (7.11b), it would be very reason- fish. We are therefore led to consider the statistical

For operational use, however, well-automated methods of analysis and processing are required. All possible variations of catchability cannot be considered, but it is very important to check for trends with time. This is because, regrettably, much fishing effort and therefore CPUE data are not well standardized for possible changes of fishing power. Such standardization is not easy to do well (see e.g. Gulland 1983), and even if one had done a good job of allowing for changes in the composition of the fleet, in the engines and gears used by the different vessels and so on, there is still the problem of changes of catchability caused by natural shifts in the spatial distribution of the fish, or shifts in the spatial distribution of fishing effort within the strata of the sampling scheme in use.

Because of the irregularity described, abun- dance indices from carefully controlled series of surveys by research vessels, covering the whole distribution of the stocks, are much the preferred form of data. However, the sampling levels at- tained on surveys may not be sufficient to keep the level of sampling errors low enough for very reli- able analysis, especially on the older and less abun- dant age groups. It is therefore usually necessary also to use and analyse data from commercial sources, in which changes of catchability are more likely.

A useful practical procedure is to examine such data for changes of catchability with time, not just because such changes are the likely consequence of imperfect standardization of the effort data, but because other changes, such as those due to cli- matic changes or changes of distribution and abun- dance, which include density-dependent changes (Myers, Chapter 6, Volume 1) are likely also to show up as time trends, even if time itself is not the underlying causal variable.