Original Article

Standardizing compositional data for stock assessment

James T. Thorson*Fisheries Resource Assessment and Monitoring Division, Northwest Fisheries Science Center, National Marine Fisheries Service, National Oceanic andAtmospheric Administration, 2725 Montlake Blvd. East, Seattle, WA 98112, USA

*Corresponding author: tel: +206 302 1772; fax: +206 860 6792; e-mail: james.thorson@noaa.gov

Thorson, J. T. 2014. Standardizing compositional data for stock assessment. – ICES Journal of Marine Science, 71: 1117–1128.

Received 12 August 2013; accepted 26 November 2013; advance access publication 9 January 2014.

Stock assessment models frequently integrate abundance index and compositional (e.g. age, length, sex) data. Abundance indices are gen-erally estimated using index standardization models, which provide estimates of index standard errors while accounting for: (i) differencesin sampling intensity spatially or over time; (ii) non-independence of available data; and (iii) the effect of covariates. However, compositionaldata are not generally processed using a standardization model, so effective sample size is not routinely estimated and these three issues areunresolved. I therefore propose a computationally simple “normal approximation” method for standardizing compositional data andcompare this with design-based and Dirichlet-multinomial (D-M) methods for analysing compositional data. Using simulated datafrom a population with multiple spatial strata, heterogeneity within strata, differences in sampling intensity, and additional overdispersion,I show that the normal-approximation method provided unbiased estimates of abundance-at-age and estimates of effective sample sizethat are consistent with the imprecision of these estimates. A conventional design-based method also produced unbiased age compositionsestimates but no estimate of effective sample size. The D-M failed to account for known differences in sampling intensity (the proportion ofcatch for each fishing trip that is sampled for age) and hence provides biased estimates when sampling intensity is correlated with variationin abundance-at-age data. I end by discussing uses for “composition-standardization models” and propose that future research developmethods to impute compositional data in strata with missing data.

Keywords: age composition, composition-standardization models, Dirichlet-multinomial, integrated model, length composition likelihoodweights, sampling intensity, stock assessment, strata.

IntroductionPopulation dynamics models for marine fish (“stock assessmentmodels”) synthesize biological and fishery information to estimatethe current and historical status of fish populations, as well as eval-uate the likely consequences of alternative management actions in-cluding projected changes in fishery catches (Hilborn and Walters,1992). Assessment models generally approximate the average abun-dance and dynamics of a population over large spatial and temporalscales, and reconcile these estimates with aggregate removals toupdate expectations regarding the productivity and sustainableharvest levels for a given stock.

The spatial and temporal aggregation of typical biological andfishery data poses several challenges. First, data may be more avail-able for some components of the stock than others, due to spatial ortemporal differences in sampling rates. These differences in “sam-pling intensity” imply that data must be weighted to accurately rep-resent the total stock, similar to the area-weightings used in the

analysis of stratified sampling designs. Second, data may not be stat-istically independent, e.g. all fish captured in a trip might be the sameage, and hence not independent samples from the age compositionof the stock. This statistical non-independence implies that “effect-ive” sample sizes (e.g. for age-composition data) may be lower thanthe total number of fish counted. In stock assessment models,however, compositional data must be weighted by their effectivesample size to avoid a situation in which compositional data havea greater influence on estimated abundance trends than abundanceindex data (Francis, 2011), and thus “effective sample sizes” must beestimated from available data to downweight data that are not stat-istically independent. Third, data may vary as a function of categor-ical factors of continuous variables, e.g. catch rates may varybetween strata, in which case the inclusion of categorical factors(e.g. strata) or continuous variables (e.g. depth) may explain alarge portion of residual variance. Including these covariates inthe analysis may then either improve the precision of resulting

Published by Oxford University Press 2014. This work is written by a US Government employee and is in the public domain in the US.

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estimates or allow for extrapolation to partially observed or unob-served components of the stock.

For these three reasons and others, catch rate data are often stan-dardized before inclusion in assessment models (Maunder andPunt, 2004). This “index standardization” generally accounts forcovariates (#3 above) such as the effect of survey or fishing effort(e.g. area swept, number of hooks) and may account for differencesin fishing power among fishing vessels, spatial strata, and the depthof the fishing activity (Robins et al., 1998; Helser et al., 2004; Bigelowand Maunder, 2007). Index standardization also must account forspatial differences in sampling intensity (#1 above), for example,when inferring stock abundance from spatially changing fisherycatch rate data (Walters, 2003; Carruthers et al., 2011). Finally,index standardization may implicitly account for data-weightingissues (#2 above), for example, when extreme catch events aregiven lower statistical leverage than they would otherwise(Thorson et al., 2011). Standardization models generally estimatethe precision of the resulting abundance index, and using a stand-ardization model will often result in more accurate estimates ofabundance indices (Ye and Dennis, 2009) and their standarderrors (Helser et al., 2004; Thorson et al., 2011). These modelshave been an active area of research in fisheries science for nearly60 years (Beverton and Holt, 1957).

In addition to abundance index data, stock assessments routinelyincorporate compositional data representing the distribution ofages, length, sex, or other characteristics of the stock (Maunderand Punt, 2013). Compositional data generally arise from fisheryor survey sampling and thus represent a sample from a localized sci-entific survey or fishing activity. Compositional data must then beaggregated to the scale of the stock assessment model while account-ing for differences in sampling intensity (#1), statistical non-independence arising from biological or sampling characteristics(#2), and covariates such as spatial strata (#3). For example, compos-itional samples from a fishery are used to inform fishery selectivityestimates and, for this purpose, are interpreted as a random samplefrom fishery catches. Therefore, fishery compositional data must beaggregated in some manner to represent fishery catches overall.

