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valuation of tag mixing assumptions in western Pacific Oceankipjack tuna stock assessment models
ale Kolodya,b,∗, Simon Hoylea
Oceanic Fisheries Programme, Secretariat of the Pacific Community, BP D5, 98848 Noumea CEDEX, New CaledoniaCSIRO Marine and Atmospheric Research (Wealth from Oceans Flagship), GPO Box 1538, Hobart TAS 7001, Australia
r t i c l e i n f o
rticle history:eceived 3 June 2013eceived in revised form 24 February 2014ccepted 8 May 2014vailable online xxx
eywords:ark-recapture
ag mixingtock assessmentiffusionkipjack
a b s t r a c t
Fish population estimators (for abundance, mortality and movement) from conventional tagging (mark-recapture) programmes require the assumption that tagged individuals are equally vulnerable torecapture as the (relevant portion of the) untagged population. Most commercial tuna species are dis-tributed over millions of square km, with poorly understood stock structure and movement dynamics,while tags tend to be released in much smaller areas that are selected for logistical reasons. If the tags donot mix sufficiently with the general population this could substantially bias tuna stock assessments. Weoutline two simple quantitative methods for evaluating tag mixing, and apply them to western PacificOcean skipjack tuna (Katsuwonis pelamis). The first approach compares, in a discrete time window, thespatial distributions of tag recoveries from pairs of release events that were separated in space and/ortime. Significant differences between the recovery distributions indicate that tags from the two releaseevents are not fully mixed with each other, such that one or both release events cannot be fully mixedwith the general population either. The second approach tests whether tag density (number of recoveredtags per unit of catch) is spatially homogeneous for an individual release event. Both tests focus on detec-
ting incomplete mixing and neither can prove that full mixing has been achieved. The analyses providestrong evidence that tag mixing assumptions are not being met in recent western Pacific skipjack tunastock assessments. Simple diffusion models were fit to the tag density observations and used to simulatethe potential magnitude of mixing biases using simple estimators for abundance and natural and fishingmortality. Options for reformulating the skipjack assessment are discussed.
Since Petersen (1892), tagging (or mark-recapture) experimentsave been used to make inferences about fish populations, withethodological extensions and applications since developing into
vast literature (e.g. see Schwarz and Seber, 1999, and references
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
herein). In the analysis of most large-scale mark-recapture pro-rammes, there are a number of fundamental assumptions, one ofhich requires that the tagged individuals are equally vulnerable
Abbreviations: CUSTARD, Comparison of Synchronous Tag Recovery Distribu-ions; TART, Tag Ratio Trend analysis; WCPO, Western and Central Pacific Ocean;SAP, Skipjack Survey and Assessment Programme; RTTP, Regional Tuna Taggingrogramme; PTTP, Pacific Tuna Tagging Programme; NNP, nearest neighbour per-utation; PS, Purse Seine.∗ Corresponding author at: CSIRO Marine and Atmospheric Research (Wealth fromceans Flagship), GPO Box 1538, Hobart TAS 7001, Australia. Tel.: +61 3 6232 5121.
to recapture as the untagged population (where the relevant pop-ulation may be a subset of the total population, e.g. a particularcohort). In the case of tuna, this usually translates into the assump-tion that the tagged fish have had an opportunity to fully mix withthe untagged population before recapture. The major commercialtuna populations are distributed over millions of square km, withpoorly understood sub-population structure and movement pat-terns. However tuna stock assessment models are either spatiallyaggregated, or disaggregated at a relatively low spatial resolution,such that it is rarely clear whether tag mixing assumptions arevalid. If individual fish are not very mobile relative to the popu-lation range, and tags are released in a restricted sub-region, thentag-based abundance estimates will be biased low (or high), if thefishery happens to operate near (or far) from the release location.
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
Thus violations of the tag mixing assumption can cause substantialestimator biases and have an important influence on the scientificadvice provided to fisheries managers. We note that there are anumber of other assumptions that underpin tag-based estimators
hich have been given more consideration in the literature, andhich are not addressed here (e.g. tag reporting rates, tag-relatedortality, tag shedding).Most modern tuna stock assessments are conducted with
n integrated statistical modelling framework that attempts tostimate population and management-related parameters byimultaneously fitting to observations of fisheries catch, size/ageomposition, catch per unit effort relative abundance indices,nd tag recoveries if available (e.g. Fournier and Archibald, 1982;aunder and Punt, 2013). Within these integrated models, the tag
ynamics are usually represented as a form of attrition model, sim-lar to tag analyses that can be conducted independently of thether data sources (e.g. Ricker, 1975; Seber, 1982) and subject to theame assumptions, but with the potential advantage that param-ters are shared with, and informed by, other components of theodel. Tuna stock assessment scientists invoke a number of argu-ents to justify model formulation decisions in relation to the tagixing assumption, including:
Individual tagged fish are often observed moving large distancesover short time periods. Some rapid movements are essential forrapid mixing over large areas, but it is the overall distribution ofthe tags that is important, and a few wide-scale dispersals mightbe misleading.Spatial disaggregation in assessment models (e.g. Goethel et al.,2011) is often used to partition the tuna population and fisheryinto smaller units, each of which should generally be more homo-geneous than the overall population. At a small enough scale, theprobability of capturing a tagged fish will be essentially propor-tional to the proportion of tagged fish in the region. However,tuna model spatial sub-units are usually large, often defined onthe basis of convenient management units and constrained by therequirement for computationally tractable estimators. Tag mix-ing assumptions are not usually explicitly considered as part ofthe formulation process.There is typically an explicit mixing period of 1–4 quarters, dur-ing which time the influence of the tags on parameter estimationis reduced (but not eliminated). Sensitivity analyses may be usedto illustrate the effect of different mixing period assumptions tothe assessment results, or statistical tests can be applied to com-pare model fits with different mixing period assumptions (e.g.Hoenig et al., 1998). However, these approaches do not directlyconfront the possibility that none of the candidate assumptionsunder consideration might be appropriate.
There is clearly a need to better understand the interactionetween tag mixing dynamics, the structural assumptions in tunatock assessment models, and the implications for estimator biases.
In this paper, we develop two methods for evaluating the tagixing assumption. Both involve examining temporal trends in tag
ecovery distributions at a spatial resolution that is much higherhan what is typically assumed in assessment models. The Compari-on of Synchronous Tag Recovery Distributions (CuSTaRD) analysislosely resembles the approach described in Latour et al. (2001).hey used a chi-square test to compare the spatial distributionf tag recoveries from consecutive striped bass (Morone saxitilis)eleases, concluding that tag recoveries in the year of releaseppeared to have a similar distribution to tag recoveries one orore years previously. Our CUSTARD approach differs qualitatively
rom Latour et al. (2001) in two respects: (i) we compare recover-es from tag release events separated in space as well as time, andii) we used a nearest neighbour permutation (NNP) test to com-
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
are distributions. Our second approach, the Tag Ratio Trend (TaRT)nalysis, evaluates whether tag density (number of tags per catch)s spatially homogeneous. Langley and Million (2012) used a simi-ar approach to help select an appropriate mixing period for Indian
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Ocean yellowfin tuna (Thunnus albacares) stock assessment. Theynoted that their argument was primarily qualitative, and wouldhave benefited from formal quantitative methods. We attempt toaddress this need by explicitly testing for gradients in tag densityrelative to the release point. The CUSTARD and TART approaches arequantitative in the sense that they can use formal statistical testsof the hypothesis that tags are mixed (with rejection of the nullhypothesis indicating incomplete tag mixing). However, we alsoprovide graphical summaries of the tag recovery distributions, andanalytical extensions based on parametric diffusion models, whichmay be more valuable than the statistical tests. The methods areapplied to the Western and Central Pacific Ocean (WCPO) skipjacktuna population (Katsuwonis pelamis) using the spatial/temporalstructure of the most recent stock assessment (Hoyle et al., 2011).The analyses demonstrate that the recent WCPO skipjack assess-ment models are not meeting the tag mixing assumptions andsimulations based on simple diffusion models suggest that largeestimator biases may result. Options for improving the assessmentmodel structure are discussed.
