\rfoot{Assessment}

Assessment {#sec:assessment}

Modeling history {#sec:assessment-modeling-history}

In spite of the relatively short history of fishing, r sp have surely been subject to a larger number of stock assessments than any marine species off the west coast of the U.S.A.~and Canada. These assessments have included a large variety of age-structured models. Initially, a cohort analysis tuned to fishery CPUE was used [@FrancisSwartzmanEtAl1982]. Later, the cohort analysis was tuned to National Marine Fisheries Service (NMFS) triennial acoustic survey estimates of absolute biomass at age [@HollowedAdlersteinEtAl1988]. Since 1989, Stock Synthesis models (or base versions of it) fit to fishery catch-at-age data and acoustic survey estimates of population biomass and age composition have been the primary assessment method.

While the general form of the age-structured assessment has remained similar since 1991, modeling procedures have been modified in a variety of ways. There have been alternative data choices, post-data collection processing routines, data-weighting schemes, structural assumptions for the stock assessment model, MCMC sampling algorithms, and control rules (Table~\@ref(tab:main-assessment-changes-tab)). Analysts are constantly trying to improve the caliber and relevance of the assessment by responding to new scientific developments related to statistics and biological dynamics, policy requirements, and different or new insights brought up during the peer review process to ensure a robust stock assessment.

Data processing, filtering, and weighting choices have been modified several times since the first assessment. For example, modifications to the target-strength relationship used to scale acoustic data changed in 1997 [@DornSaunders1997], and kriging was implemented to account for the spatial correlation in the acoustic data in 2010 [@StewartHamel2010]. While survey data have been the key index for biomass since 1988, surveys that have been used have varied considerably. The Alaska Fisheries Science Center/Northwest Fisheries Science Center West Coast Triennial Shelf Survey was used from 1988 before being discarded from the 2009 assessment [@HamelStewart2009]. Acoustic surveys from the years prior to 1995 were used for assessments in the early 1990s, but @StewartForrestEtAl2011 reviewed these early surveys and deemed that sampling was insufficient to be comparable with more recent data. Several recruitment indices have been considered but ultimately none were identified as adding appreciable contribution to model results [@HelserDornEtAl2002; @HelserFleischerEtAl2005; @StewartHamel2010], except for the fishery-independent acoustic-based relative age-1 index that has been included in the base model since the 2022 assessment. The process for generating fecundity-at-age from the combination of weight-at-age and maturity data changed in 2019 from using time-invariant to year-specific weight-at-age values. In 2024, time-varying maturity ogives were also added to the calculation of fecundity-at-age (see Section~\@ref(sec:data-maturity) for details). Even where data have been consistently used, the weighting of these data in the statistical likelihood has changed through the use of various emphasis factors [e.g., @Dorn1994; @DornSaundersEtAl1999], a multinomial sample size on age compositions [e.g., @DornSaundersEtAl1999; @HelserDornEtAl2002; @HelserFleischerEtAl2005; @StewartForrestEtAl2011], internal estimations of effective sample size using the Dirichlet-multinomial distribution [@JTC2018], and assumptions regarding year-specific survey variance. Since 2021, a more computationally efficient Bayesian MCMC sampler [No-U-Turn Sampler; NUTS; @HoffmanGelman2014] was used to estimate posterior distributions [@Monnahan2018; @Monnahan2019], a change from previous assessments that used the random walk Metropolis Hastings (rwMH) sampler [details described in @JTC2021]. The list of changes discussed above is for illustrative purposes only and represents a small fraction of the different choices analysts have made and that reviewers have required.

The structure of the assessment models has perhaps had the largest number of changes. In terms of spatial models, analysts have considered spatially explicit forms [@Dorn1994; @Dorn1997], spatially implicit forms [@HelserStewartEtAl2006], and single-area models [@JTC2012]. Predicted recruitment has been modeled by sampling historical recruitment [e.g., @Dorn1994; @HelserFleischerEtAl2005], using a stock--recruitment relationship parameterized using maximum sustainable yield (MSY) and the fishing mortality rate estimated to produce the MSY [r f_msy; @Martell2010], and using several alternative steepness priors [@JTC2012; @JTC2013]. Selectivity has also been modeled in several ways, invariant [@JTC2012; @JTC2013], time-varying with [@HelserDornEtAl2002] and without [@Dorn1994; @DornSaunders1997; @JTC2012; @JTC2013] a random walk, alternative levels of allowable deviation through time [@JTC2013; @JTC2017], age-based [@Dorn1994; @DornSaunders1997; @JTC2012; @JTC2013], and length-based [@HelserMartell2007].

Several harvest control rules have been explored for providing catch limits from stock assessment output. r sp stock assessments have presented decision makers with constant F, variable F, and the following hybrid control rules: r fspr_35, r fspr_40, r f_40_40_10, r fspr_45, r f_45_40_10, and r fspr_50 [e.g., @Dorn1996; @JTC2013]. Changes to policies such as the United States' National Standards Guidelines in 2002 and the r f_40_40_10 harvest control rule in the Agreement (Appendix~\@ref(sec:app-glossary)) have required specific changes to control rules.

In addition to the examples given above and changes documented in stock assessments, there have been many more investigations conducted at review panel meetings. Starting in 2013, the addition of the MSE [@JTC2013; @JacobsenEtAl2021] facilitated investigating changes to the modeling procedure in terms of pre-specified objectives that aim for a sustainable coast-wide fishery.

Description of base model {#sec:assessment-base-model}

The r assess_yr base model has the same general population dynamics structure as the r last_assess_yr assessment's base model. The statistical-catch-at-age model assumes that the r sp population is a single coast-wide stock subject to one aggregated fleet with combined male and female population dynamics. Stock Synthesis [@MethotWetzel2013] version r ss_version was the modeling platform used. The largest changes between the r last_assess_yr and r assess_yr stock assessments are the addition of another year of fishery and survey data, an age-1 index data point, a model-based method for empirical weight-at-age data, and time-varying maturity information integrated into the base model.

The r assess_yr base model includes a time series (1995 to r last_survey_yr) of acoustic age-2$+$ biomass estimates and acoustic estimates of relative numbers of age-1 fish (see Section~\@ref(sec:data-acoustic-survey) for more details on the age-1 index). Maturity is assumed to be time-invariant prior to 2009, and time-varying, with the integration of annual maturity ogives informed by sea temperature at depth, since 2009 (see Section~\@ref(sec:data-maturity)). Fecundity is time-varying as defined by annual weight-at-age multiplied by annual maturity ogives (r start_yr_age_comps--r last_data_yr; additionally see Section~\@ref(sec:data-weight-at-age)). The D-M likelihood approach [@ThorsonEtAl2017Dirichlet] is used to estimate the weights associated with age-composition data, rather than iteratively tuning the sample size multiplier as in 2017 and earlier assessments (see Section~\@ref(sec:data-age-comp-likelihood)). Time-varying fishery selectivity is retained in the r assess_yr base model with the magnitude of the allowable deviations unchanged from the r last_assess_yr base model (see Section~\@ref(sec:data-variability-selectivity)). The general parameterization of selectivity was retained, although additional parameters were required to estimate an additional year of deviations. The selectivity of the acoustic survey is assumed to be time invariant. Selectivity curves were modeled as non-parametric functions estimating age-specific values for each age beginning at age two for the index of age-2$+$ biomass and age one for the fishery until a maximum age of 6, after which all ages are assumed to have the same selectivity. Selectivity for the relative age-1 index was set to one for age one and zero for all other ages.

Prior probability distributions are used for a select few parameters and fixed values are used for several parameters. For the base model, the instantaneous rate of time-invariant natural mortality (M) is estimated with a r m_prior[1] prior having a median of r f(exp(as.numeric(m_prior[2])), 2) and a standard deviation (in log-space) of 0.1 (see Section~\@ref(sec:data-natural-mortality)). The stock--recruitment relationship is a Beverton--Holt parameterization, with the log of the mean unexploited recruitment (log r r_0) freely estimated. This assessment uses the same beta-distributed prior for stock--recruitment steepness (h), based on @MyersBowenEtAl1999, that has been applied since 2011 [@StewartForrestEtAl2011]. Year-specific recruitment deviations were estimated from r main_recdev_early--r main_recdev_end. The standard deviation, $\sigma_r$, of recruitment variability serves as a recruitment deviation constraint and is fixed at r base_model$sigma_R_in in this assessment. This value is based on consistency with the observed variability in the time series of recruitment deviation estimates and is the same as assumed in assessments from 2013 to r last_assess_yr (Table~\@ref(tab:main-assessment-changes-tab)). Catchabilities associated with the biomass index (q~b~) and with the relative age-1 index (q~1~) were calculated analytically as per @LudwigWalters1981 for each sample of posterior parameters, resulting in a distribution of catchability for each.

Statistical likelihood functions used for data fitting are typical of many stock assessments. The biomass index was fit via a lognormal likelihood function, using the observed (and extra 2009) sampling variability, estimated via kriging, as year-specific weightings. The relative age-1 index was specified as having a Student's t-distribution for its error structure with the number of degrees of freedom equal to one less than the number of available data points. An additional constant and additive standard deviation on the log-scale component is included for both the biomass index and the relative age-1 index, which were freely estimated to accommodate unaccounted-for sources of process and observation error. A D-M likelihood was applied to age-composition data, with input sample sizes equal to the sum of the number of trips and hauls sampled across all fishing fleets or the number of trawl sets in the research surveys (see Section~\@ref(sec:data-age-comp-likelihood)).

Model results and statistical inference were based on r f(num_mcmc_samples) MCMC samples [using the \texttt{adnuts} R package; @Monnahan2018] compiled across 8 chains, each with a 250 sample burn in period, to describe posterior distributions for model parameters and derived quantities. The number of samples used for bridging models, sensitivity models, and retrospective models was also r f(num_mcmc_samples). Medians (50% quantiles) are reported together with the bounds of 95% credibility intervals calculated as the 2.5% quantile and the 97.5% quantile of posterior distributions from the MCMC samples, to give equal-tailed intervals. A full explanation of the NUTS algorithm and the \texttt{adnuts} package, including an analysis with the r sp stock can be found in @Monnahan2019.

Response to r end_yr - 1 Scientific Review Group (SRG) review {#sec:assessment-response-review}

The Scientific Review Group (SRG) meeting was held from February 7--10th, 2023, at the Graduate Hotel, Seattle, WA, U.S.A.

The following are the 'SRG Recommendations and Conclusions for the Stock Assessment' from the 2023 SRG report and the associated responses from the JTC:

  1. Pacific Hake dynamics are highly variable even without fishing mortality. The SRG applauds the efforts of the JTC and the MSE Working Group to add capabilities for estimating dynamic reference points within the assessment and MSE, and encourage those groups to jointly develop alternative reference points, including dynamic reference points, for future SRG consideration.