However, standardizing compositional data has received rela-tively little attention relative to the large literature on index stand-ardization. I therefore compare several methods for aggregatingcompositional data that account for variable sampling intensity,non-independence among trips, and systematic differences amongstrata, while simultaneously estimating the effective sample size ofthe aggregated data. In the following, I refer primarily to age-composition samples from a fishery, but note that results can beapplied to other types of compositional data. I first explain severalmethods for standardizing compositional data, including a novelmethod that uses a normal approximation to compositional data.I use simulation modelling to explore the performance of thesecompositional-standardization methods given different data char-acteristics and compare their estimates of the aggregated stock com-position and effective sample sizes with known, “true” values. Iconclude by discussing potential applications and future researchideas for the standardization of compositional data.

MethodsMethods for standardizing compositional dataAssumptions regarding data availabilityAge-composition data for marine fish generally result from fishbeing extracted from the water, a random subsample of these

caught fish being selected, an otolith or other hard part beingextracted from each selected fish, and the age being “read” (i.e. in-ferred using some validated method) for each otolith. Therefore, age-compositionsamplesgenerallyarise fromadiscretesampleofni otolithsfor the ith of N total sampling occasions (parameters are defined inTable 1). The composition data are represented by D, where Da,i repre-sents the number of aged fish of age a in sample i, and fish are assumedto have ages ranging from age-0 recruits to a maximum age amax (andhence

∑amax

a=0 Da,i = ni). I assume that there exists some informationabout sampling intensity li for each trip i. One source of this infor-mation is when the total quantity of catch Ci (in weight) for the ithsample is known, whereas only a subset of caught fish ci (in weight)were aged. In this case, ci/Ci represents the portion of total catch intrip i that was aged, and this contributes to differences in samplingintensity whenever li ¼ ci/Ci differs among trips. In circumstanceswhere the total catch for each trip is unknown, a value of l must beassumed, i.e. fixing l at its median value for a given stratum andyear. I additionally assume that all compositional samples can be

Table 1. Variable names, symbols, and whether it is used in theestimation models (E), simulation model (S), or both (E, S)

Name Symbol Use

VariablesCompositional data (counts) Da,i E, SCompositional data (proportions) Pa,i E, STotal catch in each trip Ci E, SSampled catch in each trip ci E, SSampling intensity for each trip li E, SSampled catch in each stratum ls E, STotal catch in each stratum Ls E, SEstimated proportion at age pa E, SEstimated proportion at age (aggregated across

all strata)�pa E, S

Maximum observed age amax E, SEstimated effective sample size at age N∗

a EEstimated effective sample size for all ages N* EEstimated effective sample size at age (aggregated

across all strata)N∗∗

a E

Estimated effective sample size for all ages(aggregated across all strata)

N** E

Estimated overdispersion ta EEstimated age-specific variability among trips s2

a ETrue abundance by age, year, stratum, and region Aa,t,s,r STrue proportion at age by year, aggregated across

strata and regionpa S

True proportion at age pa SNumber of sampled trips per stratum N SExpected catch for each trip mC SNatural mortality rate for each stratum Ms SMovement rate from inshore region to offshore

regionV S

Proportion of recruitment in inshore region u SAverage recruitment R0 SVariability in recruitment s2

R SRecruitment in each year Rt SOverdispersion caused by the sampling process v S

IndicesAge a E, STrip i E, SStrata s E, SRegion r E, SYear t S

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assigned to a particular stratum S (often defined spatially, by the lo-cation of survey catch or fishery landings) and that the true age dis-tribution may differ among strata. In this case, the composition canbe estimated separately for each stratum, then aggregated as aweighted average among strata. I again assume that an estimate ofthe weighting proportion is available for each stratum, forexample, the ratio of sampled landings ls in stratum s and total land-ings Ls in that stratum. This weighting proportion will vary depend-ing on study goals. For example, when aggregating fisheryage-composition samples to represent total fishery removals, theweighting may be derived from the proportion of total landings ineach stratum. In contrast, the weighting for a scientific survey,where age-composition samples are interpreted to represent avail-able abundance, will be derived from the proportion of total abun-dance that is available to the survey in each stratum.

The analyst then seeks to aggregate available compositional dataD to estimate the proportion p (where pa is the true proportion atage a) and the effective sample size N* that represents the number ofindependent and identically distributed samples that would havethe same information content as the available data. This estimatedproportion and sample size N∗p can then be included as multino-mially distributed data in a stock assessment model, as is commonlydone in Stock Synthesis (Methot and Wetzel, 2013), or can be inter-preted as a separate proportion and effective sample size using one ofmany recently developed likelihood models for compositional data(Maunder, 2011; Hulson et al., 2012).

Design-based formulaIn the fisheries management region off the US West Coast, compos-itional data are frequently combined to produce an estimate of pro-portion at age for the population as a whole (hereafter referred to as“expanded”), but in a manner that does not give a model-based es-timate of effective sample size (Stewart et al., 2011; Gertseva andThorson, 2013). This design-based method is in many wayssimilar to that used by the DATRAS database in ICES (Larsenet al., 2006; Beare, 2011). I start by presenting the West Coast“design-based” method.