1.1. Tag mixing and movement definitions
It is useful to clarify some of the terminology that we use in rela-tion to tag mixing and movement, as some of these terms and ideasare not always used consistently in the fisheries literature. Stockassessment models tend to be parameterized in terms of bulk trans-fer coefficients that describe instantaneous fish movement amongdiscrete regions (e.g. Goethel et al., 2011). The original theory forthis approach (e.g. Beverton and Holt, 1957 and references therein)was derived from physical models (e.g. for gas movement, heattransfer, etc.), in which continuous differential equations describediffusion and advection/convection. These deterministic equationsprovide a statistical description of the large number of interactionstaking place at the molecular scale. It is assumed that these samemodels can also provide a useful approximation for the statisti-cal characteristics of large numbers of fish movements. When thecontinuous models are discretized into difference models (com-monly used in fisheries stock assessment) with a finite number ofregions, it is assumed that each region is internally homogeneous(the difference approximation more closely resembles the contin-uous equations with higher spatial and temporal resolution). Inthe context of this paper, we refer to movement as fish displace-ment by any means, within or among stock assessment regions.We refer to mixing and diffusion as a random form of movementthat changes the spatial relationship among individuals over time(e.g. by physical analogy, laminar flow can cause movement of gasparticles without changing the relative spatial pattern among par-ticles, while turbulent flow causes mixing by changing the relativespatial pattern among particles). We generally assume that mixingis a permanent process in which individuals become more mixedover time (e.g. increasing entropy). However, we recognize that fishare not particles, and biological features such as multiple migrat-ing stocks with homing tendencies could result in apparent mixingthat is reversible.
Spatial phenomena are often described in terms of first orderand second order effects (e.g. Bailey and Gatrell, 1995). First ordereffects relate to large-scale trends (spatial variation in mean val-ues), while second order effects refer to relatively small-scalepatterns resulting from the spatial correlation in deviations aroundthe mean values. Tag recoveries are often observed at a higheror lower frequency than would be expected if tags were well-mixed and recoveries independent of each other. This is often
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
interpreted in tagging models as statistical over-dispersion (i.e.non-independence of observations which tends to increase thevariance of observations and estimators without biasing the esti-mators). There are methods to compensate for this situation
e.g. Hampton and Fournier, 2001 use a negative binomial likeli-ood term to admit that the tag recovery variance is higher thanould be expected from the Poisson process that would arise if
he recoveries were truly independent). One second-order spatialechanism that potentially causes this type of incomplete mixing
s a long-term association of tags through some process such aschool fidelity. If tags are only located within certain schools, butchools with and without tags are randomly mixed, then admittingver-dispersion in the tag recovery likelihood should be reason-ble. However, non-random spatial variability resulting from firstrder spatial processes is more problematic. If there are large-scaleifferences in the distributions of tagged and untagged fish within
region that is assumed to be homogenous (e.g. a tag density gra-ient with a higher concentration closer to the release site), thenystematic patterns in tag recoveries can be expected. This rep-esents a structural error in the model formulation which cannote accounted for by simply assuming over-dispersion in the likeli-ood. It is these systematic spatial biases that we are interested inere.
.2. WCPO skipjack tuna stock assessment
The WCPO skipjack tuna fishery was selected for this studyecause it has an extensive history of tagging programmes andnalyses dating back several decades. Hoyle et al. (2011) describehe most recent stock assessment, which uses MULTIFAN-CL inte-rated modelling software (Hampton and Fournier, 2001). This is
complicated, multi-gear, international fishery, with peak annualatches (2009) exceeding 1.6 million t. Skipjack are broadly dis-ributed throughout tropical and subtropical waters, with theargest catches coming from the western equatorial Pacific. It haseen recognized for some time that WCPO skipjack tag movementsend to be small (median lifetime displacements of 420–470 nau-ical miles estimated by Sibert and Hampton, 2003) relative to theopulation range and the size of the assessment regions. The stockssessment includes the following key features that relate directlyr indirectly to tag mixing assumptions:
The spatial domain is divided into three large regions (SA1–SA3in Fig. 1), which are linked by movement (bulk transfer co-efficients) and dynamics are iterated on a quarterly time-step.The east–west dimensions at the equator of regions SA2 andSA3 are approximately 6000 and 4500 km, respectively, and thenorth–south dimensions approximately 3300 km.The model assumes that each region is internally homogeneous(spatial heterogeneity is partially recognized through some of themodel inputs, e.g. fleet disaggregation enables fishery selectivityto admit some effects of spatial partitioning of the fishery and/orfish population).Tag dynamics are aggregated into quarterly batches and assumedto be fully mixed with the untagged population in the quarterimmediately following release (tags released on the last day of aquarter are assumed to be fully mixed on the following day).During the initial release quarter, tag natural mortality is assumedto be equivalent to the untagged population, but fishing mortalityis modified, such that the exact number of observed tag recoveries(inflated to account for reporting rates) are extracted from thetagged population, and these tag recoveries are excluded fromthe model objective function.Tag recovery reporting rates are estimated in the overall modelfitting. Reporting rates for fleets that participate in tag seed-ing experiments are estimated with informative priors (Hoyle,
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
2011). Reporting rates for the remaining fleets have a diffuseprior and are inferred primarily from the assumption that tagrecoveries should be a constant proportion of catch-at-age foreach fleet within each stock assessment region. The reporting
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rates are assumed to be constant for releases from each taggingprogramme (defined below), and uniform for each fleet within astock assessment region. Violations of the tag reporting assump-tions could potentially have a similar impact to the tag mixingassumptions, but these are not considered in this paper.
• Many parameters directly or indirectly related to tags areassumed to be constant over time, including separable fish-ery selectivity for all fleets, age-dependent seasonal movementamong regions, and age-specific natural mortality.
1.3. WCPO tuna tagging programmes
There have been several WCPO tagging programmes dating backto the 1970s, generally described in three main groups (see Leroyet al., in this issue). The Skipjack Survey and Assessment Pro-gramme (SSAP) 1978–1983 (Kearney, 1982), Regional Tuna TaggingProgramme (RTTP) 1991–1996 (Kaltongga, 1998) and Pacific TunaTagging Programme (PTTP) 2006–2012+ (Caillot et al., 2012), whichtogether have released about 473,000 and recovered about 52,000skipjack tags, along with substantial numbers of yellowfin and big-eye (Thunnus obesus) tunas. Together, the tagging programmes spana period with a 4-fold increase in total skipjack landings, corre-sponding to the rapid expansion of the Purse Seine (PS) fleet, whichis responsible for the greatest catches and the majority of recov-ered tags in recent years. We only consider the conventional darttags, which were almost all released from live bait pole and linevessels. Fig. 1 shows the spatial distribution of releases and recov-eries across all tagging programmes. The overall distribution ofreleases has been very unbalanced. Six core release areas (R1-R6 inFig. 1) were defined for the purpose of this analysis, but the largestnumbers of tags have been released near Papua New Guinea andthe Solomon Islands (R3–R4). The vast majority of tag recoveriesare reported from the equatorial region around 140–180 ◦E (oftennear the release points and often with short time-at-liberty). Inde-pendent small-scale tagging of skipjack near Japan has also beenongoing for several years, but these data were not available for thisanalysis.
