Robust Regression Approach to Estimating. Fish Mortality Rates with a Cohort-Based Model. Y. CHEN AND J. E. PALOHEIMO. Department of Zoology.
Transactions of the American Fisheries Society 123:508-518. I9S>4 c Copyright by the American Fisheries Society 1994
Robust Regression Approach to Estimating Fish Mortality Rates with a Cohort-Based Model Y. CHEN AND J. E. PALOHEIMO Department of Zoology. University of Toronto Toronto. Ontario A/551 1A1, Canada Abstract. — Natural mortality rate (A/) and fishing catchability coefficient (q) were estimated by using the Paloheimo pairwise linear regression model with simulated and real catch-effort data. The four estimation methods used were least squares (LS), least absolute value (LAV), least median of squares (LMS), and LMS-based rewcighted least squares (RLS). When catch or effort data or both contained homogeneous error variances, mean squared errors (MSE) of LS and RLS were significantly smaller than those of LAV and LMS. However, there was no significant difference in MSE between LS and RLS. When catch-effort data contained variances inconsistent among years, MSEs of the robust methods were much smaller than those of LS. Reweighted least squares substantially reduced the variances in the LS-estimated q and M. and it had the smallest MSE among the four estimation methods. We suggest using the following procedures in estimating q and A/ based on the Paloheimo pairwise regression model: (1) applying LMS to the model to identify outliers in the data; and (2) weighting the defined outliers by a factor of 0 and normal data points by a factor of 1 and applying LS to the model to estimate q and A/ and their associated variances.
Among the population parameters used in fisheries management, two components of mortality, natural and fishing mortalities, are most elusive. They have received a great deal of attention from fisheries researchers. Many methods have been developed to estimate these parameters. Some of them usually require the inclusion of some unreasonable assumptions such as known natural and fishing mortality rates or cohort size at a terminal age (Pope 1972). The estimates of mortality rate that result from these methods are commonly associated with large error variances, making the estimated parameters unreliable and nonsignificant. Many models recently developed for estimating mortality try to detail the biological realism, thereby increasing the models' statistical complexity and requiring the user to be competent in mathematical modelling (e.g., Doubleday 1976; Dcriso 1980: Fournier and Archibald 1982; Derisoetal. 1985, 1989; Greenberget al. 1991). Mortality rates are often estimated in conjunction with many other parameters, which may require some unrealistic assumptions and create artificial links between parameters such as mortality and recruitment (Deriso et al. 1985: Greenberg el al. 1991). For many large-scale and complex models, builtin constraints (e.g., mortality rates must be positive) may engender reasonable results even when the models do not fit the data well. For those complex models, too many parameters must be estimated from a few data points, perhaps resulting
in overfilling of Ihe models (Ludwig and Wallers 1989). For mosl of ihe currenlly used models, Ihe same nalural mortalily is assumed belween agegroups, which may nol be realislic for many fish slocks in praclice. A somewhal differem approach lo eslimaling ihe mortalily parameiers was laken by Paloheimo (1961). He developed a cohort-based linear regression model lo eslimale fishing calchabilily coefficienl (q) and nalural mortalily (M) by using caich-al-age and fishing effort daia. This model is based on a linearized version of ihe caich equalion; for a given cohort, Ihe decrease in logarilhmic caich per unil effort (CPUE) over Iwo conseculive years is regressed againsl ihe mean effort of ihese Iwo years (hereafter referred lo as ihe pairwise melhod). Several advanlages have been accrediled lo ihis melhod (Ricker 1975; Paloheimo 1980). Because of ils simple formulalion, parameiers q and M are easily eslimaled by ihe iradilional leasl-squares (LS) melhod. The use of only iwo age-groups in ihis melhod makes il possible lo have separale eslimaies of q and M for differem age-groups. This is particularly desirable when ihere are significanl differences in mortalily belween age-groups. Il is also possible lo make a plol lo examine graphically ihe fil of ihe model lo Ihe daia. However, ihe resullanl eslimaies of q and M are usually poor, wiih greal variances, when ihe pairwise melhod is applied lo real daia (Paloheimo 1961 ). This is aiiribuled lo ihe greal variances