In the West Coast expansion method, fishery compositional dataDi for trip i are first multiplied by the inverse of the sampling inten-sity for that trip, Ci/ci, to ensure that each composition sample iscatch-weighted. Data are then multiplied by strata weightings, cal-culated as the ratio of total landings (in numbers) and landings insampled trips in stratum s, Ls/ls. The expanded compositionaldataset �Da is then the sum of this expanded distribution of compo-sitions across all trips and strata:

�Da =∑ni

i=1

∑ns

s=1

I(Si = s) Ls

ls

( )Ci

ci

( )Da,i, (1)

where I(X ¼ x) is an indicator function that equals one if X ¼ x andzero otherwise, and (

∑ns

s=1 I(Si = s)Ls/ls)(Ci/ci) is the weightingfactor for trip i (where the indicator function is used to associatethe stratum for trip i to landings for that stratum). However, thisformula does not provide any estimate of effective sample size,which then must be calculated externally using secondary methods,e.g. bootstrapping of the original compositional data (Stewart andHamel, unpublished data). I donot simulation test this bootstrappingoption because there are multiple types that are feasible: analysts maybootstrap the individuals within a tow, the tows within a stratum, orboth. Preliminary results (not shown) indicate that this decision has

important implications for resulting estimates of imprecision, andthere are few guidelines for this process. Furthermore, bootstrappingdoes not provide for extrapolation to unobserved strata and hasdegraded performance for partially observed strata.

Normal approximationIn developing a simple model-based estimator for compositionalstandardization, I proceed by identifying the sample variance ofcompositional data. I start by imagining ideal compositional sam-pling, in which the age of each sampled fish is independent and multi-nomially distributed, with proportion equal to the population-leveldistribution p. The sampling variance for the proportion Pa,i at agea for trip i would then be:

Var[Pa,i] = VarDa,i

ni

[ ]= 1

n2i

Var[Da,i] =pa(1 − pa)

ni(2a)

However, compositional data could be overdispersed as caused bycorrelations between samples within a given trip:

Var[Pa,i] =tapa(1 − pa)

ni, (2b)

where ta accounts for age-specific variance more than the multi-nomial distribution but which still decreases with increasing samplesizes. In the extreme case, this type of overdispersion could occur ifall fish are caught in groups of ten individuals of identical age, inwhich case “effective sample size” would be n/10 and t ¼ 10.

Additionally, I account for differences in sampling intensitygiven li for each trip i:

Var[Pa,i] = lit pa(1 − pa)

ni

( ). (2c)

This is similar to the inverse-probability weighting used in the con-ventional analysis of stratified sampling. In this weighting scheme,observations arising from a small portion of a given trip’s catchhave a greater “weighting” in the model, as implemented bygiving them a smaller sampling variance relative to tows with alarger sampled proportion. Given that li ¼ ci/Ci, the term li/ni

specifies that the variance of each tow is proportional to the totalweight of catch of the target species divided by its average weightin that tow, i.e. an estimate of the total number caught in that tow.This estimate requires information about the number sampled ni

as well as the proportion sampled ci/Ci.Finally, there may be variability among trips in the expected value

of Pa. This can be caused by fish behaviours, e.g. when small andlarge fish congregate in different spatial areas. This violates theassumptions of Equation (2b), wherein sampling variance goes tozero as ni go to infinity. Thus, I next incorporate a lower boundon variance:

Var[Pa,i] = litapa(1 − pa)

ni

( )+ s2

a , (2d)

where s2a is the estimated process-error variance for age a (where the

hat-symbol is used to denote that s2a is a freely estimated parameter

for all ages a). This variance is age-specific because, in the previoushypothetical example, process errors would be greatest for ages thathave very specific habitats and would be lowest for ages that occur ina similar proportion in all habitats. It also does not decrease with

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increasing sample sizes ni or decreased sampling intensities li

because these process errors represent a lower bound on variancegiven infinite sample and catch sizes.

This final equation [Equation (2d)] represents the sampling vari-ance for proportional data that is expected given both multinomialsampling variance, known variation in sampling intensity amongtrips, and two types of overdispersion: (i) variability in the expectedproportion among trips and (ii) correlated ages within a trip. A“normal approximation” estimator can therefore be developedthat uses this sample variance combined with the assumption thatthe expected value of proportion at age is equal to the populationaverage:

E[Pa,i] = pa. (3)

The simplest likelihood to match this mean and variance uses anormal distribution for each sample proportion Pa,i:

L(p, s, t |Pa,i) =∏ni

i=1

∏A

a=1

Normal Pa,i | pa, litapa(1 − pa)

ni

(( )

+ s2a

). (4)

This likelihood model is used to estimate the expected proportion atage p and its standard errors SE(p). We estimate parameters andassociated standard errors for the Normal approximation model[Equation (4)] using ADMB (Fournier et al., 2012) called fromthe R statistical environment (R Core Development Team, 2012),while bounding pa between 0 and 1, and lower-bounding ta andsa at zero for all ages a. Process errors s2

a may not be independentamong ages a, and these correlations could be included by movingto a multivariate normal likelihood in Equation (4), although wedo not explore this here.

