Estimation of Effective Effort from Catch-at-Age Data

Estimation of Effective Effort from Catch-at-Age Data J.E. Paloheirno and Yong Chen

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Depatfrnent of Zoology, University of Toronto, Toronto, ON ha%% 1A 1, Canacda

Paloheirno, J.E., and Y. Chen. 1993. Estimation of effective effort from catch-at-age data. Can. I. Fish. Aquat. Sci. 50: 2421 -2428. We present a method for estimating effective efforts or fishing ~ ~ ~ o r t arates l i t y based on a linearized version sf the catch equation. Catch-at-age for at least two age groups over a series of years is required. The method presupposes a value for natural mortality rate (M).The method is validated using simulated data with an appropriate error structure. The algorithm always converges to a set of effective efforts that are compatible with the known catches. Nevertheless, tRe solution to the basic equations is not unique although the different solutions are typically highly correlated. If the h/B assumed by the algorithm is the same as the actual M the iterated effective efforts are typically very close to the true effective efforts or fishing mortality rates. If the assumed A4 is too high or too low the pattern of effective efforts i s still recovered to a high degree of accuracy, typically 8.90 < r M , the slope is typically less than 1. In this case the high efforts are underestimated relative to the low efforts. Combined Estimates

1

4

3

4

5

6

J

I

9

%

0

1

8

Year

FIG. 1. Estimated and true efforts for a combination of the natural mortality rate M and assumed mortality rate Me. (a) M = 0.2 and Me = 8.2; (b) M = 0.2 and Me = 0.3; (c) M = 0.3 md Me = 0.2.

To test the method in a variety of circumstances, we ran six sets of 100 mns each by generating both the efforts and cohort sizes randomly. Hn the first three sets the catches were calculated assuming a 28% variability in age-specific catchabilities. The natural mortality was set at M = 0.25 and the guessed value of M set at Me = 0.25 (Set I), Me = 0.15 (Set %), and M4 = 0.4 (Set 3). In the second series of three, we imposed an additional 20% error on the observed catches and repeated the natural mortality patterns. The exact procedure is set out in Table 1. The performance of the algorithm is shown in Tdble 2 which shows the histogram of correlation coefficients between the actual and estimated efforts done separately for each of the 188 runs. Surprisingly the correlations are uniformly quite high

When the annual effective efforts are estimated for a r ~ individual age pair the estimates subsume not only the variability at age (as they should) but also all the errors of sampling as well. TFs shed some light on the extent to which the estimated annual changes in the effective efforts by pairs of ages are true for the fishable stock as a whole, we need a combined estimate. We now have a pairs of ages. The algorithm is more or less the same as it would be for a single pair with an exception that we now have far more data points than unknowns and the algorithm must be amended to include a least squares procedure. The initial starting values given in Eq. 3 are more or less the same with the obvious change that

To show how Eq. 5 needs to change, we denote the upper rz x ( ~ 1-t I) matrix of the right-hand side by A:

Let Az' = (Az,,,, ...,Bz,,,, A.zZ3 ...,AZ,,,, ,,,, 0) and AE' as before. Also, let (1)' stand for a row vector of I 's, i.e., (I)' = (1 ,...,I). Then, if we have four age groups, Can. J . Fish. Aquat. Sci., V i l . i l , 1993

TABLE1. Pattern of generating data for testing the algorithm. Ail random numbers ei,ei. and cj are ncamaE, N(0,l). M is the value used in the catch equation md Me is the value used in the recovery algorithm.

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Set

9

h

M

Me

Emor in catch (%)

Enor in catchability (%)

0 .B

TABLE2. Corelation coefficients between the actual and estimated efforts for the data sets in Table 1. Set

FIG.2. Difference between the first and the last effort plotted against the comelation coefficient between the estimated md actual efforts.

over the average of the separate estimates is relatively small. The estimates based on marginal totals have invariably the largest bias. In real data the error variance typically increases with age. Since this can be taken into account in the variance-covariance matrix V, the least squares estimates are expected to be superior to estimates based on the individual values of z,,, +,,'s or on their marginal totals.