2. Methods
Both tag mixing analyses are based on the simple procedureof following individual release events and examining the distribu-tion of recovered tags in subsequent time windows, at a higherspatial resolution than that adopted in the assessment model.Key differences between the two approaches are summarized inTable 1 and detailed in the following sections. The CUSTARD anal-ysis is extremely simple and robust to various data problems.However, it is limited to a hypothesis test (i.e. H0: Tags recov-ered from both release events represent samples from the sametag recovery distribution), and a significant result does not neces-sarily indicate practical importance. The TART analysis can also bereduced to a simple hypothesis test (i.e. H0: There is no differencebetween the tag recovery distribution and the catch distribution),but produces other quantitative results that are intuitively easier tointerpret.
For the purposes of this paper, we define a tag release event asall tag releases in a relatively small spatial area (R1–R6 in Fig. 1)during a single quarter. Tag recovery events are defined as all tagsfrom the same release event that are recaptured within the samequarter in a particular stock assessment region. Ideally, one wouldlike to examine release events from a single point in space and time,
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
and recovery events as the corresponding recaptures from instan-taneous time windows. In practice, release and recovery eventsrequire reasonable sample sizes, so we need to aggregate observa-tions across time and space. We have limited each recovery event
4 D. Kolody, S. Hoyle / Fisheries Research xxx (2014) xxx–xxx
120E 140E 160E 180 160W
40S
20S
020
N40
N
SA1
SA2SA3R1
R2R3
R4
R5
R6
F combR sessmT
ttaw(Ae
ttbwimi
TC
ig. 1. WCPO tag release (black points) and recovery locations (grey points) for theectangles indicate the spatial structure of the most recent WCPO skipjack stock asART analyses.
o an individual stock assessment sub-region, because these arehe units that are assumed to be homogeneous within the stockssessment model. Similarly, recovery events are only included inhich the release region was within the stock assessment region
i.e. because if tags are released in area A, and poorly mixed in area, then the tag estimators for both areas A and B will likely be biased,ven if the tags are fully mixed within area B).
All recovery events were also defined in terms of fish size-class,o reduce one source of variability in tag dynamics (i.e. migra-ion characteristics and fishery vulnerability may vary substantiallyy size). We used three size classes (<38, 38–58, and >58 cm);
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
here 38 cm corresponds to the median age of maturity assumedn the assessment (Hoyle et al., 2011). Recovery size was esti-
ated for each fish, calculated as the release length plus the growthncrement for the time-at-liberty inferred from the mean growth
able 1omparison of CUSTARD and TART tag mixing analysis characteristics.
CUSTARD (Comparison of SyTag Recovery Distributions)
Number of tag recovery events required foreach observation
2
Catch and catch size composition data Not used
Tag reporting rate data Not used
Usable data All fleets and tagging programwith sufficient recoveries
Form of result Statistical test for incomplete
ined SSAP, RTTP and PTTP tagging programmes (repeated points are not indicated).ent (SA1–3), and core tag release event areas (R1–6) defined for the CUSTARD and
curve in the assessment. Reported recovery lengths were not usedbecause they are often missing or very inaccurate. Due to the min-imum size of fish that can be tagged, and the fast growth rateestimated for skipjack, very few recovery events for the smallestsize-class were available for the analyses.
An important issue that we do not explicitly deal with here,and which is commonly ignored in tag analyses, arises from errorsin reported tag recovery dates and positions (Leroy et al., in thisissue). Efforts are currently being undertaken to validate PTTP tagrecovery dates and positions with satellite vessel monitoring sys-tem tracks from the recapture vessel, but tag seeding experiments
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
suggest that the errors may still be substantial. There is a non-trivialprobability of reported capture date and vessel name errors, whichundermines the position validation. Most errors are thought toarise because recoveries at fish processing plants often depend on a
nchronous TART (Tag Ratio Trend analysis)
1
RequiredNeed to account for relative variability in reporting ratesby area/fleet
mes Only fleets and tagging programmes with sufficient tagrecoveries, appropriate reporting rate estimates andcatch/size composition data
mixing Statistical test for incomplete mixing; spatial distributionof tag density; diffusion rate estimates
omplicated chain of custody which can involve transhipment andold storage. These problems are thought to be most importantn the PTTP tagging programme, as the fish handling was moreirect in earlier years. The effect of this problem differs for the twonalyses as discussed below.
.1. Comparison of Synchronous Tag Recovery DistributionsCUSTARD) analysis
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
The CUSTARD analysis is a simple method of testing the nullypothesis that tagged fish are well-mixed. It requires only theates and positions of tag releases and recoveries (by size-class).USTARD involves the following steps:
ig. 2. Top panel – CUSTARD example showing (jittered) tag positions recovered in a sivents: from region R2 1991-Q1, R3 1991-Q1, and R3 1991-Q2; labelled 1–3 respectivelyests (see text) corresponding to the example CUSTARD events. Broken lines indicate the ms located within (or beyond) the extreme tails of the null distribution, this is interprete
ixing of the two release events).
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1. Assign each recovered tag to a unique release and recovery event.2. Identify pairs of synchronous recovery events that correspond to
different release events. Fig. 2 shows an example of synchronousrecovery distributions from three different release events, illus-trating the three types of comparison that are possible: (i) releaseevents from two different release areas in the same time period,(ii) release events from the same release area, in two differenttime periods, and (iii) release events that are separated in spaceand time.
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
3. For each pair of recovery distributions from 2, calculate the prob-ability that the observed recoveries represent samples from thesame spatial distribution. If the distributions are significantly dif-ferent, this indicates that the release events are probably not
ngle time window (1991, quarter Q4, fish >58 cm), corresponding to three release. Bottom panel – Null distributions from the mean nearest neighbour permutation
ean nearest neighbour calculation (off-scale for release events 1 & 2); if this valued as evidence that the two distributions are significantly different (i.e. incomplete
fully mixed with one another, and it follows that at least one ofthe tag releases is not fully mixed with the untagged population.
. Repeat 3 for all CUSTARDs and summarize the results in relationto explanatory variables of interest. We were most interested inidentifying whether mixing depends on the time-at-liberty andstock assessment region, but other factors could be examined(e.g. release location, size-class, tagging programme, etc.).
