Choosing the observational likelihood in state-space stock assessment models
Christoffer Moesgaard Albertsen, Anders Nielsen, Uffe Høgsbro Thygesen
CChoosing the observational likelihood in state-spacestock assessment models
Christoffer Moesgaard Albertsen ∗ , Anders Nielsen, and Uffe Høgsbro Thygesen Technical University of Denmark, National Institute of Aquatic Resources,Charlottenlund Castle, DK-2920 Charlottenlund, Denmark
Abstract
Data used in stock assessment models result from combinations of biological, ecolog-ical, fishery, and sampling processes. Since different types of errors propagate throughthese processes it can be difficult to identify a particular family of distributions for mod-elling errors on observations a priori. By implementing several observational likelihoods,modelling both numbers- and proportions-at-age, in an age based state-space stockassessment model, we compare the model fit for each choice of likelihood along withthe implications for spawning stock biomass and average fishing mortality. We proposeusing AIC intervals based on fitting the full observational model for comparing differentobservational likelihoods. Using data from four stocks, we show that the model fit isimproved by modelling the correlation of observations within years. However, the bestchoice of observational likelihood differs for different stocks, and the choice is importantfor the short-term conclusions drawn from the assessment model; in particular, thechoice can influence total allowable catch advise based on reference points.
Key words: Numbers-at-age, proportions-at-age, data weighting, state-space model,stock assessment model
Introduction
Stock assessment models often use aggregated and uncertain data such as surveys andlandings-at-age which rely on age classification of effectively few individuals (Aanes andPennington 2003). Commercial fishing and scientific surveys sample from populations thatvary according to, for example, season, sex, age and region. From this catch, samples areweighed and measured to estimate the length distribution, weight-at-length, and total catchin numbers. Additional sub-samples are taken to age classify individuals for estimatingproportions-at-age; either directly or through an age-length key. The samples consist of manyindividuals from few hauls (Aanes and Pennington 2003), which may lead to underestimated ∗ Corresponding author. Email: [email protected] a r X i v : . [ s t a t . A P ] S e p ncertainties of estimates if ignored. Finally, all this information is aggregated to numbers-at-age for each year. This aggregation may be via models including e.g. spatial location, season,gear and length effects. Even though the stock population growth process at this level ofaggregation is well described (Each year the fish age by one year, some die of natural causes,and others die from fishing) aggregating the different sources of uncertainty makes it difficultto find the optimal (or true) distributions of the observations a priori.Age-based stock assessment models can be divided into two classes depending on the waythey utilize the data. Either the data can be modelled as numbers-at-age or as proportions-at-age along with total weight or numbers. Most currently used age-based stock assessmentmodels exclusively consider either numbers- or proportions-at-age and only one or few obser-vational likelihoods (ICES 2010a). When modelling numbers-at-age, the normal distribution,parameterized to avoid too much probability on negative observations, has been used (Gud-mundsson 1994, Fryer 2002) along with the log-normal distribution (Cook 2013, Nielsen andBerg 2014) and its multivariate extension (Myers and Cadigan 1995). Although recommendedover the log-normal by Cadigan and Myers (2001), the gamma distribution is infrequentlyused to model numbers-at-age in assessment models (ICES 2010a).The multinomial distribution has been popular when modelling proportions-at-age(Fournier and Archibald 1982, Methot Jr. and Wetzel 2013, Williams and Shertzer 2015).Based on the age classification sampling, it is an intuitive choice; however, when using thetrue number of data generating samples, the variances of the modelled proportions are oftentoo small, and the correlation structure too restrictive (Crone and Sampson 1998, Aanes andPennington 2003, Francis 2014). Efforts have been made to increase the variance by estimatingan effective sample size (McAllister and Ianelli 1997, Francis 2011, Hulson et al. 2011, 2012).Nonetheless, the effective sample size must be estimated by iterative optimization (McAllisterand Ianelli 1997, Francis 2011, Maunder 2011) since the multinomial distribution is improperwhen used for continuous data (Francis 2014). Hence, the multinomial distribution will notbe considered here. To avoid iterative estimation of the effective sample size, it has beensuggested to replace the multinomial with the Dirichlet distribution (Williams and Quinn1998, Francis 2014) in which the variance is only determined by parameters.While the Dirichlet distribution is an improvement over the multinomial distribution,they both have a very restrictive variance-covariance structure that only allows negativecorrelations, which may not be appropriate (Francis 2014). Therefore distributions basedon transformations of multivariate normals, such as the additive logistic normal (Francis2014) and the multiplicative logistic normal (Cadigan 2015), have recently been proposed forproportions-at-age in stock assessment models.Although several authors have compared different proportions-at-age models (Maunder2011, Francis 2014), not much effort has been given to compare different observationallikelihoods for numbers-at-age data (Cadigan and Myers 2001), and even less has been givento compare between the proportions- and numbers-at-age. Using the R-package TemplateModel Builder (Kristensen et al. 2016), we implement 13 observational likelihoods, includingboth numbers- and proportions-at-age models, in an age-based state-space stock assessmentmodel. Using assessment data from four European stocks, we compare the model fit foreach choice of likelihood along with the implications for key outputs such as spawning stockbiomass (SSB) and average fishing mortality ( ¯ F ).2 ethods We implemented age-based state-space stock assessment models (Nielsen and Berg 2014)with 13 different observational likelihoods (Table 1) for four different European stocks. Forsimplicity the same observational likelihood was used for both commercial catch data andsurvey indices. While the process model was kept unchanged for each stock, we compared thegoodness-of-fit of the observational likelihoods by AIC. We considered models for numbers-at-age and proportions-at-age combined with total catch. We considered seven differentdistributions for numbers-at-age. When using data in the form of a total and proportions-at-age we followed the wide-spread convention of modelling total catch as univariate log-normal,but considered two alternatives where the total was either in numbers or biomass. Thesetwo alternatives for total catch were crossed with three alternative distributions for theproportions. The observational likelihoods implemented cover frequently used distributionsin fisheries stock assessments and close extensions.Table 1: Overview of the observational models used in the case studies and some properties:if zero observations are allowed; whether the Baranov catch equation determines the mean,median or location; the number of estimated observational parameters per age ( a ) andfleet ( f ); and whether a correlation parameter is estimated. The models are divided in tomodel classes: Univariate numbers-at-age (UN@A), multivariate numbers-at-age (MN@A),proportions-at-age with log-normal total numbers (P@AwN), and proportions-at-age withlog-normal total weight (P@AwW).Model Distribution Class Allows 0 Baranov Est. par.s Est. cor. M log-Normal UN@A No Median 1 a f No M Gamma UN@A Some Mean 1 a f No M Generalized Gamma UN@A Some Location 2 a f No M Normal UN@A Yes Mean 1 a f No M Left Truncated Normal UN@A Yes Location 1 a f No M log-Student’s t UN@A No Location 2 a f No M Multivariate log-Normal MN@A No Median 1 a f +1 f Yes M Additive Logistic Normal P@AwN No Location 1 a f +1 f Yes M Multiplicative Logistic Normal P@AwN No Location 1 a f + 1 f Yes M Dirichlet P@AwN No Mean 1 f No M Additive Logisitc Normal P@AwW No Location 1 a f +1 f Yes M Multiplicative Logistic Normal P@AwW No Location 1 a f + 1 f Yes M Dirichlet P@AwW No Mean 1 f No Process model
The processes described in the state-space model involved the true unobserved numbers-at-age in the stock, and the true unobserved fishing mortality (See Nielsen and Berg 2014 or Should be read: One per age per fleet. Should be read: One per age per fleet and one additional per fleet. C a,f,y . Observational models
Our model M was the log-normal distribution with its usual parameterization. The medianwas determined by ˜ C a,f,y , while a scale parameter was estimated for each age and fleet.The model M was the gamma distribution parameterized to have constant coefficient ofvariation (Cadigan and Myers 2001). The mean was determined by ˜ C a,f,y , while a coefficient ofvariation (CV) was estimated for each age and fleet. The generalized gamma distribution wasincluded as model M with the parameterization of Prentice (1974). This parameterizationwas preferred over the Stacy (1962) parameterization as it both extends it, and is numericallymore stable when reducing to the log-normal distribution (Prentice 1974, Farewell and Prentice1977). The log-location parameter was determined by log( ˜ C a,f,y ) while a shape and scaleparameter was estimated for each age and fleet. The models M and M were the normal andtruncated normal (with left truncation at zero). Both were parameterized based on the meandetermined by ˜ C a,f,y and separate CV parameters for each age and fleet (which applied to theun-truncated values for the truncated normal). The Student’s t-distribution on log-scale wasour model M . The distribution was parameterized with a log-location parameter determinedby log( ˜ C a,f,y ) along with log-scale and log-degrees-of-freedom parameters estimated separatelyfor each age and fleet.Model M was the multivariate log-normal with its usual parameterization. The marginalmedians were determined by ˜ C a,f,y , while a one-parameter AR(1) structure was used forthe correlation between ages on logarithmic scale (Pinheiro and Bates 2000, Francis 2014).Separate correlation parameters were estimated for each fleet along with scale parametersestimated for each age and fleet. Our models M and M were the additive logistic normaland the multiplicative logistic normal (Aitchison 2003) with log-normal total numbers. Forboth models, the location parameters were determined by the ˜ C a,f,y s while the scale matriceswere parameterized as M . For the log-normal distributions, the medians were determinedby P a ˜ C a,f,y , while a separate scale parameter was estimated for each fleet. The Dirichletdistribution with log-normal total numbers was our model M . The Dirichlet distributionwas parameterized with concentration parameters proportional to ( ˜ C ,f,y , . . . , ˜ C A,f,y ) T . Aproportionality parameter was estimated for each fleet. The log-normal distributions for totalnumbers were parameterized as M and M . Finally, the models M , M , and M werethe additive logistic normal, multiplicative logistic normal, and Dirichlet with log-normaltotal weight parameterized as M , M , and M respectively. All estimated observational4arameters were assumed to be constant over years. The densities and further details can beseen in Appendix A. Comparing by AIC
