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Ecological Modelling 224 (2012) 76– 89

Contents lists available at SciVerse ScienceDirect

Ecological Modelling

jo u r n al hom ep age : www.elsev ier .com/ locate /eco lmodel

erformance of a Bayesian state-space model of semelparous species fortock-recruitment data subject to measurement error

henming Sua,∗, Randall M. Petermanb

Institute for Fisheries Research, Michigan Department of Natural Resources, and University of Michigan, 212 Museums Annex Building, 1109 N. University Ave., Ann Arbor, MI8109-1084, USASchool of Resource and Environmental Management, Simon Fraser University, 8888 University Drive, Burnaby, B.C., V5A 1S6 Canada

r t i c l e i n f o

rticle history:eceived 25 January 2011eceived in revised form 26 October 2011ccepted 1 November 2011vailable online 26 November 2011

eywords:tock-recruitment analysiseasurement error

a b s t r a c t

Measurement errors in spawner abundance create problems for fish stock assessment scientists. Todeal with measurement error, we develop a Bayesian state-space model for stock-recruitment data thatcontain measurement error in spawner abundance, process error in recruitment, and time series bias.Through extensive simulations across numerous scenarios, we compare the statistical performance of theBayesian state-space model with that of standard regression for a traditional stock-recruitment modelthat only considers process error. Performance varies depending on the information content in data, asdetermined by stock productivity, types of harvest situations, and amount of measurement error. Overall,in terms of estimating optimal spawner abundance SMSY, the Ricker density-dependence parameter ˇ, and

rrors-in-variablesime-series biastate-space modelayesianarkov chain Monte Carlo

optimal harvest rate hMSY, the Bayesian state-space model works best for informative data from low andvariable harvest rate situations for high-productivity salmon stocks. The traditional stock-recruitmentmodel (TSR) may be used for estimating ˛ and hMSY for low-productivity stocks from variable and highharvest rate situations. However, TSR can severely overestimate SMSY when spawner abundance is mea-sured with large error in low and variable harvest rate situations. We also found that there is substantialmerit in using hMSY (or benchmarks derived from it) instead of SMSY as a management target.

. Introduction

The relationship between spawning stock abundance (S) andubsequent recruitment (R) is fundamental for fish populationynamics (Quinn and Deriso, 1999). Parameter estimates foruch relationships provide the basis for setting key values ofariables used in fisheries management. However, traditionaltock-recruitment analyses that use standard regression and treat

as an independent variable and R as a dependent variable areraught with estimation problems (Hilborn and Walters, 1992). One

ajor issue with these analyses is that they rarely consider erroraused by mismeasuring S (measurement or observation error).owever, mismeasurement of S is quite common in fisheries sci-nce (Hilborn and Walters, 1992; Needle, 2001). Analyses thatgnore this error may be subject to severe bias in parameter esti-

ates, which may result in serious management problems (Waltersnd Ludwig, 1981). Furthermore, large natural variation in survival

ate from eggs to recruitment (McGurk, 1986) and dynamic feed-acks in stock-recruitment systems also exacerbate the difficulty

∗ Corresponding author. Tel.: +1 734 663 3554.E-mail addresses: suz@michigan.gov (Z. Su), peterman@sfu.ca (R.M. Peterman).

304-3800/$ – see front matter © 2011 Published by Elsevier B.V.oi:10.1016/j.ecolmodel.2011.11.001

© 2011 Published by Elsevier B.V.

of estimating stock-recruitment parameters (Hilborn and Walters,1992).

To deal with the drawbacks of traditional stock-recruitmentanalysis, several researchers have developed other methods. Oneof them is an “errors-in-variables” (EV) approach (Ludwig andWalters, 1981; Schnute, 1994; Schnute and Kronlund, 2002). TheEV method explicitly represents both natural variability and mea-surement errors inherent in the data. Unfortunately, the EV methodrequires that an assumption be made about what proportion of thetotal variation is due to measurement error; this step is necessary toavoid a model identification problem, i.e., a situation where modelparameters cannot be uniquely estimated (Schnute, 1994). How-ever, we rarely have data on that proportion, thus, many authorsassume it is 0.5 (e.g., Ludwig and Walters, 1981). Nevertheless,there is little basis for this assumption, and this proportion’s valuecan potentially greatly influence the resulting parameter estimatesof the stock-recruitment function, as we show later.

Another more advanced method, state-space modeling (Harvey,1989), has been applied to stock-recruitment analysis by sev-eral researchers, including Meyer and Millar (2001), Rivot et al.

(2001), Schnute and Kronlund (2002), and Walters and Martell(2004). Here, we also used a state-space model applied to stock-recruitment data on Pacific salmon (Oncorhynchus spp.), butwe went beyond what other authors have done by developing

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version of it that contains three sources of uncertainty, i.e.,easurement error in spawning stock size, process stochastic-

ty in recruitment, and the effect of time series bias. The latterias arises due to non-random scatter of data points caused by

inkage between a given year’s recruitment anomaly and the subse-uent spawner abundance (Walters, 1985). For the fitting process,e develop a Bayesian approach via Markov chain Monte Carlo

MCMC) sampling to make inferences about the state-space modelClark, 2007). The Bayesian approach is the most general methodor fitting non-linear state-space models (Millar and Meyer, 2000),hereas other methods, such as the EV and extended Kalman fil-

er methods (Schnute and Kronlund, 2002), have more limitationsRivot et al., 2004).

Compared to our model, Walters and Martell (2004) consideredhe effects of time series bias and process error in recruitment, buthey did not consider measurement error in spawner abundance.he state-space models of Meyer and Millar (2001) and Schnute andronlund (2002) are most similar to ours, but they were formulatednly for a cohort (i.e., a single reproductive line) of a salmon popula-ion. Thus, their models cannot be readily applied to the more usualituation in which data sets on spawners and recruits arise fromultiple cohorts from the same population, which must be mod-

led simultaneously (e.g., there are 4 successive cohorts of fish forraser River sockeye salmon (Oncorhynchus nerka), because most ofhem mature at age 4, reproduce, and then die). Rivot et al. (2001)onsidered both process error in recruitment and measurementrror in spawner abundance in their state-space models, but theyssumed that recruitment did not affect the subsequent spawningtock and thus ignored possible effects of time series bias. Our state-pace model includes harvesting and the full time-dynamic lifeycle with feedback between recruitment and subsequent spawnerbundance, which thereby considers time series bias in the data were fitting.

The state-space modeling framework proposed in this paperan be valuable for fish stock-recruitment analysis and manage-ent. Often, the only available data for stock-recruitment analysis

re a single time series of estimates of the number of spawnersnd recruits. Even in this case, the cost for collecting these data isften high. Thus, it will be cost-effective if the state-space modelingpproach can provide improved estimates of parameters over theraditional ones, especially because such parameters affect man-gement decisions (Walters and Martell, 2004).

State-space modeling has been increasingly used in ecologicalpplications to quantify multiple sources of uncertainty (Petermant al., 2000; Harwood and Stokes, 2003; Buckland et al., 2004;ivot et al., 2004). However, due to their complexity, state-spaceodels are computationally very intensive and difficult to fit (dealpine and Hastings, 2002; Clark, 2007). Thus, most of these mod-ls, including the ones that we investigate, have not yet been subjecto extensive evaluation by means of simulations across wide rangesf scenarios/situations to determine their performance relative tohat of simpler standard regression models that only consider oneource of uncertainty (Walters and Martell, 2004).

This study is an attempt to fill this gap by using simulations tossess the performance of a state-space model that explicitly incor-orates measurement error and to compare it to the performancef a traditional stock-recruitment model that ignores measurementrror. We intend to examine the impact of measurement errorn spawning stock size (S) on both models under a wide rangef situations, and identify situations where the consequences ofismeasuring S are substantial and thus remedies for that impact

re essential. For this purpose, we used simulations to generate

umerous data sets with varying information content and uncer-ainty in the data about the estimated parameters, as determinedy combinations of six factors: age at maturity, productivity param-ter values, harvest situations, amount of measurement error,

odelling 224 (2012) 76– 89 77

autocorrelation in time series of annual recruitments due to envi-ronmental effects, and number of data points.

We compare performance of these two models, the traditionaland state-space versions of the Ricker model (Quinn and Deriso,1999), in terms of bias, precision of parameter estimates, and accu-racy of estimates of uncertainty (Adkison and Su, 2001; Browne andDraper, 2006). The statistical properties of model parameters andother derived quantities used for setting management targets arerelevant for conservation and management of fisheries resources(Quinn and Deriso, 1999). To cope with the large computationalburden of our extensive simulations, we developed a customizedMCMC sampling program for this study.

2. Methods

2.1. Models

2.1.1. Salmon population dynamics, fishing, and managementIn this simulation study, we use two Pacific salmon species,

pink salmon (Oncorhynchus gorbuscha) and sockeye salmon, as caseexamples to examine the effect of measurement error in spawnerabundance on stock-recruitment analysis. Because adults of Pacificsalmon all die after spawning (i.e., semelparous) and there is nooverlap of individuals between generations, their dynamics can bedescribed by relationships that connect the number of spawnersand recruits between two generations (Quinn and Deriso, 1999).

Fishing for salmon occurs on homeward-migrating adult returns(recruits) in coastal waters, and salmon management is mainly con-cerned with managing fisheries so as to allow an adequate numberof returning fish to escape the fisheries to spawn (escapement).Thus, data for salmon management mainly come from catch andescapement records. The number of spawners is usually deter-mined from estimates of spawners on the spawning grounds, whichcan be subject to errors due to counting methods and procedures,observers, and physical conditions (Cousens et al., 1982; Su et al.,2001). Catch is obtained from fishing reporting systems and stock-identification methods in mixed-stock fisheries, and is usually moreaccurately estimated than the number of spawners (Quinn andDeriso, 1999).

Pacific salmon exhibit a wide range of life histories (Groot andMargolis, 1991). Whereas pink salmon go to sea directly as fryand invariably mature and return two years after spawning, othersalmon species have various lengths of freshwater and marine resi-dence, leading to variable age at maturity (Larkin, 1988). To simplifyour analysis, we only consider the constant single-age-at-maturitysituation.

2.1.2. The state-space Ricker stock-recruitment model (SSR)As already noted, parameter estimates obtained from tradi-

tional stock-recruitment analysis (defined in Section 2.1.3) thatignores errors in estimates of spawner abundance may be mis-leading (Walters and Ludwig, 1981). To attempt to overcome thisproblem, we use state-space modeling (Harvey, 1989). State-spacemodels are time series models featuring observed variables as wellas unobserved states. The state-space spawner-recruit model (SSR)proposed here uses an observation equation to incorporate theeffects of error caused by mismeasuring spawner abundance anda system equation that deals with process uncertainty caused byenvironmental factors on recruitment. We utilize the state-spacemodel to track the population dynamics of a salmon populationwith a constant age at maturity of k (Fig. 1).