The main difference between our approach and that developedy Latour et al. (2001) relates to our inclusion of tag recovery com-arisons from release events that are separated in space. Havingultiple tag release sites is a very useful luxury that might not be
vailable from most tagging programmes. This allows for a moreowerful test of mixing by potentially identifying spatial differ-nces that might not be evident if releases occurred at a singleocation separated in time. If one is only interested in inferencesbout the particular sub-population that is tagged, then incompleteixing across different release areas might not be relevant. How-
ver most stock assessments, including WCPO skipjack, assumehat all fish in the region of interest are part of the same manage-
ent unit, so these comparisons are relevant.Conventional tag recovery distributions depend upon many fac-
ors, including the distribution of the tagged fish, the distributionf fishing effort, fishery selectivity, tag reporting rates, and repor-ing errors in tag recovery positions. Because of these complicatingactors, in most tuna fisheries it is very unlikely that the tag recov-ry distribution would provide an accurate representation of therue distribution of all the tagged fish caught, or the untagged dis-ribution still at liberty. However, in the CUSTARD analysis, bothecovery distributions are from the same recovery time window,o all of the factors above that influence the observed distribu-ions are equivalent. This means that it is meaningful to compareistributions despite these complicating factors, but each of theseactors may increase the probability of type II errors (i.e. failure todentify incomplete-mixing when the two distributions are actu-lly different). For example, if tags tended to remain resident nearwo widely separated release points and the only fishery occurred
idway between them, where the tags do happen to be well mixed,hen the CUSTARD analysis would fail to identify true differencesetween the two tag distributions. Similarly, if recapture positionrrors were large relative to the difference between recovery dis-ributions, the analysis might fail to identify a difference.
Two problems could inflate the CUSTARD type I error rate (i.e.he probability of finding significant differences between distribu-ions if the recovery events are actually samples from a single fully
ixed population). Large and frequent tag recovery date errorsotentially bias results because each observed tag recovery eventay be affected by recoveries from outside of the desired timeindow. For example, if a substantial number of recoveries from
he period before full-mixing had been achieved were misreportednto a later period in which full-mixing had been achieved, thisould cause a Type I error. This is probably not a frequent prob-em, but could have serious implications for most tag analyses if it
ere. Another potential problem could be the aggregation of tagsue to the effect of long-term fidelity to specific schools (as dis-ussed in section 1.1). The CUSTARD analysis does not distinguishhe source of incomplete mixing (though it may be evident from aisual comparison of plots such as Fig. 2).
Fig. 2 (top panel) illustrates some key features of the CUSTARDnalysis. The map shows tags recovered in the fourth quarter of991, from three release events (and labelled with correspondingumeric symbols): (1) released in region R2, 1991 first quarter, (2)
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
eleased in region R3, 1991 first quarter, and (3) released in region3, 1991 second quarter. All of these recovered tags were assumedo be fully mixed with each other and the general population inhe stock assessment model. The distributions of most tags from
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release events 2 and 3 appear to overlap (with a couple of conspic-uous outliers). However, tags from release event 1 show a denseconcentration in the archipelagic waters within the R2 release area,and are clearly not fully mixed with tags from release events 2 or3. However, a few tags from release event 1 have dispersed to mixwith the other release events in the open waters. If there had beenno fishery operating in the R2 tag release area, one might have mis-takenly concluded that all three release events were well-mixedwith each other.
Various test statistics could be used to compare the recoverydistributions. We explored two simple options with simulationtesting: (i) chi-square contingency table on a two dimensional gridof binned data (similar to Latour et al., 2001), and (ii) a Nearest-Neighbour Permutation (NNP) test (e.g. Bailey and Gatrell, 1995).The R software package spatstat (Baddeley and Turner, 2005) func-tion nncross was used for the nearest neighbour calculations. Theresults of the two tests often differed for individual CUSTARDevents, but supported qualitatively very similar conclusions whenapplied across a large number of CUSTARD events. However, thechi-square test may be sensitive to the choice of bin definitions,and more importantly, proved to be less powerful than the NNPtest for smaller sample sizes in the simulation tests (e.g. <∼30 tagsper recovery distribution). Accordingly, only the NNP results arereported. The NNP test consisted of:
(1) Calculate the mean NN for all tags from the first release group,where each individual NN represents the great circle distancebetween the first release group tag recovery position and thelocation of whichever tag from the second release group is clos-est.
(2) Generate a non-parametric null distribution for mean NN usingMonte Carlo simulations:a. Pool the observed tag recovery positions from both release
events, and then randomly reassign each tag to one of thetwo release events (maintaining the original sample sizes).
b. Calculate mean NN from the randomized tag recovery distri-butions.
c. Repeat steps a and b 1000 times, creating a mean NN fre-quency distribution.
(3) If mean NN from 1 is near the tails of the null distribution from2c, this is interpreted as evidence for incomplete mixing. In thissituation, we would generally expect the mean NN value for twodistinct tag distributions to be at the right-hand tail of the nulldistribution (i.e. consistent with a large degree of spatial sep-aration), and accordingly used a one-tailed test of significance(P∗
xy > 0.95), where Pxy indicates the P-value for a comparisonof tag recoveries from release events x and y (* is explainedbelow).
Visual inspection of the map of the three CUSTARDs shownin Fig. 2 (top panel) suggests that the distributions of recoveriesfrom release events labelled 2 and 3 are similar to each other, butboth differ from 1. The corresponding P-value calculations for theseevents can be visualized in the bottom panel of Fig. 2, where themean nearest neighbour values (broken vertical lines) fall com-pletely outside the null distributions in two cases (P∗
12, P∗13 > 1.0,
such that the null hypothesis of complete mixing is extremelyimprobable), and in the third case, the mean nearest neighbourvalue is well within the null distribution (P∗
23 = 0.81, which doesnot provide strong evidence for rejecting the null hypothesis). Forconsistency of interpretation with the other results in this paper,the mean NNP P-values are summarized as P = 1 − P*, such that
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
P < ̨ = 0.05 is interpreted as significant evidence of incomplete mix-ing. Thus the reported NNP test values are P12 < 0.001, P13 < 0.001and P23 = 0.19, which is consistent with our visual inspection of themap. Note that the result of the NNP test differs depending on which
elease event is defined as first and second in step 1 above. This maye disconcerting if one is only interested in a small number of com-arisons, but should make little difference to the overall result if
large number of comparisons are made. An alternative approachhich is robust to the initial tag grouping assignment would be
o (i) calculate the P-value as above, (ii) reverse the designation ofhe two recovery events in the NNP test and calculate a second P-alue, (iii) retain the lower of the two P-values, and (iv) report theignificance with respect to ̨ at double the level used in the individ-al tests. In simulations, and the skipjack application, we excludedecovery events with <5 tags (the simulation tests seemed reliableith five tags, but smaller sample sizes were not tested).
As a simple summary of (n) multiple CUSTARD results, wereated each individual P-value (Pi) as an independent sample from
Bernoulli process, and calculated the probability of obtaining thebserved number (k) of statistically significant (Pi ≤ ̨ = 0.05) resultsor more) as the binomial probability:
B =n∑
j=k
(n
k
)˛j(1 − ˛)n−j.
As will become evident in the results, this test statistic is suf-ciently powerful to illustrate evidence for incomplete mixing inhe case of Pacific skipjack tuna. However, this approach is discard-ng information in the sense that P = 0.001 provides much strongervidence than P = 0.05, but the two values are treated equally inhe binomial function, such that a more powerful statistical testould presumably be used. Conversely, we also note that it is notompletely legitimate to consider each trial to be completely inde-endent if recovery distributions are compared with more than onether recovery distribution.