508
REGRESSION ESTIMATION OF FISH MORTALITY
inherent in catch-effort data and inclusion of only two age-groups in estimation (Paloheimo 1980: J. E. Paloheimo and Y. Chen, unpublished). To solve the latter problem, Paloheimo (1980) developed an "extended" linear regression model that included all the fully recruited age-groups of a cohort. However, the variances associated with the LS-estimated q and A/ were still appreciably large due to great variances in the catch-effort data. If these errors have a known variance that is nearly homogeneous through time, a generalized regression model with a suitable error structure can be applied to reduce the error variances associated with q and M (Paloheimo and Chen. unpub-
lished). However, in practice, the homogeneous and known variance may be hard to find in field catch-effort data. A change in oceanographic variables such as water temperature and current direction or an improvement in fishing equipment may contribute to heterogeneous variation in catch-effort data among years. Such inconsistent variations may produce so-called outliers, defined as observations that are aberrant or unusually different from the rest of the observations (Rousseeuw and Leroy 1987). In this case, the GaussMarkov assumptions essential in the LS estimation are violated, resulting in biased estimates of q and M and unreliable estimates of uncertainties of the parameters. The sensitivity of the LS estimation method to outliers in data makes it desirable to use robust regression methods. Two such methods are the least absolute value (LAV) and the least median of squares (LMS) (Rousseeuw and Leroy 1987). Although these two robust regression techniques are quite popular in other areas such as engineering, they have been rarely applied to fisheries data. We used both the LS and robust (LAV and LMS) methods to estimate q and M in conjunction with the pairwise model developed by Paloheimo (1961). Using both simulated and field data, we examine in this paper the differences in estimating q and M by LS versus robust methods when consistent and inconsistent variances exist in catcheffort data. Procedures are suggested for improving the estimation of q and A/. Methods
Model and Estimation Methods Let C\j and C2/+1 be catches of a cohort at the first and second of two consecutive ages in years j andy -I- 1, and let.// and./J+1 be the corresponding fishing efforts. For a total of N years, Paloheimo's pairwise regression model can be written as
509
where Cj is an error term, j is from 1 to A: — 1, and M and q are the two parameters (natural mortality rate and fishing catchability coefficient) to be estimated (Paloheimo 1961). The traditional estimation method is the LS, which assumes that Cj satisfies the Gauss-Markov assumptions and minimizes the residual sums of squares (RSS): LS: minimi/c 2
c 2
j -
Two robust estimation methods used in this study are the least absolute value (LAV) and the least median of squares (LMS). Instead of minimizing the RSS, the LAV method minimizes the sum of the absolute residual values (Bloomfield and Steiger 1983; Devroye and Gyorfi 1984): LAV: minimi/c 2 \ei\:
and the LMS minimizes the median of squared residuals (Rousseeuw 1984): LMS: minimi/c (median Cj2) (j = 1. 2 . . . . . A' - 1). Because of the insensitivity of the robust methods to outliers, they can be readily detected in the analysis of residuals. Unlike the traditional regression methods, the LMS estimator cannot be expressed with an explicit formula. The algorithm proposed by Rousseeuw and Leroy (1987) for LMS is similar to the bootstrap (Efron 1979). It proceeds by repeatedly drawing subsamples of K observations (K = number of parameters to be estimated). For each subsample, parameters are calculated and the corresponding LMS objective function is determined with respect to the whole data set. For a subsample J. this means that the median of residual sum of squares (r 2 ) is calculated: S = median (r/ 2 ) = median (Yt - A'/9y)2;
/ = 1, . . . . n: n is the sample size; and 9y is the vector of regression parameters. After all possible subsamples of size AT are drawn, the 6's with minimal 5 are the LMS-estimated parameters. Based on an extensive simulation study, Rousseeuw and Leroy (1987) proposed the following criteria to define outliers in an LMS linear regression analysis: (1) calculate SQ = l.4826[I + 5/(N /0][minimal median (r 2 )] 0 - 5 ; then (2), for a data point /, if |/v/Sol > 2.5, / is defined as an outlier
510
CHEN AND PALOHEIMO
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Year FIGURE 1. —Six sets of simulated effective fishing effort (in arbitrary units).