Response -- An evaluation of alternative reference points and how they may best be used in r sp management continues to be an important area for ongoing research. Future reference point discussions stemming from simulation work, preferably through the MSE, would be beneficial. The JTC, nor the MSE Working Group, have had the capacity to initiate simulations that explore alternative reference points, including dynamic reference points, to date. Several alternative simulation-based study designs have been fleshed out but the resources to complete such research is currently the limiting factor.

  1. The SRG also recommends continuing sensitivities for steepness, natural mortality, $\sigma_R$, excluding the age-1 index, alternative standard deviations for time-varying selectivity, and using the McAllister-Ianelli method to weight fishery age- composition data.

Response -- The JTC has conducted all of the requested sensitivities and provides summaries in written (Section~\@ref(sec:assessment-sensitivity-analyses)), tabular (beginning with Table~\@ref(tab:main-parameter-estimates-sens-1-tab)), and graphical (beginning with Figure~\@ref(fig:main-biomass-base-alternative-1-fig)) formats in this document. Many other model explorations were conducted during the development and exploration of the base model to understand model performance and sensitivity to data and structural decisions.

  1. The SRG recommends that the JTC include dynamic unfished spawning biomass in the 2024 assessment as a comparator with the equilibrium unfished spawning biomass used to provide management advice. The SRG also encourages the continued outreach regarding the use of dynamic reference points to stakeholders and managers, including identifying pros and cons of using dynamic unfished spawning biomass.

Response -- Two figures have been added to the assessment document. The first shows the unfished spawning stock biomass time series relative to the standard 'fished' spawning stock biomass time series (see Figure~\@ref(fig:main-fished-unfished-biomass-fig)). The second shows a comparison of relative spawning biomass when spawning biomass in year $t$ is related to unfished equilibrium biomass, $B_0$ (static $B_0$; as in Figure~\@ref(fig:main-relative-biomass-fig)) and when spawning biomass in year $t$ is related to the unfished biomass time series in year $t$ (dynamic $B_0$; see Figure~\@ref(fig:main-fished-unfished-rel-biomass-fig)). These can be used to visually compare equilibrium versus non-equilibrium assumptions and estimated changes in population dynamics in the absence of fishing.

The JTC once again delivered a presentation on dynamic reference points at the October 2023 JMC and AP meeting as a way to provide education and outreach opportunities related to dynamic reference point methods and approaches.

  1. The SRG recommends that the JTC explore alternative ways of estimating natural mortality to update the current approach in the model, which is based on methods from more than a decade ago, since newer methods are available. Information presented during the SRG meeting implies that natural mortality on age-2 Pacific Hake is higher than currently assumed in the assessment model and should be explored more fully.

Response -- The JTC investigated estimating a vector of natural mortality parameters instead of a single age-invariant parameter. Several configurations, i.e., breakpoints at different ages, were explored but no combination of breakpoints led to natural mortality decreasing with age as hypothesized given the diet data of predators of r sp. Instead parameters were estimated near the median of the prior. When the prior was removed, natural mortality at younger ages was estimated to be lower than natural mortality at older ages largely due to the lack of information in the data. If age-specific natural mortality is desired in the future, tagging studies could be used to help inform the parameters. Or, diet data could be used to externally estimate natural mortality at age. Other modeling frameworks, such as the Woods Hole Assessment Model have more flexibility in modeling natural mortality and might also be an option for future explorations.

  1. The SRG encourages an analysis of catch and CPUE distribution for Canada and US that examines latitudinal shifts in fishing over time.

Response -- The JTC has not officially investigated CPUE data since prior to the 2013 assessment because of the many difficulties in accurately characterizing catch rates given the difficulties inherent in calculating effort within the r sp fishery [@JTC2013]. Furthermore, vessel locations for the U.S. shore-based sector must be extracted from log-book information, which is known to be problematic, and current data confidentiality agreements do not allow the sharing of vessel-specific information amongst the JTC members making it difficult to do a comprehensive analysis. Maps of CPUE for the U.S. at-sea sectors and Canadian vessels were presented in the data presentation at the SRG meeting as a first take at fulfilling this request.

  1. The SRG recommends continued work to collect ovary samples and data to develop a picture of the Pacific Hake reproductive cycle both seasonally, inter-annually and at the life-time scale based on histological and physiological measurements. Given the inter-annual variability in age-at-maturity, the SRG recommends that the JTC explore ways to incorporate time-varying maturity in the stock assessment model.

Response -- Melissa Head has been working hard to continuously collect maturity information for r sp since 2009. Up until this assessment, maturity was included in the fecundity relationship as a single ogive. Maturity-at-age ogives are now estimated for each year since 2009 and used to calculate fecundity (i.e., maturity $$ weight at age). Moreover, the relationship between age and maturity includes a coefficient for temperature (see Appendix~\@ref(sec:app-maturity-analysis)). This relationship could be enhanced in the future by including samples from Canada, samples that have previously been analyzed, collected samples that have yet to be analyzed, and samples collected during future endeavors. During the JTC meeting, it was also brought to our attention that there is the potential for samples to be collected from the U.S. shoreside sector as well. The JTC, Melissa Head, and the Sarah Nayani are investigating the feasibility of this in attempts to increase the seasonal coverage of sampling, though winter months will still remain sparsely sampled without a dedicated winter survey. *

  1. The SRG noted that the age-1 index did not include a value for 2001 because it was zero. Although this decision had negligible influence on the results because the estimate for 2000 recruitment was close to zero, the SRG noted that Stock Synthesis uses a lognormal likelihood which does not handle zero values. Given that future zero values are expected to have a bigger influence on the results in the short-term, the SRG recommends that the JTC explore likelihood forms that can fit to very low index values from the age-1 survey (e.g., robust likelihood). The SRG acknowledges that implementing new likelihoods will require changes to the Stock Synthesis platform.

Response -- A Student's t-distribution was implemented for the relative age-1 index of abundance. The Student's t-distribution cannot accommodate values of zero but it does allow for fatter tails than the lognormal distribution. Secondly, the Student's t-distribution is known to provide more accurate estimates of the variability of the sample mean for small sample sizes than the lognormal distribution when the standard deviation is unknown or imprecise. In the future, additional distributions should be investigated that can accommodate values of zero but this change was seen as a positive step forward until the survey team can estimate the variability of the survey.

  1. Given the importance of the age-1 index in estimating the size of the age-classes entering the fishery, the SRG recommends that the JTC implement updated age-1 index CVs, when they are provided by the Survey Team, in the stock assessment model.

Response -- The joint survey team continues to be short staffed. During 2023, priority was given to completing the 2023 survey, including providing the 2023 survey biomass estimate, associated age compositions, and the 2023 relative age-1 index in a timely manner for the stock assessment. Age-1 index CVs were not available for inclusion in the 2024 assessment; so, the fixed values of 0.5 were retained but a Student's t-distribution was used to fit the survey instead of a lognormal distribution (see the above response).

  1. A new at-sea sampling program in the Canadian freezer-trawler fleet was implemented in 2022 involving vessel crews sampling 50 fish per tow from two tows per trip. The SRG recommends that Canada consider sampling fewer fish from more tows to spread the sampling out and provide a more representative sample of fishery catches.

Response -- This was discussed briefly in a meeting involving industry representatives and one member of the Canadian JTC and determined to be too much to ask of crew at the time. They would have to deal with more than two bags on a trip which would lead to space limitations in the freezer that they are currently using for the purpose and organizational difficulties. The space limitations could be overcome by storing bags of fish in the galley but after a short discussion on the idea, it was determined not to be feasible for food safety and storage-space reasons. The idea will be introduced again for the 2024 fishing season.

  1. The SRG encourages the JTC to consider methods to determine the maximum input sample size for the age compositions [e.g., @StewartHamel2014].

Response -- Determining input sample sizes is important for how annual fishery and survey age compositions are initially weighted, which then provides the basis from which wholesale re-weighting of data sources (fishery or survey) is done with the Dirichlet-multinomial data weighting model parameters. The JTC has considered alternative methods, including that of @StewartHamel2014, and has determined that the first step is to explore the handling of survey age-composition data (given that initial effective sample sizes and data source weighting is relative to other data sources in the model). Currently, survey age compositions represent age structure associated with the acoustic survey as viewed through an estimated selectivity curve for the acoustic-trawl sampling net. Yet, selectivity for ages two and older with acoustics is theoretically at or near one. The JTC plans to investigate whether there is a more informative way to utilize survey age-composition information in the stock assessment model in the coming year.

  1. The SRG notes that there have been multiple strong cohorts in the stock recently compared to earlier periods where there was only one strong cohort supporting the stock, including during the period of sample collection for the ageing error matrix that supports the assessment model. Therefore, the SRG encourages the resumption of the ageing error study by the Committee of Age Reading Experts (CARE) using samples collected during the past decade.

Response -- An ageing error study in conjunction with CARE has been planned for several years but a full exchange remains on hold due to continuing difficulties with permits to send biological specimens across the U.S.--Canada border. The JTC plans to move forward with updating the ageing error analysis in the coming year with the data that are available regardless of the status of the in progress CARE study.

  1. The SRG supports the investigation of alternative selectivity functions, which may include a two-dimensional autoregressive (AR) approach, which may use information from the previous year and from adjacent year classes to predict selectivity.

Response -- The JTC agrees that investigations into alternative selectivity functions is a priority research area for r sp. The JTC would like to make incremental progress on this in the coming year as time allows, including also looking into alternative methods (e.g., random effects models) to best capture age, year, and cohort effects.

Modeling results {#sec:assessment-modeling-results}

Changes from r last_assess_yr {#sec:assessment-changes}

A set of bridging' models was constructed to evaluate the component-specific effects of the steps to change from ther last_assess_yrbase model to ther assess_yr` base model. The steps are as follows:

Stock Synthesis version r ss_version includes a number of changes since the version used by @JTC2023. However, none of the changes were specifically relevant to this assessment, and thus, the software update had no effect on assessment results (Figure~\@ref(fig:main-bridge-summary-1-fig)).

The update of pre-r last_data_yr data occurs because databases are continually updated; this yielded minor adjustments to the data. For example, samples that were recently aged but not available for the r last_assess_yr assessment were included. Updates to pre-r last_assess_yr data were small enough that they had little impact on the model results.

The addition of the r last_data_yr catch and weight-at-age data extends the model to the start of r assess_yr. Recruitment estimates and historical stock trajectory were relatively unchanged, and the new data suggest a slight decrease in female spawning biomass from r assess_yr - 1 to r assess_yr (Figure~\@ref(fig:main-bridge-summary-1-fig)).

Including the r last_data_yr fishery-independent biomass estimate led to a downward shift in the stock trajectory going back to 2017 and a similar shift in the fit to the survey (Figure~\@ref(fig:main-bridge-summary-1-fig)). The addition of the r last_data_yr survey biomass age compositions led to an estimated increase in stock biomass from 2023 to 2024 as a result of small shifts in expected recruitment, particularly for the 2020 and 2021 year classes. While the addition of the r last_data_yr relative age-1 index had a negligible effect on the stock trajectory, it did slightly adjust estimates of recent recruitment strength, in particular raising the 2022 recruitment somewhat.