.2. Tag Ratio Trend (TART) analysis
The TART analysis compares the spatial distribution of tag recov-ries to the catch distribution, and examines evidence of spatialeterogeneity within each stock assessment model sub-region. Weefer to the ratio of (number of tag recoveries)/(catch in numbers)s the tag density. As stock assessment regions are assumed to benternally homogeneous, we would expect reasonably uniform tagensity within each region. Langley and Million (2012) comparedhe spatial variation in tag density (tag rate in their terminology)or Indian Ocean yellowfin tuna based on a qualitative synthesis of
time series of maps. Our approach has the advantage of havingbjective reproducibility based on formal statistical tests, but it isore limited in the sense that we only look for evidence of a single
re-defined mixing problem (a gradient of tag density relative tohe distance from the release site).
For WCPO skipjack, only the catch and tag recoveries fromhe subset of the PS fleet with tag reporting rate estimates werencluded (otherwise, spatial variability in the reporting rates couldnfluence the inferred tag density distribution). We have assumedhat the tag reporting rates within each assessment region are iden-ical for all PS fleets that participate in the tag-seeding experiments.his is consistent with the assumptions in Hoyle et al. (2011), bute recognize that the tag reporting rate analysis may be missing
mportant sources of variability due to small sample sizes, tem-oral trends in recovery effort, and the assumption that reportingates are dependent on the vessel flag (in reality, reporting ratesre probably more closely linked to the unloading ports than flag;oyle, 2011).
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
WCPO skipjack catch and size composition data are accessiblen monthly 5 × 5◦ cells, but the data were aggregated into quarterlynits, and a lower spatial resolution (see below). This aggregationllows easier visualization of spatial patterns, because the data can
PRESSearch xxx (2014) xxx–xxx 7
be noisy (e.g. some tags appeared to be returned from cells whichhad very little, or zero catch of the expected size-class, probably dueto tag recovery position or date errors, or limited size-compositionsampling). Tag recovery events were only included in the analysisif at least 10 tags were recovered, and at least 10 cells had positiveskipjack catch (for the selected PS fleets).
We would often expect to see some persistence of a pattern oftag density declining with distance from the release site until afully mixed state was achieved. To test for this pattern, we sortedthe 5 × 5◦ catch and tag recovery observations for each recoveryevent in order of increasing great circle distance from the centre ofthe release area, and aggregated the observations into area-basedquintiles (quintiles were a somewhat arbitrary choice that ensuredthat each observation would consist of at least two 5 × 5◦ cells inthis case, and enough observations would be present to identifynon-linear relationships). Tag recoveries, catch distribution and tagdensity observations are shown in Fig. 3 for an example recoveryevent with a particularly large number of recoveries (two quar-ters after release), illustrating a sharp decline in tag density withincreasing distance from the release area.
As a statistical test of homogeneity, we regressed tag den-sity (area-based quintile) on the mean great circle distance fromthe centre of the release area using a binomial Generalized Lin-ear Model (R function glm, R Core Team, 2013). A significantslope provides evidence that the tags are not fully mixed withthe untagged population. However, the absence of a significantslope does not necessarily mean that the tags are fully mixed,e.g. the density–distance function could be dome-shaped if therewas strong advection. Alternative tests for heterogeneity could beused, but this approach seemed appropriate given the prevailingpatterns observed here. To summarize results from multiple tagdensity–distance relationships, we tabulated the results by stockassessment region and time-at-liberty, and calculated the bino-mial probability of obtaining the observed frequency of statisticallysignificant (P ≤ 0.05) results (as with the CUSTARD analysis above).
2.3. TART diffusion estimates and bias simulations
As discussed in the results, a large number of the tagdensity–distance plots resembled Fig. 3 (inset), with a shaperesembling a half-normal distribution. This is consistent with thepredictions of a simple diffusive process, and suggests that theTART analysis might be usefully extended by fitting parametricmixing models. The distribution of T tags diffusing away from therelease point (or the probability distribution, p, for the position ofan individual engaged in a random walk) can be approximated bya Gaussian distribution (discussed accessibly in Turchin, 1998):
px,t = T√4�Dt
exp
(− x2
4Dt
),
where x represents the displacement distance for a single dimen-sion, D is the diffusion rate, and t represents the tag time-at-liberty(the variance of the normal distribution, �2, is expressed as 2Dt). Intwo dimensions this extends to:
px,y,t = T
4�Dtexp
(−x2 + y2
4Dt
)= T
4�Dtexp
(− r2
4Dt
),
where x and y represent displacement distances in longitude and
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
latitude and r is the absolute distance from release. Using theserelationships, we estimated D independently for each tag recov-ery event in the TART analysis (with time-at-liberty of at least onequarter). To account for the small sample sizes and large numbers
8 D. Kolody, S. Hoyle / Fisheries Research xxx (2014) xxx–xxx
140E 160E 180
10S
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0.1
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R3
0 2000 4000 6000
02
4
Distance from release (km)
Tag
dens
ity
Fig. 3. TART example showing tag recoveries positions (circles) for the largest skipjack size-class in 2008-quarter Q3, released from area R3 in 2008-Q1. Shaded contoursr r catct ggregat dence
oa
L
wet
p
wfdmatvda(O
••
•
•
•
epresent the smoothed catch in numbers from the selected PS fleet (darker = higheag density as a function of distance from the centre of release region R3 (data are aag density for the whole PS-fished area). Error bars indicate the 95% binomial confi
f observations with zero tag density, we assumed a binomial (neg-tive log) likelihood:
= − log
(∑i
(ni
ki
)(pi)
ki (1 − pi)ni−ki
),
here i indexes the quintile observations from a single TART recov-ry event, n is the catch in numbers, k is the number of recoveredags, and the probability density for each observation is given by:
i = h exp
(− r2
i
4Dt
),
here r is the median absolute distance from the release pointor each TART quintile, and h is a scaling factor for the wholeistribution, estimated (along with D in the non-linear functioninimization) to account for the fact that we do not know the over-
ll number of tags still at liberty (or the fishing mortality). If theags are well-mixed, then D will be estimated to have a very largealue, the exact value of which is not important once it results in aensity distribution that is essentially flat at the scale of the stockssessment region. We imposed an upper bound of D = 108 km2 y−1
which corresponds to � = 7070 km after one quarter of mixing).bvious criticisms of the approach include:
It ignores directed migration (or passive advection by currents).It ignores coastal and oceanographic boundary effects, andassumes that diffusion in latitude and longitude are equivalent.It ignores individual variability in migration behaviour (e.g. asmall number of long range dispersers could lead to fat-taileddistributions).As with the linear regression model, a high value for D couldbe estimated due to model mis-specification, rather than rapid
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
diffusive mixing.The diffusion models above are treating observations of tag den-sity as equivalent to tags per unit of area. This invokes the implicitassumption that the fish population distribution is uniform. This
h); line contours represent the relative tag density (tags/catch). Inset panel showsted into area-based quintiles and tag density is expressed relative to the aggregate
intervals for the tag density.
is not strictly correct, but may be reasonably valid at the scale ofinterest.