and otherwise is a normal data point. However, the LMS-estimated uncertainties and parameters are incompatible with those of the LS method. To solve this problem, Rousseeuw and Leroy (1987) suggested using the LMS method to identify outliers in the data, and then using LS to reanalyze the data by weighting the LMS-defined outliers and normal data points with 0 and 1, respectively. They referred to this method as reweighted LS (RLS). In this study, RLS was also used to estimate q and M. Simulation Study The following procedures were used to generate the simulation data. (a) The true values of M and q were chosen as 0.2 and 0.1. respectively. (b) The value of the cohort size at the beginning of year 7. N}. was generated as a random number between 1,000 and 5,000 (j = 1, . . . , 15), and these values were kept constant for each simulation run. (c) The effective efforts, fj , were generated as in Figure I . (d) The "expected" catch-at-age data C '{j , were calculated with the catch equation (Rickcr 1975). (e) The "observed" catches at age and fishing efforts, C,j and.//, were simulated under the following three scenarios. (1) We simulated C/y and./ 7 by adding log-normal errors to CJ and f :
(2) We first generated C/y and fj as in scenario (1). and then imposed additional error on 3 years of the catch data: outlier C,
(3) We used the same catch data as those in scenario (2), but we first created effort data as in scenario (1) and then subjected 3 of the 15 effort data to the following additional error structure: outlier f =
^ Ar(0,
Two situations were considered for scenario (3). The first was that the three effort and catch values affected by large variances were selected for the same years (i.e.. catch-at-age and effort data in years 1, 2, and 3). The second was that large variances were made to occur in different years for catch-at-age and fishing effort data (effort in years 1, 2, and 3; catch at age in years 8, 9, and 10). The above design made it possible to examine the effects of the different patterns of outliers in catcheffort data on the estimates. The design of the simulation study is listed in Table 1. Six sets of effective fishing effort (Figure 1) were generated to test whether the different patterns in fishing effort over years affected the output of the simulation study.
REGRESSION ESTIMATION OF FISH MORTALITY
One hundred runs were done for each case of the simulation. Two indices were computed to measure the extent to which estimates differed from the true values of the parameters over 100 runs. The mean estimated value was computed as , 100
TABLE 1. — Design of the simulation study based on allocation of log-normal errors (c\ and c3 were applied to catch data; e^ and c* were applied to fishing effort data). Normal variation Case
and the mean squared error was calculated as 1
100
MSE(0 = — 2 (Gi - Q)2* where Q and Q represent estimates and true values of the parameters (i.e., ?and A/), respectively. The MSE(()) can be decomposed into a sum of squared bias (SB) and variance of the estimates: 1
511
Normal 1 1 2
e\ 0.05 0.05 0.05
Normal 11 3 4
0.05 0.05 0.05
Variation for outliers 0
e.
0 0 0
0 1 1
0 0 0.5
0.05 0.05 0.05
0
0 0 0.5
e2
1 1
LMS, and RLS). The differences in the estimates of q and M by different methods were compared.