To compare the different observational models, we employed the Akaike Information Criterion(Akaike 1974). However, the AIC applies to comparison between specific models, whereas eachobservational model represents an entire family of models, differing in assumed relationshipsbetween parameters for different age groups. These families include “full models” where eachage group and fleet are assigned independent parameters, a “minimal model” where all agegroups share common parameters, as well as a range of models between these two extremes.A standard application of the AIC would require that the optimal model in each family isidentified, a task that would involve estimation of parameters in thousands of models.To avoid this step, which is computationally very demanding and tangential to ourpurpose, we chose to identify an AIC interval which characterized each model family. ThisAIC interval gave an upper and a lower bound on the optimal AIC within that family. Theupper bound of the interval was attained by the AIC for the full model. The lower bound ofthe interval was calculated as the AIC that would hypothetically be obtained by the smallestpossible nested sub-model if the negative log-likelihood would not increase compared to thefull model. The difference between the upper and lower bound is thus twice the difference inthe number of parameters between the full model and the minimal model.A model family was considered clearly superior to another if the upper bound of its AICinterval was below the lower bound of the other model family’s interval (i.e., the other modelfamily had a higher interval). Clearly inferior model families could be discarded. To comparethe remaining model families, it would be possible to narrow the AIC intervals throughtesting within each model family, but for simplicity we base the comparison on the full modelin each family.Using AIC to compare the models required that the models were defined on the same data,which was not the case when we compared between numbers-at-age models and proportions-at-age models. The proportions-at-age data were, however, a one-to-one transformation ofthe numbers-at-age data. Thus, using a standard transformation of densities we derived thelog-likelihood for the numbers-at-age data that is consistent with our specified distributionsbased on proportions and totals (Appendix B). Using the transformed likelihood in the AICcalculation allowed for valid comparisons of models using numbers-at-age directly versusthose using totals and proportions-at-age. Note that a similar transformation was requiredso that models that used total weight could be compared to models that used total numberswith the proportions.
Case study
We implemented the models for four different data sets used for assessments (Table 2): TheBlue Whiting data set was the basis of the 2014 ICES advice (ICES 2014a) for Subareas I-IXand XIV; the North-East Arctic Haddock data was used for the 2014 ICES advice (ICES2014b) for Subarea IV (North Sea) and Division IIIa West (Skagerrak); The North Sea Coddata was obtained from the 2012 ICES advice (ICES 2012) for subarea IV (North Sea) and5able 2: Overview of the data sources used in the case study. Q1-Q4 indicates at whichquarter of the year the survey is conducted.Fleet First year Last year First age Last age Years with missing Process parameters
Blue Whiting
North-East Arctic Haddock
North Sea Cod
Northern Shelf Haddock
Results
In all four case studies we found that the estimated average fishing mortality (Figure 1),spawning stock log-biomass (Figure 2), and their standard errors in the final year differedbetween models. In particular, we found that for North Sea Cod, the highest fishing mortalitywas 2 times the lowest fishing mortality, and the widest confidence interval was 1.7 times the6arrowest. For Northern Shelf Haddock, the confidence interval of the estimated final yearspawning stock biomass for model M , which had the highest estimate, did not overlap withthe confidence interval for model M , which had the lowest estimate. . . . . . F - A . . . F - B . . . . . F - M M M M M M M M M M M M M C . . . F - M M M M M M M M M M M M M D Model code
Figure 1: Last year fishing mortalities with 95% confidence intervals for models M to M (Table 1) in the case studies: Blue Whiting (A), North-East Arctic Haddock (B), North SeaCod (C), and Northern Shelf Haddock (D). Vertical dashed grey lines separates the modelsin model classes (Table 1). Subscripts to ¯ F indicates the ages the average is over. All agesare weighed equally in the average.We found that the models including correlation parameters obtained better fit to thedata for the full models than models without correlation parameters within each modelclass (Figure 3); the AIC for the full model (upper bound of the interval) was lower for themultivariate log-normal than for the univariate numbers-at-age, and similarly the logisticnormals had better model fits than the Dirichlet distribution, which provided one of thehighest AIC intervals of all models in all case studies. In the North Sea Cod and NorthernShelf Haddock cases, the lower bounds of the AIC intervals for the Dirichlet distribution wereclearly separated from the upper AIC bounds of all other models with an AIC difference ofmore than 190.For North Sea Cod, the multivariate log-normal achieved the lowest AIC for the fullmodel. The AIC interval for this distribution (upper bound: 6407.32) only barely overlapped7 . . . . . A . . . . . B . . . M M M M M M M M M M M M M C . . . M M M M M M M M M M M M M D l og ( SSB ) l og ( SSB ) Model code
Figure 2: Last year spawning stock biomass logarithmic (log(SSB)) for models M to M (Table 1) in the case studies: Blue Whiting (A), North-East Arctic Haddock (B), North SeaCod (C), and Northern Shelf Haddock (D). Vertical dashed grey lines separates the modelsin model classes (Table 1). 8 A B M M M M M M M M M M M M M C M M M M M M M M M M M M M D A I C A I C Model code
Figure 3: AIC intervals for models M to M (Table 1) in the case studies: Blue Whiting(A), North-East Arctic Haddock (B), North Sea Cod (C), and Northern Shelf Haddock (D).The horizontal grey lines indicate AIC differences of 50 starting at the lowest lower bound ofthe models. Vertical dashed grey lines separates the models in model classes (Table 1).9ith the intervals for the generalized gamma (lower bound: 6405.52). Hence among theobservational likelihoods we considered, the multivariate log-normal was the most appropriatefor the North Sea Cod data and this particular process model. The multivariate log-normalalso had the lowest AIC for the full model for North-East Arctic Haddock, whereas it wasthe multiplicative logistic normal with total weight for Northern Shelf Haddock and theadditive logistic normal with total numbers for Blue Whiting. However, in these cases the AICintervals of the multivariate log-normal, the additive logistic normals, and the multiplicativelogistic normals all overlapped. For the two haddock cases, the AIC interval of the generalizedgamma also overlapped with the AIC interval of the model with the lowest AIC intervalupper bound. We further found that the AIC intervals for proportions-at-age models usingtotal weight overlapped with the corresponding model using total numbers-at-age, except forthe North Sea Cod where the total weight models had lower AIC intervals. In addition, theintervals for the additive and multiplicative logistic normals overlapped for all four data sets.Overall, the trends in estimated fishing mortality (Figure 4) and spawning stock biomass(Figure 5) were similar between the models (See also supplementary material). However, therewere noticeable differences in single years. For Blue Whiting, the estimated fishing mortalityand spawning stock biomass for multivariate log-normal, multiplicative logistic-normal, andDirichlet distribution followed each other closely. The largest difference in average fishingmortality between the multiplicative logistic-normal and the multivariate log-normal was 0.07(16%), and the difference in spawning stock biomass was up to 12%. In the North Sea Codcase, the multivariate log-normal and the logistic normals had larger differences in fishingmortality and spawning stock biomass. The largest difference in mortality was 0.12 (11.2%),while the spawning stock biomass differed as much as 23%. The resulting confidence intervalsalso differed between the models. For North Sea Cod the standard errors of the estimatedaverage fishing mortality were up to 76.6% larger for the Dirichlet model, which had thehighest AIC, compared to the multivariate log-normal, which had the lowest AIC. Althoughthe trends were similar to the other models, the spawning stock biomass was estimated to be3.8 to 14.2 times higher for the left truncated normal than for the other models . Likewise,the average fishing mortality was estimated to be lower. For both North Sea Cod andNorthern Shelf Haddock , the logistic normals provided less volatile estimated time series offishing mortality and spawning stock biomass than other models. For these models the CVsfor commercial catch were estimated to be higher than for the other data sets .For Blue Whiting, North-East Arctic Haddock, and Northern Shelf Haddock, estimatedCVs were similar for the two logistic normals . The CVs were estimated to be between 0.09and 0.37. For North Sea Cod, the logistic normals with total numbers had higher CVs (0.32for commercial catch; 0.27 for survey) than the logistic normals with total weight (0.15 forcommercial catch; 0.20 for survey). For the multivariate log-normal, the estimated marginalCVs were estimated to be between 0.01 and 1.86 . The CVs were typically estimated to behigher for the first and last ages. Figure S31 Figures S34, S35, S37, S38 Figures S47, S48, S50, S51 Table S5 Table S5 Table S1-S4 . . . . . F - . . . . F - . . . . . F - . . . F - YearYear