First, the unobserved true number of spawners St at year t isrelated to the measured number of spawners (escapement) Et byan observation equation, where t is a relative year index with t = 1referring to the first year of data collection. As is traditional (Walters

78 Z. Su, R.M. Peterman / Ecological Modelling 224 (2012) 76– 89

α,β,τ

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Fig. 1. Illustration of the population dynamics of an example pink salmon popula-tion with a constant age at maturity of 2 years. Dashed lines connect quantities foryears t, t + 2, t + 4, etc., and solid lines connect quantities for years t + 1, t + 3, etc. Theseeven- and odd-year lines (cohorts) of pink salmon thus have separate populationdynamics and are reproductively isolated. Et is the measured number of spawners,Sos

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Fig. 2. The Ricker spawner (S)–recruit (R) curve and related management-orientedquantities in arbitrary units. R = S along the diagonal replacement line, henceS′ = unfished equilibrium abundance of R and S, i.e., carrying capacity. Maximumsustainable yield or catch (MSY) occurs at SMSY , the spawner abundance at which

t is the unobserved true number of spawners, Rt is the unobserved true numberf recruits, and Ct is the catch. Greek symbols refer to estimated parameters of thetate-space Ricker stock-recruitment model, as defined later in the text.

nd Martell, 2004), we assume a log-normal probability distribu-ion for Et:

t = St exp(vt), t = 1, . . . , T (1a)

here vtiid∼N(0, �2) and denotes measurement error, �2 is the mea-

urement error variance, and T is the last year in the data set.Second, we use two system equations to describe the transitions

f spawners and recruitment between two generations of the sameohort of a salmon population. The first equation defines a stochas-ic stock-to-recruitment process that relates the true spawners St−kwhere k is age at maturity in years) to the true recruitment Rt pro-uced by St−k. This is the Ricker stock-recruitment function (Quinnnd Deriso, 1999; see Fig. 2):

t = ˛St−k exp(−ˇSt−k + wt−k), t = k + 1, . . . , T (1b)

here is a productivity parameter reflecting the number ofecruits per spawner at low spawner abundance (slope at the ori-in), measures the strength of density-dependence and describesow quickly the recruits-per-spawner decrease as spawning stock

ncreases, wtiid∼N(0, �2) is process error, and �2 is the process-error

ariance. The initial state variables are S1 to Sk for the k cohorts. Therror terms vt and wt are assumed to be uncorrelated.

The second system equation defines the transition from theecruitment Rt at t to the spawners St in the presence of fishing,hich occurs only on returning fish (Rt) before they spawn. As noted

bove, catch from many salmon fisheries is usually relatively well-easured compared to escapement (Quinn and Deriso, 1999), soe assume that catch, Ct, is measured without error. In this case, St

s then:

t = Rt − Ct (1c)

arameters of this state-space model (Eqs. (1a)–(1c)) are � = {˛, ˇ,2, �2, S1, . . ., Sk}. This state-space model is illustrated in Fig. 1.

there is the maximum vertical distance between R and the replacement line, indi-cated by the tangent to the curve (thin solid straight line) at RMSY . Peak recruitment(Rp) occurs at spawner abundance Sp = 1/�.

It is more convenient to work with an additive normal errorstructure, so we take the natural logarithms of both sides ofEqs. (1a)–(1c) and let yt = loge(Et), xt = loge(St), rt = loge(Rt), andar = loge(˛), thus transforming Eqs. (1a)–(1c) to the following:

yt = xt + vt , t = 1, . . . , T (2a)

rt = ar + xt−k − exp(xt−k) + wt−k, t = k + 1, . . . , T (2b)

xt = loge(exp(rt) − Ct) (2c)

Parameters of the state-space model in Eq. (2) are � = {ar, ˇ, �2, �2,x1, . . ., xk} and data are Y = {yt, Ct}. Here Ct is a control variable andits values are treated as fixed constants used to update xt. We definethe state vector as r = {rk+1, . . ., rT}. In Eq. (2b), rt implicitly dependson rt−k through Eqs. (2b) and (2c), and the transition of rt from rt−kfollows a first-order Markov process.

2.1.3. Traditional Ricker stock-recruitment model (TSR)Most traditional stock-recruitment analyses assume that

spawning stock size S is measured without error, i.e., �2 = 0 in Eq.(1a). In this case, only process error in recruitment enters the anal-ysis. Based on this assumption, the traditional stock-recruitmentmodel (TSR) is defined as:

Robst = ˛Et−k exp(−ˇEt−k + εt−k), t = k + 1, . . . , T (3)

where the measured number of recruits Robst = Et + Ct, εt∼N(0, �2

ε )is model error, and �2

ε is the model error variance. Dividing bothsides of Eq. (3) with Et−k and taking logarithms yields:

loge

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)= ar − ˇEt−k + εt−k, t = k + 1, . . . , T (4)

This is a simple linear regression model of loge(Robst /Et−k) on Et−k.

Given pairs of data {Et−k, Robst }, standard regression techniques are

often used to obtain parameter estimates for this model (Hilbornand Walters, 1992).

There are two potential estimation problems with Eq. (4).First, using the error-prone Et−k rather than the true and error-free St−k as an independent variable (X) violates one important

assumption of standard regression (no measurement error in theX variable), and that may lead to biased estimates of parameters(Hilborn and Walters, 1992). Second, due to the nature of dynamicstock-recruitment systems, spawning stock size is not a controlled

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ndependent variable (unlike what standard regression assumes),ut instead is stochastic because it is generated by previous recruit-ent, which is also a random variable. This can cause what Walters

1985, 1990) called time series bias in parameter estimates.Biases caused by both measurement error and the time series

ffect often co-exist in stock-recruitment data for fishes andre inseparable, so both sources of bias should be consideredimultaneously (Caputi, 1988). Traditional stock-recruitment (TSR)nalysis does not do this, whereas a state-space stock-recruitmentSSR) analysis does. Here, we compare the performance of thesewo approaches.

.1.4. Deterministic relationships and management-orienteduantities

Management of most Pacific salmon fisheries is based on quan-ities derived from two deterministic relationships between stocknd recruitment at equilibrium conditions: R = ˛S exp(− ˇS) and

= R − C. The spawning stock size (SMSY) that produces a maximumustainable yield (catch) (MSY) can be found as the solution tohe equation: (1 − ˇS) exp(− ˇS) = 1 (Fig. 2). Recruitment at MSYs then RMSY = ˛SMSY exp(− ˇSMSY) and MSY = RMSY − SMSY (Fig. 2).arvest rate at MSY (hMSY) is obtained by hMSY = MSY/RMSY. To sim-lify calculations, Hilborn and Walters (1992) provide the followingood approximations to hMSY and SMSY respectively in the range of

< ar < 3:

MSY ≈ ar(0.5 − 0.07ar) (5a)

MSY ≈ ar(0.5 − 0.07ar)ˇ

(5b)

e use these two equations to calculate hMSY and SMSY from thestimates of ar and ˇ, and evaluate their statistical properties alongith those for other model parameters.

Other quantities we use are the unfished spawningtock size (carrying capacity) S′ = loge(˛)/ˇ, peak recruitmentp = ˛/( exp(1)), and spawning stock size at peak recruitmentp = 1/ˇ (Quinn and Deriso, 1999; see Fig. 2).

.2. Parameter estimation by Bayesian methods

.2.1. Estimation for the state-space modelBayesian inferential methods for the state-space model (Eqs.

2a)–(2c)) are based on the posterior distribution of parameters �nd states r given the data Y, p(�, r|Y) (Clark, 2007). To make infer-nces for �, one needs to construct a marginal posterior distributionf � by integrating out the states r in p(�, r|Y), i.e., p(�|Y) =

∫rp(�,

|Y) dr (Schnute, 1994). However, it is difficult to evaluate p(�|Y)irectly by integration. MCMC simulation techniques bypass theeed to evaluate the high dimensional integral in p(�|Y) by gener-ting dependent draws from the posterior distribution p(�, r|Y) viaarkov chains (Gelman et al., 2004). From these simulated values,

osterior summaries or the marginal posterior distributions of anyarameters (or functions of the parameters) can be evaluated byimply using the simulation draws for these parameters.

To derive the joint posterior distribution p(�, r|Y) for MCMCampling, first we observe that p(�, r|Y) ∝ p(�, r)p(Y|�, r) based onayes’ theorem (Gelman et al., 2004), where p(Y|�, r) is the likeli-ood of the data given the states and the parameters, and p(�, r)

s the joint prior distribution of � and r. Based on the conditionalndependence assumption made for yt (Eq. (2a)), the likelihood isiven by

(Y |�, r) =T∏

t=1

N(yt |xt, �2) (6)

odelling 224 (2012) 76– 89 79

The prior p(�, r) can be further factored into the product of the priordistribution of the parameters, p(�), and the hierarchical prior dis-tribution of the states given �, p(r|�). Based on the Markov propertyof the system equation (Eq. (2b)) p(rt|xi, . . ., xt−2k, xt−k) = p(rt|xt−k),where xi is one of {x1, . . ., xk}, the joint prior distribution of thestates r given �, p(r|�) = p({rk+1, ..., rT} |�), can be derived as (Petriset al., 2009):

p(r|�)= p(rk+1|x1, �)p(rk+2|x2, �) . . . p(rT |xi, . . . , xT−2k, xT−k, �)= p(rk+1|x1, �)p(rk+2|x2, �) . . . p(rT |xT−k, �)

=T∏

t=k+1

N(rt |ar + xt−k − exp(xt−k), �2)

(7)

Substituting Eqs. (6) and (7) into p(�, r|Y), we obtain the joint pos-terior distribution p(�, r|Y) as:

p(�, r|Y ) ∝ p(�)p(r|�)p(Y |�, r)

= p(�)T∏

t=k+1

N(rt |ar + xt−k − exp(xt−k), �2)T∏

t=1

N(yt |xt, �2) (8)

To perform Bayesian estimation, we need to specify prior distri-butions for the parameters �. First, we assume that the initial statevariables xt, t = 1, . . ., k, follow a normal distribution, N(m0, V0), withm0 = 0 and V0 = 103. Because xt values are in a log scale, these pri-ors are very diffuse. We assign uniform prior distributions for theRicker ar = loge(˛) and parameters (Millar, 2002): ar ∼ U(0, ∞) and

∼ U(0, ∞). We also assign uniform priors for � and �: � ∼ U(0, ∞)and � ∼ U(0, ∞) (Gelman and Hill, 2007) because the often-usednon-informative prior p(�2) ∝ 1/�2 or inverse gamma distributionIG(ε, ε) with small ε can create problems for hierarchical prior vari-ance parameters (such as �2) with values near zero (Gelman andHill, 2007).