The final step of adding r last_data_yr data involved incorporating fishery age-composition information, which shifted the ending year of the deviations in the selectivity parameters from r last_data_yr - 1 to r last_data_yr. These data had relatively little impact on the historical biomass estimates but did shift recent estimates of spawning biomass upwards (Figure~\@ref(fig:main-bridge-summary-1-fig)). Recruitment increased in 2021 and 2022, while the 2020 cohort was reduced. The increase in 2021 and 2022 recruitment contributes to the increase in female spawning biomass by the start of r assess_yr, as these fish are considered mostly mature at the start of r assess_yr. Despite both fishery age compositions and the relative age-1 index pointing towards above average cohorts in 2020, 2021 and, to a lesser extent, 2022, estimates of r assess_yr female spawning biomass remain highly uncertain (Figure~\@ref(fig:main-bridge-summary-1-fig)).

Structurally changing the error distribution for the relative age-1 index from a lognormal to a Student's t-distribution (with 13 degrees of freedom) had negligible effect on assessment results (Figure~\@ref(fig:main-bridge-summary-2-fig)). Nonetheless, the t-distribution is better suited for these data given the low sample size (14) and broader distribution tails compared to a lognormal when sample sizes are less than 30.

Input weight-at-age data were constructed using a model-based approach (see Section~\@ref(sec:data-weight-at-age) for details) to better inform changes in weight-at-age for years and ages when there is little or no data. Model predictions were based on a smoothed fixed effect for age and random effects for year and cohort. During periods of more consistent sampling protocols (since the mid-1990s), there was little effect on overall stock size or status (Figure~\@ref(fig:main-bridge-summary-2-fig)). However, there were refinements to stock size and status during the 1970s and 1980s, a period when sampling protocols were not as well documented (e.g., from foreign vessels).

Lastly, model-based estimates of time-varying maturity ogives were incorporated into the calculation of time-varying fecundity (Section~\@ref(sec:data-maturity) and Appendix~\@ref(sec:app-maturity-analysis)). This is in addition to using time-varying weight-at-age inputs, which improves the assessment model inputs that translate total biomass to spawning biomass. The addition of time-varying maturity resulted in annual differences in the age at which r sp mature, such that there were minor shifts in the translation of total biomass to spawning biomass by one year in some cases within the time series (Figure~\@ref(fig:main-bridge-summary-2-fig)), but the general trend in spawning biomass and population status remained largely the same.

Assessment model results {#sec:assessment-model-results}

Model Fit

Stationarity of the posterior distribution for model parameters was assessed via a suite of standard single-chain and multi-chain diagnostic tests via graphical summaries and interactive web applications (ShinySTAN; Appendix~\@ref(sec:app-mcmc-diagnostics)). All estimated parameters showed good mixing during sampling, no evidence for lack of convergence, and acceptable autocorrelation (results for some key parameters are shown in Figures~\@ref(fig:app-mcmc-diag-m-r0-fig)--\@ref(fig:app-mcmc-diag-dm-fig)). Correlation-corrected effective sample sizes were sufficient to summarize the posterior distributions and neither the Geweke nor the Heidelberger and Welch statistics for these parameters exceeded critical values more frequently than expected via random chance (Figure~\@ref(fig:app-mcmc-diag-hists-fig)). The Gelman-Rubin multi-chain diagnostic test, which compares within-chain variance to among-chain variance, further indicated that convergence was adequately achieved (examined via \texttt{ShinySTAN}). Correlations among key parameters were generally low, with the exception of $M$ and $\log R_0$ (Figure~\@ref(fig:app-mcmc-pairs-fig)). Estimates of recruitment in 2014 and 2016 were correlated with the derived quantity of catch from the default harvest rule in r end_yr, as to be expected given the dependencies among these quantities (Figure~\@ref(fig:app-mcmc-pairs-fig)). An examination of deviations in recruitment (log-scale differences between estimated and expected recruitment values) from recent years (Figure~\@ref(fig:app-mcmc-pairs-recruit-devs-fig)) indicates the highest correlation (r f(base_model$mcmcvals$rec_cor_2014_2016, 2)) was between the 2014 and 2016 recruitment deviations. This is the same as in the last assessment despite the fact that each cohort has been observed for an additional year.

Regarding the Dirichlet-multinomial parameter $\theta_{\text{fish}}$, the estimate (median and 95% credible interval) for $\log \theta_{\text{fish}}$ was r f(log_theta_fishery_median, 3) (r f(log_theta_fishery_lower, 3)--r f(log_theta_fishery_upper, 3)), giving an effective sample size multiplier $\theta_{\text{fish}} / (1 + \theta_{\text{fish}})$ of r f(dm_weight_fishery_median, 3) (r f(dm_weight_fishery_lower, 3)--r f(dm_weight_fishery_upper, 3)). The related log of the survey age-composition parameter $\theta_{\text{surv}}$, i.e., $\log \theta_{\text{surv}}$, was r f(log_theta_survey_median, 3) (r f(log_theta_survey_lower, 3)--r f(log_theta_survey_upper, 3)), and the resulting effective sample size multiplier $\theta_{\text{surv}} / (1 + \theta_{\text{surv}})$ of r f(dm_weight_survey_median, 3) (r f(dm_weight_survey_lower, 3)--r f(dm_weight_survey_upper, 3)).

The base model fit to the acoustic survey biomass index (Figure~\@ref(fig:main-survey-index-fit-mcmc-fig)) remains similar to the r last_assess_yr base model, up to 2017. The low r last_survey_yr survey biomass pulls down the last few years of estimated biomass, such that the fit to the 2019 data point is very good (for the r last_assess_yr assessment it was overestimated), the 2021 fit is underestimated (for the r last_assess_yr assessment it was very good). The median of the posterior distribution for the analytically-derived catchability associated with the acoustic survey biomass index (q~b~) was r f(median(base_model$extra_mcmc$q_vector), 3) (Figure~\@ref(fig:main-catchability-density-fig)).

The 2023 biomass index is the third lowest in the series (Table~\@ref(tab:main-survey-history-tab)), and is well below the model estimate, similar to the 2001 index that has always been below model estimates [@JTC2023]. While no direct cause for the 2001 index anomaly is known, the survey did begin earlier that year than all other surveys between 1995 and 2009 (Table~\@ref(tab:main-survey-history-tab)), which may explain some portion of the anomaly, along with age structure. For 2023, the survey timing is not anomalous. The estimated biomass increase from 2023 to 2024 is driven by the addition of 2023 survey age-composition data (Figure~\@ref(fig:main-bridge-summary-1-fig)). The addition of the 2023 relative age-1 index suggested an above-average 2022 cohort (and also increased the 2021 relative age-1 index compared to the previous assessment due to a previous omission; Section~\@ref(sec:data-acoustic-survey)).

The relatively stable estimated biomass from 2013--2019 is unchanged from the previous assessment. The underestimation of the 2009 and 2023 biomass estimates are larger than the underestimation of any other year. The uncertainty of the 2009 value (both modeled and actual) is high because of the presence of large numbers of Humboldt Squid during the survey. Humboldt Squid have similar target strength to hake which could introduce bias in the biomass estimate for that year, which also likely influenced the population dynamics of r sp through predation in that year. Future data will reduce the large uncertainty in the 2023 biomass estimate, which may reduce the underestimation.

Differences between the median posterior density estimates from the fit to the survey index are likely due to slight differences in what the fishery composition data and survey composition data, when considered independently, would otherwise suggest as population trends. Additionally, the population has undergone recent high, but declining, catch levels and produced a couple of above-average cohorts that are now mature.

The base model fit to the relative index of age-1 fish highlights an overall general confirmation of relative cohort strength (Figure~\@ref(fig:main-age1-index-fit-mcmc-fig)). In particular, the 2008, 2014, and 2018 cohorts were estimated to be less than the index, while the 1994 and 2016 cohorts were estimated to be larger than indicated by the index. The 2011 value (the large 2010 cohort) was closely fit. Age-1 fish in 2021 (2020 cohort) were estimated slightly below the index value (in last year's assessment they were estimated slightly above) and, being so young, include a large amount of uncertainty. The median of the posterior distribution for the analytically-derived catchability associated with the age-1 index (q~1~) was r f(median(base_model$extra_mcmc$q_vector_age1), 3) (Figure~\@ref(fig:main-catchability-density-age1-fig)).

Fits to the age-composition data continue to show close correspondence to the dominant and small cohorts observed in the data when the data give a consistent signal (Figures~\@ref(fig:main-age-comp-fits-fishery-fig) and \@ref(fig:main-age-comp-fits-survey-fig)). Because of the time-varying fishery selectivity, the fit to commercial age-composition data is particularly good, although models with time-invariant selectivity used in previous years also fit the age compositions well. In the r last_data_yr fishery, r top_coh(base_model, last_data_yr, 3, cap = FALSE). Age compositions from the r survey_end_yr acoustic survey suggest a similar age structure for older fish.

The 2020 cohort has now been observed by, and is well fit by, the acoustic survey (Figure~\@ref(fig:main-age-comp-fits-survey-fig)), with the survey's inclusion decreasing its estimated size (Figure~\@ref(fig:main-bridge-summary-1-fig)). Combined, the 2015--r last_data_yr fishery age-composition data and the 2017--r survey_end_yr acoustic survey age-composition data suggest that 2014 was a strong recruitment year, and the model was able to adequately fit to these observations (Figure~\@ref(fig:main-age-comp-fits-survey-fig)). The 2016 cohort, which has been observed three times by the survey, still appears to be smaller than the 2014 cohort.

The r survey_end_yr survey was the first to sample the r survey_end_yr - 2 cohort, suggesting that it was a large contingent of the population (r survey_a2_prop% of the r survey_end_yr survey catch). The 2020 cohort, which has now been observed by the acoustic survey, is expected to be above average in size. Residual patterns to the fishery and survey age data do not show patterns that would indicate systematic bias in model predictions (Figure~\@ref(fig:main-age-comp-pearson-fig)).

The median estimates for numbers, biomass, exploitation rate, and catch (in numbers and in biomass) for each age class in each year are given in Tables~\@ref(tab:main-est-numbers-at-age-tab)-- \@ref(tab:main-est-catch-at-age-biomass-tab). For the major cohorts, the resulting estimated age-specific catch, natural mortality, and surviving biomasses are given in Table~\@ref(tab:main-cohort-effects-tab). For example, at age-5 the catch weight of the 2014 cohort was slightly more than that of the 2010 cohort, and the resulting surviving biomass of the 2014 cohort was approximately half of the surviving biomass of the 2010 cohort.