Figure 4: Estimated average fishing mortality, ¯ F , for Multivariate log-Normal (dark grey line),Multiplicative Logistic Normal with log-Normal Weight (dashed black line), and Dirichletwith log-Normal Weight (dotted black line) in the case studies: Blue Whiting (A), North-EastArctic Haddock (B), North Sea Cod (C), and Northern Shelf Haddock (D). Horizontal dashedgrey lines show the management plan reference point. Subscripts to ¯ F indicates the ages theaverage is over. All ages are weighed equally in the average.11 . . . . . . . . . . . . . l og ( SSB ) l og ( SSB ) YearYear
Figure 5: Natural logarithm of estimated spawning stock biomass, log(
SSB ), for Multivariatelog-Normal (dark grey line), Multiplicative Logistic Normal with log-Normal Weight (dashedblack line), and Dirichlet with log-Normal Weight (dotted black line) in the case studies:Blue Whiting (A), North-East Arctic Haddock (B), North Sea Cod (C), and Northern ShelfHaddock (D). Horizontal dashed grey lines show the management plan reference point.12 iscussion
When modelling highly aggregated stock assessment data, the optimal observational likelihoodto use can not be known a priori. We provide an objective method for limiting the numberof candidate models. By fitting each model with the largest possible number of parameters,upper and lower bounds for the lowest attainable AIC can be calculated. After discardingmodels with AIC intervals that does not intersect with the lowest interval, the remaining AICintervals can be narrowed by combining parameters until a single observational likelihood isleft. Once a final stock assessment model is found, model validation tools such as residuals andretrospective analysis should be used. AIC was used to determined the optimal observationallikelihood, since the AIC estimates the Kullback-Liebler divergence between the candidatemodel and the true data generating system (Akaike 1974), i.e. the information lost by usingthe candidate model instead of the true data generating system (Burnham and Anderson2002). The Kullback-Liebler divergence can not be used directly because the calculationrequires full knowledge of the true data generating system. Although the AIC was used here,other criterias allowing multivariate data could be used instead.Choosing the best possible observational likelihood for the data is vital for the short-termmanagement and conservation of fish stocks (Figure 1, Figure 2). In 2012, the ICES advicefor Northern Shelf Haddock (ICES 2012), based on XSA (Shepherd 1999), suggested a 15%increase in the total allowable catch. The suggested increase was based on the differencebetween the estimated average fishing mortality in the last year and the management referencepoint. XSA can be seen as a special case of the univariate log-normal (ICES 2010b) byfixing parameters in a suitable way. Here we saw that the multivariate log-normal andmultiplicative logistic normal were more suitable for the Northern Shelf Haddock data thanthe univariate log-normal, which in turn provides a better fit than the XSA. The estimatedaverage fishing mortality from both the multivariate log-normal and the multiplicative logisticnormal are above the reference point. Hence, using one of these models would have suggesteda 19% decrease of the total allowable catch to get the fishing mortality at a sustainable level(Figure 4; panel D). Thus, the choice of observational likelihood in an assessment model canhave a substantial effect on the advice brought forward to the fisheries management system.All models implemented in this study fall in to one of two categories; either they areformulated for numbers-at-age or proportions-at-age with total catch. Most assessment toolsonly consider one of these categories. We have shown how to compare models between thesetwo ways of using the stock assessment data, and that models from both categories can besuitable, depending on the specific stock. Besides the choice between modelling numbersor proportions there are other differences between the models in, e.g., tail probabilities andskewness, but there are also more subtle differences. When we choose between the log-normaldistribution and the gamma distribution, we also choose between whether the Baranov catchequation should model the median or the mean observed catch. Some likelihoods can bere-parameterized to link the Baranov catch equation to either the mean, median or modalobserved catch. Although subtle, this difference is important for prediction and interpretationof the results. These choices can be compared objectively by including them in the analysis.Accounting for the correlations in the data is also important for more reliable stockassessment models. In all four case studies, the distributions with correlation parameters, inparticular the multivariate log-normal, performed well. However, the correlation structure13ust be flexible enough to mimic the data, unlike the Dirichlet distribution, which performedpoorly, even compared to the models assuming independence between ages. The Dirichletdistribution arises as the distribution of proportions of gamma distributed numbers, wherethe gamma distributions have equal scale parameters. Stock assessment data is often believedto have constant CV (Cadigan and Myers 2001), which leads to a parameterization of thegamma distribution where the scale parameters are not equal. Therefore the covariancestructure of the Dirichlet distribution does not match the appropriate structure for thegamma distributions. This corresponds with previous findings, that the additive logisticnormal generally is more suitable than the Dirichlet distribution in describing the correlationstructure in stock assessment data (Francis 2014). For simplicity we restricted ourselves tothe simple AR(1) structure for correlations in this study. The correlation structure can easilybe exchanged for other structures such as a linear, AR(2), ARMA( p , q ), compound symmetric,or unstructured covariance matrix (Pinheiro and Bates 2000, Francis 2014), and some ofthese may be even more suitable than the AR(1) structure (Berg and Nielsen 2016).We only compared frequently used distributions in fisheries stock assessments and closeextensions of them, yet any conceivable distribution can be used in the framework we presented,as long as they are all comparable. Correlations between age classes could be introduced inthe univariate models through copulas or multivariate extensions such as the multivariatet-distribution or multivariate gamma distributions. Likewise, different ways of handling zeroobservations, such as zero inflating the models, could be included. For simplicity, years withmissing data were removed to compare between univariate and multivariate models. Formultivariate numbers-at-age models, missing data may be handled by finding the marginaldistribution of the remaining ages. We also restricted the study to use the same likelihoodfor both commercial catches and surveys. This can be relaxed by including combinations ofmodels (such as Cadigan 2015), or if the survey index generation is well-understood, a suitableobservational likelihood may be derived a priori. Further, the analyses were made conditionalon the process models specified in the assessments from which the data was collected. Asimilar analysis could be made to choose the most appropriate process model conditional onthe observational model, or the analyses could be combined. This may influence the specificchoice of observational likelihood, as the observational model can, somewhat, compensate formisspecification in the process model and vice versa. Finally, these methods could just aswell be used for, e.g., length-based models.Statistical assessment modelling involves a choice of observational likelihood, and currentpractice is often to make this choice arbitrarily and subjectively. This applies particularly tothe choice of whether the data inputs should be numbers-at-age or proportions-at-age alongwith total catch in numbers. Here, we have provided methods for an objective choice, bycorrecting the AICs so that they can be compared between these two families, and we haveoutlined a computationally efficient method for choosing between families of distributions,by bounding the AICs, which avoids elaborate hypothesis testing within each family, anddemonstrated that the best fitting family depends on the particular case. These resultswill allow stock assessment modellers to choose objectively between these representationsof uncertainty on observations, thereby improving model fit and ultimately allowing moreaccurate assessments. 14 cknowledgements The authors wish to thank Casper W. Berg and three anonymous reviewers for their valuablecomments to improve the presentation of this manuscript.
References
Aanes, S., and Pennington, M. 2003. On estimating the age composition of the commercialcatch of Northeast Arctic cod from a sample of clusters. ICES Journal of Marine Science:Journal du Conseil (2): 297–303. doi: 10.1016/S1054-3139(03)00008-0.Aitchison, J. 2003. The Statistical Analysis of Compositional Data. The Blackburn Press,Caldwell, N.J.Akaike, H. 1974. A new look at the statistical model identification. Automatic Control,IEEE Transactions on (6): 716–723. doi: 10.1109/TAC.1974.1100705.Berg, C.W., and Nielsen, A. 2016. Accounting for correlated observations in an age-basedstate-space stock assessment model. ICES Journal of Marine Science: Journal du Conseil (7): 1788–1797. doi: 10.1093/icesjms/fsw046.Burnham, K.P., and Anderson, D.R. 2002. Model Selection and Multimodel Inference: APractical Information-Theoretic Approach. In (3): 560–567. doi: 10.1139/f01-003.Cook, R. 2013. A fish stock assessment model using survey data when estimates of catch areunreliable. Fisheries Research : 1–11. doi: http://dx.doi.org/10.1016/j.fishres.2013.01.003.Crone, P., and Sampson, D. 1998. Evaluation of assumed error structure in stockassessment models that use sample estimates of age composition. In Fishery Stock AssessmentModels.