We estimate model parameters and states for the state-spacemodel (Eq. (2)) by sampling from the posterior distribution in Eq.(8) using a Markov chain Monte Carlo method (MCMC) (Gelmanet al., 2004). Although WinBUGS (Lunn et al., 2000) can be usedto make inferences for our state-space model, it is not feasibleto use it for this extensive simulation study because of the largecomputational burden for the nearly 20,000 simulated datasets weexamined. To overcome this computational problem, we developedour own MCMC sampling program based on the full conditionaldistributions derived from the posterior distribution (Eq. (8)). InAppendix A, we provide two full conditional distributions, one forupdating the vector (ar, ˇ) in a block, and the other for updatingthe individual recruitment rt. The full conditional distributions ofthe other parameters can be derived easily and are available fromthe first author.

2.2.2. Estimation for the traditional Ricker stock-recruitmentmodel

For the traditional Ricker model (Eq. (4)), we have the followingposterior distribution:

p(�|Y ) ∝ p(�)T∏

t=k+1

N(loge(Robst )|ar + loge(Et−k) − ˇEt−k, �2

ε ) (9)

where � = {ar, ˇ, �2ε }. As for the SSR model, we assume ar ∼ U(0, ∞)

and ∼ U(0, ∞), and �ε ∼ U(0, ∞). We use a MCMC sampler toupdate ar and by a bivariate normal conditional distribution simi-lar to Eq. (A1) and �2

ε by an inverse-gamma conditional distribution.

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� Values Notes

˛* 2, 4, 8 These values correspond to the 10thpercentile, median, and 90th percentileof values obtained from data on 43pink salmon stocks by Su et al. (2004).

ˇ* 1 determines the absolute level ofpopulation size, as shown by theunfished equilibrium carrying capacityS′ = loge(˛)/ˇ. Different populationshave different abundance levels, butwe can always scale their abundance toa unit that makes the magnitude of ˇone. Therefore, in the simulations, weset the true value of to one. Thisproduces populations with size nearone unit of fish, e.g., one unit might be104 fish for a Pacific salmon population.

�*2 0.5 A relatively large process errorvariance

�*2 0.0, 0.17, 0.5, 1.5 �*2 = 0 is used to test the independenteffects of time series bias on parameterestimates in the absence ofmeasurement error.

�* 0, 0.25, 0.5, 0.75 �* = �*2/(�*2 + �*2)k 2, 4 k = 2 and 4 are used to simulate pink

and sockeye salmon, respectively.� 0, 0.5 The first-order autocorrelation

0 Z. Su, R.M. Peterman / Ecolo

.3. Simulation analyses

.3.1. The simulation modelWe simulate escapement, recruitment, and catch data for a

ypothetical Pacific salmon stock with a constant age at maturity k.e generate each simulated data set in two phases. In phase 1, the

nitialization phase, starting from t = −25k, we set St to the unfishedarry capacity R′ = S′ = loge(˛*)/ˇ* for the first k years. Here we use ansterisk to denote parameters used in the simulation model to dif-erentiate them from the parameters used in the estimation modelEq. (1) or (2)). Then we sequentially generate new recruits (Rt+k),atch (Ct+k), and spawners (St+k) up to t = k by

Rt+k = ˛∗St exp(−ˇ∗St + wt)wt = �wt−1 + ut, ut∼N(0, �∗2)Ct+k = ht+kRt+k

St+k = Rt+k − Ct+k

(10)

here ht is set to a constant initial harvest rate h1, which we specifyn Section 2.3.2. To allow for possible autocorrelated environmen-al effects on recruitment, we assume that wt follows a first-orderutoregressive process (AR(1)), and denote � as the autocorrelationoefficient. At the end of phase 1, we obtain realized values of S1 tok and keep them as the initial state values for phase 2. Other val-es from this phase are discarded. The initial states thus generatedre random and depend on the historical fishing level h1 (whiche explore later with different values), thus addressing the related

oncerns of Kope (1988).In phase 2, the data generation phase, starting from t = 1 and

sing the realized values of S1 to Sk, we generate Rt+k, Ct+k, and St+ksing Eq. (10) up to t = T − k. More details for ht are given in Section.3.2. We also calculate Et = St exp(vt) with vt ∼ N(0, �*2) for t = 1,

. ., T. Finally, {Et}Tt=1 and {Ct}T

t=k+1 are kept as the data used for eachimulation.

Because we use time-dynamic models to produce the simulatedata to which we apply the two estimation models, time series bias

mplicitly contributes to the resulting estimation errors, along witheasurement and process errors.

.3.2. Simulation scenariosWe consider a wide range of conditions for six factors that may

ffect performance of our two models: (1) values of the true Ricker ˛arameter: ˛*; (2) types of harvest situations; (3) proportion of theotal error variance (V = �*2 + �*2) that is due to measurement error:* = �*2/V; (4) values of age at maturity, k; (5) values of the auto-orrelation coefficient in process errors, �; and (6) number of dataoints in the time series, T. The true parameter values considered

n the simulations are listed in Table 1. Each simulation scenarioorresponds to one combination of these six sets of conditions.

We consider three types of harvest situations in our simula-ions. First, we generate harvest rates that vary over time using

piecewise linear equation:

t ={

h1+(h2 − h1)(t − k)/(0.5T), k+1 ≤ t ≤ k+0.5Th2+(h1 − h2)(t−k−0.5T)/(0.5T), k+0.5T < t ≤ T

(11)

or t = k + 1, . . ., T, where h1 is the initial harvest rate used in phase of the simulation model (Eq. (10)), and h2 is a maximum har-est rate. We set h1 = 0.2hMSY, and h2 = 1.0hMSY, and refer to thisarvest situation as “variable harvests”, where hMSY is the truearvest rate at MSY. This harvest situation produces a relativelyigh-contrast fishing history, which corresponds to a fishery start-

ng data collection when exploitation is very low (0.2hMSY), then

ith a harvest rate that increases linearly to 1.0hMSY at t = k + 0.5T,

nd decreases to a small value at the end of the fishing history. Thetock-recruitment data generated by this harvest situation con-ain data points for both high and low spawner abundance, thus

coefficientT 16, 28, 60 Number of data points

providing information for potentially estimating and reason-ably well.

We also generate two fixed-harvest rate situations using harvestrates of 0.2hMSY or 1.15hMSY, which we denote as “low harvests” and“high harvests”, respectively. The low harvest situation results in afishery that starts its data collection when exploitation is low andthis low harvest rate continues through time, i.e., ht = 0.2 × hMSY,for t = −25k to T. The stock-recruitment data generated by this har-vest situation contain data points at both low and high spawningabundance, thus providing information for estimating as wellas (Fig. 3a and b). In contrast, the high harvest situation withht = 1.15 × hMSY for t = −25k to T results in a fishery starting data col-lection with a highly exploited population and with persistent highharvest rates through time. The stock-recruitment data generatedby this harvest situation only contain data points at low spawn-ing abundance, thus providing little information for estimating ˇ(Fig. 3c and d). All three situations have many real-world analoguesfor a variety of fish species.

We determine statistical properties of the two estimation meth-ods across Monte Carlo simulations over a broad range of scenariosrepresented by numerous combinations of these six factors. Ofthese scenarios, 36 are composed of combinations of three levels of˛*, three types of harvest situations, and four levels of �* for k = 2,�*2 = 0.5, � = 0, and T = 28 (Table 1). We treat these 36 simulations asour baseline scenarios. In numerous sensitivity analyses, we evalu-ate the effect on baseline performance measures of age at maturity(k), autocorrelation in recruitment (�), and number years of data T.

2.3.3. Performance measuresFor each simulation scenario, we generate 300 sets of catch

and escapement data using the simulation model (enough to pro-duce stable performance metrics based on trials). For each data set,we run the two estimation models, SSR and TSR, separately and

obtain posterior median (�), 2.5th (�2.5), 25th (�25), 75th (�75), and97.5th (�97.5) percentiles from the posterior distribution of eachparameter (�) from the MCMC sampling draws. For each of the 300simulation runs of each scenario for a model, we record whether

Z. Su, R.M. Peterman / Ecological Modelling 224 (2012) 76– 89 81

05

1015

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ruits

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)

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45

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ruits

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)

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

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0 5 10 15 20

05

1015

20

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Spawners (S)

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Fig. 3. Examples of single Monte Carlo runs of the simulation model that generated four simulated stock-recruitment data sets for combinations of two levels of ˛* (trueRicker parameter) and two types of harvest situations (fractions of hMSY , “Low” = 0.2hMSY and “High” = 1.15hMSY), given that �*2 = 0.5, �* = 0.75, � = 0, and T = 28 for stockswith age at maturity at 2. Solid dots represent true spawner and recruit points. Open circles represent observed data points. A dotted straight line indicates displacement ofan observed point from a true point. A thick solid curved line represents the true stock-recruitment curve. A long-dashed curved line represents the stock-recruitment curveestimated from the state-space Ricker model (SSR). A dashed curved line represents the stock-recruitment curve estimated from the traditional Ricker stock-recruitmentm those

avpRviC

si7ocvtiiriewoa

2v

p

used N = 0.43 million, B = 30,000, and a thinning interval of 50. Wealso provided the results for a sensitivity analysis of different priorspecifications for �2 and �2 in Appendix D.

Table 2Averages of 20 posterior median values for each of four parameters from theBayesian state-space model (SSR) or the WinBUGS program (BUGS) for k = 2, ˛* = 4,�*2 = 0.5, �* = 0.25, � = 0, T = 28, and the low or high harvest situation. � stands for“parameter”.

� Low High

SSR BUGS SSR BUGS

odel (TSR). Inset graphs in panels (a) and (c) are close-ups of lower-left corners of

95% posterior interval (�2.5, �97.5) contained the true parameteralue (�*), and calculate the percentage relative bias (RB) of theosterior median of each parameter compared to its true value asB = 100(� − �∗)/�∗. We also calculate the coefficient of quartileariation (CQV) using (�75 – �25)/(�75 + �25) to quantify uncertaintyn parameter estimates. These two quantities, posterior median andQV, are robust to outliers.

To summarize results for an estimation model across the 300imulated data sets of a scenario, we calculate the median andnterquartile range (IQR, or the difference between the 25th and5th percentiles of a distribution) from the empirical distributionf the 300 RBs, and median of 300 CQVs for each parameter. We alsoalculate the actual coverage probability of a 95% probability inter-al from SSR and TSR as the percentage of intervals that containhe true parameter values. Because of the increasing importancen recent years of taking into account estimates of uncertaintiesn parameters and observed state variables related to conservationisks, such a coverage probability is an extremely important, yetnfrequently used performance measure for evaluating parameterstimation methods (Adkison and Su, 2001). For instance, if theidth of the probability distribution of a quantity is either under-

r over-estimated, this will then bias calculations of risk in a stockssessment.