10
°
MSE(G) = (G - G)2 + T^ 2 (G/ - G)2. Therefore, it is possible to examine whether the between-run variance or bias was the main contributor to the estimation errors. Simulated data set I was used to show the detailed comparisons among the estimation methods. To test whether the results were consistent with different patterns of fishing effort, another five data sets were simulated. For each case of the simulation, MSE values were ranked among the different estimation methods for each of six generated data sets: the smallest MSE ranked 1, the second smallest 2, and so on. A nonparametric test was employed on the ranked MSEs to test the significance of the error differences among the four estimation methods. The association of rank ordering of MSE values among estimation methods was measured nonparametrically by the Kendall
coefficient of concordance, W. The distribution of X 2 was considered to approximate the x2 distribution with a - \ degrees of freedom (Zar 1984). For this study, factors of the test (i.e., a) were the selected estimation methods, whereas blocks were the data sets. Pairwise comparisons of the MSE values were also conducted between estimation methods. Field Data Catch-at-age and effort data for lake trout Salvelinus namaycush in Lake Opeongo were obtained from Fry (1949). An analysis of covariance (ANCOVA) was done to test whether there were significant differences in q and M between agegroups. The data were then fitted to the Paloheimo pairwise model with the four methods (LS, LAV,
Results For the simulated data set I (Figure 1), the means of the LS-estimated q and M were almost the same as those estimated with the robust methods under the normal error situations (i.e., normal I and II, Table 2). The MSE and variance increased when the robust methods were used, but not dramatically. Small values of squared bias indicated that the bias was negligible and that variances made up most of the MSE. The MSE and SB of LS in normal II were slightly greater than those in normal I. For MSE and SB associated with the robust estimates. RLS < LAV < LMS. When heterogeneous variances existed in catch data (simulation cases 1 and 3, Table 1), MSEs of both LS-estimated q and M increased greatly, but SB of LS only increased slightly, compared with those in normal I and II. For the robust methods, there were no substantial differences in M and q between the simulations with constant error variance (normal I and II) and those with heterogeneous variance. The MSEs estimated with robust methods were much smaller than those of LS. However, differences in SB were small between the robust and LS methods. The MSE and SB of LAV tended to be larger than those of LMS, which were in turn larger than those of RLS (Table 1). In the case when there were heterogeneous variances in both catch and effort data (cases 2 and 4), two scenarios were considered with respect to the distribution of atypical values in catch and effort data. The first scenario was that atypical values in catch data occurred in the same years as those in effort data. In this case, the LS-estimated q and M differed significantly from the true values of0.1 and 0.2, respectively (P < 0.05, /-test). They
512
CHEN AND PALOHEIMO
TABLE 2. —Estimates of fishing catchability coefficient (q) and natural mortality (A/) by methods of least squares (LS), least absolute value (LAV), least median of squares (LMS), and reweighted least squares (RLS). Large error variances in catch and in fishing effort occurred in the same years. Cases are defined in Table 1. All values of mean squared error (MSE) and squared bias (SB) have been multiplied by 1.000. LAV
LS SB
Mean
0.06 2.6 3.6
0.001 0.009 0.680
0.097 0.098 0.094
0.099 0.097 0.074
0.09 2.6 3.7