Estimated standard errors can then be used to calculate themultinomial sample size N* given ideal (independent and identical-ly distributed) sampling that would have equivalent variance:

Var[Pa,i] =pa(1 − pa)

N∗a

= SE(pa)2 (5)

which upon simplification yields:

N∗a = pa(1 − pa)

SE(pa)2 , (6)

where N∗a is the estimated effective sample size for age a. This esti-

mate of effective sample size at age is then summarized across agesto estimate an overall effective sample size:

N∗ = median[N∗a ] (7)

where median(X) is the median of vector X, and I use the mediangiven that the distribution of N∗

a may be highly skewed.As one last complexity, I also seek to estimate a proportion p that

is aggregated across multiple strata, where strata have individualestimates of the proportions p s and contain a different proportionof the overall abundance or catch. In this case, the proportion is esti-mated separately for each stratum (p s), then averaged according to

the respective strata weightings:

�pa =∑ns

s=1

Ls∑ns

s′=1 Ls′pa,s

( ), (8)

where weighting is equal to the fraction of landings in stratum s. Thiscorresponds to a combined model in which p , s , and t are esti-mated as fixed effects for each strata in a single model. In this case,the strata-aggregated proportion will have variance derived fromtwo components: (i) the estimation variance of each stratum-specific proportion SE[pa,s]2, and (ii) the among-strata variancein proportions (pa,s − �pa)2/N∗, where both are weighted by thesquare of the proportion of total landings in that stratum:

Var[�pa] =∑ns

s=1

Ls∑ns

s′=1 Ls′

( )2

SE[pa,s]2 + (pa,s − �pa)2

N∗

( )(9)

which upon simplification yields:

N∗∗a = �pa(1 − �pa)∑ns

s Ls/∑ns

s′=1 Ls′( )2(SE[pa,s]2 + (pa,s − �pa)2/N∗)

(10)

where N∗∗a is the age-specific estimate of aggregate effective sample

size, and N** is fixed at the median of N∗∗a .

Dirichlet-multinomial modelFor comparison with the compositional standardization methodproposed above, I also apply a Dirichlet-multinomial (D-M)model for compositional data. This D-M model has been developedelsewhere (Hrafnkelsson and Stefansson, 2004), but I briefly sum-marize it here.

The D-M model is a hierarchical model that approximates multi-nomial sampling when there exists some variability in the expectedproportions among sampling occasions, caused by process errors oroverdispersion. In this context, the D-M model estimates two quan-tities. The first is a vectorp representing the true distribution of agesin a particular stratum, and the second is a positive numberb repre-senting variability among sampling occasions (where high b signi-fies low variability among sampling occasions and vice versa).Each realization of the Dirichlet distribution returns a vector pi,whose elements sum to 1 and which represents the expected fre-quency of a subsequent multinomial sample. Each sampling occasionhas a different expected frequency pi which is estimated as a randomeffect, and hence the marginal distribution for p and b accounts forthis between-sample variability. Conveniently, there exists a closed-form equation for the D-M likelihood when marginalizing acrossall trip-specific random effects (Mosimann, 1962):

L(p, b |D) =∏Ni=1

G(b)G(N + b)

∏amax

a=1

G(Di,a + pa)G(pa)

(11)

whereG(X) is the gamma function of X. The standard errors ofp canthen be used to estimate an aggregated effective sample size thataccounts for among-strata variability [Equations (5)–(10)], identi-cally to using our proposed method.

Simulation modelI next seek to use simulation modelling to explore the accuracy ofresulting estimates of the age distribution of a hypothetical

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population, as well as the estimation accuracy for effective samplesize. To do this, I simulate population dynamics from an unexploit-ed fish population that includes multiple strata, areas with differentsampling intensity and age distributions within each stratum, andadditional overdispersion (Figure 1).

I start by simulating a population in unfished equilibrium, whererecruitment Rt in year t fluctuates randomly around a constantmean:

Rt = R0 exp Normal −s2R

2,s2

R

( )[ ](12)

where R0 is the average recruitment, s2R the variance of recruitment

variability, and we use conventional bias correction for the log-normal distribution. Recruitment is then evenly divided among

three strata, which are envisioned as three different spatial areas(e.g. Washington, Oregon, and California coasts) that havecommon recruitment but have different age structure due to differ-ences in mortality rate among strata. Each stratum also has tworegions, envisioned as inshore and offshore habitats, and recruit-ment is divided between regions where u is the proportion in thefirst region (u ¼ 1 means that all recruitment enters region r ¼ 1,while u ¼ 0.5 means that recruitment is evenly split amongstrata). Thus, abundance Aa ¼ 0,t,s,r for age-0 recruits in year t,stratum s, and region r is:

Aa=0,t,s,r =

uRt

nsif r = 1

(1 − u)Rt

nsif r = 2,

⎧⎪⎪⎨⎪⎪⎩ (13)

where the number of strata ns ¼ 3. Individual fish experience twodemographic processes: natural mortality M and movement V.The natural mortality rate varies among strata, and movement isone-directional from inshore (region r ¼ 1) to offshore (regionr ¼ 2). Therefore, abundance for non-recruits (a . 0) follows:

Aa,t,s,r =Aa−1,t−1,s,r=1(1−M −V) if r = 1Aa−1,t−1,s,r=1V(1−M) + Aa−1,t−1,s,r=2(1−M) if r = 2

{(14)

I assume that the stratum for each sample is always known (a sam-pling trip can be traced to a given state) and hence can be used to thestratify the model. In contrast, the region for a sample is unknown(agivensampling tripmay have occurred ineitheronshore oroffshorehabitats), and hence region cannot be used to stratify the analysis.

