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 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 Vb1b: Faroe Plateau
ICES Vb2: Faroe Bank
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
100
Variability of 2 years-old length at age
is similar for both stocks when reared
in similar conditions
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
The same variability of 2-year old fish that
hatched in different days
Hatching date 21/4/94
8
0
2
4
Frequency
6
8
23/4/94
6
0
2
4
Frequency
4
2
0
Frequency
6
8
22/4/94
50 55 60 65 70 75 80
50 55 60 65 70 75 80
50 55 60 65 70 75 80
y[x1 == x[i]]
y[x1 == x[i]]
y[x1 == x[i]]
Length (cm)
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
100
In mesocosm, the variability of
size at age seems the same
over ages
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
100
Reared in the same conditions,
fish from both stocks have
similar growth patterns
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
100
In the wild variability of size at age
seems to decrease at older ages
(fewer recoveries also)
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
100
Growth rates
seem different by
area released
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
100
Wild (fished) X Mesocosm (unfished)
Apparently different growth pattern
and variability of size at age
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
Full data set
100
90
80
length (cm)
70
60
50
40
Faroe Plateau Mesocosm
30
Faroe Plateau Wild
20
Faroe Bank Mesocosm
10
Faroe Bank Wild
0
0
500
1000
1500
age (days)
2000
2500
Variability of size at age
CV length at age
CV length at age
Wild
Coefficient of Variation
Standard Deviation (cm)
Mesocosm
20
18
16
14
12
10
8
6
4
2
0
Mesocosm
Average
8.16%
0
20
18
16
14
12
10
8
6
4
2
0
20
Wild
0
20
40
60
40
60
mean length at age (mm)
Mean length at age (cm)
80
100
mean length at age (mm)
80
100
Variability of size at age
5.0
4.0
Faroe Bank / Mesocosm
3.0
2.0
Log. (Faroe Bank /
Mesocosm)
1.0
0.0
0
500
1000
1500
2000
2500
age (days)
7.0
Wild
6.0
Faroe Bank / Wild
5.0
4.0
Faroe Plateau / Wild
3.0
2.0
Poly. (Faroe Plateau /
Wild)
1.0
0.0
0
500
1000
1500
2000
2500
age (days)
Age (days)
CV length at age
SD length at age (mm)
SD length at age (mm)
Standard Deviation (cm)
Faroe Plateau /
Mesocosm
Coefficient of Variation
Mesocosm
6.0
CV length at age
7.0
20
18
16
14
12
10
8
6
4
2
0
0
500
1000
1500
2000
2500
age (days)
20
18
16
14
12
10
8
6
4
2
0
0
500
1000
1500
age (days)
2000
2500
Growth pattern:
Expected size at age
Mesocosm (Unfished)
100
90
data
Specialized VB
Monomolecular growth
Exponential Growth
Richards
Generalized VB
Smith
Blumberg
Generic growth
Second order exp polynomial
Gompetz
Generalized Gompetz
Generalized Logistic Growth
80
length at age (cm)
70
60
50
40
30
20
10
0
0
500
1000
1500
age (days)
Exponential Monomolecular Generalized Specialized
Growth
growth
VB
VB
alpha
1
0
1.38
0.67
beta
1
1
1
0.33
gama
1
1
1
1
K
0.00
0.09
0.00
0.03
Linf
70.53
78.93
70.06
71.49
L0
11.35
0.00