Edited by
Q. Funk F. Alaska Sea Grant College Program Report No. AK-SG-98-01,University of Alaska Fairbanks. pp. 355–370.Farewell, V.T., and Prentice, R.L. 1977. A Study of Distributional Shape in Life Testing.Technometrics (1): 69–75.Fournier, D., and Archibald, C.P. 1982. A General Theory for Analyzing Catch atAge Data. Canadian Journal of Fisheries and Aquatic Sciences (8): 1195–1207. doi:10.1139/f82-157.Francis, R.I.C.C. 2011. Data weighting in statistical fisheries stock assessment models.Canadian Journal of Fisheries and Aquatic Sciences (6): 1124–1138. doi: 10.1139/f2011-025.Francis, R.I.C.C. 2014. Replacing the multinomial in stock assessment models: A firststep. Fisheries Research (0): 70–84. doi: http://dx.doi.org/10.1016/j.fishres.2013.12.015.Fryer, R. 2002. TSA: is it the way? In Report of Working Group on Methods of FishStock Assessment, Dec. 2001. ICES CM 2002/D:01. pp. 86–93.15udmundsson, G. 1994. Time Series Analysis of Catch-At-Age Observations. Journal ofthe Royal Statistical Society. Series C (Applied Statistics) (1): pp. 117–126.Hulson, P.-J.F., Hanselman, D.H., and Quinn, T.J. 2011. Effects of process and observationerrors on effective sample size of fishery and survey age and length composition using varianceratio and likelihood methods. ICES Journal of Marine Science: Journal du Conseil (7):1548–1557. doi: 10.1093/icesjms/fsr102.Hulson, P.-J.F., Hanselman, D.H., and Quinn, T.J. 2012. Determining effective samplesize in integrated age-structured assessment models. ICES Journal of Marine Science: Journaldu Conseil (2): 281–292. doi: 10.1093/icesjms/fsr189.ICES. 2010a. Report of the Workshop on Reviews of Recent Advances in Stock AssessmentModels Worldwide: “Around the World in AD Models” (WKADSAM), 27 September - 1October 2010, Nantes, France. ICES CM 2010/SSGSUE:10.ICES. 2010b. Report of the Working Group on Methods of Fish Stock Assessment(WGMG), 20–29 October 2009, Nantes, France. ICES CM 2009/RMC:12.ICES. 2012. Report of the ICES Advisory Committee 2012. ICES Advice 2012, Book 6.ICES. 2014a. Report of the ICES Advisory Committee 2014. ICES Advice 2014, Book 9.ICES. 2014b. Report of the ICES Advisory Committee 2014. ICES Advice 2014, Book 3.Kristensen, K., Nielsen, A., Berg, C.W., and Bell, H.S.B. 2016. TMB: AutomaticDifferentiation and Laplace Approximation. Journal of Statistical Software (1): 1–21. doi:10.18637/jss.v070.i05.Maunder, M.N. 2011. Review and evaluation of likelihood functions for compositiondata in stock-assessment models: Estimating the effective sample size. Fisheries Research (2–3): 311–319. doi: 10.1016/j.fishres.2011.02.018.McAllister, M.K., and Ianelli, J.N. 1997. Bayesian stock assessment using catch-age dataand the sampling - importance resampling algorithm. Canadian Journal of Fisheries andAquatic Sciences (2): 284–300. doi: 10.1139/f96-285.Methot Jr., R.D., and Wetzel, C.R. 2013. Stock synthesis: A biological and statisticalframework for fish stock assessment and fishery management. Fisheries Research : 86–99.doi: http://dx.doi.org/10.1016/j.fishres.2012.10.012.Myers, R.A., and Cadigan, N.G. 1995. Statistical analysis of catch-at-age data withcorrelated errors. Canadian Journal of Fisheries and Aquatic Sciences (6): 1265–1273. doi:10.1139/f95-123.Nielsen, A., and Berg, C. 2014. Estimation of time-varying selectivity in stock assessmentsusing state-space models. Fisheries Research : 96–101. doi: 10.1016/j.fishres.2014.01.014.Pinheiro, J.C., and Bates, D.M. 2000. Mixed-effects models in S and S-Plus. Springer,New York, USA.Prentice, R.L. 1974. A Log Gamma Model and Its Maximum Likelihood Estimation.Biometrika (3): 539–544.Shepherd, J.G. 1999. Extended survivors analysis: An improved method for the analysisof catch-at-age data and abundance indices. ICES Journal of Marine Science: Journal duConseil (5): 584–591. doi: 10.1006/jmsc.1999.0498.Stacy, E.W. 1962. A Generalization of the Gamma Distribution. The Annals of Mathe-matical Statistics (3): 1187–1192. Institute of Mathematical Statistics.Williams, E.H., and Quinn, T.J. 1998. A Parametric Bootstrap of Catch-Age CompositionsUsing the Dirichlet Distribution. In Fishery Stock Assessment Models.
Edited by
Q. Funk F.16laska Sea Grant College Program Report No. AK-SG-98-01, University of Alaska Fairbanks.pp. 371–384.Williams, E. H., and Shertzer, K.W. 2015. Technical documentation of the BeaufortAssessment Model (BAM). NOAA Technical Memorandum, U.S. Department of Commerce.doi: 10.7289/V57M05W6.
Appendix A
Process model
The process model is identical to Nielsen and Berg (2014). For a model including age groupsfrom 1 to A + (where age group A + contains all ages from A and up), the fishing mortality ismodelled by a multivariate random walk, where the oldest modelled ages may be groupedtogether, indicated by using A ∗ rather than A + . Let F y = ( F ,y , F ,y , . . . , F A ∗ ,y ) T be a vectorof age specific fishing mortalities in year y . Thenlog F y = log F y − + (cid:15) y , where (cid:15) y ∼ N (0 , Σ). The covariance matrix, Σ, is parameterized by and AR(1) structure,Σ i,j = ρ | i − j | σ i σ j .Given the fishing mortalities, the population is modelled by an exponential decay model,log N ,y = log ( R ( w ,y − , . . . , w A + ,y − , p ,y − , . . . , p A + ,y − , N ,y − , . . . , . . . , N A + ,y − )) + η ,y , log N a,y = log N a − ,y − − F a − ,y − − M a − ,y − + η a,y , ≤ a < A + , log N A + ,y = log (cid:16) N A + − ,y − e − F A + − ,y − − M A + − ,y − + N A + ,y − e − F A + ,y − − M A + ,y − (cid:17) + η A + ,y , where all error terms are assumed independent normal distributed. The natural mortalities( M a,y ), the age specific weight in stock ( w a,y ), and the proportion mature ( p a,y ) are all assumedto be known. The function R describes the relationship between recruitment and spawningpopulation. For North Sea Cod, R was modelled by a Beverton-Holt curve, R ( w ,y − , . . . , w A + ,y − , p ,y − , . . . , p A + ,y − , N ,y − , . . . , N A + ,y − ) = a · SSB y − b · SSB y − , a, b > SSB y = P A + a =1 p a,y w a,y N a,y , whereas for the other stocks, R was modelled by a randomwalk R ( w ,y − , . . . , w A + ,y − , p ,y − , . . . , p A + ,y − , N ,y − , . . . , N A + ,y − ) = N ,y − . Observational models
Univariate numbers-at-age models
We consider six univariate observational models for numbers-at-age. Their densities are listedbelow for each age each year. The joint density for the vector of catches-at-age each year isthe product of the age-wise densities. Unless otherwise noted, x > µ > σ > τ ∈ R is a shape parameter.17 og-normal distribution f ( x ; µ, σ ) = 1 √ πσ exp − (log( x ) − log( µ )) σ ! x − The mean of the log-normal distribution is µ exp( σ / µ and the variance is(exp ( σ ) −
1) exp (2 log( µ ) + σ ). Gamma distribution f ( x ; µ, σ ) = 1Γ( σ ) (cid:16) µσ (cid:17) σ x σ − exp ( − xσ/µ )The gamma distribution has mean µ and variance µ σ . Generalized gamma distribution f ( x ; µ, σ, τ ) = | τ | ( τ − ) τ − exp (cid:16) τ − (cid:16) τ log( x ) − log( µ ) σ − exp (cid:16) τ log( x ) − log( µ ) σ (cid:17)(cid:17)(cid:17) / ( σx Γ( τ − )) τ = 0(2 π ) − / exp (cid:16) − (log( x ) − log( µ )) σ (cid:17) ( σx ) − τ = 0Note that f ( x ; µ, σ,
0) = f ( x ; µ, σ ), and f ( x ; µ, σ, σ ) = f ( x ; µ, σ − ) for σ > µ ( τ − ) τ − Γ( τ − ) ( τ ) σ τ +1 τ Γ (cid:16) σ τ +1 τ (cid:17) τ < µ exp( σ / τ = 0 µ Γ( τ − ) τ στ Γ (cid:16) σ τ +1 τ (cid:17) τ > − µ (Γ( τ − )) (cid:18) ( τ − ) τ − ( τ ) σ τ +1 τ (cid:16) Γ (cid:16) σ τ +1 τ (cid:17)(cid:17) − ( τ ) σ τ +1 τ ( τ − ) τ − Γ (cid:16) σ τ +1 τ (cid:17) Γ ( τ − ) (cid:19) τ < σ ) −
1) exp (2 log( µ ) + σ ) τ = 0 µ (Γ( τ − )) τ στ (cid:18) Γ (cid:16) σ τ +1 τ (cid:17) Γ ( τ − ) − (cid:16) Γ (cid:16) σ τ +1 τ (cid:17)(cid:17) (cid:19) τ > Normal distribution f ( x ; µ, σ ) = 1 q π ( µσ ) exp − ( x − µ ) µσ ) ! For the normal distribution x ∈ R and µ ∈ R . The mean is µ and the variance is µ σ runcated normal distribution f ( x ; µ, σ ) = f ( x ; µ, σ )1 x ≥ ( x )1 − R −∞ f ( y ; µ, σ ) dy For the truncated normal distribution x ≥ µ ≥