0.001 0.009 0.680
0.099 0.097 0.092
Normal I 1 2
0.199 0.213 0.376
3.7
64.5 145.1
0.001 0.169 31.000
0.212 0.209 0.234
Normal II 3
0.203 0.207 0.378
4.9 65.1 144.5
0.009 0.049 31.680
0.202 0.209 0.239
Case
Mean
Normal I 1 2
0.099 0.097 0.074
Normal II 3 4
4
MSE
RLS
LMS Mean
MSE
SB
Mean
MSE
SB
Catchability coefficient 0.14 0.009 0.096 0.097 0.31 0.004 0.34 0.036 0.098
0.21 0.30 0.30
0.016 0.009 0.002
0.098 0.099 0.101
0.09 0.10 0.11
0.002 0.001 0.001
0.097 0.104 0.095
0.21 0.29 0.54
0.009 0.016 0.025
0.098 0.101 0.097
0.10 0.14 0.20
0.004 0.001 0.009
MSE
0.18 0.31 0.59
SB
0.001 0.009 0.064
Natural mortality 8.3 0.144 19.1 0.081 15.2 1.160 10.2 13.3 25.2
had much larger values of MSE and SB than those of robust methods. There was no substantial difference in the LS-estimated Mand q between cases 2 and 4. The LAV method had greater MSE and SB for both M and q than LMS. which had greater MSE than RLS. There was a large increase in SB of LAV when variance in fishing effort was inconsistent compared with the simulation cases when variance in effort was consistent among years. No great differences in SB of robust estimates of q and \t between cases 2 and 4 were observed. However, MSEs in case 4 were substantially greater than those in case 2 for the robust methods (Table 2). The second scenario was that atypical values in catch data occurred in different years from those in effort. Estimates of M and q were similar to those in the first scenario for all methods in simulation cases 2 and 4 (Table 3). However. MSEs of both M and q in the second scenario were great-
0.004 0.081 1.520
0.209 0.211 0.215
12.8 16.4 13.2
0.081 0.121 0.225
0.206 0.205 0.205
5.1 6.2 5.7
0.036 0.025 0.025
0.207 0.205 0.209
10.4 12.3 19.4
0.049 0.025 0.081
0.201 0.205 0.203
5.5 6.1 9.2
0.001 0.025 0.009
er than those of the first scenario for all estimation methods (Tables 2, 3). Six data sets were simulated to test whether the comparisons among estimation methods were consistent under the different patterns of fishing effort (Figure 1). The MSE ranks for both M and q were LS > RLS > LAV > LMS for normal I, RLS > LS > LMS > LAV for normal II. and RLS > LMS > LAV > LS for simulations 1, 2, 3, and 4 (Table 4). The nonparametric test with Friedman's \r statistic indicated a significant difference in the rank ordering of MSEs among the four estimation methods for both M and q in all six simulation cases. Kendall's coefficient of concordance ranged from 0.7 to 1 (Table 5), indicating good agreement in the MSE ranks among methods in all six cases. The following results were derived based on the paired comparisons between the estimation methods: (1) for normal I and normal II,
TABLE 3. —Estimates of fishing catchability coefficient (q) and natural mortality (A/) by four methods when large variation (i.e., ^3 and e* in Table 1) occurred in different years for catch and effort data. Conventions are as for Table 2. LS Case Mean
MSE
LAV
SB
Mean
2 4
0.078 0.078
5.9 6.0
0.484 0.484
0.083 0.088
2 4
0.343 0.346
461.7 463.1
20.450 21.300
0.330 0.287
MSE
RLS
LMS
SB
Mean
Catchability coefficient 2.7 0.289 0.094 3.2 0.144 0.092
MSE
SB
1.63 0.036 3.00 0.064
Mean
MSE
0.102 0.098
0.72 1.45
SB 0.004 0.004
Natural mortality
201.0 235.4
16.900 7.570
0.175 0.174
194.3 220.6
0.625 0.676
0.187 0.211
73.2 0.169 109.4 0.121
REGRESSION ESTIMATION OF FISH MORTALITY
TABLE 4.—Sum of mean squared error (MSE) ranks for each estimation method under different simulation scenarios. Conventions are as for Tables 2 and 3. LAV
LS
RLS
LMS
Simulation case
M
Q
A/
q
M
0 O
S
105
(5
.g
0.700.
OJ
55
0.31
0.35 -
0.00 4-5
6-7
5-6
7-8
8-9
9-10
10-11
11-12
Age-group FIGURE 4.— The LS- and RLS-eslimated standard errors of q and A/ for lake trout of different age-groups in Lake Opeongo.