15.68
0.98
sd
5.23
5.51
5.19
5.23
NLL
917
934 916.3616405
919
npar
4
4
5
4
AIC
1842.0
1875.8
1842.7
1845.5
delta AIC
4.68
38.47
5.40
8.14
Richards
1
2.953
1
0.00
69.66
17.60
5.19
916
5
1842.2
4.83
Smith
0.473
1.00
1.00
0.02
71.29
1.90
5.23
918
4
1844.7
7.42
2000
Blumberg
-0.198
1
0.316
0.12
68.95
0.10
5.13
913
6
1837.7
0.35
2500
Generic
growth
0.036
7.643
1.126
0.04
69.68
11.39
5.19
916
6
1844.2
6.87
Generalized
Gompetz
1
0.0000001
0.603
25.54
69.00
14.21
5.16
914
5
1838.9
1.53
Gompetz
1
0.0000001
1
23302.81
71.21
4.38
5.23
918
4
1844.4
7.10
Second order
exp polynomial
1
0.0000001
0.5
4.11
68.91
17.32
5.16
915
4
1837.3
0.00
Generalized
Logistic Growth
-0.098
0.176
0.314
0.13
68.95
0.10
5.13
913
7
1839.592222
2.27
Recoveries from the wild
(Fareau Plateau) follow similar
growth patterns for middle
ages…
100
90
80
data
Series2
Specialized VB
Monomolecular growth
Exponential Growth
Richards
Generalized VB
Smith
Blumberg
Generic growth
Second order exp polynomial
Gompetz
Generalized Gompetz
length at age (cm)
70
60
50
40
30
20
10
0
0
500
1000
1500
age (days)
2000
2500
But not on the extremes
100
90
data
Series2
Specialized VB
Monomolecular growth
Exponential Growth
Richards
Generalized VB
Smith
Blumberg
Generic growth
Second order exp polynomial
Gompetz
Generalized Gompetz
80
length at age (cm)
70
60
50
40
30
20
10
0
0
500
1000
1500
age (days)
2000
2500
100
Wild (Fished)
90
data
Specialized VB
80
Monomolecular growth
length at age (cm)
70
Exponential Growth
Richards
60
Generalized VB
50
Smith
Blumberg
40
Generic growth
30
Second order exp polynomial
Gompetz
20
Generalized Gompetz
10
Generalized Logistic Growth
0
0
alpha
beta
gama
r_
Linf
L0
sd
NLL
npar
AIC
delta AIC
500
1000
1500
age (days)
Exponential Monomolecular Generalized Specialized
Growth
growth
VB
VB
1
0
0.256
0.667
1
1
1
0.333
1
1
1
1
0.00
0.10
0.05
0.03
57.77
64.60
59.30
58.92
12.07
0.00
0.02
3.11
3.98
691
4
1390
0.83
4.00
692
4
1391
1.96
3.97
691
5
1391
1.88
3.97
691
4
1389
0.00
Richards
2000
Smith
2500
Blumberg
1
0.029
1
0.10
58.56
6.53
0.473
1.000
1.000
0.02
58.68
4.41
0.001
1
0.651
0.09
57.29
0.10
3.97
691
5
1391
2.22
3.97
691
4
1389
0.12
3.97
691
6
1393
4.25
Generic Generalized
growth
Gompetz
0.794
1
0.352
0.0000001
1.585
0.882
0.05
3957.36
64.83
57.61
0.05
8.73
3.96
690
6
1393
3.63
3.98
691
5
1392
2.49
Gompetz
1
0.0000001
1
29202.77
58.58
6.29
3.97
691
4
1389
0.20
Second order
Gen
exp polynomial Logis
1
0.0000001
0.5
5.26
55.66
17.32
4.01
692
4
1392
3.30
Exploring variability parameterizations
with best expected value model
Hypotheses:Linear models
Explanatory/var
sd
cv
Length at age
Option 1
Option 3
age
Option 2
Option 4
Delta AIC
Explanatory/var
Mesocosm (Unfished)
sd
cv
Length at age
0.0
0.0
age
4.3
0.01
Explanatory/var
Wild (Fished)
Lowest AIC 1817.1, one sd AIC 1837.3
Lowest AIC 1364.4 one sd AIC 1389
sd
cv
Length at age
0.0
0.3
age
0.0
11.6
Mesocosm (Unfished)
Best: sd linear with
mean length at
Option 1
age
100
90
2 with age
Worst:
linear
Worst sd Option
100
90
80