Posterior distributions for both steepness and natural mortality are influenced by priors (Figures~\@ref(fig:main-prior-posterior-fig)-- \@ref(fig:main-prior-posterior-truncated-fig)). The posterior for steepness is only slightly updated by the data, as expected given the low level of information available to inform steepness as found in previous hake assessments. The posterior of natural mortality, on the other hand, is shifted to the right of the prior distribution and the prior may be constraining the posterior distribution from shifting further. Broadening the prior distribution by increasing the prior standard deviation for the natural mortality parameter is examined in sensitivity runs (see Section~\@ref(sec:assessment-sensitivity-analyses)). Other parameters showed updating from diffuse priors to posterior distributions, including $\log \theta_{\text{fish}}$ and $\log \theta_{\text{surv}}$ (as outlined in Section~\@ref(sec:data-age-comp-likelihood)).

The r assess_yr base model specified the same level of variation (standard deviation of $\Phi =$ r sel_phi_val) associated with time-varying fishery selectivity as the r last_assess_yr base model, effectively allowing the model flexibility (i.e., a lower penalty on the overall likelihood) to fit to data that suggests high variability among years for each age. This level of variation led to results that remained consistent with the r survey_end_yr acoustic survey age-composition data (but not the biomass index) and gave reasonable fits to the fishery age-composition data, given that there is considerable uncertainty associated with spatial changes in fish availability (due to movement) and recent variability in oceanographic conditions. Estimated selectivity deviations for age-3 and age-4 fish are larger from 2010 to 2012 than in subsequent years until 2020 when the deviation for age-4 was large again (Figures~\@ref(fig:main-tv-selex-fig) and \@ref(fig:main-tv-selex-uncertainty-fig)). The median selectivity peaks at age-4 in 2010, 2012 and 2020 and at age-3 in 2011 suggesting targeting (or generally higher availability) of the younger cohorts in those years. This pattern is consistent with the 2008 cohort appearing strong in the fishery age compositions initially, but decreasing in prominence from 2013 onward (Figure~\@ref(fig:main-age-comp-fits-fishery-fig)). Fishery selectivity on age-2 fish was at its highest in 2016. Fishery selectivity for r last_data_yr was characterised by a local peak at age 3 rather than a logistic pattern (Figure~\@ref(fig:main-tv-selex-uncertainty-fig)), likely as a result of increased availability of the above-average 2020 cohort. Even though the survey selectivity is time invariant, the posterior shows a broad band of uncertainty between ages 2 and 5 (Figure~\@ref(fig:main-tv-selex-posteriors-fig)). The decline in survey selectivity between ages 3 and 4 may be an artifact of the interaction between large cohorts and the biennial timing of recent surveys, with the 2010, 2014, 2016, and 2020 cohorts occurring in the survey at ages 3 and/or 5 but not age 4. Fishery selectivity is likewise very uncertain (Figures~\@ref(fig:main-tv-selex-uncertainty-fig) and \@ref(fig:main-tv-selex-posteriors-fig)), but in spite of this uncertainty, changes in year-to-year patterns in the estimates are still evident, particularly for age-2, age-3, and age-4 fish, though these patterns might also reflect time-varying mortality processes.

Stock biomass

The base stock assessment model indicates that, since the 1960s, r sp female spawning biomass has ranged from well below to above unfished equilibrium (Figures~\@ref(fig:main-biomass-fig) and \@ref(fig:main-relative-biomass-fig) and Tables~\@ref(tab:main-median-posterior-tab) and \@ref(tab:main-ci-posterior-tab)). The model estimates that it was below the unfished equilibrium in the 1960s, at the start of the assessment period, due to lower than average recruitment. The stock is estimated to have increased rapidly and was above unfished equilibrium in the mid-1970s and mid-1980s (after two large recruitments in the early 1980s). It then declined steadily to a low in 1999. This was followed by a brief increase to a peak in 2003 as the very large 1999 year class matured. The 1999 year class largely supported the fishery for several years due to relatively small recruitments between 2000 and 2007. With the aging 1999 year class, median female spawning biomass declined throughout the late 2000s, reaching a time-series low of r median_bio_min million~t in r median_bio_min_yr. The assessment model estimates that median female spawning biomass then peaked again in 2014 due to a very large 2010 year class and an above-average 2008 year class. The subsequent decline from 2014 to 2016 is primarily from the 2010 year class surpassing the age at which gains in weight from growth are greater than the loss in weight from mortality (growth-mortality transition). The 2014 year class is estimated to be large, though not as large as the 1999 and 2010 year classes, resulting in an increased biomass in 2017. The estimated biomass mostly declined from 2018 to 2022 due to the 2014 and 2016 year classes moving through the growth-mortality transition during a period of high catches. The increase in female spawning biomass in 2023 and 2024 is due to the expected above-average 2020 and potentially large 2021 cohorts entering maturity, and the recent declining trend in catch.

The median estimate of the r end_yr relative spawning biomass (spawning biomass at the start of r end_yr divided by that at unfished equilibrium, r b_0) is r curr_depl_median%. However, the uncertainty is large, with a 95% posterior credibility interval from r curr_depl_lower% to r curr_depl_upper% (Tables~\@ref(tab:main-median-posterior-tab) and \@ref(tab:main-ci-posterior-tab)), partly due to remaining unknowns about the size of the potentially large 2021 cohort because the acoustic survey has only provided one year of information about it.

The median estimate of the r end_yr female spawning biomass is r curr_bio_median million~t (with a 95% posterior credibility interval from r curr_bio_lower to r curr_bio_upper million~t). The current estimate of the r end_yr - 1 female spawning biomass is r prev_bio_median (r prev_bio_lower--r prev_bio_upper) million~t, giving less uncertainty than the estimate from the r end_yr - 1 assessment of r prev_bio_median_last_assess (r prev_bio_lower_last_assess--r prev_bio_upper_last_assess) million~t. The current median is reduced from last year, partly due to the tail of the distribution being greatly curtailed (upper end of the interval is much lower than it was in the r end_yr - 1 assessment), and a slight lowering of the lower end of the interval. The decrease appears to be due to the addition of the age-2$+$ biomass index pulling down the estimated biomass for recent years, plus the addition of the survey age compositions lowering the estimated 2020 recruitment (Figure~\@ref(fig:main-bridge-summary-1-fig)).

Recruitment

The new data for this assessment do not significantly change the general pattern of recruitment estimated in recent assessments. However, estimates of absolute recruitment for the most recent years can change with new data, and the improved methods for modeling temporal weight-at-age and spatio-temporal maturity can slightly change some historical estimated recruitments.

The estimate of 2020 recruitment in last year's assessment was based on only two years of data and thus was highly uncertain; it suggested the cohort could potentially be huge (95% credible interval: r f(last_yr_base_model$mcmccalcs$rlower["2020"], 1)-- r f(last_yr_base_model$mcmccalcs$rupper["2020"], 1)~billion fish). But with the extra data in this year's assessment the 2020 cohort looks to be above average but not huge (95% interval: r f(base_model$mcmccalcs$rlower["2020"], 1)-- r f(base_model$mcmccalcs$rupper["2020"], 1)~billion fish). The median has consequently fallen from r f(last_yr_base_model$mcmccalcs$rmed["2020"], 1) to r f(base_model$mcmccalcs$rmed["2020"], 1)~billion fish between the two assessments.

Similarly, median estimates of 2019 recruitment have changed by r f(((((recruitment_med_to_compare - prev_assess_recruitment_med)["2019"]) / (prev_assess_recruitment_med)["2019"]) * 100), 0)% (which is only r f(-(recruitment_med_to_compare - prev_assess_recruitment_med)["2019"], 1) ~billion fish because 2019 was already estimated to be a small year class).

The 2021 recruitment is now estimated to be potentially large, whereas it was estimated to be below average in last year's assessment (for which the only information was the proportions of age-1 fish caught in the 2022 commercial fishery). The 95% credible interval in the r end_yr - 1 assessment was r f(last_yr_base_model$mcmccalcs$rlower["2021"], 2)-- r f(last_yr_base_model$mcmccalcs$rupper["2021"], 2)~billion fish), expanding in the current assessment to r f(base_model$mcmccalcs$rlower["2021"], 1)-- r f(base_model$mcmccalcs$rupper["2021"], 1)~billion fish). Consequently, the median has increased by r f(((((recruitment_med_to_compare - prev_assess_recruitment_med)["2021"]) / (prev_assess_recruitment_med)["2021"]) * 100), 0)% (r f(((recruitment_med_to_compare - prev_assess_recruitment_med)["2021"]), 1) ~billion fish). The general notion remains that recent recruitment is highly uncertain, and estimates for recent years (based on limited data) can change substantially.

r sp have low average recruitment with occasional large year classes (Figures~\@ref(fig:main-recruitment-fig) and \@ref(fig:main-recruitment-devs-fig), Tables~\@ref(tab:main-median-posterior-tab) and \@ref(tab:main-ci-posterior-tab)). Very large year classes in 1980, 1984, and 1999 supported much of the commercial catch from the 1980s to the mid-2000s. From 2000 to 2007, estimated recruitment was at some of the lowest values in the time series, but this was followed by an above average 2008 year class and a very strong 2010 year class. Above average year classes occurred in 2014 and 2016, which have been sustaining the fishery in recent years (Figure~\@ref(fig:main-age-comp-fits-fishery-fig)).

The current assessment estimates a strong 2014 year class (Figure~\@ref(fig:main-numbers-at-age-fig)) comprising r top_coh(base_model, 2016, ret_cohort = 2014)% of the 2016 catch, r top_coh(base_model, 2017, ret_cohort = 2014)% of the 2017 catch, r top_coh(base_model, 2018, ret_cohort = 2014)% of the 2018 catch, r top_coh(base_model, 2019, ret_cohort = 2014)% of the 2019 catch, r top_coh(base_model, 2020, ret_cohort = 2014)% of the 2020 catch, r top_coh(base_model, 2021, ret_cohort = 2014)% of the 2021 catch, r top_coh(base_model, 2022, ret_cohort = 2014)% of the 2022 catch, and r top_coh(base_model, 2023, ret_cohort = 2014)% of the 2023 catch.

The 2016 cohort also appears to be strong, comprising r top_coh(base_model, 2018, ret_cohort = 2016)% of the 2018 catch, r top_coh(base_model, 2019, ret_cohort = 2016)% of the 2019 catch, r top_coh(base_model, 2020, ret_cohort = 2016)% of the 2020 catch, r top_coh(base_model, 2021, ret_cohort = 2016)% of the 2021 catch, r top_coh(base_model, 2022, ret_cohort = 2016)% of the 2022 catch, and r top_coh(base_model, 2023, ret_cohort = 2016)% of the 2023 catch.