.3.4. Preliminary results and MCMC sampling algorithmerification

First, we evaluated the implementation of our MCMC sam-ler for the Bayesian state-space model (SSR) by comparing

panels.

its parameter estimates with those obtained from a WinBUGSprogram (Appendix B) for 20 data sets from the low or high harvestsituation with k = 2, ˛* = 4, �*2 = 0.5, �* = 0.25, � = 0, and T = 28. Theyproduced almost identical results (Table 2).

For our MCMC samplers, we conducted extensive pilot runsbefore the simulations to determine by means of MCMC diagnostictests (Gelman et al., 2004) appropriate MCMC sampling details suchas the total number of iterations (N), the length of burn-in period(B), and the thinning interval. Detailed descriptions of the diagnos-tics are given in Appendix C. Based on these diagnostics, we usedN = 1.5 million, B = 0.5 million, and a thinning interval of 100 for allour calculations for the state-space model. In contrast, for TSR, we

˛ 4.67 4.66 5.26 5.27ˇ 1.06 1.06 1.23 1.24�2 0.42 0.41 0.32 0.32�2 0.24 0.24 0.80 0.80

8 gical M

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3

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2 Z. Su, R.M. Peterman / Ecolo

. Results

.1. Influence of stock productivity, harvest situations, andmount of measurement error

For our 36 baseline results, given true values of age at matu-ity k = 2, process error variance �*2 = 0.5, first order autocorrelationoefficient � = 0, and the number of years T = 28, we comparearameter estimates of the two models (traditional Ricker modelnd the state-space version of it) based on three performance mea-ures for four model parameters (˛, ˇ, �, and �) and two derivedanagement parameters (SMSY and hMSY).

.1.1. Bias in ˛, ˇ, SMSY, and hMSYFirst, in the absence of measurement error (�* = 0), there is little

ias in parameter estimates in most cases except for estimates of ˇnd SMSY with the high harvest rate situation at ˛* = 2 or 8 (Fig. 4).hese latter cases reflect the effects of time series bias.

Measurement error in observed spawner abundance can cause

iased parameter estimates for both the Bayesian state-space (SSR)nd traditional Ricker stock-recruitment (TSR) models (Fig. 4).lthough a moderate amount of measurement error (�* ≤ 0.25)ay not induce much bias in most scenarios, a large amount can

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SY)

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0 .25 .5 .75 0 .25 ρ* = Propo rtion of total error varia

ig. 4. Interquartile range (IQR: 25th to 75th percentiles, shown as vertical lines) and mstimates of four parameters of two models, i.e., ˛, ˇ, hMSY , and SMSY (one parameter in eacnd T = 28, and different combinations of ˛* (2, 4, 8), harvest situations (Low, Variable, atate-space model (SSR), “X” = traditional Ricker model (TSR). The SSR methods have som

odelling 224 (2012) 76– 89

cause substantial bias in many situations (Fig. 4). The impacts ofmeasurement error vary widely depending on stock productivity˛*, historical harvest patterns, amount of measurement error asindicated by �*, models considered, and parameters examined, sowe categorize results into various cases of these conditions andhighlight key results below.

Stock productivity ˛* and historical harvest patterns have a sub-stantial effect on both the sign and magnitude of the bias, whereasthe amount of measurement error mainly affects the magnitude ofbias in estimates of all four estimated quantities, ˛, ˇ, SMSY, andhMSY (Fig. 4). First, SSR overestimates for low-productivity stocks(˛* ≤ 4) (Fig. 4a–f, first row). This bias increases with decreasing˛*, increasing �*, and increasing average harvest rates. As a result,

is severely overestimated by SSR with larger values of �* (>0.5)at ˛* = 2 with all three harvest situations (Fig. 4a–c). In contrast,in those same situations, TSR produces much less (but still posi-tively) biased estimates of than SSR at large values of �* at ˛* = 2(Fig. 4a–c) or at ˛* = 4 with the variable or high harvest situation(Fig. 4e and f). In other cases, estimates of from TSR are either

more negatively biased (Fig. 4d, g and h) or more positively biased(Fig. 4i) than those produced by SSR.

SSR overestimates by a large amount at large values of �*at ˛* = 2 (Fig. 4a–c, second row), produces just slightly biased

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nce due to measurement error

edians (symbols on lines) of 300 estimates of percentage relative biases (RBs) inh row, respectively, from top to bottom). Results are shown for k = 2, �*2 = 0.5, � = 0,nd High), and �* (0, 0.25, 0.5, 0.75). Models are indicated by solid dots = Bayesiane RB(ˇ) values >270% in the second row of the vertical panel (c).

Z. Su, R.M. Peterman / Ecological M

Table 3Medians of 300 estimates of percentage relative biases (RBs) for � and � from theBayesian state-space model for k = 2, �*2 = 0.5, � = 0, and T = 28, and different com-binations of ˛* (2, 4, 8), harvest situations (Low, Variable, High), and �* (0.25, 0.5,0.75). Values greater than |40|% are highlighted with boldface.

Harvest situation �* ˛* = 2 ˛* = 4 ˛* = 8

� � � � � �

Low 0.25 28 −17 23 −12 4 00.5 13 −26 16 −25 3 −30.75 5 −46 9 −49 8 −26

Variable 0.25 4 −5 26 −10 18 −60.5 12 −10 23 −25 23 −150.75 9 −49 16 −54 28 −44

High 0.25 −3 −2 93 −21 492 −61

eaom(busˇ

t(t�v

aa(vaaba

btTab

3

m�˛o(u

3

lcp�rw

estimates.

0.5 7 −4 75 −39 273 −660.75 18 −46 54 −61 146 −68

stimates of at ˛* = 4 (Fig. 4d–f), but underestimates (Fig. 4g–i)t ˛* = 8. With the low and variable harvest situations at each valuef ˛*, TSR mostly underestimates at large values of �*, and theagnitude of underestimation increases with increasing ˛* and �*

Fig. 4a–b, d–e, and g–h, second row). This leads to more negativelyiased estimates of from TSR than those from SSR at larger val-es of �* at ˛* = 4 or 8 (Fig. 4d–e and g–h). With the high harvestituation, TSR tends to overestimate ˇ, and in this case, the bias in

estimates is largest at intermediate levels of �* (Fig. 4c, f and i).The pattern of bias of estimates of hMSY across scenarios (Fig. 4,

hird row) is very similar to that of ˛. However, the relative biasesRBs) of hMSY are much smaller than those of ˛. Additionally, despitehe large bias in the estimates of hMSY from both SSR and TSR when* > 0.25 at ˛* = 2, bias of the hMSY estimates from both models isery slight at ˛* = 4 or 8 (Fig. 4).

The magnitude of RBs of SMSY from SSR (Fig. 4, fourth row) islso much smaller than that of and ˇ. RB(SMSY) from SSR is smallnd not sensitive to �* except for cases at ˛* = 8 with variablefor �* = 0.75, Fig. 4h) and high (for all levels of �*, Fig. 4i) har-est rates. Compared to SSR, TSR can severely overestimate SMSYt larger values of �*(0.75) with the low and variable harvest situ-tions (Fig. 4a–b, d–e and g–h). TSR also causes highly negativelyiased estimates of SMSY at ˛* = 8 with the high harvest situation atll levels of �* (Fig. 4i).

As noted above, at ˛* = 8 with the high harvest situation (Fig. 4i),oth models have difficulties with estimating and SMSY becausehe data contain little information on at all levels of �* (Fig. 3d).SR overestimates by about 45–85% and underestimates SMSY bybout −25% to −42%. In contrast, SSR underestimates by −32%ut overestimates SMSY by about 40–55%.

.1.2. Bias in � and �SSR tends to overestimate � but underestimate �, and esti-

ates of � become more negatively biased with the increase of* (Table 3). With the low and variable harvest situations for all* and the high harvest situation at ˛* = 2, SSR produces estimatesf � and � with relatively small bias (except for � with �* = 0.75)Table 3). However, SSR severely overestimates � at ˛* = 4 or 8 andnderestimates � at ˛* = 8 with the high harvest situation (Table 3).

.1.3. Measures of uncertaintyCoefficients of quartile variation (CQVs) from SSR are usually

arger than those from TSR, but SSR produces close to nominal 95%overage probability more often than TSR (Table 4b). The coveragerobability of TSR is especially poor for SMSY at ˛* = 4 and 8 and when

* ≥ 0.5 with the low or variable harvest situation. High harvestates also lead to deteriorated coverage probability for both modelshen �* ≥ 0.5 (Table 4b).

odelling 224 (2012) 76– 89 83

3.2. Influence of age at maturity k and autocorrelation

Both k and � have little effect on relative bias of parameter esti-mates for hMSY and SMSY for both SSR and TSR (Table 5, compare RBsfor different k (or �) at the same � (or k) and �* for the same model).Analogous effects of k and � are found on RB of other parameters(results not shown).

3.3. Influence of the number of data points T

Increasing the number of data points (T) leads to less biasedestimates (for ˛* = 2 and 4) or no change (for ˛* = 8) in the estimatesof ˛, ˇ, hMSY, and SMSY of SSR, but leads to more biased estimates ofSMSY at all ˛* and ˛, ˇ, hMSY at ˛* = 4 and 8 for TSR (Table 6).

3.4. Using known � in the estimation of SSR

As observed in Section 3.1, SSR has difficulties with fitting datafrom the high harvest situation, especially at ˛* = 8. For these data,we evaluate whether fixing � to 0.5 (i.e., assuming that half the totalvariance is due to measurement error) can ease the problem. Com-pared to our baseline case where � is a free parameter, SSR with� = 0.5 has a substantial effect on estimates of and SMSY at ˛* = 8with the high harvest situation (Table 7). This case reverses thesign of bias for the estimates of and SMSY. At ˛* = 2 and �* = 0.75,fixed � reduces the bias in the estimates of ˛, ˇ, and hMSY substan-tially.

4. Discussion

4.1. Conclusions

This paper makes three new contributions to the analysis ofone of the most fundamental relationships in modeling of fishpopulation dynamics, the stock-recruitment relationship. First, weextended previous Bayesian state-space stock-recruitment mod-els for salmon populations (Meyer and Millar, 2001; Schnute andKronlund, 2002) to take into account both measurement error andprocess variation together in a time-dynamic model that gener-ated time-series bias as well. We also developed full Bayesianestimation procedures and a computer program for this nonlinearstate-space model. Second, by evaluating our Bayesian state-space stock-recruitment model across broad ranges of simulationconditions, we contributed to the understanding of nonlinearstate-space models that consider multiple sources of uncertainty,in particular the effects of measurement error. Third, we providednew insights into the effects of measurement error on the tradi-tional stock-recruitment models under combinations of conditionsnot considered previously.