Expected value
70
Expected
value
70
40
length at age (cm)
length at age (cm)
80
30
data
60
60
mean-1.96sd
mean-1.96sd
50
50
mean+1.96sd
mean+1.96sd
40
30
20
20
10
10
0
0
0
500
1000
1500
age (days)
2000
data
0
2500
500
1000
1500
age (days)
2000
2500
1500
2000
4.000
4
3.000
3
2.000
2
1.000
1
0.000
0
0
-1
0
500
1000
1500
2000
2500
-1.000
-2
-2.000
-3
-3.000
-4
-4.000
500
1000
2500
Wild (Fished)
Worst:
Worst CV linear with age
Best: sd linear with mean length at
Option 1
age or age
100
80
length at age (cm)
Expected value
60
mean-1.96sd
40
mean+1.96sd
20
data
0
0
500
-20
1000
1500
2000
2500
age (days)
4
3
2
1
0
-1
-2
-3
0
500
1000
1500
2000
2500
Artic trout
Zeta Lake
7106’ N, 10634’ W
Campana et al 2008, CJFA 65: 733-743
Artic trout
• Fish collected in 2003
• Validation of ring interpretation using bombradiocarbon method
Reference chronologies for
several artic species NWA
Reference chronologies
for a freshwater artic
species , compared
Atomic bomb testing
with atmosphere and
1958:Peak of bomb testing
NWA
14C for Artic char  and
cores of old lake trout o
Artic trout
100 m
Annual growth
increment of a
29 year old (56
cm) artic trout
100 m
Trout
1200
Artic lake trout Salvelinus namaycush
True
“outliers”
Female Juvenile
1000
Female Adult
fork length (mm)
Male Juvenile
Male Adult
800
600
400
200
0
0
10
20
30
age (years)
40
50
60
70
Linear-von Bertalanffy hybrid model
Combines linear growth for juveniles with von Bertalanffy growth for
adults.
𝐿 𝑡 = 1 − 𝑝 𝑡 𝑎 + 𝑏𝑡 + 𝑝 𝑡 𝑎 + 𝑏𝑐 + 𝑉𝐵 𝑡
𝑉𝐵 𝑡 = 𝐿∞ − 𝑎 + 𝑏𝑐
1 − 𝑒𝑥𝑝 −𝐾 𝑡 − 𝑡0
𝑝 𝑡 = 1 + 𝑒𝑥𝑝 −𝑙𝑛 19 𝑡 − 𝑡50 / 𝑡95 − 𝑡50
−1
t95 could be fixed at t50 + 0.1 to make an abrupt change
c 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)
Best fit for male trout
Uncertain area, no data
1200
1.00
Arctic trout MALES
0.80
800
0.60
600
0.40
400
0.20
200
0
0.00
0
10
20
30
age ( years)
40
50
60
proportion mature
length (mm)
1000
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 lengthat-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 lengthat-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
Two rarely available data sets (because of no or minimal ageing error)
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
90
90
80
80
90
80
60
50
40
60
40 mean+1.96sd
20
20
10
10
0
0
0
500
1000
1500
age (days)
2000
2500
mean-1.96sd
50
30
30
length at age (cm)
length at age (cm)
length at age (cm)
70 Expected value
70 Expected value
70
Option 4
100
Option 3
100
Option 2
100
60
mean-1.96sd
50
40 mean+1.96sd
30
data
data
20
10
0
0
500
1000
1500
age (days)
2000
0
2500
500
1000
1500
age (days)
2000
2
4.000
4.000
4.000
3.000
3.000
3.000
2.000
2.000
2.000
1.000
1.000
1.000
0.000
0
500
1000
1500
2000
2500
0.000
0.000
0
500
1000
1500
2000
2500
-1.000
-1.000
-1.000
-2.000
-2.000
-2.000
-3.000
-3.000
-4.000
-4.000
-3.000
-4.000
0
500
1000
1500
2000
250
100
Individual variability
40
60
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80
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500
1000
age (days)
1500
2000