Using the parametric diffusion models, we provide a generalindication of the magnitude of population estimator bias that mightbe expected due to incomplete tag mixing under a range of plausibleconditions. We use a simple Brownie–Petersen model (nomencla-ture proposed by Polacheck et al., 2006) to estimate populationsize, N, natural mortality, M, and fishing mortality, F, under a rangeof conditions. The Brownie element of the model (Brownie et al.,1985) consists of tagging and recapturing the same cohort in con-secutive time intervals. The estimator is based on the Baranov catchequations:
Tr,t+1 = Tr,t exp(−Mt − Ft),
where T is the number of tags at liberty from release event r, atthe beginning of timestep t, and the number of recoveries, R iscalculated:
Rr,t = Tr,tFt
Ft + Mt(1 − exp(−Mt − Ft)),
Knowing the number of tag releases from the same cohort in twoconsecutive timesteps (T1,1,T2,2) and the number of tag recoveries(R1,1, R1,2, R2,2), it is possible to estimate M1, and F1 (in this caseusing an iterative least squares approach to minimize the differ-ence between predicted and observed tag recoveries). If the catchis known, the Petersen-type estimator can be used to estimateabundance:
N1 = C1F1 + M1
F1(1 − exp(−M1 − F1)).
These tag dynamics are the same, or very similar to, the tag-ging component of most integrated stock assessment models. TheM estimator above is equivalent to the tag estimator in the WCPO
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
skipjack assessment (aside from the error distribution assump-tions). The simulated F and N estimates are not equivalent to theskipjack assessment, because F1 corresponds to the skipjack mix-ing period, and would not be considered to be informative about
D. Kolody, S. Hoyle / Fisheries Research xxx (2014) xxx–xxx 9
Table 2Evidence of incomplete tag mixing from CUSTARD and TART analyses for WCPO skipjack tuna, partitioned by stock assessment region and tag time-at-liberty (0 = tag releasedand recaptured in the same quarter). Proportion incomplete mixing refers to the proportion of observations in which the analysis resulted in significant evidence (P ≤ 0.05)of incomplete mixing, and binomial P-values are the probability of attaining at least that many significant results by chance. In the CUSTARD analysis, the time-at-libertyrefers to the lesser of the two release events.
USTARD and TART tag mixing analysis results for western Pacific skipjack tuna.
shing mortality on the general population in the assessment.urthermore, in the assessment, the N estimate would often benformed by multiple observations, accumulating additional infor-
ation from subsequent timesteps which would have had a longerixing period. More realistic simulations could be achieved by
ncreasing the number of release events, extending the recaptureuration and adding movement among areas, but comprehensiveimulations were beyond the scope of this paper.
For the simulations, we assumed that all of the standard taggingssumptions were met (e.g. 100% reporting of recovered tags, noag shedding, etc.), except for the tag mixing assumption. N, M and
estimates were generated from the spatially aggregated data at low temporal resolution (range of timesteps from one month tone year). The underlying simulation dynamics were calculated at
much higher spatial and temporal resolution, with the followingssumptions:
Dynamics iterated with 10 sub-intervals per timestep, to approx-imate continuous fishing and diffusion.The general population (Ns, where subscript s indicates high reso-lution spatial subunits) is normally distributed with SD = 2000 km(latitude and longitude), centred within, and truncated at thebounds of, a 6000 km square (approximate longitudinal widthof region SA2). The shape of the population distribution does notchange over time.The fishing effort distribution (Es), is related to the fish popu-lation density, Es = Ng
s , where the exponent g represents effortconcentration (targeting efficiency), i.e. g = 0 represents uniformfishing effort, g = 1 represents effort distributed proportional topopulation density (g = 0.5 and 2 are presented to represent weakand strong targeting). An additional parameter, u, defines an areafrom which fishing is excluded. Two options are presented: u = 0(whole area fished), and u = 0.05 (5% of the area correspondingto the region with highest tag concentration is not fished). Thelatter option is intended to illustrate the potential importance oftag releases in regions which do not have a fishery (or reliable tag
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
reporting).Tags are released at a single point directly west of the centre ofthe region (0–3000 km presented), and diffuse at the median rateestimated for WCPO skipjack. Tags are reflected at the boundaries.
1 0.00 1.00
The shape of the tag distribution (and general population) ignoresthe effects of previous fishery removals (probably only realisticat low F).
tively; F varies with the scenario).• Observed catches and numbers of tag recoveries are deterministic
outputs from the catch equations.
Estimator performance is summarized in a series of contourplots of observed bias, expressed as 100% × (Estimated − True)/True.
3. Results
3.1. WCPO skipjack CUSTARD analysis
The CUSTARD analysis identified 553 CUSTARDs in stock assess-ment region SA2, 15 in region SA3 and 0 in region SA1 (Table 2). Theresults for region SA2 indicate strong evidence for incomplete mix-ing for at least six quarters following release. Only two CUSTARDswere identified with time-at-liberty exceeding six quarters, so theanalysis is not informative about longer mixing periods withinregion SA2. For region SA3, the evidence of incomplete mixing isstrong 0–1 quarters after release. For time-at-liberty of two quar-ters or more there are only five CUSTARD observations in total, andevidence for incomplete mixing is not compelling.
3.2. WCPO skipjack TART analysis
The TART analysis yielded 152 observations for stock assess-ment region SA2, and 15 for SA3 (Table 2). The TART results aresimilar to the CUSTARD analysis for region SA2, with strong evi-dence of incomplete mixing for time-at-liberty up to and includingfive quarters (and only four observations for longer times-at-liberty). The evidence for incomplete mixing in region SA3 isnot compelling when partitioned by time-at-liberty (although the
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
probability of attaining five significant results out of the 15 obser-vations aggregated over time-at-liberty yields a P-value < 0.001).
In region SA2, tag density generally declines very steeply withdistance from the tag release site (Fig. 4A). Tag density within
10 D. Kolody, S. Hoyle / Fisheries Research xxx (2014) xxx–xxx
0 1000 2000 3000 40000
12
34
5R
elat
ive
tag
dens
ity
A) Stoc k assessment region SA2
0 1000 2000 3000 4000
01
23
45
Mean distance from release (km)
Rel
ativ
e ta
g de
nsity
B) Stoc k assessment region SA3
Spatially aggregated tag density Tag density of area−based quintile LOWESS Smoothers:1−2 Quarters after release3−4 Quarters after release5−6 Quarters after release
Fig. 4. WCPO skipjack TART results partitioned by assessment model region and number of quarters from release (lines represent LOWESS smoothers). Each point representsthe tag density for one quintile of the purse-seine fished area relative to the mean tag density of the entire recovery event. For legibility, 30 points are off of the y-axis scale,a of obs
∼tatteosmgepp
dd(
3
ddttecfliteG
point of this option is to illustrate that having no fishery in a
nd binomial 95% confidence intervals are only shown for a random sample of 10%
1200 km of the release location tends to be higher than the meanag density (within each individual release event), and tag densityt greater distances tends to be less than the mean. The slope of theag density gradient declines with time, but is still evident 5–6 quar-ers after release (there were no observations with time-at-libertyxceeding six quarters). The LOWESS smoothers suggest that theverall density–distance relationship is not linear, but roughly con-istent with a fat-tailed half normal distribution. However, theseean relationships represent a combination of multiple hetero-
eneous density–distance relationships (e.g., the TART recoveryvents span different spatial ranges due to the different releaseoints and variable fleet distributions, and different release areasresumably have different mixing dynamics).
In region SA3, there is also a general decline in tag density withistance (Fig. 4B), but the relationship is less consistent, the gra-ient is less steep, and a trend over time-at-liberty is not obviouspresumably due in part to the small sample sizes).