From our results, we found that measurement error in spawn-ing stock size might lead to bias in parameter estimates for boththe Bayesian state-space Ricker stock-recruitment model (SSR) andthe traditional Ricker stock-recruitment model (TSR). In addition,performance of the two models varies considerably with harvestsituations, stock productivity, and amount of measurement errorin data. All three of these factors can alter the information contentabout parameters in the data, as discussed in Sections 4.2 and 4.3below, thereby affecting the performance of the models at estimat-ing those parameters. Increasing the number of data points may bebeneficial to the state-space model but may lead to more biasedparameter estimates for the traditional Ricker model. On the otherhand, age at maturity and autocorrelation are less influential to the

To summarize the results, we divide our data into two typesbased on information contrast in spawner abundance (S) as deter-mined by stock productivity and harvest situations. First is a

84 Z. Su, R.M. Peterman / Ecological Modelling 224 (2012) 76– 89

Table 4(a) Medians of 300 coefficients of quartile variation (CQVs), and (b) Actual coverage probability of 95% intervals (CVRG) for hMSY and SMSY , under ˛* (2, 4, 8), �* (0, 0.5, 0.75), andthree harvest situations (Low, Variable, High), with k = 2, �*2 = 0.5, � = 0, and T = 28. Models are: SSR: Bayesian state-space model; TSR: traditional Ricker stock-recruitmentmodel. � stands for “parameter”.

Harvest � Model ˛* = 2 ˛* = 4 ˛* = 8 Average

0 0.5 0.75 0 0.5 0.75 0 0.5 0.75

(a) CQV (%)

Low hMSY SSR 23 25 24 9 15 17 4 7 10 15TSR 19 22 27 8 14 21 4 11 20 16

SMSY SSR 15 19 24 8 15 21 5 13 19 15TSR 13 17 24 6 11 19 4 10 19 14

Variable hMSY SSR 20 22 19 9 11 12 4 6 8 12TSR 18 19 21 8 11 13 4 7 10 12

SMSY SSR 16 20 22 11 16 19 7 16 20 16TSR 15 16 20 8 11 15 5 9 14 13

High hMSY SSR 16 16 15 9 10 8 8 8 7 11TSR 15 15 16 7 7 7 3 3 4 9

SMSY SSR 25 23 23 26 23 22 28 25 25 24TSR 22 18 19 21 15 16 30 20 18 20Average 18 19 21 11 13 16 9 11 14 15

(b) CVRG (%): values ≤85% are highlighted in boldface

Low hMSY SSR 97 96 92 98 99 99 97 99 99 97TSR 98 98 98 97 88 92 97 72 65 89

SMSY SSR 98 96 97 100 99 99 98 96 96 98TSR 97 97 93 96 83 68 94 58 48 82

Variable hMSY SSR 94 86 71 94 96 96 95 97 99 92TSR 93 92 93 95 96 98 92 88 86 93

SMSY SSR 98 97 96 99 93 88 99 78 38 87TSR 96 90 74 96 56 18 94 29 8 62

High hMSY SSR 94 74 54 97 88 78 92 95 94 85TSR 95 80 77 95 80 77 94 76 69 83

SMSY SSR 96 82 78 97 92 90 81 75 60 83TSR 95 81 72 96 77 49 93 70 60 77Average 96 89 83 97 87 79 94 78 69 86

Table 5Medians of 300 estimates of percentage relative biases (RBs) of hMSY and SMSY from the Bayesian state-space (SSR) and the traditional Ricker stock-recruitment (TSR) modelsfor two levels of age at maturity k (2, 4), two levels of first-order autocorrelation coefficients (� = 0 or 0.5), and four levels of �* (0, 0.25, 0.5, 0.75) with ˛* = 4, �*2 = 0.5, T = 28,and the variable harvest history. � stands for “parameter”.

� Model k � = 0 � = 0.5

�* = 0.0 �* = 0.25 �* = 0.5 �* = 0.75 �* = 0.0 �* = 0.25 �* = 0.5 �* = 0.75

hMSY SSR 2 4 9 16 22 1 3 11 224 5 6 11 21 6 7 12 21

TSR 2 2 0 −3 −4 −2 −4 −4 −44 2 −1 −3 −2 2 1 −1 0

SMSY SSR 2 −1 4 12 24 2 6 14 224 −1 7 17 28 −1 6 16 32

ls˛tos

TMr

TSR 2 −1 10 36

4 −2 9 30

ow-information type that exhibits low contrast in S, including dataets at a productivity of ˛* = 2 with all harvest situations and at* = 4 or 8 with high harvest rates. Second is a high-information

ype that has high contrast in S, including data sets generated by lowr variable harvest rates at ˛* = 4 or 8. Overall, the Bayesian state-pace model (SSR) works best for informative data sets. With the

able 6edians of 300 estimates of percentage relative biases (RBs) in four estimated parame

ecruitment model (TSR), for three levels of the number of data points (T), with ˛* = (2, 4,

Model � ˛* = 2 ˛* = 4

T = 16 T = 28 T = 60 T = 16

SSR 72 45 16 38

ˇ 75 43 17 6

hMSY 63 44 19 16

SMSY −13 −8 2 7

TSR ˛ 32 19 11 0

ˇ 22 4 −5 −20

hMSY 34 21 14 0

SMSY 2 16 23 20

133 2 9 30 119110 −3 6 29 109

low or variable harvest situation, it is the best performer in termsof estimating SMSY (for all ˛*), (for ˛* = 4 or 8), and and hMSY (for˛* = 8) with the smallest relative bias, and it gives the most accurate

estimate of uncertainty in parameter estimates (i.e., 95% coverageprobabilities). For data sets with poor information (e.g., ˛* = 2), thesimpler model, TSR, often outperforms SSR in terms of estimating

ters from the Bayesian state-space model (SSR) and the traditional Ricker stock- 8), k = 2, �* = 0.5, � = 0, and the “variable” harvest situation.

˛* = 8

T = 28 T = 60 T = 16 T = 28 T = 60

36 22 5 6 21 −3 −24 −25 −25

16 10 1 2 112 13 33 32 34−5 −10 −18 −22 −30

−30 −35 −33 −38 −44−3 −6 −6 −8 −1236 42 34 46 56

Z. Su, R.M. Peterman / Ecological M

Table 7Median of 300 estimates of percentage relative biases (RBs) of estimated parameters(�) from the Bayesian state-space model with � as an estimated parameter (free) andthe same model except with � fixed to 0.5 (fixed), for k = 2, �*2 = 0.5, � = 0, T = 28, anddifferent combinations of ˛* (2, 8) and �* (0.25, 0.75) with the high harvest situation.

� � ˛* = 2 ˛* = 8

�* = 0.25 �* = 0.75 �* = 0.25 �* = 0.75

˛ Fixed 18 64 −5 24Free 19 161 10 17

ˇ Fixed 47 95 25 45Free 55 164 −34 −34

hMSY Fixed 20 58 −1 6Free 22 103 3 4

˛m

4s

mctastrscgorSleirpr

aleAhet

dcecefebfiTepdt

SMSY Fixed −15 −18 −18 −27Free −20 −27 45 55

and hMSY. However, TSR can severely overestimate SMSY when S iseasured with large error in low and variable harvest situations.

.2. Stock productivity, harvest situation, and contrast inpawner abundance

Broadly speaking, the performance of the two models is deter-ined by the information about the model parameters that is

ontained in the data (Adkison, 2009), which is a function ofhe magnitude of measurement error and variation in spawnerbundance, S. This general point is well known for traditionalpawner-recruit analysis (Hilborn and Walters, 1992). The varia-ion in S is in turn influenced by stock productivity (˛) and harvestates, as shown in Fig. 3. In the absence of harvesting, a spawningtock of a salmon population varies around its unfished carryingapacity S′ = loge(˛)/ based on the Ricker model. Therefore, for aiven ˇ, the range of S increases with increasing value. On thether hand, a high harvest rate has the effect of restricting theange of S. The combined effect of ˛ and harvesting is to restrict

of low-productivity stocks (˛* = 2) to very low values even at theow harvest situation (Fig. 3a). Similarly, the range of S from highlyxploited stocks (1.15hMSY) is also severely restricted to low spawn-ng abundance, even at large values of ˛* (Fig. 3d). In contrast, a wideange of S, including high spawner abundance, is produced by high-roductivity stocks with a low (Fig. 3b) or variable exploitationate.

The effects of contrast in S are apparent in our results. In ournalyses of stocks with low productivity (˛* = 2), contrast in S isow, and parameter estimates for ˛, ˇ, and hMSY have the largeststimation biases compared to scenarios with higher productivity.s well, for a given ˛*, estimation bias is the largest with the higharvest situation, and coverage probabilities are the most under-stimated. Therefore, scenarios with high harvest rates producedhe worst performance measures.

Although the important influence of contrast in spawner abun-ance is very well known (Hilborn and Walters, 1992), theontribution to that contrast of different values has not beenxplored thoroughly, even though it clearly influences how much San change over time. For instance, Kehler et al. (2002) explored theffect of different ˛, but only for the standard estimation methodor the basic Ricker model (our TSR). Our results for TSR regardingffects of the range of S on estimates of are consistent with theirs,ut they concluded that the underlying true is not an importantactor, unlike our study. The reason for these different conclusionss that the simulation model of Kehler et al. (2002) differs from ours.hey first generated S based on uniform distributions and then gen-

rated R according to the Ricker model. This static method ignoresossible time-series effects that we simulated here with our time-ynamic model. Thus, we conclude that the magnitude of bias dueo measurement error is influenced by true productivity (˛*).

odelling 224 (2012) 76– 89 85

4.3. Effects of measurement error

We can understand the effects of measurement error in spawnerabundance (S) on parameter estimates for both models by exam-ining how measurement error affects the scatter of the observedstock/recruitment (SR) points relative to those of the true datapoints (Fig. 3). Measurement error has the effect of displacingobserved data points (Et, Robs

t+k) from corresponding true points

(St, Rt+k) both horizontally and vertically (Fig. 3) (Schnute andKronlund, 2002). The direction and magnitude of such a displace-ment depend on where the true data point falls relative to the stocksize at the peak recruitment Sp = 1/ˇ (Kehler et al., 2002). A true datapoint with S < Sp tends to be displaced more vertically, whereasa true data point with S > Sp tends to be displaced more horizon-tally (Kehler et al., 2002). The overall effect of measurement errorwill thus be influenced by where the bulk of true data points islocated relative to Sp on the x-axis, which is in turn affected byharvest rates and true value, as described in the previous sec-tion.

Lightly exploited stocks have more data points with S > Sp (Fig. 3aand b) than heavily exploited stocks (Fig. 3c and d). For thesedata, measurement error has a larger chance of producing datapoints with large horizontal and vertical displacements and tendsto spread the true SR points right- and upward (Fig. 3a and b).This produces observed SR points that appear to have lower ini-tial slope (˛) and lower density-dependent effect (ˇ) values thanthe true SR points. In contrast, for spawning stocks that mainlyfall below Sp, such as those with the high harvest rates, mea-surement error tends to shift the true SR points left and upward,producing observed SR points that appear to have larger initialslope (˛) and larger ˇ values than the true SR points (Fig. 3c andd).