Sampling is then conducted randomly by strata in the final simu-lated year T. There are N sampling trips in each strata. Given the ithtrip in stratum s, a region ri is randomly chosen where the probabilityof selecting region r depends on the proportion of total abundancein that region:

Pr[ri] =∑amax

a=1 Aa,t=T,s,r∑2r=1

∑amax

a=1 Aa,t=T,s,r

(15)

The number of fish caught follows a Poisson distribution withexpected number mC, and the number of caught fish that aresampled for age (ni) follows a binomial distribution with probabilityli. Differences among strata in the proportion of caught fish that aresampled for age (l) constitutes a known type of sampling intensity.This is incorporated into design [Equation (1)] and normal-approximation [Equation (4)] estimators, but not into the D-Mmodel (which has not previously been developed to use this typeof information).

The distribution of ages for these ni fish is multinomial with afixed degree of variance inflation v:

D†,i,s = v · MultinomialA†,t=T,s,r=ri∑amax

a=1Aa,t=T,s,r=ri

, ni

⎛⎜⎜⎝

⎞⎟⎟⎠ (16)

where D†,i,s is a vector representing the sample for trip i in stratum s(the dot in the subscript in place of subscript a signifies that it is avector of all ages, and this is similarly defined for A†,t=T,s,r=ri

Figure 1. Schematic representing the hierarchical sampling designenvisioned in the simulation model, with three known sampling strata(north, central, and south). Each stratum has multiple samples fromboth shallow and deep regions, where the total catch Ci (area withineach circle) and sampled catch ci (black area within each circle) areknown for each sample, and hence their ratio li ¼ ci/Ci represents aknown difference in sampling intensity among samples. In the casedepicted here, the sampling intensity is twice as high in shallow thandeep regions. The “true” age distribution may differ between shallowand deep regions, and hence ignoring differences in sampling intensitywill in this case cause “shallow” depths to be over-represented in theaggregated age distribution. This overweighting of regions with highsampling intensity is rectified by including sampling intensity li in thevariance calculation of the “normal-approximation” estimator.

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which is a vector of abundance over all ages). Variance inflationv .

1 represents a scenario in which, every time a fish is randomlyselected, the sampler “sees” v fish of that age. It represents asimple way to simulate overdispersed data where the “true”sample size of independent fish is easily seen to be ni, whereas thenominal sample size is vni, and hence, is useful to compare esti-mated and “true” sample size. An unbiased estimate of the aggre-gated age distribution will be proportional to total abundanceacross regions and strata

∑3s=1

∑2r=1 Aa,t,s,r .

This simple set of equations [Equations (12)–(16)] generatesmany different possible age distributions and types of overdispersion.Strata and regions can have identical biological (u ¼ 0.5, V ¼ 0,Ms ¼ 1 ¼Ms ¼ 2 ¼Ms ¼ 3) and sampling processes (lm ¼ 1 ¼ lm ¼ 2

andv ¼ 1), resulting is no difference in age distribution and no over-dispersion. In this case, each age measurement Da,i,s is independentand identically distributed, and hence the total sample size is 3N.When natural mortality varies among strata, each stratum has a dif-ferent age distribution. In this case, the “effective” sample size willbe lower than 3N, because data include variability more than a multi-nomial distribution with independent and identical samples.Similarly, differences in the age-distribution among regions withina given strata are caused by varying u and V. For example, u ¼ 1and V ¼ 0.2 causes all fish torecruit to inshore habitats (region 1), fol-lowed byontogenicmovement offshorewith increasing age and henceresults in large differences in age distribution among regions. Thisagain causes overdispersion and hence will decrease “effective”sample size relative to nominal sample size. Finally, variation in sam-pling intensity, e.g. lr ¼ 2 ¼ 5lr ¼ 1, will cause a naıve average of theavailable data to resemble the age distribution of whichever regionhas greater sampling intensity. In this case, properly accounting forsampling intensity is necessary to return an unbiased estimate ofage distribution in each stratum.

ConfigurationsI explore performance of three methods for standardizing compos-itional data. Specifically, I apply the “design-based” estimator[Equation (1)], the normal-approximation estimator [Equation

(4)], and the D-M estimator [Equation (11)] to each simulation rep-licate. Moreover, I explore two alternative versions of the “normalapproximation” estimator. The first (“NA-0”) does not explicitlyinclude variability among trips [i.e. uses Equation (4) while fixingsa ¼ 0 for all ages a]. The second (“NA-1”) estimates age-specificvariability among tows [i.e. uses Equation (4) and estimates sa].

All four models are applied to three different scenarios of thesimulation model, where each scenario has three different levels ofoverdispersion (listed in Table 2). Specifically, I explore a baselinescenario (“Scenario 1”) with no difference among regions withina given strata (V ¼ 0 and u ¼ 0.5). Against this baseline, Icompare two scenarios with different age composition in the tworegions, caused by inshore recruitment coupled with ontogenicmovement offshore (V ¼ 0.2 and u ¼ 1). This situation is exploredeither without (“Scenario 2”) or with (“Scenario 3”) differences insampling intensity between regions (without different sampling in-tensity: l1 ¼ l2 ¼ 0.6; with different sampling intensity: l1 ¼ 1.0and l2 ¼ 0.2). Within each scenario, I also explore the impact ofadditional overdispersion, i.e. no overdispersion (v ¼ 1), moderateoverdispersion (v ¼ 5), and high overdispersion (v ¼ 20). In allconfigurations, I assume that natural mortality varies amongstrata (M1 ¼ 0.35, M2 ¼ 0.3, M3 ¼ 0.25), such that the aggregateage distribution varies among strata, and simulate N ¼ 100 tripsper stratum, where the catch for each trip i follows a Poisson distri-bution with expected value mC ¼ 25, and hence the expectednumber of age composition reads per stratum is NmCl. I simulatevariability in the number of age reads per trip because trips willoften have variable catch rates, and because this variation is neces-sary to distinguish between overdispersion that does and does notvary as a function of sample size [t2

a and s2a, respectively, from

Equation (4)].