where A1,A2, m d A3 all have the same structure as A in Eq. 6. The elements of A, are calculated using the selectivities s, and s2,those sf A2 are calculated using the selectivities s2and x3, and so on. In Eq. 7, there are 3n + 1 equations and only rz + 1 U ~ ~ O W ~ S . Estimating Selectivities To estimate them, we take a recourse to a least squares procedure We have so far assumed that all 5,'s have known values. If this is not the case, they may be estimated if M is indeed constant for all ages. We found that it is better to estimate the selectivities and effective efforts separately rather thaw simultaneously. As the simultaneous estimation procedure is quite complicated and where V denotes an appropriate variance-csvariance matrix. often fails to give reasonable estimates even at moderate ensr This is given in the Appendix. levels, we will not go into it further, Alternatively, we can get a combined estimate of efforts by To estimate the selectivities, we sum Eq. 2 over all years for averaging the estimates obtained separately for each age pair. each age pair. With four age groups, we get This, as shown below, is better than first getting the marginal totals of z,,, over the ages (i) and then subjecting the margindl averages to the algorithm for two age groups as suggested by Pope and Shepherd (1982) (dong with a different algorithm). Generalized least squares estimates, estimates based on the marginal totals, and estimates based on the average of separate estimates for each age pair were compared using six data sets, simulated with the same initial conditions as in Table I for four age groups. Only I0 mns were made at each setting. The results where we have ignored the term (In El - In E,, ,,). for data set 1 are shown in Fig. 4 where we have plotted the average difference between the true and estimated efforts. The The above equations show how observed average mortality other data sets gave qualitatively s i ~ l aresults. r rates are affected by the selectivities. If we know two of them, From Fig. 4, it appears that by and large the least squares say s, = s4 = 1, the other two can be solved by an iterative procedure seems to give the best results, although the advantage procedure that is similar to the one used to solve for Ei9s. + 1 J 7 ~

Can. S.Fish. Aquut. Sci., Val. 50,1993

TABLE3. Correlation coefficients between the actual and estimated efforts by years over all BOO mns. Initial conditions as in Table 1. Year

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Set

0.W

0.10

1

0.20

3

2

0.M

4

0.40

5

6

7

8

9

10

11

0.50

Actual effort

RG.3. Relationship between the actual and estimated effort when the assumed natural mortality ride is less than the actual rate. Data are the same as in Fig. 16. A diagonal line is drawn for comparison.

We begin the iterative procedure by chgosing reasonable initial values for S , and s, and by calculsting E from the last equation in Eq. 9 which reduces to G4 = qE + M when s, = s, = 1. Next, we calculate

Year

FIG.4. Emor in the estimated efforts for four age groups using three different methods. GLS refers to generalized Beast squares method, ASE to average of estimates done separately for each age pair, and MTE is based on the overdl mortality rate for age groups (marginal totals).

0.5 and 0.75. The results based on 100 runs of each combination are shown in Table 3. For each data set the estimated values sf s, and s, were close to the true values (8.5 and 0.75) with standard errors of the estimates ranging between 0.039 and 0.054 for s, and between 0.044 m d 8.064 for s, when error either in catches or in catches and selectivities was set around 26%. The results are independent of the initial values of s, and 5,. Application

We now look for corrections on the initial estimates of sl and s2 that make the estimated A?,, and AF2, zero. To that end, we replace s, and s2 with sI(l 9 As1) and s,(l i-AS,) and make use of the approximation In(] + Asi) = As, to linearize the contributions sf h i ' s in Eq. 10. The result, written in matrix format, is

1 I I where a, = -qs, E + 1, h2 = -qs2 E - 1. and c, = 5qs2 E + 1. We

2

2

now solve for Asl awd Ay2.Using s,(l + As,) and s2(l + As,) as our next estimates of selectivities, we recalculate &i,i+l's as before and then As,'s again, and so on. The iterative procedure converges fairly rapidly. The iterative procedure was tried on six simulated data sets. We added an error component on the catchabilities and on the catches as in Table 1. The age-specific catchabilities were set at