0. The mean of the distribution is µ + µσ f (0; µ, σ )1 − R −∞ f ( y ; µ, σ ) dy and the variance is µ σ − µ σ f (0; µ, σ ) (cid:16) − R −∞ f ( y ; µ, σ ) dy (cid:17) Student’s t-distribution on log-scale f ( x ; µ, σ, τ ) = Γ (cid:16) τ +12 (cid:17) σx √ τ π Γ (cid:16) τ (cid:17) x ) − log( µ )) /σ ) τ ! − ( τ +1) / For the Student’s t-distribution on log-scale, τ > τ → ∞ the distribution converges to a log-normal distribution. The distribution does not have meanand variance. Multivariate numbers-at-age models
We consider one multivariate observational model for numbers-at-age. In the density listedbelow, A is the number of ages, x > is the observed vector of catches (or survey indices), µ > is the calculated catches (or survey indices) based on the Baranov catch equation (orproportional to the total abundance), and Σ is an A × A symmetric positive-definite scalematrix. Multivariate log-normal distribution f ( x ; µ, Σ ) = (2 π ) − A/ | Σ | − / exp (cid:18) −
12 (log( x ) − log( µ )) T Σ − (log( x ) − log( µ )) (cid:19) ( x · · · x k ) − For the multivariate log-normal distribution, the scale matrix is the covariance matrix ofthe logarithm of the observations. When the scale matrix is diagonal, the distribution reducesto univariate log-normals. The marginal means are µ i exp (cid:16) Σ ii (cid:17) , and the variance/covarianceis µ i µ j exp (cid:16) (Σ ii + Σ jj ) (cid:17) (exp(Σ ij ) − Proportions-at-age models
We consider three multivariate observational model for proportions-at-age. In the densitylisted below, A is the number of ages, x > with P Ai =1 x i = 1 is a vector of A observed catchproportions (or survey proportions), µ > with P Ai =1 µ i = 1 is a vector of A calculated catchproportions (or survey proportions) based on the Baranov catch equation (or proportional tothe total abundance), and Σ is an A − × A − − A denotes the vector without the A th element.19 dditive logistic-normal distribution f ( x ; µ, Σ ) = (2 π ) − ( A − / | Σ | − / exp (cid:18) −
12 ( α ( x ) − α ( µ )) T Σ − ( α ( x ) − α ( µ )) (cid:19) ( x · · · x A ) − Here, α is the additive logratio transformation α ( x ) = log (cid:16) x − A x A (cid:17) . The scale matrix is thecovariance of the additive logratio transformed observations. Note that if a numbers-at-agevector y follows a multivariate log-normal, then the proportions y / P i y i follows an additivelogistic-normal distribution (Aitchison 2003). The mean and variance does not have simpleforms (Aitchison 2003). Multiplicative logistic-normal distribution f ( x ; µ, Σ ) = (2 π ) − ( A − / | Σ | − / exp (cid:18) −
12 (m( x ) − m( µ )) T Σ − (m( x ) − m( µ )) (cid:19) ( x · · · x A ) − Here, m is the multiplicative logratio transformationm( x ) = log (cid:18) x − x (cid:19) , . . . , log x A − − x − · · · − x A − !! . The scale matrix is the covariance of the multiplicative logratio transformed observations.The mean and variance does not have simple forms (Aitchison 2003).
Dirichlet distribution f ( x ; µ, σ ) = Γ (cid:16)P Ai =1 σµ i (cid:17)Q Ai =1 Γ( σµ i ) A Y i =1 x σµ i − i For the Dirichlet distribution, σ > µ i , the variances are µ i − µ i σ +1 , and the covariances are − µ i µ j σ +1 .The Dirichlet distribution is related to the gamma distribution, since if each element ofa numbers-at-age vector follows a gamma distribution where the scale parameters in theusual parameterization are equal, then the vector of proportions y / P i y i follow a Dirichletdistribution. Note that f does not have the same scale parameters for all ages, as theydepend on the mean value. References
Aitchison, J. 2003. The Statistical Analysis of Compositional Data. The Blackburn Press,Caldwell, N.J.Cox, C., Chu, H., Schneider, M.F., and Muñoz, A. 2007. Parametric survival analysis andtaxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine (23): 4352–4374. doi: 10.1002/sim.2836.Nielsen, A., and Berg, C. 2014. Estimation of time-varying selectivity in stock assessmentsusing state-space models. Fisheries Research : 96–101. doi: 10.1016/j.fishres.2014.01.014.20 ppendix B Transformation of densities for proportions-at-age models
To compare the AIC of models for numbers-at-age with models for proportions-at-age, thedata must be on the same scale. We note that we can transform the numbers-at-age data toproportions-at-age with total catch in numbers by the function g (cid:16) ( x , . . . , x A ) T (cid:17) = x P Ai =1 x i , . . . , x A − P Ai =1 x i , A X i =1 x i ! T , with inverse function h (cid:16) ( y , . . . , y A − , y total ) T (cid:17) = y · y total , . . . , y A − · y total , (1 − A − X i =1 y i ) · y total ! T . Hence, if Y is a random vector of proportions-at-age with total catch in numbers withdensity f , then the corresponding numbers-at-age is X = h ( Y ). By a change of variable, P ( X ∈ B ) = P ( Y ∈ g ( B ))= Z g ( B ) f ( y ) dy = Z A f ( g ( x )) | det( Dg ) | dy, the density for the numbers-at-age data is f ( g ( y )) | det( Dg ) | . Hence, to compare the AICbetween the (natively) numbers-at-age and proportions-at-age models, the log-likelihoods ofthe proportions-at-age models must be corrected by the logarithm of the absolute determinantof the Jacobian of g , log | det( Dg ) | . The entries of ( Dg ) are ( Dg ) A,j = 1 for all j , ( Dg ) i,i = P Ak =1 x k − x i ( P Ak =1 x k ) for all i < A and ( Dg ) i,j = − x i ( P Ak =1 x k ) otherwise. If the total is in weight,where the weight-at-age is assumed to be known, then g is adjusted to g (cid:16) ( x , . . . , x A ) T (cid:17) = x P Ai =1 x i , . . . , x A − P Ai =1 x i , A X i =1 w i x i ! T hoosing the observational likelihood in state-spacestock assessment models Supplementary materialChristoffer Moesgaard Albertsen ∗ , Anders Nielsen, and Uffe Høgsbro Thygesen Technical University of Denmark, National Institute of Aquatic Resources,Charlottenlund Castle, DK-2920 Charlottenlund, Denmark ∗ email: [email protected] a r X i v : . [ s t a t . A P ] S e p stimated coefficients of variation Table S1: Estimated marginal coefficients of variation for models M , M , M , M for BlueWhiting. Standard errors of the estimates (in parentheses) are derived by the delta method. M and M are not included as they do not have constant CV. For M , τ is set to 0 in thecalculation for | ˆ τ | < . M M M M M Commercial 1 0.40 (0.08) 0.37 (0.07) 0.35 (0.07) 0.32 (0.06) 0.50 (0.11)Commercial 2 0.25 (0.06) 0.24 (0.06) 0.23 (0.02)* 0.21 (0.05) 0.38 (0.09)Commercial 3 0.07 (0.11) 0.06 (0.13) 0.01 (0.15)* 0.00 (0.01) 0.15 (0.07)Commercial 4 0.18 (0.05) 0.18 (0.05) 0.20 (0.01)* 0.17 (0.05) 0.01 (0.09)Commercial 5 0.14 (0.06) 0.14 (0.06) 0.11 (0.04)* 0.12 (0.06) 0.23 (0.07)Commercial 6 0.12 (0.05) 0.12 (0.05) 0.13 (0.02)* 0.12 (0.05) 0.25 (0.05)Commercial 7 0.06 (0.10) 0.06 (0.11) 0.07 (0.12)* 0.00 (0.09) 0.23 (0.05)Commercial 8 0.05 (0.16) 0.05 (0.16) 0.04 (0.71)* 0.00 (0.03) 0.28 (0.06)Commercial 9 0.31 (0.08) 0.29 (0.07) 0.31 (0.02)* 0.26 (0.06) 0.42 (0.08)Commercial 10 0.38 (0.08) 0.37 (0.07) - ( - ) 0.32 (0.06) 0.48 (0.08)Survey 1 3 0.40 (0.11) 0.35 (0.09) 0.29 (0.80) 0.31 (0.08) 0.44 (0.12)Survey 1 4 0.23 (0.07) 0.22 (0.07) 0.21 (0.58) 0.21 (0.06) 0.22 (0.07)Survey 1 5 0.31 (0.09) 0.29 (0.08) 0.22 (8.23) 0.24 (0.07) 0.29 (0.08)Survey 1 6 0.17 (0.06) 0.16 (0.06) 0.18 (0.12) 0.17 (0.06) 0.20 (0.07)Survey 1 7 0.25 (0.07) 0.25 (0.07) 0.26 (0.02)* 0.26 (0.08) 0.24 (0.07)Survey 1 8 0.39 (0.10) 0.37 (0.09) 0.39 (0.02)* 0.37 (0.10) 0.38 (0.10)2able S2: Estimated marginal coefficients of variation for models M , M , M , M for North-East Arctic Haddock. Standard errors of the estimates (in parentheses) are derived by thedelta method. M and M are not included as they do not have constant CV. For M , τ isset to 0 in the calculation for | ˆ τ | < . M M M M M Commercial 3 0.60 (0.08) 0.51 (0.03) 0.49 (0.02) 0.43 (0.06) 0.68 (0.09)Commercial 4 0.30 (0.05) 0.28 ( - ) 0.29 (0.01)* 0.25 (0.04) 0.35 (0.05)Commercial 5 0.25 (0.04) 0.24 ( - ) 0.23 (0.04) 0.22 (0.03) 0.28 (0.04)Commercial 6 0.20 (0.04) 0.20 (0.03) 0.20 (0.01)* 0.18 (0.04) 0.24 (0.04)Commercial 7 0.24 (0.04) 0.23 (0.04) 0.23 (0.01)* 0.21 (0.03) 0.27 (0.04)Commercial 8 0.00 (0.01) 0.00 ( - ) 0.00 (0.00)* 0.00 (0.03) 0.01 (0.07)Commercial 9 0.73 (0.09) 0.50 (0.05) 0.40 (0.03) 0.35 (0.05) 0.73 (0.09)Commercial 10 0.39 (0.07) 0.35 (0.05) 