The large size of the 2014 and 2016 cohorts is informed by observations from several years of fishery data and the acoustic survey. For all other years from 2011 to 2019, the model currently estimates small year classes (median recruitment below the mean of all median recruitments). As noted above, the 2020 cohort is estimated to be somewhat smaller than in last year's assessment (though last year's estimate was highly uncertain), due to the introduction of new information from the 2023 age-2$+$ biomass index and survey age-composition data (Figure~\@ref(fig:main-bridge-summary-1-fig)). The 2021 cohort strength is only informed by fishery data and the 2023 biomass survey, and is estimated to be potentially large with a median and 95% credible interval of r recruitment_med_in_2021 (r recruitment_lower_in_2021-- r recruitment_upper_in_2021)~billion fish.

The 2022 cohort was observed by the age-1 index in 2023, suggesting it is average to below average (Figures~\@ref(fig:main-survey-age1-fig) and~\@ref(fig:main-recruitment-fig)), and it will not be observed as part of the age-2$+$ survey index until 2025. There is no information in the data to estimate the sizes of the r end_yr-1 and r end_yr year classes. Retrospective analyses of year class strength for young fish have shown the estimates of recent recruitment to be unreliable prior to at least model age-3 (observed at age-2 the previous year) without a survey in the most recent year and age-2 (observed at age-1) with a survey.

From Figure~\@ref(fig:main-recruitment-fig) it looks as though the 2014 recruitment could be as large as the 2010 recruitment. However, the assessment model estimates a r prob_percent_2014_rec_gt_2010_rec% chance that this could be the case. The overlapping of the credible intervals in Figure~\@ref(fig:main-recruitment-fig) is because large MCMC estimates of 2010 recruitment are associated with large estimates of 2014 recruitment (presumably with large estimates of r r_0). By scaling all recruitments by the 2010 recruitment, Figure~\@ref(fig:main-recruitment-scale-all-fig) provides an intuitive way to compare recruitment across years [see @JTC2022 for motivation and methods]. It shows that only the 1980 recruitment is probably larger than 2010 (median relative values $>1$), and the 1984 recruitment has a small chance of being as large as 2010. Whereas Figure~\@ref(fig:main-recruitment-fig) suggests that 1967, 1973, 1977, 1999, 2014, and 2020 could also possibly be larger than in 2010, giving an over-optimistic impression of how often we can expect cohorts the size of the 2010 cohort to occur. The 2021 cohort is still very uncertain but has a small chance of exceeding the 2010 cohort (Figure~\@ref(fig:main-recruitment-scale-all-fig)). Participants in the r sp process have an intuition that the 2010 is a very large recruitment event -- Figure~\@ref(fig:main-recruitment-scale-all-fig) shows how it is the largest in the last 30~years, and that such large cohorts are rarer than is inferred from Figure~\@ref(fig:main-recruitment-fig).

The estimated recruitments with uncertainty for each year and the overall stock--recruitment relationship are provided in Figure~\@ref(fig:main-stock-recruitment-fig). Extremely large variability about the expectation and about the joint uncertainty of individual recruitment and female spawning biomass pairs are evident. High and low recruitments have been produced throughout the range of observed female spawning biomass (Figure~\@ref(fig:main-stock-recruitment-fig)). The standard deviation of the time series of median recruitment deviation estimates for the years 1970--r last_data_yr - 1, which are informed by the age compositions and the age-1 index, is r sd_med_recr_dev_estimates.

Exploitation status

The median estimated relative fishing intensity on the stock is below 1.0 r median_intensity_above_1_text (Figure~\@ref(fig:main-fishing-intensity-fig) and Tables~\@ref(tab:main-median-posterior-tab) and \@ref(tab:main-ci-posterior-tab)). It was closest to 1.0 in 1999 and 2008, but catch in 2008 did not exceed the catch limit that was specified, based on the best available science and harvest control rules in place at the time; however, catch did exceed the catch limit in 1999 (Table~\@ref(tab:main-landings-tac-tab)). Exploitation fraction (catch divided by biomass of fish of age-2 and above) has shown relatively similar patterns (Figure~\@ref(fig:main-exploitation-fraction-fig) and Tables~\@ref(tab:main-median-posterior-tab) and \@ref(tab:main-ci-posterior-tab)). Although displaying similar patterns, the exploitation fraction does not necessarily correspond to fishing intensity because fishing intensity more directly accounts for the age-structure of both the population and the catch. Median relative fishing intensity is estimated to have declined from r median_intensity_2010% in 2010 to r median_intensity_2015% in 2015, and then leveled off around 73--80% from 2016 to 2019 before declining to r median_intensity_2023% in 2023. The median exploitation fraction has increased from a recent low of r exploitation_med_2012 in 2012 to r exploitation_med_2021 in 2021 then declined to r exploitation_med_2023 in 2023. Although there is a considerable amount of imprecision around these recent estimates due to uncertainty in recruitment and spawning biomass, the 95% posterior credibility interval of relative fishing intensity was below 100% from 2012--2016 and again from 2020--2023 (Figure~\@ref(fig:main-fishing-intensity-fig)).

Management performance

Over the last decade (r end_yr - 10--r end_yr - 1), the mean coast-wide utilization rate (i.e., proportion of catch target removed) has been r tot_last_10_yrs_attainment% and catches have been below coast-wide targets (Table~\@ref(tab:main-landings-tac-tab)). From r end_yr - 5 to r end_yr - 1, the mean utilization rates differed between the United States (r us_last_5_yrs_attainment%) and Canada (r can_last_5_yrs_attainment%), though Canada's rate was higher than the U.S.'s in 2020. In 2015, the utilization rate for the coast-wide fishery was the lowest of the previous decade (r tot_2015_attainment%) due, in part, to difficulties locating aggregations of fish and possibly economic reasons. Before 2015, the under-utilization in the United States was mostly a result of unrealized catch in the tribal apportionment, while reports from stakeholders in Canada suggested that hake were less aggregated in Canada and availability had declined. In 2016, the utilization rate increased but remained below pre-2015 levels, despite the total 2016 catch being one of the highest of the preceding years. This is in large part due to increasing catch targets as biomass continues to increase. While the total utilization rate between 2017--2021 was relatively steady, it decreased to r tot_penult_yr_attainment% in r last_data_yr - 1 and to r tot_last_yr_attainment% in r last_data_yr. This is due to the utilization rate in Canada steadily declining since 2020 to a time-series low of 16.5% in 2023, and also a fall in the U.S. utilization rate to 59.7% in 2023. Country-specific quotas (or catch targets) in 2020 and 2021 were specified unilaterally, due to the lack of an agreement on coast-wide 2020 and 2021 TACs. The usual r us_allotment_percent% and r can_allotment_percent% allocation of coast-wide TAC, as specified in the Joint U.S.-Canada Agreement for r sp, was once again implemented in 2022 and 2023. Total landings last exceeded the coast-wide quota in 2002 when utilization was 112%.

As noted above, the median relative fishing intensity was below 100% (i.e. median fishing intensity below r fspr_40) in all years. The median relative spawning biomass was above 40% (the r b_40 reference point) in all years except 2007--2011 (Table~\@ref(tab:main-median-posterior-tab) and Figure~\@ref(fig:main-relative-biomass-fig)). These are also shown on a phase plot of the joint history of relative spawning biomass and relative fishing intensity (Figure~\@ref(fig:main-phase-fig)). Relative spawning biomass increased from the lows in 2007--2012 with above average recruitment in 2008, 2010, 2014, 2016, and 2020. Correspondingly, median relative fishing intensity has remained below 100%, and total catch has been declining since the time series high in 2017. While there is large uncertainty in the r end_yr-1 estimates of relative fishing intensity and relative spawning biomass, the model estimates a r joint_percent_prob_above_below% joint probability of being both above the r fspr_40 fishing intensity in r end_yr - 1 and below the r b_40 spawning biomass level at the start of r end_yr.

Model uncertainty {#sec:assessment-model-uncertainty}

The base assessment model integrates over the substantial uncertainty associated with several important model parameters including: biomass index and age-1 index catchabilities (q~b~ and q~1~, respectively), the magnitude of the stock (via the r r_0 parameter for equilibrium recruitment), productivity of the stock (via the steepness parameter, h, of the stock--recruitment relationship), the rate of natural mortality (M), annual selectivity for key ages, recruitment deviations, and survey and fishery data weights (via the Dirichlet-multinomial parameters $\theta_{\text{fish}}$ and $\theta_{\text{surv}}$).

The medians of the key parameters from the posterior distribution are generally similar to those in last year's base model (Table~\@ref(tab:main-parameter-estimates-tab)). The largest change was a reduction of the 2020 recruitment by more than half, as discussed above, leading to a fall in the estimated median relative spawning biomass at the start of 2023. Medians of the 2014 and 2016 recruitment also both decreased, by about 10% from those estimated in the r assess_yr - 1 assessment.

The r sp stock displays a very high degree of recruitment variability, perhaps the largest of any west coast groundfish stock, resulting in large and rapid biomass changes. This volatility, coupled with a dynamic fishery that potentially targets strong cohorts (resulting in time-varying selectivity) will in most circumstances continue to result in highly uncertain estimates of current stock status and even less-certain projections of the stock trajectory. This is particularly true for female spawning biomass estimates in r assess_yr and throughout the current forecast period, because there is considerable uncertainty associated with the absolute size of the, now mostly mature, 2020 and 2021 year classes that propagates into forecasts. Although the 2023 acoustic survey helped to refine these estimates and reduce uncertainty, further observations of these year classes will improve estimates. The inclusion of the age-1 index in this assessment will, in some cases, also help to reduce this uncertainty (as it currently does in this case; see Figure~\@ref(fig:main-recruitment-base-alternative-2-fig) discussed later). However, further work is needed to improve upon the characterization of uncertainty in the age-1 index itself, which is based on a time invariant assumption about index observation error and catchability.

Uncertainty measures in the base model underestimate the total uncertainty in the current stock status and projections, because they do not account for alternative structural models for hake population dynamics and fishery processes (e.g., recruitment, selectivity, or spatial fleet or population structure), the effects of alternative data-weighting choices, survey catchability, and the scientific basis for prior probability distributions. To address structural uncertainties, the JTC investigated a range of alternative models, and we present the key sensitivity analyses along with other informative sensitivity analyses using full MCMC results (Section~\@ref(sec:assessment-sensitivity-analyses)).

The JTC continues to be committed to advancing MSE analyses, by coordinating research with the r sp MSE Working Group and other scientists in the region engaged in similar research. Incorporating feedback from the Working Group and stakeholders will ensure that operating models will be able to provide insight into the important questions defined by interested parties. Specifically, the development of MSE tools will evaluate major sources of uncertainty relating to data, model structure and the harvest policy for this fishery, and will compare potential methods to address them. In the coming years, this will include a host of research evaluations (see Section~\@ref(sec:assessment-response-review) and Section~\@ref(sec:research)), including evaluating the utility of incorporating environmentally-driven age-0 recruitment indices into the stock assessment.

Reference points {#sec:assessment-reference-points}

The term 'reference points' is used throughout this document to describe common conceptual summary metrics. The Agreement specifically identifies r fspr_40 as the default harvest rate and r b_40 as a point where the 40:10 TAC adjustment is triggered (see the Glossary in Appendix~\@ref(sec:app-glossary)).