The method was applied to data on the lake mitefish (Csrego~zlasclupeufirmis) fishery in eastern Lake Ontario from 1953 to 1971. The data were obtained from J. Christie and J. Casselmm, Glenora Fisheries Station, Pictsn, Ont. The fishery is a giklnet fishery and the likelihood of capture depends on the size of the fish. Based on existing gillnet selectivity data and observed size-at-age frequencies, the selectivities of three major age groups (ages 4,5, and 6) were calculated for each yea. As there were substantial differences in growth from year to year, the selectivities changed both with age and year. The separability condition was not sbviously met. However, since the selectivities are known, we can still estimate the effective effort. Eq. 7 is now written as

Can. J. Fish. Aquaf. Sci., Voi.50,1993

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Both A, and A2 have the same structure as A in Eq. 6. The individual elements are calculated slightly differently The elements of A,, for q = I, are given as

+,

are replaced with and those in A, as above, except that s,,, ,s, 'zJ' $3, + I * For this fishery. we also have data on total effort. Two measures of effort are available. One is based on total yards of net fished each year and the other is "corrected yardage, corrected for assumed gear saturation in years of high fishing effort. We have estimated the annual effective efforts, calculated by iteration from Eq. 7'. Interestingly, our effective effort (= estimated fishing mortality) correlates better with the "corrected" yardage than with the total- In Fig. 5. we have plotted our effective effort against the total corrected yardage.

Discussion We have given a simple algorithm to estimate effective effort from catch-at-age data. The algorithm presupposes that the average log decline in catch-at-age is an estimate of the true average mortality rate, corrected if necessary for differences in selectivities at age. It turns out that this log decline is a biased 1 estimate of the average mortality rate. the bias being ; (In El In En where n is the number of years minus I and E, and En+, are the efforts in the first and last year- Alternatively, we could have done as Pope and Shepherd (1982) did and arbitrarily fixed the effective effort (or the fishing mortality) at the final year and then solved Eq. 5 (with appropriate modification) iteratively. While our feeling is that equating the log decline of catch-at-age with the average mortality is not nearly so arbitrary, there may be little to chose between the two methods if data are available for two ages only. However, with three or more age groups, our method has been extended to a generalized least squares procedure that permits inclusion of an appropriate error stmcture. The method was verified using simulated data with a specified error structure. We generated catches-at-age from randomly distributed population sizes at the age of entry to the fishery using randomly distributed effective efforts. The age-specific selectivities were allowed to fluctuate with a 20% error limit. In addition, we let the observed catches differ from the actual catches by up to 20%. The equations used for determining effective effort do not have a unique solution. Moreover the estimated values depend on an assumed natural rnortdlity rate which in the examples considered has been at times 68% above or 40% below the actual or true value. Nevertheless, and in spite of all this, our estimates of the effective effort correlate well with the actual effective efforts used in generating the basic data. We therefore believe that in the majority of cases the proposed algorithm does recover, if not the actual effective efforts, at least estimated efforts that we closely correlated with m e fishing mortalities. When the true M is higher than the assumed M ( M e ) the effective efforts or fishing mortalities are overestimated and Can. J. Fish. Aquut. Sci., Lbl. 50, 1993

0

4 o

IOOB

am

3640

4m

mao

towl

D TOW

Total effort

FIG. 5. Relationship betvileen the estimated and total (= corrected yardage) effort in the Lake Ontario lake whitefish fishery.

conversely, when M < Me, they are underestimated. This is expected because the algorithm partitions the total decline in catchat-age from one year to the next to its components. Similarly, changes in selectivities can also be "absorbed into because the constant tern in Q. 2 is M + In s, - In s, +,.In theory, if M is in fact constant (and known) over all ages in the database the selectivities can be estimated as shown. However, their values are dependent on how good our guess of M is. Too high a value of Me will tend to overestimate the differences between s,'s, and conversely, a low value of Me will tend to even out the real differences. Moreover, if M is not constant over age, then any changes in true M will be reflected in our estimates of the selectivities. A good indication of whether M and/or selectivities are changing with age is obtained by comparing the annual effective efforts estimated separately for each age pair. If, for instance, the estimated effective efforts increase with age, this may be caused by either a change in M with age or a change in the "selectivities." It will be difficult to distinguish between changes in M and in the selectivities without some additional information. We applied the algorithm to estimate the effective efforts (==fishing mortalities) in the lake whitefish fishery in eastern Lake Ontario. When the estimates were compared with the "field" data, we observed a nonlinear relationship between the estimated fishing mortality and the observed effort, akin to the findings of Crecco and Overholz (1990). It appears that in this fishery,low levels of effort generate proportionally larger fishing mortality rates than higher levels. However7the low effort levels we usually associated with low fish abundance. It may well be that when the abundance is low, fish are more concentrated (occupy smaller area) than in years of high abundance and hence are easier to catch. relatively speaking.