consecutive years and the dependent variable was the average effort of these two years. Therefore, it might not be necessary' for an outlier in catch or fishing effort to be an outlier in the dependent or TABLE 8.—Age-groups in which the LS- and RLS-estimated q and Mdiffered significantly from 0 (inequality signs, P < 0.05) and equaled 0. Age-groups
Parameter
Comparison
LS
Q
0
6-8 4-6. 8-1 1 11-12
6-8 4-6,8-9 9-12
0
11-12 4-11 None
5-6, 11-12 4-5.6-7.8-9. 10-11 7_gt 9_IO
\f
RLS
independent variables. Alternatively, an outlier in catch-effort data may not necessarily mean an outlier in the dependent or independent variables. An example could be found in Figure 5, showing the result of one of the 100 runs at normal II (Table 1): whereas the error variances in catch and effort were constant, two points were identified by LMS as outliers. Similar examples could be found in 23 of the 100 runs for normal II. This might explain why RLS had a smaller MSE than LS at normal II (Table 4). It can be expected that higher proportion of the simulated data sets will contain outliers if the assumed homogeneous variance of log(effort) (case normal II, Table 1) is higher than that in the simulation. In practice, variances associated with fishing effort may be much higher than those used in the simulation. Thus, biases of
REGRESSION ESTIMATION OF FISH MORTALITY
517
1.4
1.2 -
^
1.0 1
03
tr
|
0.8 H
Is -2
c
0.6
03
^
0.4 H 0.2 -
Normal
LS
LAV
LMS
— - RLS
El
Outlier
0.0
6
8
10
12
Mean effort FIGURE 5. —Plot of the logarithmic catch per unit effort for a cohort over two consecutive years (mean Z) against the mean effort in these two years (mean/) for a run of catch-effort data simulated at normal II.
the LS estimates of q and A/ may be greater than those reported in Table 2. However, the RLS-estimated q and M may have similar biases as outliers are removed from the analysis. For the LS estimation, one assumption is that the independent variable is error-free. In the Paloheimo pairwise model, this requires that fishing effort be error-free. This is certainly not the case in practice. Large variances are commonly associated with fishing effort data as well as with catch data. This was found as one of the main sources of variance in LS estimates of q and M when the Paloheimo pairwise model was used for real data (Paloheimo and Chen, unpublished). If the variances in fishing effort and catch were homogeneous and known, a generalized least-squares (GLS) regression method that considers variances in both fishing effort and catch could be applied to the pairwise model to reduce the estimated variation in q and M (Paloheimo and Chen, unpublished). In practice, however, variances in fishing effort and catch data are usually heterogeneous. Thus, GLS may not be applicable. In this case, LMS can be used for preliminary analysis to identify outliers. After defined outliers are excluded, LS or GLS can be used to estimate q and M. This two-stage procedure in estimating the Palo-
heimo pairwise model can greatly improve the estimation of q and M. as can be seen in Figure 3. Indeed, as shown in Figure 5, it is still possible to have outliers in estimation even if variances in catch and effort data are constant. Therefore, it is advantageous to adopt this two-stage procedure for the Paloheimo pairwise model to improve the estimates of q and M. With the real data (Fry 1949), standard errors of q and M estimates were substantially lower with RLS than with LS (Figure 4), indicating that the two-step procedures improved the estimation of q and M. The decline in logarithmic CPUE (dependent variable of pairwise model) from 1937 to 1938 (data point 2) was defined as an outlier for most of the age-groups (Figure 3). This implied that the variance inherent in these data differed greatly from that of the remaining years in the analysis. Various sources of error in the census were detailed by Fry (1949). Special geographic peculiarities of Lake Opeongo might also have contributed some error variances (Fry 1949). However, the survey began in 1936, and inexperience of the survey staff might have been one of the main error sources for the first several years. The reliability of aging Lake Opeongo lake trout
in this early time has been questioned (Shuter et
518
CHEN AND PALOHEIMO
