Probability reporting of spawning
biomass estimates derived from daily
egg production surveys
Andrew Penney
March 2016
Pisces Australis (Pty) Ltd
© 2016 Pisces Australis (Pty) Ltd
All rights reserved
Ownership of Intellectual property rights
Unless otherwise noted, copyright and any other intellectual property rights in this publication are owned by
Pisces Australis Pty Ltd.
This publication and any information sourced from it should be attributed to:
Penney A.J. (2016) Probability reporting of spawning biomass estimates derived from daily egg
production surveys. Pisces Australia (Pty) Ltd, Report to the Australian Fisheries Management
Authority, 20 pp.
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Disclaimer
The authors do not warrant that the information in this document is free from errors or omissions. The authors
do not accept any form of liability, be it contractual, tortious, or otherwise, for the contents of this document
or for any consequences arising from its use or any reliance placed upon it. The information, opinions and
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Contact Details
Name:
A. Penney
Address:
PO Box 997, Belconnen ACT2616
Phone:
0401-788289
Email:
[email protected]
Acknowledgements
DEPM parameter estimates and ranges used in this paper were obtained from reports prepared by Prof T. Ward
and his co-authors.
DEPM probability-based reporting
Contents
Summary
3
1.
3
Introduction
2.
2.1
2.2
2.3
2.4
3.
Methods
Input parameters
Derivation of input parameter probability distributions
Derivation of estimated biomass probability distributions
Probability-based advice
Results
3.1
Base-case probability distributions
3.2
Comparison of published and estimated probability distributions
3.3
Probability-based biomass estimates
3.4
Exploratory alternative scenarios for Jack Mackerel
4
4
5
6
6
7
7
9
11
12
Alternative Scenario 1: Extended spawning area
12
Alternative Scenario 2: Uncertain spawning fraction
13
Alternative Scenario 3: Increased spawning fraction
14
4.
Discussion and Recommendations
16
5.
References
18
6.
Appendix A - Dashboards
19
Tables
Table 1. Input mean value and 95% confidence interval for P0, and minimum, mean and maximum values for W,
R, F and S, used to generate DEPM-based spawning biomass estimates for Jack Mackerel (from Ward
et al. 2015b), showing reported spawning biomass and the estimate derived using the mean
parameter values. .................................................................................................................................. 4
Table 2. Input mean values and 95% confidence intervals for parameters used to generate DEPM-based
spawning biomass estimates for Blue Mackerel (from Ward et al. 2015a) , showing reported
spawning biomass and the estimate derived using the mean parameter values. ................................. 5
Table 3. Maximum estimated spawning biomass within the >50% (mean), >75% and >90% probability ranges
under the Fixed area base case, Extended spawning area, Uncertain spawning Fraction and Increased
spawning fraction scenarios. ................................................................................................................ 15
Figures
Figure 1. Randomly re-sampled probability distributions for the base-case input parameters: spawning area
(top left), daily egg production (top right), female weight (centre left), sex ratio (centre right), batch
fecundity (bottom left) and spawning fraction (bottom right), used to estimate spawning biomass for
east coast Jack Mackerel. ....................................................................................................................... 7
Figure 2. One realisation of the probability distribution of east coast Jack Mackerel spawning biomass estimates
(t) derived using the base-case probability distributions for the input parameters shown in Figure 1.