We report estimates of the base reference points (e.g., r fspr_40, r b_40, r b_msy, and MSY) with posterior credibility intervals in Table~\@ref(tab:main-reference-points-tab). The median of the female spawning biomass at r fspr_40 (namely the median of r bspr_40) and the median yield at r fspr_40 have remained about the same as estimates in the r last_assess_yr assessment (Table~\@ref(tab:main-parameter-estimates-tab)).

As part of the DFO Sustainable Fisheries Framework, @DFO2009 defined a limit reference point as being a biomass below which serious harm is believed to be occurring to the stock, and an upper stock reference point above which the stock is considered to be healthy. These would equate to the Agreement reference points of r b_10 and r b_40 (the female spawning biomass being 10% and 40%, respectively, of the unfished equilibrium female spawning biomass). The probabilities of the female spawning biomass at the start of r assess_yr being above each of these points are P(B~r assess_yr~ > r b_10) = r probs_curr_b10% and P(B~r assess_yr~ > r b_40) = r probs_curr_b40% such that the stock is estimated to be in the 'healthy zone' (above the upper stock reference point of r b_40). This probability is slightly higher than in last year's assessment, where the equivalent calculation was P(B~r assess_yr - 1~ > r b_40) = 98.1%. Note that a probability of '100%' (or '0%') is based on the MCMC results, and is not meant to imply that something definitely occurs (or definitely does not occur).

With respect to DFO's provisional limit reference point of 0.4r b_msy and provisional upper stock reference point of 0.8r b_msy, the probabilities are P(B~r assess_yr~ > 0.4r b_msy) = r f(dfo_probs_curr[1, 2] |> pull(), 0)% and P(B~r assess_yr~ > 0.8r b_msy) = r f(dfo_probs_curr[1, 3] |> pull(), 0)% such that the stock is estimated to be in the provisional healthy zone'. For completeness, we note that P(*B*~r assess_yr~ >r b_msy) =r f(dfo_probs_curr[1, 1] |> pull(), 1)`%.

Reference levels of stock status that are used by the U.S. Pacific Fisheries Management Council (PFMC) for r sp include r b_40 and a Minimum Stock Size Threshold (MSST) of r b_25. For r min(forecast_yrs), the estimated posterior median relative spawning biomass is r curr_depl_median%, such that the female spawning biomass is well above r b_40 and r b_25. The probability that female spawning biomass at the beginning of r assess_yr is above r b_40 is P(B~r assess_yr~ > r b_40) = r probs_curr_b40% (as noted above), and of being above r b_25 is P(B~r assess_yr~ > r b_25) = r probs_curr_b25%.

Model projections {#sec:assessment-model-projections}

The catch limit for r min(forecast_yrs) based on the default r f_40_40_10 harvest policy has a median of r ct_limit_quantiles["median"]~t and a wide range of uncertainty (Figure~\@ref(fig:main-projected-catch-density-fig)), with the 95% credibility interval being r ct_limit_quantiles["lower"]-- r ct_limit_quantiles["upper"]~t.

Decision tables give projected population status (relative spawning biomass and relative fishing intensity) under different catch alternatives for the base model (Tables~\@ref(tab:main-decisions-biomass-tab) and \@ref(tab:main-decisions-spr-tab)). The tables are organized such that the projected outcome for each potential catch level and year (each row) can be evaluated across the quantiles (columns) of the posterior distribution. Table~\@ref(tab:main-decisions-biomass-tab) shows projected relative spawning biomass outcomes, and Table~\@ref(tab:main-decisions-spr-tab) shows projected fishing intensity outcomes relative to r fspr_40.

Population dynamics and governing parameters assumed during the forecast period include random recruitment; selectivity, weight-at-age and maturity (and thus fecundity) averaged over the five most recent years (r assess_yr - 5-- r assess_yr - 1); and all estimated parameters constant (at their estimates for each particular MCMC sample).

Relative fishing intensity exceeding 1 (or 100% when shown as a percentage) indicates fishing in excess of the r fspr_40 default harvest rate limit. A slight exceedance can happen for the median relative fishing intensity in r forecast_yrs[1:(length(forecast_yrs) - 2)] and r forecast_yrs[(length(forecast_yrs) - 1)] because the r fspr_40 default harvest-rate catch limit is calculated using baseline selectivity-at-age (1966--1990; prior to time-varying deviations), whereas the forecasted catches under the default harvest-rate are removed using selectivity averaged over the last five years. Recent changes in selectivity could be reflected in the projection of slight over- or under-fishing relative to the desired r fspr_40 rate.

Key management metrics are presented for r forecast_yrs[2:(length(forecast_yrs) - 1)] and r forecast_yrs[length(forecast_yrs)] projections (Tables~\@ref(tab:main-risk-year-1-tab)-- \@ref(tab:main-risk-year-3-tab) and Figures~\@ref(fig:main-risk-year-1-fig)-- \@ref(fig:main-risk-year-3-fig)). These metrics summarize the probability of various outcomes from the base model given each potential management action. Although not linear, probabilities can be interpolated from this table for intermediate catch values in r assess_yr (Table~\@ref(tab:main-risk-year-1-tab) and Figure~\@ref(fig:main-risk-year-1-fig)). However, interpolation may not be applicable for all catches in r assess_yr + 1 and r assess_yr + 2 because they are conditional on catch levels from the previous year or years. This explains why a few probabilities decline (rather than rise) with increased r assess_yr + 1 and r assess_yr + 2 catch levels in Tables~\@ref(tab:main-risk-year-2-tab) and~\@ref(tab:main-risk-year-3-tab) and Figures~\@ref(fig:main-risk-year-2-fig) and \@ref(fig:main-risk-year-3-fig).

Figure~\@ref(fig:main-depletion-comparison-fig) shows the projected relative spawning biomass trajectory through r forecast_yrs[length(forecast_yrs)] for several of these management actions. With zero catch for the next three years, the biomass has a r zero_catch_prob_bio_down_1% probability of decreasing from r forecast_yrs[1] to r forecast_yrs[2] (Table~\@ref(tab:main-risk-year-1-tab) and Figure~\@ref(fig:main-risk-year-1-fig)), a r zero_catch_prob_bio_down_2% probability of decreasing from r forecast_yrs[2] to r forecast_yrs[3] (Table~\@ref(tab:main-risk-year-2-tab) and Figure~\@ref(fig:main-risk-year-2-fig)), and a r zero_catch_prob_bio_down_3% probability of decreasing from r forecast_yrs[3] to r forecast_yrs[4] (Table~\@ref(tab:main-risk-year-3-tab) and Figure~\@ref(fig:main-risk-year-3-fig)).

The probability of the female spawning biomass decreasing from r end_yr to r end_yr + 1 is greater or equal to r prob_all_catch_greater% for all catch levels examined other than zero (Table~\@ref(tab:main-risk-year-1-tab) and Figure~\@ref(fig:main-risk-year-1-fig)). The probability is r prob_decl_last_yr_catch% for the r end_yr catch level similar to that for r end_yr - 1 (catch alternative~r letters[ct_actual_ind]). For all explored catches, the maximum probability of the female spawning biomass dropping below r b_10 at the start of r end_yr + 1 is r prob_max_below_10_yr1%, at the start of r end_yr + 2 is r prob_max_below_10_yr2%, and at the start of r end_yr + 3 is r prob_max_below_10_yr3% (Tables~\@ref(tab:main-risk-year-1-tab)--\@ref(tab:main-risk-year-3-tab) and Figures~\@ref(fig:main-risk-year-1-fig)-- \@ref(fig:main-risk-year-3-fig)). The similar maximum probability of dropping below r b_40 at the start of r end_yr + 1 is r prob_max_below_40_yr1%, at the start of r end_yr + 2 is r prob_max_below_40_yr2%, and at the start of r end_yr + 3 is r prob_max_below_40_yr3%.

It should be noted that forecasted biomass is not only influenced by catch levels. As the above average 2014 and 2016 cohorts continue to age, total biomass of these cohorts even without fishing mortality will continue to decrease (Tables~\@ref(tab:main-est-biomass-at-age-tab) and \@ref(tab:main-cohort-effects-tab)) as losses from mortality outweigh increases from growth. The above-average 2020 cohort entered this growth-mortality transition period around 2023 (Tables~\@ref(tab:main-est-biomass-at-age-tab) and \@ref(tab:main-cohort-effects-tab)). During 2024, the age-3 2021 cohort will likely begin the growth-mortality transition where a net increase in total biomass is less likely (note that fecundity will increase which will influence the exact change in female spawning biomass, Figure~\@ref(fig:main-maturity-ogive-fig)). The estimated above-average (yet still highly uncertain) 2020 and 2021 cohorts will continue to play a large role in determining female spawning biomass during the forecast years presented here. The below-average 2015, 2018, and 2019 cohorts will contribute much less to forecasted spawning biomass than the larger cohorts.

The age composition (in numbers) of the catch in r forecast_yrs[1] is projected to be (using MCMC medians) r fore_catch_prop[["3"]]% age-3 fish from the r forecast_yrs[1] - 3 cohort, r fore_catch_prop[["4"]]% age-4 fish from the r forecast_yrs[1] - 4 cohort, r fore_catch_prop[["8"]]% age-8 fish from the r forecast_yrs[1] - 8 cohort, r fore_catch_prop[["2"]]% age-2 fish from the r forecast_yrs[1] - 2 cohort, and r fore_catch_prop[["10"]]% age-10 fish from the large r forecast_yrs[1] - 10 cohort (Figure~\@ref(fig:main-age-comp-forecast-fig)). However, those estimates are highly uncertain with the 95% credibility interval for the age-3 fraction spanning r f(fore_catch_prop_age3_lower)%-- r f(fore_catch_prop_age3_upper)%.

Due to the higher average weight of older fish compared to younger fish, the median expected proportion of the r forecast_yrs[1] catch by weight is r f(fore_catch_prop_wt_age3_median)% for the age-3 r forecast_yrs[1] - 3 cohort (compared to r fore_catch_prop[["3"]]% by numbers) and r f(fore_catch_prop_wt_age4_median)% for the age-4 r forecast_yrs[1] - 4 cohort (compared to r fore_catch_prop[["4"]]% by numbers; Figure~\@ref(fig:main-age-comp-forecast-fig)).

With respect to the DFO reference points, with the largest r assess_yr catch of r largest_next_catch~t given in Table~\@ref(tab:main-risk-year-1-tab), at the start of r assess_yr + 1 the stock is expected to be above the critical zone with a probability of P(B~r assess_yr + 1~ > r b_10) = r prob_next_over_b10% and in the healthy zone with a probability of P(B~r assess_yr + 1~ > r b_40) = r prob_next_over_b40%. With respect to the DFO provisional reference points (based on r b_msy), the stock is expected to be above the provisional critical zone with a probability of P(B~r assess_yr + 1~ > 0.4r b_msy) = r dfo_prob_next_over_40bmsy%, in the healthy zone with a probability of P(B~r assess_yr + 1~ > 0.8r b_msy) = r dfo_prob_next_over_80bmsy%, and above r b_msy with a probability of P(B~r assess_yr + 1~ > r b_msy) = r dfo_prob_next_over_bmsy% for this catch.