al. 1987), and inaccurate ages may be another reason that outliers occurred in the LMS analysis. Whenever an age-group showed any defined outlier, data point 2 (1937-1938) was one of them (Figure 3), indicating an uncommon inherent variance for the data of those years. In this study, the LMS-based RLS was superior to the traditional LS when catch and effort data had heterogeneous variances. The difference between RLS and LS was not significant when catch and effort contained no outliers. This makes the RLS more desirable than LS in the regression analysis. This conclusion applies not only to use of the Paloheimo pairwise model with catch-effort
data but also to any linear regression analysis of other fisheries data. In practice, outliers in catcheffort data often go unnoticed because many fisheries data now are analyzed by computers without careful screening and inspection, especially for the data sets with large number of observations. This makes the LMS-based RLS more desirable than LS in the regression analysis in fisheries. In conclusion, we suggest implementing a twostep procedure to estimate q and M for the Paloheimo pairwise model: (1) use of LMS to identify outliers and (2) application of LS estimation after defined outliers have been excluded. If there are estimates for the variance of catch-effort data without outliers, a GLS can be employed to further improve the estimation of q and M. Acknowledgments We thank D. Jackson, T. A. EdsalL R. M. Dorazio, R. J. Conser, and K. Ing for their helpful comments. This work was financially supported by a Canadian Natural Science and Engineering Research Council (NSERC) operating grant to J.E.P. and an NSERC postgraduate scholarship to Y.C. References Bloomfield, P., and W. L. Stciger. 1983. Least absolute deviations: theory, applications, and algorithms. Birkhauser Verlag, Boston. Deriso, R. B. 1980. Harvesting strategies and parameter estimation for an age-structured model. Canadian Journal of Fisheries and Aquatic Sciences 37: 268-282. Deriso, R. B., P. R. Neal, and T. J. Quinn II. 1989. Further aspects of catch-age analysis with auxiliary information. Canadian Special Publication of Fisheries and Aquatic Sciences 108:127-135. Deriso, R. B., T. J. Quinn II. and P. R. Neal. 1985. Caich-age analysis with auxiliary information. Ca-
nadian Journal of Fisheries and Aquatic Sciences 42:815-824. Devroye. L., and L. Gyorfi. 1984. Nonparametric density estimation: the L] view. Wiley-Interscience. NewYork. Doubleday, W. G. 1976. A least squares approach to analyzing catch at age data. International Commission of Northwestern Atlantic Fisheries Research Bulletin 12:69-81. Efron, B. 1979. Computers and the theory of statistics: thinking the unthinkable. SIAM (Society for Industrial and Applied Mathematics) Journal of Applied Mathematics 21:460-480. Fournier, D., and C. P. Archibald. 1982. A general theory for analyzing catch at age data. Canadian Journal of Fisheries and Aquatic Sciences 39:11951207. Fry, F. E. J. 1949. Statistics of a lake trout fishery. Biometrics 5:27-67. Greenberg. J. A.. S. C. Matulich, and R. C. Mittelhammer. 1991. A system-of-equations approach to modeling age-structured fish populations: the case of Alaskan red king crab, Paralithodes camtschaticus. Canadian Journal of Fisheries and Aquatic Sciences 48:1613-1622. Ludwig, D.. and C. J. Walters. 1989. A robust method for parameter estimation from catch and effort data. Canadian Journal of Fisheries and Aquatic Sciences 46:137-144.
Paloheimo, J. E. 1961. Studies on estimation of mortalities I. Comparison of a method described by Beverton and Holt and a new linear formula. Journal of the Fisheries Research Board Canada 18:645662. Paloheimo, J. E. 1980. Estimating mortality rates in fish populations. Transactions of the American Fisheries Society 109:378-386. Pope. J. E. 1972. An investigation of the accuracy of virtual population analysis using cohort analysis. International Commission of Northwestern Atlan-
tic Fisheries Research Bulletin 9:65-74. Ricker, W. E. 1975. Computation and interpretation of biological statistics of fish populations. Bulletin of the Fisheries Research Board of Canada 191. Rousseeuw, P. J. 1984. Least median of squares regression. Journal of the American Statistical Association 79:871-880. Rousseeuw, P. J.. and A. M. Leroy. 1987. Robust regression and outlier detection. Wiley. New York. Shuter, B., J. E. Matuszek. and H. A. Regier. 1987. Optional use of creel survey data in assessing population behaviour: Lake Opeongo lake trout (Salvelinus namaycush) and smallmouth bass (Micropterus dolomieui) 1936-83. Canadian Journal of Fisheries and Aquatic Sciences 44 (Supplement 2): 229-238. Zar. J. H. 1984. Biostatistical analysis. 2nd edition. Prentice-Hall, Englewood Cliffs, New Jersey. Received June 3, 1993 Accepted November 7, 1993