The change in observed scatter caused by measurement errorcan have a direct impact on the parameter estimates of TSR becauseTSR tries to fit the observed recruitment data directly (Eq. (9)). Forstocks subject to low harvest rates (Fig. 3a and b), observed SRpoints can greatly shift the fitted SR curve of TSR toward the right,causing large underestimates of and ˇ. On the other hand, forstocks subject to very high exploitation, TSR tries to fit the narrowrange of observed data and overestimates and (Fig. 3d).

Unlike TSR, SSR takes measurement error into account in theparameter estimation (Schnute and Kronlund, 2002). Thus, forinformative data, as from ˛* = 8 with the low harvest situation, thefitted SR curve from SSR is closer to the true curve than that ofTSR, and SSR improves parameter estimates in this case over TSR(Fig. 3b).

The effects of measurement error on the traditional Rickermodel have been investigated via simulation by several early stud-ies (e.g., Walters and Ludwig, 1981; Caputi, 1988). However, thosestudies only explored a limited range of scenarios compared toour analyses. For instance, Walters and Ludwig (1981) only testeda single value of ˛* = 2.72. They found that TSR overestimated˛, ˇ, and hMSY, but underestimated SMSY for stocks subject towidely different initial exploitation rates in a closed-loop simu-lation. Although these biases are observed in some of our resultsfor similar scenarios (˛* = 2), we found, for productive stocks with˛* = 4 or 8, that TSR is more likely to produce negatively biasedestimates of ˛, ˇ, and hMSY, and positively biased estimates ofSMSY when harvest rates are either persistently low or widely vari-able.

4.4. Management implications

Our results also contribute useful new insights for discussionsabout the relative merits of management targets based on optimalescapement, SMSY, compared to targets based on optimal percent

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arvest rate, hMSY (Walters and Martell, 2004). Theoretically, aim-ng to achieve SMSY will produce the largest sustainable catch overhe long term. However, in practice, SMSY, which is a function of ˛nd 1/ˇ, is frequently difficult to estimate with confidence due toarge errors in estimating spawner abundance and poor estimatesf due to insufficient data at high S. Furthermore, due to practi-al difficulties of fine-tuning harvesting in small increments, SMSY isifficult to achieve even if it were known perfectly. These disadvan-ages of SMSY also apply to more biologically conservative targetshat are some multiple of SMSY. Walters and Martell (2004, p. 59)ote the advantages of managing on the basis of setting a harvestate such as hMSY rather than setting a fixed escapement goal, SMSY.he harvest rate, hMSY, is only a function of the Ricker parameternd fishing effort can be turned off and on in space and time suchhat a maximum percentage of fish can be caught by only allowingshing for a limited time per week (Su and Adkison, 2002).

Our findings suggest that there is substantial merit in consid-ring moving toward using hMSY (or benchmarks derived fromt) instead of SMSY as a management target. For moderately toighly productive stocks with ˛* = 4 or 8, hMSY is more reliablystimated by both models than SMSY, as indicated by its smallerelative bias and its much narrower interquartile range across sim-lated data sets. As well, 95% coverage probabilities for hMSY areloser to 95% for the widely used TSR method than SMSY. Bias instimates of hMSY decreases rapidly with increasing ˛*, and it isegligible at ˛* = 8 with variable or high harvest rates. However,

n our low-productivity case with ˛* = 2, SMSY is generally esti-ated better than hMSY based on relative bias; at ˛* = 2, SSR and

SR produce hMSY estimates that are highly positively biased atarger values of �*. Such large positive bias in hMSY may lead to over-xploitation of fish stocks and create severe conservation risks. Theain options for managers in this case are to improve accuracy

f spawner abundance measurement and/or reduce target har-est rates by some safety margin. Low-productivity stocks are alsorone to being overfished in mixed-stock fisheries and are hard toebuild (Quinn and Deriso, 1999). For all ˛*, TSR always has smallerositive bias or larger negative bias in hMSY than SSR, so is moreiologically conservative than SSR. This implies that, for fisherieshat use hMSY as target, using TSR would lead to lower biologicalisk.

Nevertheless, most fisheries for pink, sockeye, and chumOncorhynchus keta) salmon currently have escapement goals basedn SMSY or a target derived from it (Quinn and Deriso, 1999; Sund Adkison, 2002), rather than being managed for a target basedn hMSY. For estimating SMSY, we prefer the Bayesian state-spaceodeling approach to TSR. At large values of measurement error

*, relative biases of SMSY from the SSR model are much smallerhan those from TSR with low and moderate harvest rates. How-ver, at larger ˛* and �*, TSR and SSR tend to over-estimate SMSY.ositive bias in SMSY implies under-utilization of resources (leav-ng more spawners than necessary) and a biologically conservative

anagement policy. This may be beneficial for fish populations butay lead to long-term foregone social and economic benefits from

arvesting.For fisheries scientists involved in recovery planning in a con-

ervation situation where a population is currently depleted, it isritical to obtain good quality estimates of to estimate both theaximum percentage harvest rate that could be tolerated, as well

s the potential rate of recovery if the cause of depletion is removed.imilar to the previous case with hMSY, the best method of data anal-sis depends on ˛* and harvest histories. For lower productivitytocks (˛* = 2 or 4), TSR is preferred. In contrast, for higher produc-ivity stocks (˛* = 8), SSR is preferred. However, if the emphasis isn obtaining good quality estimates of SMSY for identifying what is

alled a lower limit reference point on spawner abundance (e.g.,0% of SMSY), SSR should be used.

odelling 224 (2012) 76– 89

4.5. Other considerations and future research

To attempt to improve parameter estimates, it is possible to useprior information about the Ricker parameter that is availablein Su et al. (2004) for 43 pink salmon populations and in Mueteret al. (2002) for 40 chum and 37 sockeye populations. Althoughincorporation of such an informative prior distribution for orother parameters could reduce the overall average bias for the twoestimation methods considered, that step would create biased esti-mates of for individual stocks with below- or above-average dueto the shrinkage effect of such estimators (Gauch, 2006).

Alternative prior specifications for ˇ, or equivalently using dif-ferent parameterizations for the Ricker model, may have a stronginfluence on the Bayesian estimation methods. The seemingly rea-sonable non-informative priors used for in this study, or for hMSYand SMSY used by Schnute and Kronlund (2002), may bring potentialunexpected information to one’s analysis (Rivot et al., 2001). A keyneed for future research is to use habitat information as covariatesto predict unfished carrying capacity (loge(˛)/ˇ). Simulation testswith this approach for Skeena River salmon data indicate that it cangreatly help to reduce biases in both hMSY and SMSY (Carl Walters,Fisheries Centre, 2202 Main Mall, University of British Columbia,Vancouver, B.C., Canada V6T 1Z4, personal communication).

Our state-space stock-recruitment model explicitly representsboth measurement error in spawner abundance and process errorin recruitment. It is an extension of the state-space model pro-posed by Walters and Martell (2004, pp. 167–173), which did notconsider measurement error in spawner abundance. Both of thesestate-space models incorporate a stochastic stock-to-recruitmentprocess and a deterministic recruitment-to-stock process. There-fore, both can deal with time series bias (Walters, 1985, 1990).However, Walters and Martell (2004, pp. 167–173) demonstratedthat state-space models may be not helpful for correcting timeseries bias for individual stocks. Thus, the biases in parameter esti-mates observed in this study are partially due to time series biasas indicated by the bias in and SMSY estimates in some of thehigh harvest rate situations when �* = 0. Time series bias may havea large effect on stocks with smaller ˛*, few data points, and fixedharvest rates. However, this bias diminishes with increasing ˛* andfor variable harvest rates (Korman et al., 1995). Simulation analyseslike ours need to be done to compare Walters’ (1990) bias correctionmethod with our results. Walters and Martell (2004) and Walterset al. (2008) develop and apply another possible solution to timeseries bias based on correlated variation across fish stocks. Specifi-cally, their state-space models can reduce bias in and estimatesfor a set of stocks that have been subject to strong and shared envi-ronmental effects such as those found in pink, chum, and sockeyesalmon in the Northeastern Pacific (Pyper et al., 2005).

In this paper, we assumed that catch is measured without error.Although that error is often relatively small compared to obser-vation error on spawner abundance, some measurement error incatch undoubtedly exists in real fisheries. Scientists can incorpo-rate this error into analyses by adding an observation equation forcatch to the state-space model, as proposed by Meyer and Millar(2001). We expect similar or even greater difficulties in separatelyestimating three error variances, rather than just two as in our SSRmethod.

Acknowledgments

This study was funded partially by Federal Aid Project F-80

Research. Funding was also provided by a Strategic Projects grant toR.M. Peterman, C.C. Wood, M.J. Bradford, and D.M. Ware from the

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atural Sciences and Engineering Research Council of Canada, andhe Canada Research Chairs Program to R.M. Peterman. We thank

ilo Adkison, Rob Kronlund, Marc Mangel, Steve Martell, Terranceuinn II, and Carl Walters for reviewing an earlier draft manuscript,nd two anonymous reviewers for many helpful comments on thisanuscript.

ppendix A. Full conditional distributions

.1. Updating the vector ϕ = (ar, ˇ)′

First we derive the full conditional distribution of the vector = (ar, ˇ)′ by making use of the conditionally linear property oficker ar = log(˛) and (rt is a linear function of ar and ˇ):

et X =

⎛⎝ 1 −S1

.

.

....

1 −ST−k

⎞⎠

(T−k)×2

, Y =

⎛⎝ loge(Rk+1) − loge(S1)

.

.

.loge(RT ) − loge(ST−k)

⎞⎠

(T−k)×1

nd based on the uniform prior distributions for ar and ˇ:r ∼ U(0, ∞) and ∼ U(0, ∞), we obtain the following full condi-ional distribution of ϕ, which is used to update ar and in alock:

(ϕ|·) ∝ N(ϕ, V ) (A1)

here V = (�−2(X ′X))−1

, ϕ = V (�−2X ′Y ).

