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

*cminte@iattc.org

Outline of the TalkIntroduction

– 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 workRecommendations

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?

Methods1. 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.• 8408 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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

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

in similar conditions

21/4/94

y[x1 == x[i]]

Freq

uenc

y

50 55 60 65 70 75 80

02

46

8

22/4/94

y[x1 == x[i]]

Freq

uenc

y

50 55 60 65 70 75 80

02

46

8

23/4/94

y[x1 == x[i]]

Freq

uenc

y

50 55 60 65 70 75 80

02

46

8

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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

Growth rates seem different by area released

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

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

0 500 1000 1500 2000 25000

10

20

30

40

50

60

70

80

90

100

Faroe Plateau Mesocosm

Faroe Plateau Wild

Faroe Bank Mesocosm

Faroe Bank Wild

age (days)

leng

th (c

m)

Full data set

Variability of size at age St

anda

rd D

evia

tion

(cm

)

Mesocosm

Wild

Mean length at age (cm)

02468

101214161820

0 20 40 60 80 100

CV l

engt

h at

age

mean length at age (mm)

Faroe Plateau /Mesocosm

Faroe Bank / Mesocosm

Linear (Faroe Bank /Mesocosm)

02468

101214161820

0 20 40 60 80 100CV

leng

th at

agemean length at age (mm)

Faroe Bank / Wild

Faroe Plateau / Wild

Linear (Faroe Plateau /Wild)Co

effici

ent o

f Var

iatio

n

Mesocosm

Wild

Average8.16%

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 500 1000 1500 2000 2500

SD le

ngth

at a

ge (m

m)

age (days)

Faroe Plateau /Mesocosm

Faroe Bank / Mesocosm

Log. (Faroe Bank /Mesocosm)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 500 1000 1500 2000 2500

SD le

ngth

at a

ge (m

m)

age (days)

Faroe Bank / Wild

Faroe Plateau / Wild

Poly. (Faroe Plateau /Wild)

02468

101214161820

0 500 1000 1500 2000 2500

CV l

engt

h at

age

age (days)

Faroe Plateau /Mesocosm

Faroe Bank / Mesocosm

Linear (Faroe Bank /Mesocosm)

02468

101214161820

0 500 1000 1500 2000 2500CV

len

gth

at a

geage (days)

Faroe Bank / Wild

Faroe Plateau / Wild

Expon. (Faroe Plateau /Wild)

Variability of size at age St

anda

rd D

evia

tion

(cm

)

Mesocosm

Wild

Age (days)

Coeffi

cien

t of V

aria

tion

Growth pattern:Expected size at age

Exponential Growth

Monomolecular growth

Generalized VB

Specialized VB

Richards Smith BlumbergGeneric growth

Generalized Gompetz

GompetzSecond order

exp polynomialGeneralized

Logistic Growthalpha 1 0 1.38 0.67 1 0.473 -0.198 0.036 1 1 1 -0.098beta 1 1 1 0.33 2.953 1.00 1 7.643 0.0000001 0.0000001 0.0000001 0.176gama 1 1 1 1 1 1.00 0.316 1.126 0.603 1 0.5 0.314K 0.00 0.09 0.00 0.03 0.00 0.02 0.12 0.04 25.54 23302.81 4.11 0.13Linf 70.53 78.93 70.06 71.49 69.66 71.29 68.95 69.68 69.00 71.21 68.91 68.95L0 11.35 0.00 15.68 0.98 17.60 1.90 0.10 11.39 14.21 4.38 17.32 0.10sd 5.23 5.51 5.19 5.23 5.19 5.23 5.13 5.19 5.16 5.23 5.16 5.13NLL 917 934 916.3616405 919 916 918 913 916 914 918 915 913npar 4 4 5 4 5 4 6 6 5 4 4 7AIC 1842.0 1875.8 1842.7 1845.5 1842.2 1844.7 1837.7 1844.2 1838.9 1844.4 1837.3 1839.592222delta AIC 4.68 38.47 5.40 8.14 4.83 7.42 0.35 6.87 1.53 7.10 0.00 2.27

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

dataSpecialized VBMonomolecular growthExponential GrowthRichardsGeneralized VBSmithBlumbergGeneric growthSecond order exp polynomialGompetzGeneralized GompetzGeneralized Logistic Growth

Mesocosm (Unfished)

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

dataSeries2Specialized VBMonomolecular growthExponential GrowthRichardsGeneralized VBSmithBlumbergGeneric growthSecond order exp polynomialGompetzGeneralized Gompetz

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

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

dataSeries2Specialized VBMonomolecular growthExponential GrowthRichardsGeneralized VBSmithBlumbergGeneric growthSecond order exp polynomialGompetzGeneralized Gompetz