This research was funded by an NSERC g m t to 3.E. Paloheirno and m NSERC postgraduate scholarship to Y. CChen. We also thank John Casselman of Glenora Fisheries Station, Picton, Ont., for access to lake whitefish data and Jack Christie, also from Glenora, for support m d help

im the data analysis.

References CRECCO,V., AND W.J. OVERZPOLZ. 1990. The causes of density-dependent catchability for Gesrges Bank haddock .Melanogrummus neglefinus. Cm. J. Fish. Aquat. Sci. 47: 385-394.

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BERISO, R.B., T.3. QUINNII. .$ED P.R. NEAL. 1985. Catch-age analysis with auxiliary information. Can. J. Fish. Aqmt. Sci. 42: W 15-824. DOUBLEDAY, W.G. 1976. A least squares approach to analyzing catch-&age data. Res. Bull. Int. Comm. Northwest Atl. Fish. 12: 69-81. FOLTPUIJER, B.. AND C.P. ARCHIBALD. 1982. A general theory for analyzing catch-at-age data. Can. J. Fish. Aquat. Sci. 39: 1195-1207. PALOHEIMO, J.E. 1961. Studies on estimation of mortalities: 1: comparison of a method desci-ikd by Beverton and Hoit and a new linear formula. J. Fish. Wes. Board Can. 18: 645-652. PALOHEIMO, J.E. 1980. Estimation of mowality rates in fish populations. Trans. Am. Fish. Sw. 109: 378-386. PALOHEIMO, J.E.. AND L.M. Dacw~.1964. Abundance and fislning success. Cons. Int. Explor. Mer Rapp. P.-V. 152-163. POPE,J.G. 1972. An investigation of the accuracy of virtual population analysis using cohort analysis. Res. Bull. Hnt. Comm. Northwest At]. Fish. 9: 65-74. Pow. J.G., AND J.G. SHEPHERD. 1982. A simple method for the consistent interpretationof catch-at-age data. I. Cons. Int. Explor. Mer 40: 176-184. SHEPHERD, J.G., AND M.D. NHCHOLSON. 1991. Multiplicative modeling of catch-at-agedata, and its application to catch forecasts. J. Cons. lnt. Explor. Mer 47: 284-294.

We write Eq. 2 in a somewhat more general form by including the possible changes in annual age-specific catchabilities:

From the above, we can see that the errors in e,., can be w&en approximately as

+, jt

cd1 them

Denoting the variances of catches at age i by a,,,and the variance of catchabilities by o,,, it follows from above that Var (e,) = occi + o,,i+l + (I

+ 21 q E,)? cSs

Cov (ed,eL7)= O whenever i - i 5 1,j -j ' > 1 . Note that the vdue of V, the elements of which were given above, depends on the parameters to be estimated (El's) as well as on the unknown values sf the errors in catches and catchabilities. However, the estimates are not very sensitive to the actual values of the elements in V as long as the pattern is more or less correct in the sense that the elements on the diagonal =are positive and those on the subdiagonals are negative. In the simulations that we ran the V matrix was not updated to account for changes in Ei9s.~ o ~ e o v eo,.,,'s r , were also taken as constant. To accommodate the inevitable increase in variance with age in real data, we suggest that oCci'sbe made proportional to the logarithm of the percent marginal catches-at-age. The V matrix is a 3n + I x 3n + I matrix, The variances and covariances defined above give us the first 3n rows and columns. The last row and column are zeros on the account sf the condition ZAE] = 8. To avoid the numerical somersaults needed to get around the singular V matrix, we have replaced the last zero on the principal diagonal by a small number (e.g., 0.001). This means that the condition XAE, = 0 is given a substantial weight but not made into an absolute condition.

Carl. J. Fish. Ayuat. Sci., Vol.50,1993