With respect to PFMC stock size reference points, a level of r assess_yr catch consistent with the Agreement default harvest control rule (r next_treaty_catch~t) has a r pfmc_prob_next_yr_below_b40% estimated probability of resulting in the biomass going below r b_40 at the start of r assess_yr + 1 (and r pfmc_prob_next_yr_below_b25% probability of going below r b_25; Table~\@ref(tab:main-risk-year-1-tab)). If catches in r assess_yr and r assess_yr + 1 are the same as in r assess_yr - 1 (r same_catch_as_last_yr~t, catch scenario~r letters[ct_actual_ind]) then the probability of the biomass going below r b_40 is r same_catch_prob_next_year_below_b40% for the start of r assess_yr + 1 and r same_catch_prob_yr_after_next_below_b40% for the start of r assess_yr + 2.

Sensitivity analyses {#sec:assessment-sensitivity-analyses}

Sensitivity analyses were conducted to investigate influence of data inputs and structural uncertainty of the base model by investigating how changes to the model affected the estimated values and derived quantities. All sensitivity analyses compared MCMC posteriors with the same number of posterior samples as the base model. Several key underlying structural model assumptions were identified that have persisted across many previous hake assessments, and thus warrant revisiting annually as a set of reference sensitivity examinations to new base models. Many additional sensitivity runs were conducted while developing and testing the r assess_yr base model. Here we focus on the main sensitivities, relative to the base model, which are as follows:

  1. Consideration of higher standard deviations on the prior distribution for natural mortality;

  2. Consideration of an alternative prior distribution (mean and standard deviation) for natural mortality based on the @Hamel2015 and @HamelCope2022 life history meta-analytic method;

  3. Consideration of an alternative prior distribution and a fixed value for steepness, to change the resiliency of the stock;

  4. Consideration of higher and lower variation about the stock--recruitment relationship ($\sigma_r$);

  5. Removal of the age-1 index as a data source;

  6. Downweighting the fishery age-composition data; and

  7. Consideration of alternative standard deviations for time-varying selectivity.

None of the sensitivities resulted in a substantial departure from the main population dynamics of the base model (Tables~\@ref(tab:main-parameter-estimates-sens-1-tab)-- \@ref(tab:main-parameter-estimates-sens-3-tab) and Figures~\@ref(fig:main-biomass-base-alternative-1-fig)-- \@ref(fig:main-index-base-alternative-4-fig)). All sensitivity models showed large estimated increases in female spawning biomass in the early- to mid-2010s that continues to be driven by the 2010, 2014, and 2016 cohorts, followed by a period of general decline (2018--2023) before increasing again due to the above average 2020 and 2021 cohorts. All sensitivity models indicate that r assess_yr relative spawning biomass is above r b_40. The overall scale of the population was impacted by various alternative assumptions, and the highly uncertain size of the recent cohorts were more variable across sensitivity analyses than earlier cohorts that have been observed for more years.

The standard deviation of the prior distribution on natural mortality was increased from the base model value of 0.1 to 0.2, 0.3, and 0.31. Note that the median of the prior was also changed for the latter sensitivity. Each of these sensitivities led to an increase in estimates of natural mortality relative to the base model. The medians of the MCMC posteriors for natural mortality increased from r f(nat_m[2], 3) to r f_nat_m_02[2], r f_nat_m_03[2], and r f_nat_m_hamel[2], respectively. The 95% credibility intervals also increased, with the largest differences in the upper rather than the lower credible interval. Credible intervals were r f(nat_m[1], 3)--r f(nat_m[3], 3) for the base model, r f_nat_m_02[1]--r f_nat_m_02[3] for the sensitivity run with the prior standard deviation set to 0.2, r f_nat_m_03[1]--r f_nat_m_03[3] for the sensitivity run with the prior standard deviation set to 0.3, and r f_nat_m_hamel[1]--r f_nat_m_hamel[3] for the sensitivity run with the @HamelCope2022 prior (Table~\@ref(tab:main-parameter-estimates-sens-1-tab)). In addition to increased estimates of natural mortality, results from these sensitivity models also showed an increase in the overall scale of the population, the estimated stock status relative to r b_0 prior to 1990, the uncertainty in female spawning biomass on both absolute and relative scales, roughly halved estimated relative fishing intensity in r end_yr - 1, and more than doubled equilibrium yield at r bspr_40 (Table~\@ref(tab:main-parameter-estimates-sens-1-tab) and Figures~\@ref(fig:main-biomass-base-alternative-1-fig) and \@ref(fig:main-status-base-alternative-1-fig)).

The mean of the prior distribution on steepness was decreased from 0.777 (base) to 0.5 and, separately, steepness was fixed at 1.0. The decrease in the mean of the prior resulted in a decrease in the MCMC estimate of steepness from a median of r f(steep[2], 3) with a 95% credible interval of r f(steep[1], 3)--r f(steep[3], 3) to a median of r f_steep_prior_05[2] with a 95% credible interval of r f_steep_prior_05[1]--r f_steep_prior_05[3] (Table~\@ref(tab:main-parameter-estimates-sens-1-tab)). However, neither steepness sensitivity analysis had a large impact on the overall model results (Figures~\@ref(fig:main-biomass-base-alternative-1-fig) and \@ref(fig:main-status-base-alternative-1-fig)), because r sp female spawning biomass has remained above levels where changes in steepness would appreciably influence the stock--recruitment relationship (Figure~\@ref(fig:main-stock-recruitment-fig)).

Input values of $\sigma_r$ were changed from r sigma_r (base) to alternative high (r sigma_r_sens[2]) and low (r sigma_r_sens[1]) states. Both sensitivities were similar to the base model in that the calculated standard deviation of recruitment deviations (from the main period) was higher than the input $\sigma_r$, i.e., r sigma_r_lo_main and r sigma_r_hi_main when $\sigma_r$ was r sigma_r_sens[1] and r sigma_r_sens[2], respectively. The calculated standard deviation of recruitment deviations from the base model was intermediate at 1.70. These calculated standard deviations should match the input if the vectors of deviations were from the 'population' of values rather than just a sample. However, this systematic bias to be larger than the input value indicates that the standard deviation of recruitment deviations is accounting for more variability than just variability in recruitment. The high $\sigma_r$ model led to a larger difference between the female spawning biomass at unfished equilibrium and the female spawning biomass at the initial year of the model than the low $\sigma_r$ model (Figure~\@ref(fig:main-biomass-base-alternative-1-fig)). Similar to previous assessments, estimates of unfished equilibrium recruitment and relative spawning biomass are sensitive to $\sigma_r$, whereas absolute estimates of female spawning biomass are relatively insensitive. The method @MethotTaylor2011 proposed to tune $\sigma_r$ was developed in the context of maximum likelihood estimation and not Bayesian inference, where the latter potentially allows for estimating $\sigma_r$ using random effects, and thus, this proposed method is not used here to tune the fixed input value.

The sensitivity of the base model to the removal of the relative age-1 index provides a method to evaluate how the information about juvenile fish is propagated through the model. Estimates of female spawning biomass throughout most of the time series are similar between models with and without the relative age-1 index but diverge near the end of the time series (Table~\@ref(tab:main-parameter-estimates-sens-1-tab), Figures~\@ref(fig:main-biomass-base-alternative-2-fig) and \@ref(fig:main-status-base-alternative-2-fig)). The r end_yr estimates of relative spawning biomass are r f(100 * bratio_curr[2], 1)% for the base model (95% credible interval of r f(100 * bratio_curr[1], 1)-- r f(100 * bratio_curr[3], 1)%) and r f(100 * bratio_age1[2], 1)% for the model where the relative age-1 index is removed (95% credible interval of r f(100 * bratio_age1[1], 1)--r f(100 * bratio_age1[3], 1)%). This difference is due to the relative age-1 index providing additional information on recruitment for cohorts associated with recent age-1 indices (i.e., 2020 and 2022 cohorts detected in the 2021 and 2023 age-1 indices). In particular, the base model with the relative age-1 index indicates slightly larger 2020 and 2021 year classes than the model removing the age-1 index (Figure~\@ref(fig:main-recruitment-base-alternative-2-fig)). Similarly, recruitment in 2022 is estimated to be slightly above average when the model is fit to the relative age-1 index compared to slightly below average without the index. Removing the relative age-1 index led to minor changes in fit to the age-2$+$ survey biomass index, with 2021 showing a slight improvement and 2023 a deterioration compared to the base model (Figure~\@ref(fig:main-index-base-alternative-2-fig)).

The base model includes a Dirichlet-multinomial likelihood component that includes two estimated parameters to automatically weight each of the fishery and survey age compositions. The base model was compared to a model that downweighted the fishery age compositions relative to the survey age compositions. This downweighting was based on the McAllister--Ianelli method, which requires manual iterative adjustments to the input sample sizes using a derived multiplier. The McAllister--Ianelli method, which was used in assessments prior to 2018 (Table~\@ref(tab:main-assessment-changes-tab)), attempts to make the arithmetic mean of the input sample size approximately equal to the harmonic mean of the effective sample size. Here, this was accomplished with weighting factors of 0.14 and 0.46 (ratio of 0.30) for fishery and survey age compositions, respectively. These weighting factors are not specific to this year's base model, rather they are values calculated from previous maximum likelihood estimates. The median estimate from the Dirichlet-multinomial method used in the base model was r f(dm_weight_fishery_median, 3) and r f(dm_weight_survey_median, 3) (ratio of r f(as.numeric(dm_weight_fishery_median) / as.numeric(dm_weight_survey_median), 2)). Downweighting fishery composition data led to minor changes in relative spawning biomass, recruitment estimates, and increased uncertainty in estimates of early recruitments compared to the base model (Figures~\@ref(fig:main-status-base-alternative-2-fig) and \@ref(fig:main-recruitment-base-alternative-2-fig)). The largest changes in the time series occurred prior to the availability of survey data.

