Exploratory analysis using trial VPAs

7.3.2 Exploratory analysis using trial VPAs

The retrospective ‘convergence’ property of VPA means that, even without additional information, it may often be used to obtain some idea about fish- ing mortality a few years ago, and the size of all but the most recent year classes, provided fishing mortality is not too small. Indeed, there are few professional analysts who, given a suitable set of catch-at-age data, would not carry out some form of trial VPA just to get some rough idea of what is, or rather was, going on. Such a trial analysis must usually be based on rather arbitrary assumptions concerning terminal F. We recommend setting trial terminal Fs to a high value (of the order of 1) in such an analysis, to ensure rapid convergence. This can conveniently be achieved, especially when using cohort analysis, by setting the termi- nal population equal to the terminal catch num- ber, or some fraction thereof. The value of terminal

F so created depends on the level of natural mortal- ity, but is satisfactorily high (0.644 when M is taken as 0.2). Examples of such a trial analysis, for a

131 Table 7.1 North Sea cod: catch number at age, 1990–5.

Dynamic Pool Models I: Interpreting the Past

Table 7.2 North Sea cod: population estimates from trial cohort analysis with high terminal F (0.644). Age

Table 7.3 North Sea cod: fishing mortality estimates setting the terminal populations equal to double from trial cohort analysis with high terminal F (0.644).

the final catch number, are given in Tables 7.4 and

Age

The time trends of average fishing mortality 1990

Year

and year-class strength from these analyses are plotted in Figs 7.2 and 7.3, which illustrate the con-

siderable difference between these equally feasible 2 0.972

interpretations of the same data, particularly (but 3 0.974

not only) for recent years.

7.3.3 The Judicious Averaging Method (JAM)

If (and only if) one is prepared to assume that things set of ICES data for the North Sea cod (Gadus such as fishing mortality and the exploitation pat- Morhua ) stock from 1990 to 1995 (Anon. 1991) are tern have not changed very much recently, a trial given in Tables 7.1, 7.2 and 7.3, using the fairly analysis can also give some idea of what is going on high estimate of terminal F obtained in this way.

now, as well as in the past, but the validity of the The results of an alternative trial analysis, interpretation depends entirely on the assumption using a lower estimate of terminal F, obtained by of constancy. One is really relying on additional

Chapter 7

Table 7.4 North Sea cod: population estimates from trial cohort analysis with low terminal F (0.373). Age

Table 7.5 North Sea cod: fishing mortality estimates tively rerun the VPA, inserting terminal Fs based from trial cohort analysis with low terminal F (0.373).

on the average over several (usually about five) pre- vious years, for each age in the last year, and the

Age

average over (say) the five next younger age groups 1990

Year

for the oldest age group in each year. This formalizes the assumptions being made, 1 0.205

i.e. approximate constancy of F in recent years, and 2 0.949

approximate flatness of the exploitation pattern 3 0.905

on the oldest ages, and creates a workable method 4 0.738

of analysis sometimes known as the JAM method. 5 0.475

The acronym may be expanded either informa- tively as the Judicious Averaging Method, or cyni- cally as Just Another Method, according to taste. An example of such an analysis on the same North

information from a stronger model, based on the Sea cod data set is given in Tables 7.6 and 7.7, and assumption of more or less constant fishing mor- the resulting fishing mortality and year-class tality, as described above. This procedure can of strength estimates are included in Figs 7.4 and course be automated quite easily. One can itera-

7.5. This third interpretation of the data is less 0.9

0.4 F (ages 2 to 6) 0.3

Mean 0.2 0.1

Fig. 7.2 Fishing mortality 0 estimates from two trial VPA

analyses. 䊏 represents higher F;

Year

䉬 represents lower F.

Dynamic Pool Models I: Interpreting the Past

Population number at age 1

Fig. 7.3 Year-class strengths from two trial VPA analyses.

䉭 represents low F; 䊏 represents

1993 1994 1995 high F.

Table 7.6 North Sea cod: population estimates from JAM method. Age

Table 7.7 North Sea cod: fishing mortality estimates not thereby guaranteed to be closer to the truth. from JAM method.

This will only happen if the assumptions made happen to be correct. The validity of these assump-

Age

tions is not known and cannot be tested on these 1990

Year

data alone.