0.32 (0.04) 0.29 (0.05) 0.38 (0.07)Commercial 11 0.36 (0.06) 0.31 (0.05) - ( - ) 0.25 (0.04) 0.35 (0.06)Survey 1 3 0.35 (0.07) 0.35 (0.06) 0.32 (0.04) 0.29 (0.06) 0.48 (0.10)Survey 1 4 0.42 (0.08) 0.38 (0.07) - ( - ) 0.47 (0.11) 0.42 (0.07)Survey 1 5 0.52 (0.10) 0.47 (0.07) 0.45 (0.03) 0.41 (0.07) 0.53 (0.10)Survey 1 6 0.35 (0.06) 0.32 (0.06) 0.32 (0.05) 0.32 (0.06) 0.35 (0.06)Survey 1 7 0.58 (0.11) 0.46 (0.07) 0.41 (0.05) 0.41 (0.08) 0.52 (0.09)Survey 2 3 0.24 (0.06) 0.24 (0.05) 0.21 (0.59) 0.23 (0.06) 0.26 (0.06)Survey 2 4 0.28 (0.06) 0.27 (0.06) - ( - ) 0.28 (0.07) 0.29 (0.06)Survey 2 5 0.32 (0.08) 0.29 (0.06) 0.29 (0.02)* 0.28 (0.08) 0.33 (0.07)Survey 2 6 0.37 (0.09) 0.36 (0.08) 0.37 (0.05) 0.34 (0.08) 0.35 (0.08)Survey 2 7 0.52 (0.13) 0.49 (0.10) 0.44 (1.65) 0.43 (0.09) 0.57 (0.13)Survey 3 3 0.31 (0.06) 0.29 (0.05) 0.27 (0.05) 0.30 (0.07) 0.28 (0.06)Survey 3 4 0.34 (0.07) 0.34 (0.06) 0.32 (0.04) 0.32 (0.06) 0.28 (0.05)Survey 3 5 0.42 (0.09) 0.41 (0.08) - ( - ) 0.48 (0.12) 0.36 (0.07)Survey 3 6 0.40 (0.09) 0.39 (0.07) 0.41 (0.02)* 0.40 (0.08) 0.35 (0.07)Survey 3 7 0.36 (0.09) 0.36 (0.08) 0.28 (1.16) 0.32 (0.07) 0.51 (0.14)Survey 3 8 0.71 (0.15) 0.59 (0.10) 0.63 (0.06) 0.56 (0.13) 0.77 (0.18)Survey 4 3 0.41 (0.12) 0.39 (0.10) 0.42 (0.04)* 0.38 (0.12) 0.34 (0.08)Survey 4 4 0.29 (0.08) 0.29 (0.07) 0.28 (0.02)* 0.27 (0.08) 0.24 (0.06)Survey 4 5 0.12 (0.06) 0.12 (0.05) - ( - ) 0.13 (0.07) 0.12 (0.04)Survey 4 6 0.09 (0.07) 0.09 (0.07) 0.08 (0.06)* 0.10 (0.06) 0.17 (0.09)Survey 4 7 0.32 (0.09) 0.31 (0.09) 0.33 (0.03)* 0.32 (0.10) 0.39 (0.13)Survey 4 8 0.40 (0.12) 0.39 (0.10) 0.37 (0.13) 0.40 (0.12) 0.56 (0.19)3able S3: Estimated marginal coefficients of variation for models M , M , M , M for NorthSea Cod. Standard errors of the estimates (in parentheses) are derived by the delta method. M and M are not included as they do not have constant CV. For M , τ is set to 0 in thecalculation for | ˆ τ | < . M M M M M Commercial 1 0.81 (0.12) 0.65 (0.07) 0.80 (0.01)* 0.58 (0.09) 0.82 (0.12)Commercial 2 0.22 (0.04) 0.21 (0.03) 0.20 (0.05) 0.19 (0.03) 0.20 (0.04)Commercial 3 0.08 (0.03) 0.08 (0.03) 0.09 (0.08) 0.07 (0.03) 0.10 (0.03)Commercial 4 0.08 (0.02) 0.08 (0.02) 0.07 (0.01)* 0.07 (0.03) 0.09 (0.02)Commercial 5 0.07 (0.03) 0.07 (0.03) - ( - ) 0.05 (0.04) 0.08 (0.03)Commercial 6 0.09 (0.03) 0.09 (0.03) 0.08 (0.01)* 0.09 (0.04) 0.10 (0.03)Commercial 7 0.15 (0.03) 0.15 (0.03) 0.15 (0.00)* 0.13 (0.03) 0.15 (0.03)Survey 1 1 0.71 (0.12) 0.60 (0.08) 0.70 (0.02)* 0.61 (0.11) 0.71 (0.12)Survey 1 2 0.29 (0.05) 0.28 (0.04) 0.29 (0.01)* 0.26 (0.04) 0.28 (0.04)Survey 1 3 0.23 (0.03) 0.23 (0.03) 0.23 (0.00)* 0.22 (0.03) 0.23 (0.03)Survey 1 4 0.30 (0.04) 0.29 (0.04) - ( - ) 0.30 (0.05) 0.29 (0.04)Survey 1 5 0.33 (0.05) 0.32 (0.04) - ( - ) 0.35 (0.05) 0.31 (0.04)4able S4: Estimated marginal coefficients of variation for models M , M , M , M forNorthern Shelf Haddock. Standard errors of the estimates (in parentheses) are derived bythe delta method. M and M are not included as they do not have constant CV. For M , τ is set to 0 in the calculation for | ˆ τ | < . M M M M M Commercial 0 1.71 (0.35) 1.01 (0.10) 1.73 ( 0.00)* 0.84 (0.14) 1.86 (0.39)Commercial 1 0.71 (0.12) 0.56 (0.07) 0.76 ( 0.01)* 0.38 (0.06) 0.79 (0.13)Commercial 2 0.34 (0.05) 0.31 (0.05) 0.34 ( 0.01) 0.26 (0.05) 0.32 (0.05)Commercial 3 0.25 (0.05) 0.24 (0.05) 0.24 ( 0.03) 0.20 (0.04) 0.27 (0.05)Commercial 4 0.18 (0.05) 0.20 (0.04) 0.20 ( 0.00) 0.21 (0.04) 0.21 (0.04)Commercial 5 0.00 (0.00) 0.00 ( - ) 0.00 ( 0.00)* 0.00 (0.27) 0.07 (0.09)Commercial 6 0.00 ( - ) 0.00 (0.09) 0.08 ( 0.09)* 0.00 (0.00) 0.13 (0.07)Commercial 7 0.27 (0.06) 0.23 (0.06) - ( - ) 0.17 (0.07) 0.30 (0.05)Commercial 8 0.19 (0.08) 0.19 (0.08) 0.19 ( 0.02)* 0.00 (0.01) 0.22 (0.06)Survey 1 0 0.53 (0.14) 0.46 (0.11) - ( - ) 0.36 (0.10) 0.53 (0.13)Survey 1 1 0.24 (0.06) 0.23 (0.06) - ( - ) 0.23 (0.06) 0.25 (0.06)Survey 1 2 0.16 (0.04) 0.16 (0.04) 0.14 ( 1.10) 0.17 (0.05) 0.24 (0.05)Survey 1 3 0.36 (0.09) 0.32 (0.07) 0.33 ( 0.08) 0.27 (0.07) 0.36 (0.07)Survey 1 4 0.23 (0.06) 0.20 (0.06) - ( - ) 0.15 (0.06) 0.28 (0.07)Survey 1 5 0.40 (0.08) 0.38 (0.07) - ( - ) 0.39 (0.09) 0.48 (0.12)Survey 1 6 0.94 (0.24) 0.74 (0.13) 0.91 ( 0.04)* 0.74 (0.17) 1.15 (0.34)Survey 2 0 0.78 (0.17) 0.63 (0.10) 0.51 ( - ) 0.55 (0.12) 0.67 (0.13)Survey 2 1 0.15 (0.05) 0.14 (0.06) - ( - ) 0.13 (0.06) 0.14 (0.04)Survey 2 2 0.34 (0.06) 0.33 (0.06) 0.30 ( 0.00)* 0.32 (0.07) 0.31 (0.06)Survey 2 3 0.36 (0.07) 0.34 (0.06) 0.35 ( 0.00)* 0.31 (0.06) 0.38 (0.07)Survey 2 4 0.40 (0.08) 0.36 (0.07) - ( - ) 0.30 (0.07) 0.42 (0.08)Survey 2 5 0.51 (0.10) 0.49 (0.08) - ( - ) 0.47 (0.10) 0.56 (0.10)Survey 2 6 1.22 (0.31) 0.75 (0.12) 1.25 ( 0.05)* 0.58 (0.12) 1.53 (0.43)Survey 3 0 0.45 (0.10) 0.42 (0.09) - ( - ) 0.40 (0.11) 0.56 (0.13)Survey 3 1 0.19 (0.05) 0.19 (0.05) 0.19 ( 5.99) 0.17 (0.05) 0.29 (0.07)Survey 3 2 0.00 (0.00) 0.00 ( - ) - ( - ) 0.08 (0.06) 0.14 (0.05)Survey 3 3 0.27 (0.06) 0.25 (0.06) 0.25 ( 0.03) 0.23 (0.06) 0.29 (0.06)Survey 3 4 0.30 (0.07) 0.28 (0.06) 0.24 ( - ) 0.25 (0.06) 0.33 (0.07)Survey 3 5 0.33 (0.07) 0.31 (0.06) 0.33 ( 0.01)* 0.31 (0.07) 0.36 (0.07)Survey 3 6 0.44 (0.09) 0.42 (0.08) 0.44 ( 0.02)* 0.39 (0.08) 0.43 (0.08)Survey 4 0 1.01 (0.28) 0.64 (0.12) 0.42 ( - ) 0.44 (0.10) 0.99 (0.27)Survey 4 1 0.27 (0.06) 0.27 (0.06) 0.24 ( 0.00)* 0.28 (0.06) 0.29 (0.06)Survey 4 2 0.21 (0.06) 0.21 (0.06) 0.20 ( - ) 0.20 (0.06) 0.26 (0.06)Survey 4 3 0.19 (0.06) 0.19 (0.06) - ( - ) 0.22 (0.06) 0.21 (0.06)Survey 4 4 0.37 (0.09) 0.32 (0.07) 0.36 (-0.09)* 0.25 (0.06) 0.37 (0.09)Survey 4 5 0.45 (0.10) 0.42 (0.08) - ( - ) 0.49 (0.13) 0.46 (0.10)Survey 4 6 0.50 (0.12) 0.48 (0.10) 0.53 ( 0.03)* 0.44 (0.10) 0.53 (0.13)Survey 5 0 0.28 (0.07) 0.28 (0.06) 0.26 ( - ) 0.32 (0.06) 0.28 (0.06)Survey 5 1 0.34 (0.05) 0.33 (0.05) - ( - ) 0.33 (0.06) 0.30 (0.04)Survey 5 2 0.34 (0.05) 0.32 (0.05) - ( - ) 0.30 (0.05) 0.34 (0.05)Survey 5 3 0.31 (0.05) 0.29 (0.05) - ( - ) 0.28 (0.05) 0.36 (0.06)Survey 5 4 0.36 (0.06) 0.33 (0.05) 0.34 ( 0.00)* 0.31 (0.05) 0.47 (0.08)5able S5: Estimated coefficient of variation for totals with models M − M . Standarderrors of the estimates (in parentheses) are derived by the delta method from the estimatedlogarithm of the scale parameter.Fleet M M M M M M Blue Whiting
Com. 0.15 (0.11) 0.16 (0.09) 0.16 (0.09) 0.14 (0.12) 0.15 (0.10) 0.16 (0.10)Surv. 1 0.13 (0.04) 0.13 (0.04) 0.13 (0.04) 0.13 (0.04) 0.12 (0.04) 0.13 (0.04)
North-East Arctic Haddock
Com. 0.11 (0.05) 0.11 (0.05) 0.05 (0.07) 0.10 (0.05) 0.09 (0.05) 0.06 (0.06)Surv. 1 0.37 (0.06) 0.34 (0.06) 0.39 (0.07) 0.37 (0.06) 0.35 (0.06) 0.39 (0.07)Surv. 2 0.22 (0.05) 0.21 (0.05) 0.25 (0.06) 0.23 (0.05) 0.22 (0.05) 0.25 (0.06)Surv. 3 0.22 (0.05) 0.26 (0.06) 0.25 (0.06) 0.22 (0.05) 0.26 (0.06) 0.27 (0.06)Surv. 4 0.27 (0.07) 0.28 (0.08) 0.25 (0.08) 0.22 (0.06) 0.23 (0.07) 0.21 (0.07)
North Sea Cod
Com. 0.32 (0.05) 0.32 (0.05) 0.03 (0.05) 0.15 (0.03) 0.15 (0.02) 0.04 (0.04)Surv. 1 0.27 (0.05) 0.27 (0.05) 0.34 (0.05) 0.20 (0.04) 0.20 (0.04) 0.21 (0.03)
Northen Shelf Haddock
Com. 0.26 (0.05) 0.28 (0.05) 0.13 (0.06) 0.18 (0.06) 0.17 (0.05) 0.15 (0.06)Surv. 1 0.34 (0.08) 0.34 (0.08) 0.29 (0.07) 0.23 (0.06) 0.25 (0.06) 0.25 (0.06)Surv. 2 0.22 (0.05) 0.21 (0.05) 0.24 (0.05) 0.19 (0.04) 0.19 (0.04) 0.19 (0.04)Surv. 3 0.24 (0.06) 0.24 (0.06) 0.26 (0.06) 0.19 (0.05) 0.20 (0.05) 0.17 (0.05)Surv. 4 0.15 (0.05) 0.17 (0.05) 0.10 (0.04) 0.11 (0.03) 0.12 (0.04) 0.07 (0.03)Surv. 5 0.29 (0.05) 0.30 (0.05) 0.29 (0.05) 0.27 (0.04) 0.27 (0.04) 0.25 (0.04)6 stimated fishing mortality and spawning stock biomass