.3. TART diffusion estimates and simulations
Qualitatively, the normal density–distance relationship pre-icted by the diffusion model seemed to provide a reasonableescription of the tag density observations for the majority ofag recovery events (at least at the level of aggregated quin-iles). The distribution of D estimates was approximately normal,xcept for a mode on the upper bound (∼5% of estimates). Theases on the upper bound corresponded to a range of situations:at density–distance relationships (consistent with full-mixing),
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
ncreasing density with distance, or an irregular pattern (the latterwo cases are not consistent with simple diffusion). The medianstimate of diffusion (D = 8.51 × 105 km2 y−1) corresponds to aaussian spatial distribution with � = 652 km after 1 quarter of
ervations.
mixing (3690 km after two years of mixing), 10th–90th percentiles1.77 × 105 − 6.80 × 106 km2 y−1.
We note the following generalizations from the N, M and F biassimulations (Fig. 5). If the whole area is vulnerable to fishing (u = 0):
• M and F bias contours tend to be qualitatively similar, and ofopposite direction to the N biases (mortality overestimated andabundance underestimated if the release point is close to thecentre of the population/fishery).
• At some intermediate release point location, individual estima-tors may appear to be unbiased (i.e. because even a stopped clockshows the correct time twice per day), but this point is not thesame for all estimators.
• The biases tend to decrease slightly with increasing timestepduration (i.e. presumably due to mixing).
• Factors that increase the spatial variability in tag density or con-centrate the range of the fishery tend to increase the magnitude ofthe estimator biases. This is shown for two levels of the targetingfactor (g = 0.5, 2), and the exclusion of the fishery from some partof the population range (u = 0, 0.05), but was also evident withalternative values of the diffusion rate and population variance(not shown).
If the fishery is excluded from 5% of the region with the highesttag density (u = 0.05), the effect is essentially equivalent to increas-ing the distance between the tag release site and fishery/populationmode, i.e. N over-estimated, M and F under-estimated. The key
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
relatively small area near the release site can be particularly prob-lematic. Conversely, excluding the fishery from 25% of the regionwith the lowest tag density had only a minor effect on the biasesrelative to the fully fished scenarios (not shown).
D. Kolody, S. Hoyle / Fisheries Research xxx (2014) xxx–xxx 11
Weak targeting All areas fished
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00.
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N Bias M Bias F Bias
Distance from fishery centre to tag release point (km)
Tim
este
p (y
)
F Petero
3
sWetiyfmptssfs
ig. 5. Contours of N, M and F bias (100% × (estimated − true)/true) from the Brownie–ther simulation parameters are defined in the text.
.4. Discussion
The CUSTARD and TART analyses were sufficient to demon-trate that the mixing assumptions are not being met in recent
CPO skipjack assessments and probably cannot be met by simplyxtending the initial mixing period. The simulations indicate thathe magnitude of the incomplete mixing is potentially sufficient tontroduce considerable bias into population estimators. These anal-ses can be extended in various directions, e.g. examining differentunctional relationships in the TART density–distance relationship
ight be insightful, more specific questions could be posed (can weredict tag mixing characteristics in relation to oceanographic fea-ures?), and more statistical sophistication might lead to greater
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
ensitivity to detect mixing problems. However, in this discus-ion we focus on the practical implications of the current resultsor WCPO skipjack assessment. We consider whether these resultshould be interpreted as being of practical importance, and discuss
sen simulations. Movement was simulated using the median diffusion rate estimate,
options for reducing the incomplete tag mixing problem in tunastock assessments.
Table 2 indicates that WCPO skipjack tags are not meetingthe rapid mixing assumption, but statistical significance, per se,does not necessarily indicate that the problem is serious enoughto undermine the assessment. Many statistical models in ecologyrely upon assumptions that are difficult to test, or known not tobe strictly valid, but which may not greatly affect conclusions.However, there is a growing recognition that uncertainties due tofisheries model mis-specification are often considerable and meritfurther attention (e.g. Punt, 2008), and the TART and CUSTARDgraphics (Figs. 2–4) suggest that the magnitude of the problem isdifficult to ignore in this case. The simulations represent an initial
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
attempt to roughly quantify the biases arising from the tag mix-ing problem. The most optimistic scenarios from Fig. 5 suggest thatincomplete mixing might often result in N, M or F estimates thatare within ±20% of the true value. This would probably represent
tolerable stock assessment problem. However, the simulationslso illustrate plausible situations with much higher biases (and0% of the diffusion estimates would result in situations that areore extreme than those shown).In evaluating the likely bias of incomplete mixing on the WCPO
kipjack assessment, it may be important to consider the otherotential mitigating factors. Tag analyses are based on the assump-ion that there is an equal probability of capturing tagged andntagged fish. In addition to the rapid mixing of individuals, thisssumption could be achieved through a random spatial distribu-ion of releases, random spatial distribution of fishing effort, orome combination of these factors. The tag release events are farrom random, but they are more widespread than the areas definedor this mixing analysis (Fig. 1), and within the assessment, simulta-eous releases from the R1-R6 sub-regions are pooled at the level ofhe assessment regions. It seems unlikely that the release distribu-ion has much potential to compensate for the limited tag mixing.owever, it would be a worthwhile sensitivity analysis to comparessessment models which systematically exclude release eventsrom different sub-regions. It is also unlikely that the spatial dis-ribution of the fishery would compensate for limited mixing. TheS fishery (the component with informative tag reporting rate esti-ates) has to balance several competing objectives including the
esire for high catch rates, proximity to landing ports, access con-traints, etc., none of which are randomly distributed with respecto the fish population (e.g. we know that there are areas of sub-tantial fish abundance that are not frequented by the PS fleet,ecause there are other fleets that operate outside of the PS region).mproved simulations based on detailed analyses of fish distribu-ions and fleet behaviour, and more realistic population estimatorse.g. spanning longer time series, with information shared acrosselease events) would help to refine our understanding of the esti-ator biases. These refinements will not change the conclusion thatCPO skipjack mixing rates are limited, however, they might assist
n understanding the mitigating effects of concurrent release eventsnd the targeting efficiency of the fleets in reducing these biases.
There are two obvious ways in which the current WCPO skip-ack assessment modelling framework could be modified to reducehe incomplete mixing problem. The first approach would simplynvolve extending the mixing period. Ideally, we would have foundo evidence of incomplete mixing after a reasonably short time-t-liberty. Unfortunately, the CUSTARD and TART analyses suggesthat the appropriate mixing period for the current WCPO skipjackssessment region definitions would be longer than six quarterspotentially much longer if the diffusion estimates are indicative).his would result in a relatively small number of informative tagecoveries for this short-lived species (and would not remove the
biases). This solution would not be much better than simplyxcluding the tag data altogether. Given the expense of the taggingrogramme and the limitations of other sources of assessment dataparticularly for skipjack tuna), improved methods for using theseata should be pursued.