Evaluation criteriaI explore the performance of these three methods in estimating the“true” distribution at age for all strata combined and across regions,because it is this distribution that would represent the abundance-at-agein a non-spatial assessment model. To evaluate performance, I compute

Table 2. Parameter values for simulation model configurations

Parameter values common to all ScenariosMs ¼ 1 ¼ 0.35, Ms ¼ 2 ¼ 0.30, Ms ¼ 1 ¼ 0.25N ¼ 100mC ¼ 25

Scenario 1 Scenario 2 Scenario 3“Identical regions” “Different regions” “Different regions and sampling intensities”

No overdispersion u ¼ 0.5 u ¼ 1 u ¼ 1V ¼ 0 V ¼ 0.2 V ¼ 0.2lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ 1.0v ¼ 1 v ¼ 1 lr ¼ 2 ¼ 0.2

v ¼ 1Mild overdispersion u ¼ 0.5 u ¼ 1 u ¼ 1

V ¼ 0 V ¼ 0.2 V ¼ 0.2lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ 1.0v ¼ 5 v ¼ 5 lr ¼ 2 ¼ 0.2

v ¼ 5Strong overdispersion u ¼ 0.5 u ¼ 1 u ¼ 1

V ¼ 0 V ¼ 0.2 V ¼ 0.2lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ lr ¼ 2 ¼ 0.6 lr ¼ 1 ¼ 1.0v ¼ 20 v ¼ 20 lr ¼ 2 ¼ 0.2

v ¼ 20

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the ratio of estimated and true abundance-at-age for each age andcompute the median and 80% simulation intervals for this ratio (an un-biased model will have a median ratio of 1, and a precise model will havenarrow 80% simulation intervals).

I also explore the estimated effective sample size from the normalapproximation and D-M estimators (the design-based method doesnot estimate effective sample size). To do so, I compare effectivesample size estimates when aggregating across all three strata withthe true variance, computed from the estimated and true proportionat age for each replicate. Specifically, an “ideal” multinomial samplewith expected proportions equal to the aggregated estimate �pa willhave variance as follows:

Var[�pa] =�pa(1 − �pa)

N(equiv)a

(17)

where N (equiv)a is the sample size of the “ideal” multinomial sample.

Solving for N (equiv)a yields:

N (equiv)a = �pa(1 − �pa)

Var[�pa](18)

Replacing Var[�pa] with the squared difference between estimatedand true proportions (i.e. the sampling variance given that the esti-mator is unbiased) and taking its average across ages yields:

N (equiv) = 1

amax + 1

∑amax

a=0

�pa(1 − �pa)( pa − �pa)2 (19)

where N(equiv) is the sample size that would have (on average acrossages) the same variance as the estimation variance in the simulationexperiment. I then compare N** with N(equiv) for each estimator,across each replicate and configuration.

ResultsSimulation model outputThe simulation model generates identical age distribution for both“shallow” and “deep” regions of each stratum in Scenario 1 (i.e.abundance-at-age is identical for both regions of each strata in theleft column of Figure 2). However, abundance-at-age differsamong strata in all scenarios, where stratum 3 has the greatest pro-portion of old fish due to having the lowest natural mortality rate. InScenarios 2 and 3, in contrast, the abundance-at-age differs betweenshallow and deep regions, because the shallow region is the source ofall recruitment but fish migrate from shallow to deep waters as theyage. This leads to the majority of old fish being in the deep region forboth Scenarios 2 and 3 and represents a type of “process error” (i.e.variability among sampling occasions within a given stratum).

Estimated age distributionScenario 1 (identical regions) with no overdispersion (v ¼ 1) repre-sents the simplest case for our simulation model (top-left panel ofFigure 3). In this case, all models have essentially identical impreci-sion (0.04) and are unbiased. The presence of overdispersion (v ¼ 5or 20; middle- and bottom-left panels of Figure 3) causes the D-Mmodel to have increasing bias. This is a nearly 220% bias in age10 fish given high overdispersion. However, other models continueto perform similarly and are unbiased.

Scenario 2 (different abundance-at-age in each region, but con-stant sampling intensity) with no overdispersion again results in

bias for the D-M model (although in this case it is a positive biasfor old fish), and unbiased estimates for other models. As overdis-persion is added and increased, the D-M is the only biased model,and its bias increases with increased overdispersion. However,Scenario 3 (different abundance-at-age in each region and lowersampling intensity in region 2) is substantially different. In thiscase, the D-M model is strongly biased. This is not surprising,given that it has no way to account for known differences in sam-pling intensity among regions, and it hence under-weights region2, where all the old fish are located. Surprisingly, however, the“normal approximation” estimator that includes process errorsalso has a small negative bias (20.012). This small but appreciablebias remains with increasing overdispersion. In summary, onlythe design-based and “normal approximation” model withoutbetween-trip variability is unbiased for all scenarios and levels ofoverdispersion.

