guidance for modelling the variability of length-at-age: lessons from datasets with no aging error...
TRANSCRIPT
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 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• 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.
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
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y[x1 == x[i]]
Freq
uenc
y
50 55 60 65 70 75 80
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y[x1 == x[i]]
Freq
uenc
y
50 55 60 65 70 75 80
02
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y[x1 == x[i]]
Freq
uenc
y
50 55 60 65 70 75 80
02
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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
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
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80
90
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
0
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30
40
50
60
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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• 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
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
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30
40
50
60
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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
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age (days)
len
gth
(cm
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