The second approach would involve changing the spatial (andossibly temporal) structure of the assessment model to be moreonsistent with tag mixing assumptions. An initial step would be toxamine some of the obvious potential boundaries between fish-ng regions, which may be associated with barriers to movement.or example, a boundary at 130 ◦E would separate the westernhilippines–Indonesia region from the rest of region SA2. Releaseshould mix within this smaller region more rapidly than they mixeyond the boundary, and this should reduce the impact of onef the worst potential situations, i.e. limited mixing in a region
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
hat is largely unexposed to the informative PS fishery. In this case,ncomplete mixing may seriously affect the tag reporting rate esti-
ates as well, and this problem is not easy to resolve if the fleets doot have sufficient spatial overlap. Skipjack may also have smaller
PRESSearch xxx (2014) xxx–xxx
movement rates in coastal areas than in the offshore pelagic waters.Quantifying this latter effect was not an objective of this study, butsupporting evidence was observed. CUSTARDs consisting of releaseevents from SA2 and recovery events in SA3 were identified butexcluded from Table 2. These few events showed no evidence forincomplete mixing. Similarly, the fat-tailed distributions in Fig. 3are consistent with a mixture of low diffusion near the coastalrelease locations, and high diffusion in oceanic regions. The mix-ing problem might be substantially reduced if the spatial structurehad higher resolution near the coastal regions, and lower resolu-tion in the open ocean. It should be possible to define a combinationof regions, release events, and mixing periods which would greatlyreduce the violation to the assumption of equal vulnerability oftagged and untagged fish.
However, increasing the skipjack assessment model complex-ity raises other concerns. Given the unbalanced tag release design(an inevitable consequence of logistical difficulties and finiteresources), it is doubtful that movement rates (and confoundedparameters such as recruitment patterns, fishery selectivity and tagreporting rates) could be reliably estimated with a more compli-cated MULTIFAN-CL modelling framework. However, by modellingat a higher resolution and imposing a range of alternative plausiblemovement assumptions, the more complicated framework shouldbe more appropriate for expressing the uncertainty in the system.
More complicated options for improving the assessment mightinvolve a two-stage process in which high resolution models wereused to estimate key parameters (or ad hoc bias corrections) thatcould be input to the low resolution assessment model (e.g. as pri-ors). Higher resolution models have already been developed forPacific skipjack populations. Sibert et al., 1999 (and Sibert andHampton, 2003) describe a high resolution advection–diffusionreaction model that estimates movement and mortality on the basisof tag recoveries and the effort distribution. This model should notbe badly affected by the tag-mixing problem (but is affected by theunbalanced tag release design and poor reporting rate estimates).Alternatively, SEAPODYM (e.g. Lehodey et al., 2008) is another highresolution tuna model that has a qualitatively different approach todescribing population dynamics. SEAPODYM includes mechanisticmodelling of tuna behaviour in relation to dynamic environmentalfields and may be able to compensate for unbalanced tag releasedesigns in a way that the simpler tagging models cannot (i.e. if thehabitat effects are estimable and consistent among regions, theycan be assumed to apply to regions that do not have tag obser-vations). But we note that neither of these approaches attempt todescribe stock structure, which could result in similar individualsin the same region having very different movement characteristics.
In deciding how to improve the spatial structure in the WCPOskipjack assessment (and other tuna assessments more generally),it is not clear which refinements would be most effective. Any solu-tion will involve a compromise, but will also depend very muchon the question that one is attempting to answer. Simulationswould be the best option for understanding how tag mixing, andother assumption violations (e.g. tag reporting rate estimates anddate/position errors), will bias assessment results. High resolutionmechanistic models, such as SEAPODYM, might represent the bestoption for plausible operating models. Simulations in the context ofManagement Strategy Evaluation (e.g. Punt and Donovan, 2007) canbe used to identify management options that are robust to manyof these inevitable types of biases and uncertainties. This would beuseful to help distinguish model performance questions that mat-ter for fisheries management, from those that may be of interestfor other theoretical reasons.
tag mixing assumptions in western Pacific Ocean skipjack tunaj.fishres.2014.05.008
We encourage similar mixing analyses for other tuna popula-tions, as the issue is not unique to Pacific skipjack tuna. Yellowfintuna displacements may be similar to (Sibert and Hampton, 2003)or less than (Caillot et al., 2012; Hoyle et al., 2013) those of skipjack
n the WCPO, so the mixing problems would presumably be simi-ar or more severe (though greater longevity should allow a longer
ixing period to be used). Langley and Million (2012) noted sensi-ivity to the yellowfin tuna mixing period in the Indian Ocean, andonducted an analysis suggesting that (within the existing modeltructure) the mixing period for free-school PS sets should be ateast one year (while adequate mixing was never observed foroating-object sets). Similarly, average displacements for bigeyeuna in the Pacific appear to be lower than for skipjack (Sibert et al.,003), with most recaptures near the point of release (Hampton andunn, 1998; Schaefer and Fuller, 2005). Sensitivity analyses for the
ecent WCPO bigeye tuna assessment suggested that stock statusesults may be very sensitive to tag dynamics both in the relativelymall (in terms of abundance) south-western region of the WCPOnd elsewhere (Ianelli et al., 2012; Hoyle et al., 2013). We expecthat these sorts of problems may be more common, and of greater
agnitude, than assessment analysts have tended to realize (e.g.olody and Hoyle, 2013, applied the CUSTARD analysis to skipjack,ellowfin and bigeye tuna populations in the western Pacific andndian Oceans, and identified a range of minimum mixing periods).
Finally, it is worth re-emphasizing that the issue of tuna tagixing and movement should be carefully considered in the design
hase of future tagging programmes. A smaller number of widelyispersed conventional tags would probably be more informativehan a large number of tags released at a single location, not onlyecause of the mixing problem, but also more generally because thenbalanced release design limits the estimation of movement. Andhe potential benefits of other tagging technologies (e.g. electronicosition-recording tags, or genetic mark-recapture techniques)hould be considered in the cost-benefit analysis.
. Conclusions
1) We present two methods for identifying when tagged fish arenot completely mixed with the general population, which (fortuna fisheries at least) generally indicates a violation to the fun-damental tag estimator assumption that tagged and untaggedfish are equally vulnerable to the fishery. The methods cannotidentify when full-mixing has been achieved.
2) Application of the methods to western Pacific skipjack tunatags clearly demonstrated that mixing rates are not consistentwith the assumptions of recent stock assessments. The testsdo not directly quantify the magnitude of the estimator biasesexpected due to incomplete tag mixing, and the effects of poten-tial mitigating factors (e.g. the degree of randomness in taggingrelease events or fishing effort) were not explicitly evaluated.
3) Simulations based on the estimated diffusion rates and a rangeof fishery assumptions demonstrated that the tag-based esti-mators could be introducing large biases to tuna assessments.It is noted that increased spatial (and possibly temporal) resolu-tion in tuna assessment models will generally reduce tag-basedestimator biases caused by poor mixing, but increasing struc-tural complexity introduces other challenges.
cknowledgements
The Papua New Guinea National Fisheries Authority fundedhis work, with additional contributions from the Secretariatf the Pacific Community and CSIRO Marine and Atmosphericesearch and Wealth from Oceans Flagship. The authors wisho thank the vast numbers of participants in the various
Please cite this article in press as: Kolody, D., Hoyle, S., Evaluation ofstock assessment models. Fish. Res. (2014), http://dx.doi.org/10.1016/
ag release and recovery programmes who have made thisork possible. John Hampton, Simon Nicol, Tony Lewis and
ylvain Caillot provided a wealth of insight into historical anal-ses and data. Adam Langley, Pierre Kleiber, Paige Eveson,
PRESSearch xxx (2014) xxx–xxx 13
Rich Hillary and two anonymous reviewers provided usefulsuggestions for the manuscript. Thanks to MannyG and Whuberat gis.stackexchange.com for suggesting the NNP test to somebodyelse for an unrelated problem.
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