Effective sample sizeA similar model ranking emerges when comparing estimates of ef-fective sample size, although the design-based estimator does notprovide any estimate of effective sample size and hence fails topropagate information regarding the information content of com-positional data. All other estimators have an accurate estimate ofsample size in Scenario 1 and given no overdispersion (Figure 4).However, the D-M model progressively overestimates effectivesample size given increasing overdispersion (v ¼ 5 or 20), whenthere is unmodelled heterogeneity in the true population(Scenario 2), and when there is variability in sampling intensity(Scenario 3). In contrast, the two “normal approximation” estima-tors yield accurate estimates of effective sample size in all scenariosand levels of overdispersions and perform similar to one another.The two “normal approximation” estimators have similar precision,with an interquartile simulation interval ranging from �0.66 to 1.5across all scenarios and levels of overdispersion.

DiscussionSynthesizing differences in model performanceWe have contrasted four different estimators for aggregating com-positional data across known strata and unknown “regions” repre-senting differences in proportion at age among trips. Severalincorporate known differences in sampling intensity and accountfor overdispersion more than that expected from a multinomialsample. This comparison shows that the “design-based” estimatorused on the US West Coast is unbiased in its estimates of averageage distribution. However, this design-based estimator fails toprovide any estimate of effective sample size and thus provides noguidance regarding the proper weighting of compositional datarelative to abundance index data, prior information, or other contri-butions to a stock assessment objective function (Francis, 2011).The effective sample size of such a design-based estimator can beestimated using computation-intensive bootstrap estimators(Stewart and Hamel, unpublished data; Larsen et al., 2006).However, non-parametric bootstrap methods for hierarchical andnested data are still an active area of research in statistics andecology (Ren et al., 2010) and have not been previously testedusing simulation.

The “normal approximation” method that does not explicitlyaccount for between-trip variability provides unbiased estimatesof proportion at age, similar to the design-based estimator. Unlikethe design-based estimator, however, it also provides a model-based

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estimate of effective sample size. This “normal approximation”method was derived to match the mean of multinominally distrib-uted data and to provide an estimate of effective sample size. Thiseffective sample size estimate is consistent with the estimation vari-ance in the simulation model and hence appears to be robust to thescenarios explored in the simulation model, i.e. unmodelled differ-ences in abundance-at-age within strata, differences among strata,and additional overdispersion. It is surprising that the normal ap-proximation without between-trip variability (NA-0) performedslightly better than with between-trip variability (NA-1), giventhat variability between regions within a given stratum is a source

of between-trip variability. However, perhaps this result will changegiven increased differences between regions.

Despite providing estimates of effective sample size, the normalapproximation cannot account for additional types of “processerrors”. These additional process errors arise whenever the processcontributing to compositional data is not included in the modelthat is interpreting these data. For example, variation in fishery se-lectivity over time will cause “process errors” whenever its compos-itional data are included in a model with time-constant selectivity.These process errors will require that compositional data receive adecreased weighting in the model likelihood, relative to the

Figure 2. Abundance-at-age for strata 1–3 (rows 1–3) and for all strata combined (bottom row), given either identical abundance-at-age for eachregion (left column), different abundance-at-age but constant sampling intensity for each region (middle column), or different abundance-at-ageand sampling intensities for each region (right column). In each panel, the dotted line represents the “inshore” region (r ¼ 1), the dashed linerepresents the “offshore” region (r ¼ 2), and the solid line represents the total abundance-at-age across both regions. The “identical regions”scenario (left column) has identical abundance-at-age for both regions, and hence the dashed and dotted lines are indistinguishable.

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weighting implied by effective sample sizes estimates from our pro-posed methods. We therefore envision that effective sample size esti-mates be used as a starting point in assessment models, subject tofurther up- or down-weighted during the model-building process(Maunder, 2011; Hulson et al., 2012). Such down-weightingmethods estimate effective sample size using an entirely different prin-ciple than the models presented here, i.e. by evaluating thegoodness-of-fit of compositional data to model the estimates of

abundance (McAllister and Ianelli, 1997; Magnusson, 2012). Thesetwo methods are consistent because the normal-approximationmethods uses a type of information (among-trip sampling variance)that is otherwise ignored in the design-based estimator and assessmentmodel down-weighting methods and can be further tuned using previ-ously developed methods.

I have not shown here the results for other model-based methodsfor aggregating compositional data, including the multivariate

Figure 3. The ratio of estimated and true distribution at age pa/pa for all strata combined, given either identical abundance-at-age for each region(left column), different abundance-at-age but constant sampling intensity for each region (middle column), or different abundance-at-age andsampling intensities for each region (right column) and either no overdisperison (top row), mild overdispersion (middle row), or largeoverdispersion (bottom row). In each panel, the red area represents the 80% simulation interval (10 and 90% quantiles) for the design-basedestimator, the method green area represents D-M estimator, the blue represents the normal approximation without between-trip variability (sa ¼0), and the black represents the normal approximation with between-trip variability (estimatingsa). In each panel, the top number (colour code asabove) is the median absolute relative error in estimates of age composition (a measure of imprecision, where low represents good modelperformance), and the number in parentheses is the median relative error (a measure of bias, where a number close to zero is unbiased).