.2. Update state variable rt = loge(Rt) using anndependence-chain algorithm

The conditional distribution of the state rt is

(rt |rt−k, rt+k, �, yt)

= p(rt |rt−k, ar, ˇ, �2)p(rt+k|rt, ar, ˇ, �2)p(yt |xt, �2) (A2)

here t = k + 1, . . ., T.Eq. (A2) is not a standard density function. Following Geweke

nd Tanizaki (2001), we use an independence-chain sampling algo-ithm to update rt. In this algorithm, we take the density functionf the system equation in Eq. (2b), as the proposal distribution ofhe Metropolis–Hastings algorithm (Geweke and Tanizaki, 2001):

(r∗) = p(r∗|r(i)t−k

) = N(r∗|a(i)r + x(i+1)

t−k− ˇ(i)S(i+1)

t−k, (�2)

(i)) (A3)

here i is the current iteration number of the sampler.The proposal distribution q(r*) does not depend on the current

alue of rt, r(i)t , therefore, it is called an independence-chain algo-

ithm (Chib and Greenberg, 1995). The rt can be updated using theeneral Metropolis–Hastings algorithm. To update each rt, a can-idate value r* is drawn from Eq. (A3). Then r* is accepted as anpdate for rt with a probability:

in

(1,

p(r(i)t+k

|r∗)p(yt |r∗)

p(r(i)t+k

|r(i)t )p(yt |r(i)

t )

)(A4)

therwise, we reject r* and set r(i+1)t = r(i)

t .

odelling 224 (2012) 76– 89 87

Appendix B. WinBUGS program for the state-space Rickermodel

model

{# Observation equation

for (t in 1:T)

{E[t] ∼ dlnorm(mu.logE[t], prec.E); mu.logE[t] <-

log(S[t]);

}# Prior distributions for the state variables

for (t in 1:k)

{S[t] <- S1[t]; S1[t] ∼ dlnorm(0.0, 1.0E−3)

}for (t in 1:(T − k)) {

mu[t] <- b[1] + log(S[t]) − b[2] * S[t];

R[t] ∼ dlnorm(mu[t], prec.R)I(C[t], 1.0E+9); # R and C

represent Rt+k and Ct+k, respectively

S[t + k] <- R[t] - C[t];

}# Prior distributions for model parameters

# a bivariate normal prior for a r and beta

b[1:2] ∼ dmnorm(mu.b[], prec[,])

a r <- b[1]; alpha <- exp(b[1]);

beta <- b[2]

# Priors for standard deviations

tau ∼ dunif(0, 1.0E+9); tau2 <- tau * tau; prec.R <-

1/tau2

sig ∼ dunif(0, 1.0E+9); sig2 <- sig * sig; prec.E <-

1/sig2

# Management-oriented quantities

SMSY <- hMSY/beta

hMSY <- 0.5*a r - 0.07*a r *a r

}Note: This program uses a bivariate normal prior distribution

for ar and ˇ. This instructs WinBUGS to update ar and in a blockas was done for our Bayesian state-space model (SSR). However, ar

and are not restricted to be positive as in the case for ar and ˇin SSR. Nevertheless, the results from the WinBUGS program canbe compared to those from SSR by discarding those draws withnon-positive ar and ˇ.

Here is an example data set generated from the conditions: k = 2,˛* = 4, ˇ* = 1, �*2 = 0.01, �*2 = 0.003, �* = 0.25, � = 0, T = 28, and the“variable” harvest situation:Data: list(mu.b = c(1, 1), prec = structure(.Data = c(1.0E−03,

0, 0, 1.0E−06), .Dim = c(2, 2)), T = 28, k = 2, E = c(1.062,

1.179, 1.385, 1.213, 1.176, 1.385, 1.141, 0.944, 1.087, 1.119,

0.966, 0.918, 0.793, 0.775, 0.645, 0.661, 0.548, 0.703, 0.807,

0.792, 0.9, 1.007, 1.222, 1.269, 1.001, 1.108, 1.032, 1.099),

C = c(0.165, 0.23, 0.272, 0.34, 0.385, 0.409, 0.508, 0.596,

0.609, 0.669, 0.674, 0.779, 0.708, 0.864, 0.611, 0.658, 0.631,

0.583, 0.548, 0.534, 0.556, 0.513, 0.332, 0.319, 0.235, 0.2))

Inits:

list(b = c(6, 2), tau = 2, sig = 3, R = c(1, 1, 1, 1, 1, 1, 1,

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), S1 =

c(1, 1))

list(b = c(3, 0.5), tau = 1, sig = 1, R = c(2, 2, 2, 2, 2, 2,

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2),

S1 = c(2, 2))

Results: posterior medians for (˛,ˇ,�2,�2) from WinBUGS are

(4.05,1.18,0.0071,0.0066) and those from our program are

(4.05,1.00,0.0075,0.0055).

Appendix C. MCMC diagnostics

To assess convergence of the MCMC simulation runs of theBayesian state-space model (SSR) and the traditional Ricker model(TSR), we calculated detailed MCMC diagnostic statistics for 50randomly selected data sets from the 300 data sets of each of

18 selected scenarios (Table A1). For each data set, we ran 3chains with dispersed parameter starting values for a total num-ber of N = 0.8 million (m) iterations, a thinning interval (Thin) of80 iterations and stored all sampling draws. Using the R CODA

88 Z. Su, R.M. Peterman / Ecological Modelling 224 (2012) 76– 89

Table A1Summaries of MCMC diagnostics over 50 data sets and three levels of ˛* = (2, 4, 8) forfour parameters from the Bayesian state-space model for k = 2, � = 0, and T = 28, anddifferent combinations of �* (0, 0.75) and harvest situations (Low, Variable, High).Lag 1 – Average of lag-1 autocorrelation function. Gelman (%) – Average of percentof data sets with the Gelman–Rubin’s potential scale-reduction factor greater than1.1. Geweke (%) – Average of percent of data sets with the Geweke’s diagnostic outof (−2, 2). HW Stat (%) or HW hw (%) – Average of percent of data sets failed with theHeidelberger and Welch’s stationarity test or half-width test.

Diagnostic Harvest �* = 0 �* = 0.75

ar � � ar � �

Lag1

Variable 0.1 0.2 0.6 0.2 0.3 0.4 0.3 0.4Low 0.1 0.2 0.4 0.3 0.2 0.4 0.2 0.3High 0.3 0.2 0.6 0.3 0.4 0.5 0.3 0.4

Gelman Variable 0 0 7 0 0 0 3 0Low 1 1 5 1 0 0 1 0High 1 1 3 0 0 0 0 0

Geweke Variable 0 1 3 2 2 1 0 0Low 1 1 2 2 0 0 1 2High 1 1 4 1 1 1 2 2

HWstat

Variable 0 1 3 2 1 0 1 0Low 1 1 3 2 1 0 1 1High 2 2 3 2 2 8 1 1

HW Variable 0 1 8 2 1 1 1 0

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Table A2Medians of 300 estimates of percentage relative biases (RBs) for six parameters fromthe Bayesian state-space model for k = 2, � = 0, and T = 28 and different combinationsof ˛* (2, 8), harvest situations (Low, High), and �* (0, 0.75) with three different MCMCsampling settings (N, B, Thin): Short – (0.23 m, 0.03 m, 20), where “m” is millioniterations, Median – (0.43 m, 0.03 m, 50), Long – (1.5 m, 0.5 m, 100), where N – totalnumber of iterations, B – number of initial iterations discarded (burn-in), and Thin– thinning interval. RBs for � at �* = 0 are not calculated because the true value of �is zero in this case.

˛* Harvest �* Setting ˛ ˇ � � hMSY SMSY

2 High 0 Short 7 31 −9 9 −15Median 7 31 −9 9 −15Long 7 31 −9 9 −15

0.75 Short 159 165 16 −45 102 −28Median 159 167 18 −45 102 −28Long 161 164 18 −46 103 −27

Variable 0 Short 4 9 −8 5 −3Median 4 9 −8 5 −3Long 5 9 −8 6 −3

0.75 Short 142 111 9 −49 96 −13Median 142 111 9 −49 96 −13Long 142 113 9 −49 96 −13

8 High 0 Short 0 −34 −59 0 46Median 2 −35 −59 1 48Long 1 −34 −59 0 46

0.75 Short 16 −36 146 −68 4 56Median 17 −35 146 −69 4 54Long 17 −34 146 −68 4 55

Variable 0 Short 5 0 −5 1 1Median 5 0 −5 1 1Long 6 0 −5 2 0

TMh

hw Low 1 1 3 2 1 0 1 1High 2 4 5 3 1 1 1 1

ackage (Plummer et al., 2006), we calculated the autocorrelationunction (acf), Gelman–Rubin’s potential scale-reduction fac-or (psrf), Geweke’s diagnostic, and Heidelberger and Welch’siagnostic. We also examined psrf plots (gelman.plot), which showshe evolution of psrf’s as a function of the number of iterations,o determine the burn-in period (the number of initial iterationsiscarded).

For the Bayesian state-space model (SSR), the psrf plots showedhat ˛, ˇ, hMSY, and SMSY converged quickly (mostly less than 10,000terations). However, the observation error standard deviation �howed very slow mixing in its chains for a moderate number ofata sets, especially for those generated with �* = 0 (no measure-ent error) or small �* (<0.5). The process error standard deviation

converged more quickly than �, but slower than other parameters.he diagnostic statistics in Table A1 show the same convergencessue of � for some data sets with �* = 0. However, other parame-ers at both �* = 0 and 0.75 and � at �* = 0.75 show no convergenceroblem for majority (over 96%) of the data sets examined.

As a further examination, we compared summary statistics of

arameter estimates for 8 data sets (Table A2) from three differ-nt MCMC simulation settings (N, B, Thin) for SSR: Short – (0.23 m,.03 m, 20), where “m” is millions, Median – (0.43 m, 0.03 m, 50),ong – (1.5 m, 0.5 m, 100), where B – number of initial iterations

able A3edians of 300 estimates of percentage relative biases (RBs) for six parameters from the

arvest situations, and �* = (0.25, 0.75), with the uniform prior (U(0, ∞)) for � or �, or the

Harvest �* �2 or � �2 or �

Variable 0.25 IG IG 19

IG U 26

U U 20

U IG 11

0.75 IG IG 73

IG U 73

U U 53

U IG 56

High 0.25 IG IG 16

IG U 24U U 17

U IG 16

0.75 IG IG 133

IG U 133

U U 108

U IG 106

0.75 Short −2 −45 28 −44 −1 81Median −2 −45 28 −44 −1 81Long −2 −46 28 −44 0 83

discarded (burn-in). These settings yield essentially the sameresults. Similar results are found for other summary statistics (thecoefficient of quartile variation, coverage probability) and otherdata sets. This indicates that our results are stabilized even for theshort-run setting (0.23 m, 0.03 m, 20). However, all our results forthe state-space model are based on the long runs (1.5 m, 0.5 m, 100)to minimize any effects of these simulation settings.

The traditional Ricker model (TSR) converges much morequickly than the state-space model (less than 10,000 iterations).The results for (TSR) are based on a setting: (N, B, Thin) = (0.43 m,0.03 m, 50).

Appendix D. Sensitivity analysis of prior probabilitydistributions

Estimates from the Bayesian state-space model (SSR), especiallythose of �2 and �2, are sensitive to prior specifications of �2 and�2. An inverse Gamma IG(ε, ε) prior with ε = 0.001 used for �2 and a

Bayesian state-space model for k = 2, ˛* = 4, � = 0, T = 28 for the variable and high inverse Gamma prior (IG(0.001, 0.001)) for �2 or �2.