In the absence of any other relevant informa- 1 0.208

tion, however, these assumptions may be reason- 2 0.972

able null hypotheses, to be accepted until 3 0.974

disproved. They are in fact weaker than the as- 4 0.859

sumptions made in some alternative analyses of 5 0.656

the same data. If one were to average over years, and plot and analyse the resulting catch curve (see Gulland 1983, Section 4.3.3), then one would be

arbitrary than the two trial analyses, since it is implicitly assuming that fishing mortality is con- based on clearly stated assumptions about the be- stant over the whole period in question, that the haviour of fishing mortality. It depends on a rather exploitation pattern is flat for all fully recruited more restrictive model for fishing mortality, but is ages, and that recruitment has been constant, or at

Fishing mortality (ages 1 to 4) 0.2

Fig. 7.4 Fishing mortality

Year

estimates from the JAM method.

Population numbers at ages 1 to 3 20 000

Fig. 7.5 Population numbers from

Year

the JAM method.

least trend-free for the whole period. These are vides no evidence about them. As will be discussed much stronger assumptions, which most analysts below, it is also a method which is very sensitive to would, however, probably make quite cheerfully errors in the catch data, which may seriously affect in sufficiently desperate circumstances.

the population estimates obtained. A similar inter- The strong assumptions implied by the JAM pretation of the data, based on similar assump- method need not therefore forbid its use as an ex- tions, which is less sensitive to these errors can in ploratory tool. However, one should remember fact be constructed using separable VPA, and this that it is based on assumptions concerning the is preferable, particularly if the analysis is to form things one would really like to determine, particu- the basis of a catch forecast. Finally, it should be larly the recent trend of fishing mortality, and pro- noted that it is not appropriate to carry out a catch

135 Table 7.8 North Sea cod: population estimates from cohort analysis with implausible terminal F.

Dynamic Pool Models I: Interpreting the Past

curve analysis to determine starting assumptions Table 7.9 North Sea cod: fishing mortality estimates for a VPA, whether using the JAM method or other- from cohort analysis with implausible terminal F. wise, because the assumptions made are substan-

tially stronger than those needed for the VPA itself, Year and are not more likely to be true.

7.3.4 Trial VPA – its virtues and

its vices 0.204

A trial VPA is a very simple procedure enabling a 4 0.937 0.528 0.957 0.673 1.108 5 0.204 0.308 0.930 0.316 0.964 0.378 1.033 quick preliminary analysis of a set of catch-at-age 6 1.033 0.204 1.033

0.204 1.033 data to be carried out. Even the Judicious Averag- 0.204 ing Method for setting the terminal F values relies

on weaker assumptions than alternative methods such as catch curve analysis, and the whole thing can easily be set up using a spreadsheet program

7.4 SEPARABLE VPA

on a microcomputer, particularly if one makes use

7.4.1 Separability of fishing

of Pope’s cohort analysis algorithm (equation 7.2).

mortality

The calculation can be done with only (say) 10 years and 10 ages – only 100 numbers – and survey The ‘solutions’ of ordinary VPA, based on different or fishing effort data are not necessary. One should assumptions about terminal F, all fit the data not, however, allow oneself to be seduced by all equally well (exactly), and cannot therefore be dis- this convenience. Regarded as a statistical proce- tinguished on the basis of their goodness-of-fit. dure, a simple trial ‘untuned’ VPA is completely There is no doubt, however, that some of the pos- underdetermined, and can at best be thought of as a sible solutions would be regarded by any reason- non-unique transformation of the data. The catch able person as more plausible than others. An data are also treated as though they were exact, and example of an implausible interpretation is given any sampling and measurement errors feed in Tables 7.8 and 7.9. This was constructed by in- through directly into the fishing mortality and serting alternately high and low estimates of the population estimates obtained, especially of the survivors, leading to values of approximately 1.0 survivors at the end of the final year, which is an and 0.2 alternately for terminal F. The oddity of undesirable feature.

this interpretation is obvious even in Table 7.9,

0.4 3 4 Fishing mortality at ages 1 to 4

Fig. 7.6 Fishing mortality estimates for implausible

terminal F assumptions.

Population numbers at ages 1 to 3

1996 Fig. 7.7 Population numbers for implausible terminal F

Year

assumptions.

but may be more easily appreciated graphically, as more plausible interpretation of the data than Fig. in Figs 7.6 and 7.7.

