Guidance for modelling the variability of length-at-age: lessons from datasets with no aging error C.V. Minte-Vera (1)*, S. Campana (2), M. Maunder (1) (1)Inter-American Tropical Tuna Commission (2)Bedford Institute of Oceanography

Outline of the Talk Introduction – The problem – Why it matters – What we will do to address it Methods – General overview Results – For each data set General escription of data By question: models and results Lessons learnt and future work Recommendations

Introduction Several stock assessments rely mainly on length frequencies, such as those for tropical tunas that have no or very limited age frequency data (or age conditional on length data). Because of lack of information, assumptions about how the variability of length-at-age changes with age are adopted. Most likely the parameters are fixed are “reasonable” values. The variability of length-at-age can highly influence the interpretation of the length-frequency information in the context of integrated analysis for stock assessment. Potential effects on the magnitude of the estimated derived quantities (biomass, harvest rate) and on the management advice

Introduction What assumption to choose? And why? for both the expected size at age and variability of size at age

Introduction In this study we will address these questions by taking advantage of two rarely available data sets no (or minimal) ageing error One data set with completely know age structure One data set from a pristine long-lived lake population

Specific questions to be addressed What is the best model to describe the growth trajectory in for the fished and unfished groups? What is the magnitude of variability of size at age of a cohort with other sources of variability controlled (birth date, aging error, sampling,…)? How does it varies over size or age? Does this depends on whether it was fished or not?

Methods 1. What is the best summary statistics? Computed mean size at age, standard deviations and coefficient of variation of size at age and explore relationships. For continuous age data, break the distribution into intervals. 2. What is the best functional form? a. generalized logistic function, which can metamorphose into more than 10 growth functions (Von Bertalanffy, Richard, Gompetz, …). AIC. b. introduce a new growth function: linear-Von Bertalanffy, which is also fit to maturity data. 3. What is the best of size-at-age variability assumption? Four model: linear relationship between sd or CV as a function of either mean size at age or age (SS3 assumption) More details latter…

Faroe Cod ICES Vb2: Faroe Bank ICES Vb1b: Faroe Plateau

Faroe Cod Enhancement program of the Faroese Fisheries Laboratory and the Aquaculture Research Station Stock decline and fishery collapsed in 1990 Fish caught at the two spawning grounds in 1994, held in captivity until matured Eggs and larvae reared in tanks, separated by origin With about 1 year old, tagged, released either to mesocosm or to the wild, after a couple of weeks of tagging In mesocosm, mixed in three pens, with 50%fish of each stock, subsamples taken between January and April each year released to Faroe Plateau in 1995, recovered by fishes (same for Faroe Bank, but very few recoveries) 3500 fish from Faroe Plateau and 3000 from Faroe Bank help in mixed pens until the spring of 2000.

In mesocosm, fish measured for the first time at 2 years old

Variability of 2 years-old length at age is similar for both stocks when reared in similar conditions

The same variability of 2-year old fish that hatched in different days Length (cm) Hatching date Faroe Bank fish

Fish released in the wild (recovered by fishers) of about 2years old have similar variability of size at age than mesocosm fish

In mesocosm, the variability of size at age seems the same over ages

Reared in the same conditions, fish from both stocks have similar growth patterns

In the wild variability of size at age seems to decrease at older ages (fewer recoveries also)

Growth rates seem different by area released

Wild (fished) X Mesocosm (unfished) Apparently different growth pattern and variability of size at age

Full data set

Variability of size at age Standard Deviation (cm) Mesocosm Wild Mean length at age (cm) Coefficient of Variation Mesocosm Wild Average 8.16%

Variability of size at age Standard Deviation (cm) Mesocosm Wild Age (days) Coefficient of Variation

Growth pattern: Expected size at age

Mesocosm (Unfished)

Recoveries from the wild (Fareau Plateau) follow similar growth patterns for middle ages…

But not on the extremes

Wild (Fished)

Exploring variability parameterizations with best expected value model Explanatory/varsdcv Length at ageOption 1Option 3 ageOption 2Option 4 Hypotheses:Linear models Explanatory/varsdcv Length at age0.0 age Delta AIC Mesocosm (Unfished) Explanatory/varsdcv Length at age age Wild (Fished) Lowest AIC one sd AIC 1389 Lowest AIC , one sd AIC

Best: sd linear with mean length at age Mesocosm (Unfished) Worst Worst: sd linear with age

Best: sd linear with mean length at age or age Wild (Fished) Worst Worst: CV linear with age

Artic trout Zeta Lake 71  06’ N, 106  34’ W Campana et al 2008, CJFA 65:

Artic trout Fish collected in 2003 Validation of ring interpretation using bomb- radiocarbon method Reference chronologies for several artic species NWA Reference chronologies for a freshwater artic species , compared with atmosphere and NWA  14 C for Artic char  and cores of old lake trout o Atomic bomb testing 1958:Peak of bomb testing

Artic trout Annual growth increment of a 29 year old (56 cm) artic trout

Trout True “outliers”

Best fit for male trout Uncertain area, no data

Future work Cod: Compare likelihood (normal, lognormal) Trout: Model error structure as a mixed distribution Maybe do factorial design

Recommendations For age-and-growth laboratories: Expected values – Try different growth functions using unified approach (e.g. generalized logistic model) – Combine age and growth study with maturity study – Try hybrid models when both info are available Variability – Focus not only on the estimation of the position (e.g. the growth function parameters) but also on the scale parameters (variability) when designing the sampling scheme. – Try different parameterizations for the modelling of the variability of length- at-age, report those on the papers – Explore the effect of the different assumptions related to variability on the estimation of the position parameters. –.

Recommendations For stock assessment modelers: If there is no study of the variability of size at age for the stock, take into account the life-history before setting the assumptions (e.g. outliers) Try a couple of sensitivity cases If the variability of the unfished population is to be represented assume constant CV over age (or standard deviation increase with mean size at age) If the variability of the exploited population is to represented assume CV or SD decreasing with ages When rebuilding a stock consider also revisiting the variability of size at age If linear-VB is appropriate, in SS3 use a first reference age accordingly

Thank you! And… Alex Aires-da-Silva, Cleridy Lennert-Cody, Rick Deriso (IATTC) for comments and inputs Steve Martell for help with some ADMB library issues

Modelling 1. Estimation of central tendency – Choice of available data – Choice of growth model Estimation of variability at age – Pdf: what probability density function best describes the variability of length- at-age for fished and unfished populations? – Parameter: What is the best summary statistics of the variability of length-at-age – Model: What functional form (e.g. constant with age, increasing with length-at-age) best summarized the changes of the variability of length-at-age over ages for fished and unfished populations? Model selection for same likelihood= AIC, BIC Model diagnostics residual analyses, predictive posterior distribution

Methods Cod (Gadus morua) from Faroe Islands The fish were hatched in captivity then tagged and released Two subject to fishing (released in the wild in Faroe Plateau and Faroe Bank) Two unexploited (kept in mesocosm) Artic trout (Salvinus namaycush) from Zeta Lake Never fished Minimal ageing error ( age validated with bomb-radiocarbon methods) Maturity information for each fish also available Two rarely available data sets (because of no or minimal ageing error)

Individual variability