Colours indicate the biomass estimates falling below the >90% (dark green), >75% (light green),
>50% (yellow) and < 50% (red) probability ranges. ............................................................................... 8
Figure 3. Randomly re-sampled probability distributions for the base-case input parameters: spawning area
(top left), daily egg production (top right), female weight (centre left), sex ratio (centre right), batch
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DEPM probability-based reporting
fecundity (bottom left) and spawning fraction (bottom right), used to estimate spawning biomass for
east coast Blue Mackerel. ...................................................................................................................... 8
Figure 4. One realisation of the probability distribution of east coast Blue Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters shown in
Figure 3. Colours indicate the biomass estimates falling below the >90% (dark green), >75% (light
green), >50% (yellow) and < 50% (red) probability ranges. ................................................................... 9
Figure 5. One realisation of the comparison between published and estimated probability distributions (mean
and 95% confidence intervals) for the contributory parameters and resulting spawning biomass
distribution for east coast Jack Mackerel. Also shown is the biomass distribution that would result
from a lognormal fit to the mean and estimated SD. ............................................................................ 9
Figure 6. One realisation of the comparison between published and estimated probability distributions (mean
and 95% confidence intervals) for the contributory parameters and resulting spawning biomass
distribution for east coast Blue Mackerel. Also shown is the biomass distribution that would result
from a lognormal fit to the mean and estimated SD. .......................................................................... 10
Figure 7. Jack Mackerel biomass estimates with a 50%, 75% and 90% probability that that biomass is higher
than each value. ................................................................................................................................... 11
Figure 8. Blue Mackerel biomass estimates with a 50%, 75% and 90% probability that that biomass is higher
than each value. ................................................................................................................................... 12
Figure 9. Randomly re-sampled probability distribution of spawning area (km2) for east coast Jack Mackerel,
modelled to include an exponentially declining likelihood of spawning occurring beyond the best
estimate of spawning area shown in Figure 1. The maximum value on the x-axis is the total survey
area. ..................................................................................................................................................... 13
Figure 10. One realisation of the probability distribution of east coast Jack Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters shown in
Figure 1, but incorporating the exponentially declining likelihood that spawning occurred beyond the
best estimate of spawning area, as shown in Figure 9.Colours indicate the biomass estimates falling
within the >90% (dark green), >75% (light green), >50% (yellow) and < 50% (red) probability ranges.
............................................................................................................................................................. 13
Figure 11. Randomly re-sampled probability distribution of increased uncertainty around the base-case
estimated spawning fraction for east coast Jack Mackerel shown in Figure 1. ................................... 14
Figure 12. One realisation of the probability distribution of east coast Jack Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters shown in
Figure 1, but incorporating increased uncertainty around spawning fraction, as shown in Figure 11.
Colours indicate the biomass estimates falling within the >90% (dark green), >75% (light green),
>50% (yellow) and < 50% (red) probability ranges. ............................................................................. 14
Figure 13. Randomly re-sampled probability distribution of an increase in the estimated spawning fraction for
east coast Jack Mackerel, compared with the base-case in Figure 1 . ................................................. 15
Figure 14. One realisation of the probability distribution of east coast Jack Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters shown in
Figure 1, but using the increase in spawning fraction shown in Figure 13. Colours indicate the
biomass estimates falling within the >90% (dark green), >75% (light green), >50% (yellow) and < 50%
(red) probability ranges........................................................................................................................ 15
Figure 15. Maximum estimated spawning biomass at the >50% (mean), >75% and >90% probability ranges
under the Fixed area base case (Figure 2), Extended spawning area (Figure 10), Uncertain spawning
Fraction (Figure 12) and Increased spawning fraction (Figure 14) scenarios (values from Table 3). ... 16
Figure 16. Sensitivity analysis of the effects of individual parameters on estimates of spawning biomass of Jack
Mackerel, showing estimated mean (red arrows), minimum and maximum (black arrows) values
(from Ward et al. 2015b) ..................................................................................................................... 17
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DEPM probability-based reporting
Probability reporting of spawning biomass estimates
derived from daily egg production surveys
Andrew Penney
Pisces Australis (Pty) Ltd
Summary
Difficulties in obtaining adequate adult fish samples to generate reliable estimates of key adult
biological parameters required for Daily Egg Production Method estimates of spawning
biomass led to advice that recent biomass estimates for Blue Mackerel (Scomber australasicus)
should be 'treated with caution' (Ward et al. 2015a). At the 1st meeting of the Australian
Fisheries Management Authority Scientific Panel for the Small Pelagic Fishery, a question arose
as to how such precautionary advice might be provided in a manner that would allow fisheries
managers to make decisions depending on their preferred level of risk in relation to quantified
probabilities of alternative outcomes.