Blue Whiting log-Normal . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S1: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Normal model for Blue Whiting including 95 % pointwise confidence intervals (greyarea); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75% and 95 % levels. The red lines indicate the management plan reference points while theblack point is the estimated value in the final year.7 amma . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S2: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Gamma model for Blue Whiting including 95 % pointwise confidence intervals (greyarea); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75% and 95 % levels. The red lines indicate the management plan reference points while theblack point is the estimated value in the final year.8 eneralized Gamma . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S3: Estimated average fishing mortality (A) and log spawning stock biomass (B) with theGeneralized Gamma model for Blue Whiting including 95 % pointwise confidence intervals(grey area); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50%, 75 % and 95 % levels. The red lines indicate the management plan reference points whilethe black point is the estimated value in the final year.9 ormal . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S4: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Normal model for Blue Whiting including 95 % pointwise confidence intervals (grey area);their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75 %and 95 % levels. The red lines indicate the management plan reference points while the blackpoint is the estimated value in the final year.10 eft Truncated Normal . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S5: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Left Truncated Normal model for Blue Whiting including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.11 og-Students t . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S6: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Students t model for Blue Whiting including 95 % pointwise confidence intervals(grey area); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50%, 75 % and 95 % levels. The red lines indicate the management plan reference points whilethe black point is the estimated value in the final year.12 ultivariate log-Normal . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S7: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multivariate log-Normal model for Blue Whiting including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.13 dditive Logistic Normal with log-Normal Numbers . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S8: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logistic Normal with log-Normal Numbers model for Blue Whiting including 95% pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 14 ultiplicative Logistic Normal with log-Normal Numbers . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S9: Estimated average fishing mortality (A) and log spawning stock biomass (B) with theMultiplicative Logistic Normal with log-Normal Numbers model for Blue Whiting including95 % pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 15 irichlet with log-Normal Numbers . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S10: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Numbers model for Blue Whiting including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.16 dditive Logisitc Normal with log-Normal Weight . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S11: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logisitc Normal with log-Normal Weight model for Blue Whiting including 95% pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 17 ultiplicative Logistic Normal with log-Normal Weight . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S12: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Weight model for Blue Whiting including95 % pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 18 irichlet with log-Normal Weight . . . . Year F - A 1980 1990 2000 2010 . . . . . Year l og ( SSB ) B0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) C 0.0 0.2 0.4 0.6 . . . . . F - l og ( SSB ) D Fig. S13: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Weight model for Blue Whiting including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.19 orth-East Arctic Haddock log-Normal . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S14: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Normal model for North-East Arctic Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.20 amma . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S15: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Gamma model for North-East Arctic Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.21 eneralized Gamma . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S16: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Generalized Gamma model for North-East Arctic Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.22 ormal . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S17: Estimated average fishing mortality (A) and log spawning stock biomass (B)with the Normal model for North-East Arctic Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.23 eft Truncated Normal . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S18: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Left Truncated Normal model for North-East Arctic Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.24 og-Students t . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S19: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Students t model for North-East Arctic Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.25 ultivariate log-Normal . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S20: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multivariate log-Normal model for North-East Arctic Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.26 dditive Logistic Normal with log-Normal Numbers . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S21: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logistic Normal with log-Normal Numbers model for North-East Arctic Haddockincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 27 ultiplicative Logistic Normal with log-Normal Numbers . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S22: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Numbers model for North-East ArcticHaddock including 95 % pointwise confidence intervals (grey area); their estimated trajectory(C); and confidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The redlines indicate the management plan reference points while the black point is the estimatedvalue in the final year. 28 irichlet with log-Normal Numbers . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S23: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Numbers model for North-East Arctic Haddock including 95% pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 29 dditive Logisitc Normal with log-Normal Weight . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S24: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logisitc Normal with log-Normal Weight model for North-East Arctic Haddockincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 30 ultiplicative Logistic Normal with log-Normal Weight . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S25: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Weight model for North-East ArcticHaddock including 95 % pointwise confidence intervals (grey area); their estimated trajectory(C); and confidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The redlines indicate the management plan reference points while the black point is the estimatedvalue in the final year. 31 irichlet with log-Normal Weight . . . . . Year F - A 1950 1970 1990 2010