But not on the extremes

Exponential Growth

Monomolecular growth

Generalized VB

Specialized VB

Richards Smith BlumbergGeneric growth

Generalized Gompetz

GompetzSecond order

exp polynomialGeneralized

Logistic Growthalpha 1 0 0.256 0.667 1 0.473 0.001 0.794 1 1 1 1.455beta 1 1 1 0.333 0.029 1.000 1 0.352 0.0000001 0.0000001 0.0000001 0.341gama 1 1 1 1 1 1.000 0.651 1.585 0.882 1 0.5 5.095r_ 0.00 0.10 0.05 0.03 0.10 0.02 0.09 0.05 3957.36 29202.77 5.26 0.13Linf 57.77 64.60 59.30 58.92 58.56 58.68 57.29 64.83 57.61 58.58 55.66 106.58L0 12.07 0.00 0.02 3.11 6.53 4.41 0.10 0.05 8.73 6.29 17.32 0.01

sd 3.98 4.00 3.97 3.97 3.97 3.97 3.97 3.96 3.98 3.97 4.01 3.96NLL 691 692 691 691 691 691 691 690 691 691 692 690npar 4 4 5 4 5 4 6 6 5 4 4 7AIC 1390 1391 1391 1389 1391 1389 1393 1393 1392 1389 1392 1394delta AIC 0.83 1.96 1.88 0.00 2.22 0.12 4.25 3.63 2.49 0.20 3.30 5.27

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

data

Specialized VB

Monomolecular growth

Exponential Growth

Richards

Generalized VB

Smith

Blumberg

Generic growth

Second order exp polynomial

Gompetz

Generalized Gompetz

Generalized Logistic Growth

Wild (Fished)

Exploring variability parameterizationswith best expected value model

Explanatory/var sd cv

Length at age Option 1 Option 3

age Option 2 Option 4

Hypotheses:Linear models

Explanatory/var sd cv

Length at age 0.0 0.0

age 4.3 0.01

Delta AIC

Mesocosm (Unfished)

Explanatory/var sd cv

Length at age 0.0 0.3

age 0.0 11.6Wild (Fished)

Lowest AIC 1364.4 one sd AIC 1389

Lowest AIC 1817.1, one sd AIC 1837.3

0

10

20

30

40

50

60

70

80

90

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0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

Option 1

Expected value

mean-1.96sd

mean+1.96sd

data

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

Option 2

Expected value

mean-1.96sd

mean+1.96sd

data

-4

-3

-2

-1

0

1

2

3

4

0 500 1000 1500 2000 2500

Best: sd linear with mean length at age

Mesocosm (Unfished)Worst

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

0 500 1000 1500 2000 2500

Worst: sd linear with age

Best: sd linear with mean length at age or age

Wild (Fished)WorstWorst: CV linear with age

-20

0

20

40

60

80

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

Option 1

Expected value

mean-1.96sd

mean+1.96sd

data

-4

-3

-2

-1

0

1

2

3

4

0 500 1000 1500 2000 2500

Artic trout

Zeta Lake 7106’ N, 10634’ W

Campana et al 2008, CJFA 65: 733-743

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

14C for Artic char and cores of old lake trout o

Atomic bomb testing1958:Peak of bomb testing

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

100 m

100 m

Trout

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60 70

fork

leng

th (m

m)

age (years)

Artic lake trout Salvelinus namaycush

Female Juvenile

Female Adult

Male Juvenile

Male Adult

True “outliers”

Linear-von Bertalanffy hybrid model

Combines linear growth for juveniles with von Bertalanffy growth for adults.

t95 could be fixed at t50 + 0.1 to make an abrupt changec could be set equal to t50

t0 could be set to t50 t50 could be set at the approximate age at maturity

Integrated maturity information (Binomial likelihood) and length at age information (normal likelihood)

0.00

0.20

0.40

0.60

0.80

1.00

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

prop

ortio

n m

atur

e

leng

th (m

m)

age ( years)

Arctic trout MALES

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

Modelling1. 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 selectionfor same likelihood= AIC, BIC

Model diagnosticsresidual 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)

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

Option 2

Expected value

mean-1.96sd

mean+1.96sd

data

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500le

ngth

at a

ge (c

m)

age (days)

Option 3

Expected value

mean-1.96sd

mean+1.96sd

data

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

leng

th a

t age

(cm

)

age (days)

Option 4

Expected value

mean-1.96sd

mean+1.96sd

data

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

0 500 1000 1500 2000 2500

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

0 500 1000 1500 2000 2500

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

0 500 1000 1500 2000 2500

0 500 1000 1500 2000

02

04

06

08

01

00

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len

gth

(cm

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