The degree of flexibility of annual variation in the fishery selectivity was tested using three alternative values of standard deviations ($\Phi$) (Figures~\@ref(fig:main-biomass-base-alternative-4-fig)-- \@ref(fig:main-index-base-alternative-4-fig)). The consideration of alternative $\Phi$ values is discussed earlier in Section~\@ref(sec:data-variability-selectivity). Changing $\Phi$, which controls the flexibility in time-varying selectivity, from the base model value of $\Phi = 1.40$ to 0.21, 0.70, and 2.10 did not appreciably influence the estimates, or precision, associated with recruitment in 2014 or 2016 but it did impact more recent recruitments (Figure~\@ref(fig:main-recruitment-base-alternative-4-fig)). In particular, recruitment estimates for 2020 and 2021 are linked to the choice of $\Phi$, where the smallest investigated value of $\Phi$ (0.21) led to the highest estimates of the 2020 and 2021 recruitment deviations of the investigated models (Figure~\@ref(fig:main-recruitment-devs-base-alternative-4-fig)). The high estimates of recruitment led to a large increase in female spawning biomass in recent years compared to the base model (Figure~\@ref(fig:main-biomass-base-alternative-4-fig)). When $\Phi = 0.21$, the fit to the most recent age-2$+$ survey biomass index was the worst of the three investigated models (Figure~\@ref(fig:main-index-base-alternative-4-fig)).

Retrospective analyses {#sec:assessment-retrospective-analyses}

Retrospective analyses were performed by iteratively removing the terminal years' data (going back 10 years) and estimating the posterior distribution of parameters under the assumptions of the base model. This year's base model shows similar retrospective results to last year's [Figure~\@ref(fig:main-retrospective-recruitment-fig) and @JTC2022] for the older cohorts. Uncertainties are shown for select cohorts in Figures~\@ref(fig:main-retrospective-recruitment-fig) and~\@ref(fig:main-retrospective-recruitment-2-fig). In previous years, thes figures showed only the median lines. The uncertainty is represented as credible intervals from r f(probs[1] * 100, 1)% to r f(probs[3] * 100, 1)% as shaded areas surrounding the median lines. For cohorts that have positive recruitment deviations, the uncertainty is a narrower band around the median due to a higher sampling rate over the years than the cohorts with negative recruitments.

The 2020 cohort has been estimated lower this year than in last year's assessment, which is also evident in Figure~\@ref(fig:main-retrospective-recruitment-2-fig) when excluding the 2023 data -- the uncertainty at age-3 gets reduced when including the 2023 data, shown by the age-4 intervals being tighter and also lower. The latest data no longer suggest that 2020 is a huge cohort. The 2021 cohort at age-3 has a similar median to the 2020 cohort at age-3, but with less uncertainty (narrower credibility interval in Figure~\@ref(fig:main-retrospective-recruitment-2-fig)) because it has been seen in the age-2+ biomass survey (whereas the 2020 cohort was not seen in that survey until it was another year older). Although the 2021 displays unusual behaviour in that the median is below 0 at age-2 and then above 0 at age-3, its uncertainty at age-2 was very large (Figure~\@ref(fig:main-retrospective-recruitment-2-fig)).

Some cohort recruitments are over or under-estimated at age-2. Over-estimation can be seen most clearly with the 2014, 2015, 2017, 2018, and 2020 cohorts (Figures~\@ref(fig:main-retrospective-recruitment-fig) and \@ref(fig:main-relative-retrospective-recruitment-fig)). The 2014 cohort reached high deviation after two years, then even higher after three years only to drop back down to a lower value and then stabilize at around age-4 with the addition of more data. A similar pattern can be seen with the smaller 2017 and 2018 cohorts. Even with the addition of new data, the size of the very small 2015 cohort has not fully stabilized. Under-estimation is slight, but apparent, for the 2016 cohort as recruitment estimates have risen by a small amount since the estimate at age-3. The under-estimation of the 2021 cohort stands out as it was estimated as being slightly smaller than average at age-2 in last year's assessment and then estimated to be a very large cohort at age-3 this year, though this is based on medians and the age-2 estimate was highly uncertain, as mentioned above.

Cohort strength is further informed once at least one year of age-2$+$ survey biomass index age-composition data are available for a cohort, which for even-numbered recruitment years typically does not occur until the cohort reaches age-3, due to the acoustic survey occurring in odd years; though the age-1 index does provide some information.

The stability of the recruitment estimates seen in this plot is also evident in the absolute estimates of uncertainty for each cohort. Uncertainty of the 2016--2021 cohorts has been substantially reduced compared to removing five years of data (Figure~\@ref(fig:main-biomass-recruitment-retrospective-fig), bottom figure). The uncertainty of the 2020 cohort was substantially increased with the removal of only 1 year of data. This increase was exacerbated by the removal of the 2023 survey index as well as the fishery catch, as all data sources are removed for each year of the retrospectives. Medians of various quantities of interest are given in Table~\@ref(tab:main-parameter-estimates-retro-tab).

Overall, there is little retrospective change to the relative spawning biomass trajectory up to the mid-2010s, and most retrospective change occurs in the final 5 years of the retrospective model (upper panel of Figure~\@ref(fig:main-biomass-recruitment-retrospective-fig)). In this assessment, there is very little retrospective bias, with only slight year-specific positive and negative bias in female spawning biomass, some minor adjustments to recruitment deviates, and a slight trend in r b_0 as the retrospective year increases. All of these retrospective differences are well within the range of estimation uncertainty across all retrospective years. There is no indication from retrospective evaluations that the base model is displaying a systematic bias.

Comparison with past assessments {#sec:assessment-past-assessments}

A comparison of the base models, approved for management, used in each year since r min(assess_history_df$yr, na.rm = TRUE) indicates that the variability between model results, especially early on in the estimated time series, is larger than the estimated uncertainty reported from the current base model (Figure~\@ref(fig:main-historical-assessments-fig)). There have been substantial differences in the structural assumptions of the models and, thus, results submitted each year. Prior to 2004, catchability was fixed at 1.0. This assumption was investigated between 2004 and 2007, leading to variability in model results because of the use of several different, but fixed, values of catchability. Since 2008, catchability has been freely estimated by the model ($q_b$ = r f(median(base_model$extra_mcmc$q_vector), 2)). The fixing of survey catchability had the effect of driving the estimate of initial biomass upward, which in turn scaled the entire biomass trajectory up, leading to higher estimates of relative spawning biomass than in more recent assessments. The median estimates of female spawning biomass for 2016 and 2017 have remained similar to the previous assessment, being somewhat lower than in the 2016 and 2017 assessments. In addition to more information about the 2014 and 2016 cohorts, the 2018 assessment model also included a change in the data weighting method, an update to maturity and fecundity, and a change to selectivity parameterization (Table~\@ref(tab:main-assessment-changes-tab)). The uncertainty interval associated with the r end_yr assessment brackets the majority of the historical estimates.

The level of uncertainty associated with each assessment's estimate of that year's current female spawning biomass (i.e., that used to convey current stock status and inform management advice) changes from assessment to assessment given updates in data and r sp population structure and dynamics. Uncertainty around the absolute amount of r assess_yr female spawning biomass is similar to the final-year estimates from previous assessments, with both absolute interquartile range and the relative amount of dispersion (or variability relative to the stock size; similar to a coefficient of variation) consistent with previous assessments (Figure~\@ref(fig:main-historical-dispersion-fig)).

Performance of past projections {#sec:assessment-past-projections}

Without rigorous simulation experiments it can be difficult to operationally assess the accuracy of projections in stock assessments because the truth is never known with 100% certainty. For r sp, hindsight comparisons have been conducted since 2021 [@JTC2021] to evaluate performance of projections provided in decision tables (such as Tables~\@ref(tab:main-risk-year-1-tab) and~\@ref(tab:main-risk-year-2-tab)) of past assessments relative to updated assessments. Overall, results indicate that assessment model projections give a relatively good idea of general projected trends and status.

As an example, the 2019 assessment [@JTC2019] gave the estimated probability of the female spawning biomass declining in the subsequent year, i.e., P(B~2020~ < B~2019~), for several possible catches in 2019, such as 0~t, 180,000~t, 350,000~t, 410,000~t etc. Now that we 'know' the catch in 2019 was 412,015~t, we can select the 410,000~t row (close enough to 412,015~t) in the table from the 2019 assessment to give that assessment's P(B~2020~<B~2019~) = r prob_decline_from_2019_to_2020_historic%; Figure~\@ref(fig:main-historical-1-fig). We can also calculate this probability from the current assessment model, which implicitly includes the 412,015~t catch from 2019, giving P(B~2020~ < B~2019~) = r prob_decline_from_2019_to_2020_curr%; Figure~\@ref(fig:main-historical-1-fig). We extracted similar probabilities from past assessment documents going back to 2012 and calculate analogous probabilities, P(B~t+1~ < B~t~), from the current base model [Figure~\@ref(fig:main-historical-1-fig); see @JTC2022 for full methods].

Each assessment correctly predicted whether the stock would most likely increase or decrease the following year, except for 2017 and 2023; Figure~\@ref(fig:main-historical-1-fig). Estimates from previous assessments are almost always closer to 50% than those from the current base model (Figure~\@ref(fig:main-historical-1-fig)), because the current assessment model has more information and thus provides a more definitive probability (closer to 0% or to 100%) than year $t$'s assessment model. It is desirable that the probabilities from the assessment documents are not too definitive (too close to 0% or to 100%), because they are admitting a wide range of uncertainty given unknown recent recruitments.

The 2017 and 2023 assessments incorrectly' projected that the stock would likely decline the following year (given the catch that subsequently occurred), because the current assessment model estimates a likely increase (Figure~\@ref(fig:main-historical-1-fig)). For the 2017 [@JTC2017] assessment the biomass trend was projected to be relatively flat the following year, so even slight changes in biomass could influence the binomial outcome of an 'increase' or 'decrease' in biomass, despite the overall change in biomass not being very substantial. The 2023 assessment [@JTC2023] had minimal information on the 2021 cohort and predicted the biomass would probably decline in 2024 with any non-zero 2023 catch. However, the current assessment estimates that the 2021 cohort was potentially large, which further highlights how impactful a realized large deviation from average recruitment (rather than assuming average recruitment) can be on forecasted outcomes. Similarly, the 2012 assessment had no information on the very large 2010 recruitment, and so also over-estimated the probability of decline the following year (Figure~\@ref(fig:main-historical-1-fig)). A range of catch alternatives are shown for the current assessment because realizedr assess_yrcatches are not yet known (Figure~\@ref(fig:main-historical-1-fig)), and give a mostly greater that 50% chance that the stock will decline fromr assess_yrtor assess_yr + 1`.

A similar approach was used to calculate the probability of the biomass falling below r b_40 in the subsequent year, i.e., P(B~t+1~ < r b_40); Figure~\@ref(fig:main-historical-2-fig). The 2012 assessment was the only one that gave a >50% chance of the biomass falling below r b_40 in the subsequent year, but later data determined that the 2010 year class was substantial and so in hindsight the probability of going below r b_40 was 0% (based on the current assessment). From the 2018 assessment onwards, the estimated P(B~t+1~ < r b_40) probabilities rose, until falling due to the incoming above-average 2020 cohort and lower catches (Figure~\@ref(fig:main-historical-2-fig)). The same probabilities calculated from the current base model similarly rose, but all remained lower than the previous assessments' calculations, similar to the analogous figure in the 2023 assessment [@JTC2023].

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pacific-hake/hake-assessment documentation built on July 21, 2024, 8:19 a.m.