Year l og ( SSB ) B0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) C 0.2 0.4 0.6 0.8 1.0 F - l og ( SSB ) D Fig. S26: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Weight model for North-East Arctic Haddock including 95% pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 32 orth Sea Cod log-Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S27: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Normal model for North Sea Cod including 95 % pointwise confidence intervals (greyarea); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75% and 95 % levels. The red lines indicate the management plan reference points while theblack point is the estimated value in the final year.33 amma . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S28: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Gamma model for North Sea Cod including 95 % pointwise confidence intervals (greyarea); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75% and 95 % levels. The red lines indicate the management plan reference points while theblack point is the estimated value in the final year.34 eneralized Gamma . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S29: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Generalized Gamma model for North Sea Cod including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.35 ormal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S30: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Normal model for North Sea Cod including 95 % pointwise confidence intervals (greyarea); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50 %, 75% and 95 % levels. The red lines indicate the management plan reference points while theblack point is the estimated value in the final year.36 eft Truncated Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S31: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Left Truncated Normal model for North Sea Cod including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.37 og-Students t . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S32: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Students t model for North Sea Cod including 95 % pointwise confidence intervals(grey area); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50%, 75 % and 95 % levels. The red lines indicate the management plan reference points whilethe black point is the estimated value in the final year.38 ultivariate log-Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S33: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multivariate log-Normal model for North Sea Cod including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.39 dditive Logistic Normal with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S34: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logistic Normal with log-Normal Numbers model for North Sea Cod including95 % pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 40 ultiplicative Logistic Normal with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S35: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Numbers model for North Sea Codincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 41 irichlet with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S36: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Numbers model for North Sea Cod including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.42 dditive Logisitc Normal with log-Normal Weight . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S37: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logisitc Normal with log-Normal Weight model for North Sea Cod including 95% pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 43 ultiplicative Logistic Normal with log-Normal Weight . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S38: Estimated average fishing mortality (A) and log spawning stock biomass (B)with the Multiplicative Logistic Normal with log-Normal Weight model for North Sea Codincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 44 irichlet with log-Normal Weight . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.2 0.6 1.0 F - l og ( SSB ) C 0.2 0.6 1.0 F - l og ( SSB ) D Fig. S39: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Weight model for North Sea Cod including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.45 orthern Shelf Haddock log-Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S40: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Normal model for Northern Shelf Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.46 amma . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S41: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Gamma model for Northern Shelf Haddock including 95 % pointwise confidence intervals(grey area); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50%, 75 % and 95 % levels. The red lines indicate the management plan reference points whilethe black point is the estimated value in the final year.47 eneralized Gamma . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S42: Estimated average fishing mortality (A) and log spawning stock biomass (B)with the Generalized Gamma model for Northern Shelf Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.48 ormal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S43: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Normal model for Northern Shelf Haddock including 95 % pointwise confidence intervals(grey area); their estimated trajectory (C); and confidence ellipses in the final year (D) at 50%, 75 % and 95 % levels. The red lines indicate the management plan reference points whilethe black point is the estimated value in the final year.49 eft Truncated Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S44: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Left Truncated Normal model for Northern Shelf Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.50 og-Students t . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S45: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe log-Students t model for Northern Shelf Haddock including 95 % pointwise confidenceintervals (grey area); their estimated trajectory (C); and confidence ellipses in the final year(D) at 50 %, 75 % and 95 % levels. The red lines indicate the management plan referencepoints while the black point is the estimated value in the final year.51 ultivariate log-Normal . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S46: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multivariate log-Normal model for Northern Shelf Haddock including 95 % pointwiseconfidence intervals (grey area); their estimated trajectory (C); and confidence ellipses in thefinal year (D) at 50 %, 75 % and 95 % levels. The red lines indicate the management planreference points while the black point is the estimated value in the final year.52 dditive Logistic Normal with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S47: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logistic Normal with log-Normal Numbers model for Northern Shelf Haddockincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 53 ultiplicative Logistic Normal with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S48: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Numbers model for Northern ShelfHaddock including 95 % pointwise confidence intervals (grey area); their estimated trajectory(C); and confidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The redlines indicate the management plan reference points while the black point is the estimatedvalue in the final year. 54 irichlet with log-Normal Numbers . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S49: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Dirichlet with log-Normal Numbers model for Northern Shelf Haddock including 95 %pointwise confidence intervals (grey area); their estimated trajectory (C); and confidenceellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicate themanagement plan reference points while the black point is the estimated value in the finalyear. 55 dditive Logisitc Normal with log-Normal Weight . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S50: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Additive Logisitc Normal with log-Normal Weight model for Northern Shelf Haddockincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 56 ultiplicative Logistic Normal with log-Normal Weight . . . Year F - A 1970 1990 2010
Year l og ( SSB ) B0.5 1.0 1.5 F - l og ( SSB ) C 0.5 1.0 1.5 F - l og ( SSB ) D Fig. S51: Estimated average fishing mortality (A) and log spawning stock biomass (B) withthe Multiplicative Logistic Normal with log-Normal Weight model for Northern Shelf Haddockincluding 95 % pointwise confidence intervals (grey area); their estimated trajectory (C); andconfidence ellipses in the final year (D) at 50 %, 75 % and 95 % levels. The red lines indicatethe management plan reference points while the black point is the estimated value in thefinal year. 57 irichlet with log-Normal Weight . . . Year F - A 1970 1990 2010