�2 �2 SMSY hMSY

2 −15 −20 4 95 −40 124 3 120 −19 60 4 9

−2 −3 −52 4 62 −93 41 20 262 −94 53 21 26

−3 −79 31 23 21−4 −73 20 24 2215 −32 122 −4 811 −67 436 3 1112 −38 273 −1 816 −17 −5 −8 829 −94 129 4 3725 −95 157 7 3721 −85 136 8 3323 −82 109 4 33

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niform prior used for � ∼ U(0, ∞), denoted as (�2 ∼ IG, � ∼ U), leadso the most negatively biased estimates of �2 and most positivelyiased estimates of �2 among the four prior combinations of �2 or2 under the same scenario (Table A3). This in turn produces theost positively biased estimates of and hMSY. (�2 ∼ IG, �2 ∼ IG)

as similar effects on parameter estimates as (�2 ∼ IG, � ∼ U) with* = 0.75. In contrast, (� ∼ U, �2 ∼ IG) often leads to negatively biasedr the least positively biased estimates of �2, the least negativelyiased estimates of �2, and the least positively biased estimates of

and hMSY. The parameter estimates from (� ∼ U, � ∼ U) are close tohose from (� ∼ U, �2 ∼ IG) (Table A3). In all cases, high correlationetween the estimates of �2 and �2 are observed.

Based on these results and other similar results not shown,e prefer (� ∼ U, � ∼ U) and (� ∼ U, �2 ∼ IG). However, the currentayesian literature for prior specifications warns against using IG(ε,) as a default prior for hierarchical variance parameters, such as2 (Gelman and Hill, 2007). The IG(ε, ε) distribution function isharply peaked near zero. When data provide little information forhe variance parameters �2 or �2, this prior can strongly constrainhe posterior estimates near zero as indicated by our results (neg-tively biased estimates of �2 or �2). Based on this consideration,e choose (� ∼ U, � ∼ U) as our default prior specification, although

his prior choice may give more conservative (more biased) resultshan (� ∼ U, �2 ∼ IG) in some scenarios.

eferences

dkison, M.D., 2009. Drawbacks of complex models in frequentist and Bayesianapproaches to natural resource management. Ecol. Appl. 19, 198–205.

dkison, M.D., Su, Z., 2001. A comparison of salmon escapement estimates using ahierarchical Bayesian approach versus separate estimation of each year’s return.Can. J. Fish. Aquat. Sci. 58, 1663–1671.

rowne, W.J., Draper, D., 2006. A comparison of Bayesian and likelihood-based meth-ods for fitting multilevel models. Bayesian Anal. 1, 473–550.

uckland, S.T., Newman, K.B., Thomas, L., Koesters, N.B., 2004. State-space modelsfor the dynamics of wild animal populations. Ecol. Model. 171, 157–175.

aputi, N., 1988. Factors affecting the time series bias in stock-recruitment relation-ships and the interaction between time series and measurement error bias. Can.J. Fish. Aquat. Sci. 45, 178–184.

hib, S., Greenberg, E., 1995. Understanding the Metropolis–Hastings algorithm. Am.Stat. 49, 327–335.

lark, J.S., 2007. Models for Ecological Data: An Introduction. Princeton UniversityPress, Princeton.

ousens, N.B.F., Thomas, G.A., Swann, C.G., Healey, M.C., 1982. A review of salmonescapement estimation techniques. In: Can. Tech. Rep. Fish. Aquat. Sci. No. 1108.

e Valpine, P., Hastings, A., 2002. Fitting population models incorporating processnoise and observation error. Ecol. Monogr. 72, 57–76.

auch, H.G., 2006. Winning the accuracy game. Am. Sci. 94, 133–144.elman, A., Carlin, J.B., Stern, H., Rubin, D.B., 2004. Bayesian Data Analysis. Texts in

Statistical Science. Chapman and Hall, London, UK.elman, A., Hill, J., 2007. Data Analysis using Regression and Multilevel/Hierarchical

Models. Cambridge University Press, New York.eweke, J., Tanizaki, H., 2001. Bayesian estimation of state-space models using the

Metropolis–Hastings algorithm within Gibbs sampling. Comput. Stat. Data Anal.37, 151–170.

root, G., Margolis, L., 1991. Pacific Salmon Life Histories. University of BritishColumbia Press, Vancouver.

arwood, J., Stokes, K., 2003. Coping with uncertainty in ecological advice: lessonsfrom fisheries. Trends Ecol. Evol. 18 (12), 617–622.

arvey, A.C., 1989. Forecasting, Structural Time Series Models, and the Kalman Filter.Cambridge University Press, Cambridge, UK.

ilborn, R., Walters, C.J., 1992. Quantitative Fisheries Stock Assessment and Man-agement: Choice, Dynamics and Uncertainty. Chapman and Hall, New York.

ehler, D.G., Myers, R.A., Field, C.A., 2002. Measurement error and bias in themaximum reproductive rate for the Ricker model. Can. J. Fish. Aquat. Sci. 59,854–864.

ope, R.G., 1988. Effects of initial conditions on Monte Carlo estimates of bias inestimating functional relationships. Can J. Fish. Aquat. Sci. 45, 185–187.

odelling 224 (2012) 76– 89 89

Korman, J., Peterman, R.M., Walters, C.J., 1995. Empirical and theoretical analy-ses of time-series bias in stock-recruitment relationships of Sockeye salmon(Oncorhynchus nerka). Can. J. Fish. Aquat. Sci. 52, 2174–2189.

Larkin, P.A., 1988. Pacific salmon. In: Gulland, J.A. (Ed.), Fish Population Dynam-ics: The Implications for Management. John Wiley and Sons, Chichester, pp.153–183.

Ludwig, D., Walters, C.J., 1981. Measurement errors and uncertainty in parameterestimates for stock and recruitment. Can. J. Fish. Aquat. Sci. 38, 711–720.

Lunn, D.J., Thomas, A., Best, N., Spiegelhalter, D., 2000. WinBUGS—a Bayesianmodelling framework: concepts, structure, and extensibility. Stat. Comput. 10,325–337.

McGurk, M.D., 1986. Natural mortality of marine pelagic fish eggs and larvae: therole of spatial patchiness. Mar. Ecol. Prog. Ser. 34, 227–242.

Meyer, R., Millar, R.B., 2001. State-space models for stock-recruit time series. In:George, E.I. (Ed.), Bayesian Methods with Applications to Science, Policy, andOfficial Statistics; Selected Papers from ISBA 2000: The Sixth World Meeting ofthe International Society for Bayesian Analysis, Monograph in Official Statistics,Eurostat. , pp. 361–370.

Millar, R.B., 2002. Reference priors for Bayesian fisheries models. Can. J. Fish. Aquat.Sci. 59, 1492–1502.

Millar, R.B., Meyer, R., 2000. Non-linear state-space modeling of fisheries biomassdynamic using Metropolis–Hastings within Gibbs sampling. Appl. Stat. 49,327–342.

Mueter, F.J., Peterman, R.M., Pyper, B.J., 2002. Opposite effects of ocean temperatureon survival rates of 120 stocks of Pacific salmon (Oncorhynchus spp.) in northernand southern areas. Can. J. Fish. Aquat. Sci. 59, 456–463, plus the erratum printedin Can. J. Fish. Aquat. Sci. 60, 757.

Needle, C.L., 2001. Recruitment models: diagnosis and prognosis. Rev. Fish Biol. Fish.11, 95–111.

Quinn, T.J., Deriso II, R.B., 1999. Quantitative Fish Dynamics. Oxford University Press,New York.

Peterman, R.M., Pyper, B.J., Grout, J.A., 2000. Comparison of parameter estimationmethods for detecting climate-induced changes in productivity of Pacific salmon(Oncorhynchus spp.). Can. J. Fish. Aquat. Sci. 57, 181–191.

Petris, G., Petrone, S., Campagnoli, P., 2009. Dynamic Linear Models with R. Springer,New York.

Plummer, M., Best, N., Cowles, K., Vines, K., 2006. CODA: Convergence Diagnosis andOutput Analysis for MCMC. R News 6, 7–11.

Pyper, B.J., Mueter, F.J., Peterman, R.M., 2005. Across-species comparisons of spatialscales of environmental effects on survival rates of Northeast Pacific salmon.Trans. Am. Fish. Soc. 134, 86–104.

Rivot, E., Prevost, E., Parent, E., 2001. How robust are Bayesian posterior inferencesbased on a Ricker model with regard to measurement errors and prior assump-tions about parameters? Can. J. Fish. Aquat. Sci. 58, 2284–2297.

Rivot, E., Prevost, E., Parent, E., Bagliniere, J.L., 2004. A Bayesian state-space modelingframework for fitting a salmon stage-structured population model to multipletime series of field data. Ecol. Model. 179, 463–485.

Schnute, J.T., 1994. A general framework for developing sequential fisheries models.Can. J. Fish. Aquat. Sci. 51, 1676–1688.

Schnute, J.T., Kronlund, A., 2002. Estimating salmon stock-recruitment rela-tionships from catch and escapement data. Can. J. Fish. Aquat. Sci. 59,433–449.

Su, Z., Adkison, M.D., Van Alen, B.W., 2001. A hierarchical Bayesian model for esti-mating historical salmon escapement and escapement timing. Can. J. Fish. Aquat.Sci. 58, 1648–1662.

Su, Z., Adkison, M.D., 2002. Optimal in-season management of pink salmon givenuncertain run sizes and seasonal changes in economic value. Can. J. Fish. Aquat.Sci. 59, 1648–1659.

Su, Z., Peterman, R., Haeseker, S., 2004. Spatial hierarchical Bayesian models forstock-recruitment analysis of pink salmon (Oncorhynchus gorbuscha). Can. J. Fish.Aquat. Sci. 61, 2471–2486.

Walters, C.J., 1985. Bias in the estimation of functional relationships from time seriesdata. Can. J. Fish. Aquat. Sci. 45, 185–187.

Walters, C.J., 1990. A partial bias correction factor for stock-recruitment parametersin the presence of autocorrelated environmental effects. Can. J. Fish. Aquat. Sci.47, 516–519.

Walters, C.J., Ludwig, D., 1981. Effects of measurement errors and uncertainty inparameter estimates for stock and recruitment. Can. J. Fish. Aquat. Sci. 38,704–710.

Walters, C.J., Martell, S.J.D., 2004. Fisheries Ecology and Management. PrincetonUniversity Press, New Jersey.

Walters, C.J., Lichatowich, J.A., Peterman, R.M., Reynolds, J.D., 2008. Report ofthe Skeena Independent Science Review Panel. A report to the CanadianDepartment of Fisheries and Oceans and the British Columbia Ministry of theEnvironment , May 15, 2008, p. 144, available at: http://www.psf.ca/index.php?option=com content&view=article&id=21&Itemid=49.