7.6. Why?

This shows that the exploitation pattern im- Fishing mortality is a function of both time and plied by this analysis is very ‘rough’, alternat- age – we usually write it as F(y,a). However, once ing between high and low values on adjacent fish have reached a certain age and size, they ages and cohorts, and that the positions of the mature, their growth slows down and they tend to highs and lows are reversed each year, creating behave as a group, so that fish of similar ages are

a ‘checkerboard’ or diagonally banded pattern. found together, and are of similar size. Even if they This may be compared with the equivalent dia- deliberately tried to do so, fishermen would have gram derived from the JAM analysis, shown in difficulty in applying a much different fishing mor- Fig. 7.4. Here the exploitation pattern is more tality to adjacent age groups of the same stock – and or less flat, and has a similar shape in each year. there is no obvious reason why they should try to One feels instinctively that Fig. 7.4 represents a do so. One would therefore suppose that fishing

137 mortality should not change very much between simplest and most common is simply to specify

Dynamic Pool Models I: Interpreting the Past

adjacent age groups in the same year, at least once that S is defined to have the value 1.0 at some age, the fish are mature and fully recruited. One would the age of unit for selection, denoted by a*. All the also expect that the exploitation pattern created by other S values are then defined relative to that on fishing should not normally vary haphazardly be- this age, and the F s values are then conditional on tween one year and the next. Progressive changes the exploitation pattern so defined. The choice of

over time are quite possible, as are sharp changes, a * is in principle completely free, but some practi- for example if and when a new larger or smaller cal considerations are discussed below. minimum mesh size is enforced. These changes should, however, be systematic rather than hap-

7.4.2 Indeterminacy of terminal F

hazard. It is for these reasons that one feels that the smoother and less structured pattern of fishing

and terminal S

mortality of Fig. 7.4 is more plausible than that of There are many possible ways of fitting such a Fig. 7.6. The argument is based on the internal con- model to a set of catch-at-age data. A number of sistency of the interpretation – one is favouring the methods based on direct least squares fitting have interpretation in which adjacent values of fishing been proposed (Pope 1974; Doubleday 1976; Gray mortality are more or less similar, and in which the 1977), but were not very successful. The reason for exploitation pattern is fairly consistent from year this was elucidated by Pope and Shepherd (1982). to year.

One can remove the effect of varying year-class This argument can easily be formalized. Let us strength in the catch-at-age data by taking ratios of seek an interpretation in which the exploitation adjacent catches for the same cohort, i.e. pattern (the relative fishing mortality on each age

(7.6) although it is still an unspecified function of age. The overall level of fishing mortality, denoted

group), denoted by S(a), is the same each year, CyaCy ( , ) ( + 1 , a + 1 ) .

According to the separable model, the loga- by F s (y), where the subscript s is used to identify rithms of these ratios should be well approximated this as the overall F, subject to the assumption of by a two-way analysis of variance based on the separability (see below), may change as required main effects, a year effect and an age effect, only. In from year to year. This constitutes a separable practice one finds that such an analysis of variance model for the behaviour of fishing mortality:

can indeed explain a very large part of the variance of the log : catch ratios, leaving residuals which are

Fya ( , )=()() FySa s . (7.5) usually quite consistent with expected sampling

errors.

This technique, of representing a function of The consequence of this is that one can only two variables by the product of two functions of determine, from such an analysis of variance, a set one variable, is known mathematically as separa- of values for the ratios F s (y)/F s (y + 1) and S(a)/S(a + tion of variables. Hence the common name, sepa- 1). One has, from the row and column means (the rable VPA, used for the method of analysis based on main effects) of the log : catch ratio matrix, just this model (Pope and Shepherd 1982). It should be enough constraints to specify and determine all noted that in equation (7.5) there is a degree of arbi- these ratio uniquely. Given catch data for t years trariness in the definition of F s and S. One could and g age groups there are only (t - 1) column take any set of values for them, multiply all the F s means and (g - 1) row means, and these are not all values by a constant, and divide all the S values by independent because the grand mean of both sets is the same constant, and have exactly the same set the same. One therefore has enough constraints of values for F(y,a). To remove this ambiguity it is (equations) to determine only (t + g - 3) parameters. necessary to apply some normalization constraint. The normalization constraint S(a*) = 1.0 provides There are many possible ways to do this, but the an additional equation, leaving just enough to

Chapter 7

determine the (t - 1) ratios of F s and the (g - 1) ratios with conventional VPA, it is usual to require the of S only.

analyst to provide assumptions for terminal F and The solution in terms of the F s (y) and S(a) values terminal S. This means values for F s (t) and S(g) are themselves is therefore twofold indeterminate. required where t is the last year and g the oldest age This means mathematically that attempts to group. Notice that because S has been normalized determine these parameters directly, for example to unity, the value S(g) is relative to that on the ref- by least squares, are likely to fail, because the prob- erence age a*, whilst the value of F s (t) is absolute, lem is singular, involving the inversion of a matrix and of approximately the same size as the actual which is rank-deficient of order two, or three if no individual fishing mortality on age a* in the last normalization constraint on S is used. The argu- year. The analyst therefore again has to assume ment above is based of course on several approxi- precisely those things he would most like to deter- mations, and is not exact. For this reason the mine, namely the actual level of fishing mortality singularity will not prove to be exact in practice, in the most recent year, and a parameter which but the system is likely to have two very small controls the overall shape of the exploitation pat- eigen-values (three if S(a*) is not fixed). The prob- tern. In spite of the strong assumption of separabil- lem is very similar to that of multiple regression ity, we are still left with an analysis which can using correlated (co-linear) explanatory variables. determine the details, but not the overall underly- The solution of such ill-determined systems is ing patterns. notoriously difficult. In practice all this means that in spite of the very strong model of separabil-