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normal-multinomial mixture model (Hrafnkelsson and Stefansson,2004). This model is conceptually similar to the D-M model, inthat the multivariate normal is used as a hyperdistribution for trip-specific distribution at age, then sampling within each trip isassumed to be multinomially distributed. By representing between-trip variability, the multivariate-normal distribution thus performsa similar role to the Dirichlet distribution in the D-M model.Unlike the Dirichlet distribution, however, the multivariate normalprovides a more flexible approximation to between-trip variability.The multivariate normal-multinomial method has been explored

previously (Hrafnkelsson and Stefansson, 2004) but, like other alter-native proposals (Candy, 2008; Dunn and Hanchet, 2010), it is notobvious how to incorporate the “sampling intensity” informationthat is used in the design-based estimator, and which hence repre-sents a necessary element for any replacement. Additionally, ex-ploratory analysis (Thorson, unpublished results) showed thatthe multivariate normal-multinomial model was biased in thescenario with unequal sampling intensities, in a similar mannerto the D-M model (which also failed to account for differencesin sampling intensity).

Figure 4. The ratio of estimated effective sample size for all strata combined [Equation (10)] and the average sample size with variance equal to thesimulation variance [Equations (17)–(19)] for each model [design, design-based estimator, Equation (1); D-M, Dirichlet-multinomial estimator,Equation (11); NA-0 and NA-1, normal approximation estimator, Equation (4), either with or without estimating additional process errors,respectively], given either identical abundance-at-age for each region (left column), different abundance-at-age but constant sampling intensity foreach region (middle column), or different abundance-at-age and sampling intensities for each region (right column) and either no overdisperison(top row), mild overdispersion (middle row), or large overdispersion (bottom row). In each panel, the design-based formula provides no estimate ofeffective sample size (and is hence blank), and an estimate greater than one implies that the model overestimates effective sample size.

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Future directions for standardizing compositional dataSurvey data are routinely “standardized” when estimating an indexof abundance. This standardization accounts for unequal samplingintensity, differences among strata, and observation-level covariatesthat may improve precision of the resulting index. However, thestandardization of compositional data is relatively under-studiedand represents an important area of future research.

I have attempted to develop a generic and computationallysimple platform for standardizing compositional data. The“normal approximation” model without process errors can in thefuture be estimated using standard linear modelling software, e.g.by treating the expanded data liDi as a normally distributed re-sponse variable and by specifying model variance as varying byage, stratum, and other covariates while also fixing it as a linear func-tion of sample sizes and intensities nili. This provides a computa-tionally simple method for estimating parameters in the model,even when analysing multiple years and strata simultaneously.

One possible benefit of a generic compositional-standardizationapproach would the ability to “fill in” strata and years without anycompositional data. Any stratum without any compositional datawill result in no information for the design-based estimator forthe entire year. As one concrete example, a recent assessment ofdarkblotched rockfish (Sebastes crameri) had age compositionaldata from 1980 to 2012 for the primary fishery (Gertseva andThorson, 2013). However, early years had data from only two ofthe three West Coast states (Washington, Oregon, and California),so these historical age-composition data could not be aggregatedand were hence discarded. One way to avoid discarding compos-itional data for any year without compositional data in one ormore strata is to standardize compositional data while treating theinteraction of stratum and year as a random effect. This is analogousto treating the stratum–year interaction as random in index stand-ardization models (Maunder and Punt, 2004; Thorson and Ward,2013), and it will result in a large estimate of imprecision for the esti-mated distribution at age for that stratum whenever there is substan-tial variability among strata and years. However, if the totalabundance or catch is small in the stratum that is missing data,the estimated imprecision for the aggregated abundance-at-agemay still be low. This could complicate assessment fitting (in par-ticular the analysis of model residuals), but may provide an alterna-tive to treating data from each stratum as a separate fleet (“areas asfleets”, Punt et al., in press), although the benefits and risks of thiswould require exploration and testing.

Another potential advantage to standardizing compositionaldata is the ability to explicitly model “vessel effects”. Contractvessels are used to collect catch rate and compositional data onthe US West Coast (Bradburn et al., 2011), and index standardiza-tion models routinely account for differences in catch rates amongcontract vessels (Helser et al., 2004). Importantly, however, vesselsmay also differ in the age and length distribution of their catches,due to some vessels being more or less likely to approach rocky habi-tats that are associated with different ages (Jagielo et al., 2003).Therefore, failing to account for vessel effects in compositionaldata may contribute to unmodelled variability among years in com-positional data.

For these reasons and others, I propose that the standardizationof compositional data is an important next step in the analysis ofsampling data from both randomized scientific surveys and fisheries.This standardization can proceed using the “normal approximation”developed here, which isunbiased in its estimatesof proportionat age,

and provides acomputationallysimple frameworkfor future develop-ment and testing. Regardless of the model used, however, thesemethods must account for (i) differences in sampling intensity spa-tially or over time, (ii) non-independence of available data, and (iii)the effect of covariates such as spatial strata.

AcknowledgementsThis research was greatly improved by discussions with I. Stewart,who provided constructive and clear-eyed suggestions, O. Hamel,who helped develop the formula for aggregating sample sizesamong strata [Equation (10)] and provided a helpful review, andG. Thompson, who discussed ideas for alternative estimators. It wasalso aided by V. Gertseva, B. Horness, C. Minto, and G. Thompson.Finally, it was improved by comments from two anonymousreviewers, J. Hastie, M. McClure, A. Punt, and S. Zhou.

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