7.4.3 Practical implementation of

ity, as expressed by equation (7.5), and the conse- quent reduction in the number of parameters to be

separable VPA

determined, to many fewer than the number of Given the indeterminacy of the solutions dis- data points, the problem of determining fishing cussed above, it may be questioned whether there mortality still has no unique solution. The num- is any point in carrying out separable VPA rather ber of undetermined parameters has in fact been than, say, the JAM analysis based on averaging, reduced from (t + g - 1), the number of terminal as discussed above. We believe that there is, for

F values required for conventional VPA, to 2 for several reasons. First, separable VPA provides a separable VPA, assuming that natural mortality is simple and convenient method for generating trial known, throughout, which is certainly progress, analyses, by specifying only two numbers which but not enough. The problem is structurally represent clearly defined and stated assumptions. underdetermined.

Second, the fact that the analyst is required to There are still therefore infinitely many possi- make these explicit choices forces the recognition ble interpretations based on separable VPA. It of ignorance. Third, it turns out that the surviving turns out that, because the separable model is only populations determined by separable VPA are less an approximation, these alternative solutions are sensitive to measurement errors than those of the only approximately equally good fits to the data, JAM and similar methods. Finally, separable VPA and of course none of them are exact. If one had proves to be a most useful tool for examining and exact data, it would be possible to choose among determining exploitation patterns, over several them on the basis of goodness-of-fit. In practice the fairly short periods of time required. By contrast, in differences are completely swamped by sampling the more heuristic methods such as JAM, the as- errors, and goodness-of-fit is generally useless as sumptions are less clear and less explicit. Why take

a guide to the correct solution. To obtain a com- the average itself? – why not 1.2 or 1.5 or 0.6 times pletely specified solution, it is therefore necessary the average? Why average over five years or ages – to supply two additional pieces of information, why not three or 10 instead? Also, if these choices usually by means of extra assumptions. Because of are made automatically instead of explicitly, the the convergence property of VPA, and by analogy analyst may be seduced into believing that he has

139 determined something which he has actually the earlier years, the fit may be quite slack. The

Dynamic Pool Models I: Interpreting the Past

assumed. fishing mortalities given by equation (7.7) are of

A surprisingly efficient simple algorithm for course rather smooth and well-behaved, and repre- separable VPA, invented by Pope, was described by sent some sort of averaged values. Pope and Shepherd (1982). This algorithm has been

The fitting procedure itself is based on the implemented several times, and is provided as part log : catch ratios, and leaves year-class strength un- of the Centre for Environment, Fisheries and determined. Once F¢(y,a) has been determined, the Aquaculture Sciences (CEFAS) Lowestoft and correspondence between it and the catch data may, ICES VPA program suites. It can also be executed however, be used to determine year-class strength. perfectly well using a spreadsheet package on a mi- Every pair of values F¢(y,a) and C(y,a) implies a crocomputer, if required. To run separable VPA, value for the current population size (in numbers), the analyst must of course supply a complete set of using the catch equation (7.1), and each of these catch-at-age data, for a minimum of about five may be converted using conventional VPA or co- years and five age groups in practice, together with hort analysis to a value of either initial year-class data or, more likely, conventional assumptions strength or terminal population. about the level of natural mortality. The first

Using this procedure one should not be sur- choice to be made is the age for unit selection. This prised that the fishing mortality estimates in the is in principle arbitrary and may be chosen freely as final year are just as noisy as those in previous it will not affect the essential results. In practice it years and include occasional odd high or low val- is recommended to choose the youngest age group ues. This is a natural consequence of the proce- that is likely to be fully exploited, which can usu- dure, since such values are the result of catch data ally be guessed adequately as that age which usu- in the final year which are somewhat inconsistent ally contributes most to the catch in numbers, or with previous data for the cohort, presumably as a the next older age group. This is advisable because result of sampling and ageing errors. These anom- it helps to make the results for the exploitation alous values are not usually carried forward into pattern relatively independent of the choice of the catch forecast or other subsequent calcula- fishing mortality, and vice versa.

tions, and need not be caused for great concern, al- though they may bear further investigation since