An option for expressing advice in terms of the probability distribution for biomass estimates
derived from DEPM surveys is explored in this paper, using available published information for
two small pelagic species, Jack Mackerel (Trachurus declivis) (Ward et al. 2015b) and Blue
Mackerel (Scomber australasicus) (Ward et al. 2015a). Illustrative examples are shown for how
such an approach could be used to explore alternative assumptions regarding input
parameters to provide advice addressing resulting uncertainties. Where it appears that input
parameters have complex or variable error distributions, consideration could be given to use
of Bayesian methods to derive posterior probability distributions, and Markov Chain MonteCarlo estimation to derive posterior biomass probability distributions.
1. Introduction
At the first meeting of the Australian Fisheries Management Authority (AFMA) Scientific Panel for the
Small Pelagic Fishery, a question arose regarding how the Panel might provide precautionary advice if
a Management Advisory Committee is advised to apply caution as a result of particular uncertainty in
the estimates of one or more of the contributory parameters to Daily Egg Production Method
(DEPM) based biomass estimates. This discussion arose as a result of difficulties in obtaining
adequate samples of adult fish during the recent DEPM survey for Blue Mackerel (Ward et al. 2015a),
necessitating the assumption of values from previous surveys for adult biological parameters in the
DEPM biomass estimation equation – spawning fraction, batch fecundity and sex ratio. The
uncertainty arising from the need to substitute values obtained in previous surveys led Ward et al.
(2015b) to recommend that the estimate of spawning biomass “should be treated with caution as
adult samples were not collected during the study”.
DEPM estimates of spawning biomass are generally considered to be unbiased (Ward et al. 2011).
However, assumption of sex ratio, spawning fraction or batch fecundity values significantly above or
below those that actually occurred during a survey can result in bias in biomass estimates. If values
assumed for sex ratio, spawning fraction or batch fecundity are higher than the correct values, then
the resulting biomass estimates will be underestimated. Conversely, if values for these biological
parameters are lower than the real values, then spawning biomass will be overestimated. In the
absence of a formal approach to quantifying risk, and of providing advice in terms of probabilities of
occurrence related to quantified levels of precaution, advice to 'apply caution' is not helpful to
fisheries managers. The question then arises: how can advice be provided in a way that quantifies
the level and range of uncertainty, and expresses advice in terms of probabilities and levels of
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DEPM probability-based reporting
precaution, giving managers the information they need to make a risk-based decision at their
preferred level of precaution
2. Methods
2.1
Input parameters
Daily egg production method spawning biomass estimates were derived as a product of the six
contributory parameters relating to spawning activity or adult biology, using the following form of
the DEPM equation:
SB
A
P0
W
R
F
S
- Spawning biomass (t)
- spawning area (km2)
- egg production (eggs.day-1.m-2)
- female weight (g)
- sex ratio
- batch fecundity (eggs)
- spawning fraction
Published mean estimated values and 95% confidence intervals for these contributory parameters
and the resulting spawning biomass estimates for Jack Mackerel (Trachurus declivis) were obtained
from Ward et al. (2015b) (Table 1). The mean value and 95% confidence interval was obtained for
Blue Mackerel (Scomber australasicus) egg production (P0) from Ward et al. (2015a). Due to
difficulties with obtaining adequate samples of adult fish to determine adult biological parameters,
Ward et al. (2015a) had to resort to using values derived from previous surveys, and so 95%
confidence intervals for these parameters were not available. Instead, the published minimum, mean
and maximum values from previous surveys were used for W, R, F and S (Table 2).
Table 1. Input mean value and 95% confidence interval for P0, and minimum, mean and maximum
values for W, R, F and S, used to generate DEPM-based spawning biomass estimates for Jack
Mackerel (from Ward et al. 2015b), showing reported spawning biomass and the estimate
derived using the mean parameter values.
Species:
Parameter
2
Jack Mackerel
Parameter
Input probability ranges
<95%
Mean
>95%
Spawning area (km )
Egg production (eggs.day-1.m-2)
Female weight (g)
Sex ratio
Batch fecundity (eggs)
Spawning fraction
A
Po
W
R
F
S
23,959
15.9
187.2
0.44
21,584
0.035
23,960
28.9
208.8
0.47
34,068
0.056
23,961
48.7
230.7
0.51
82,010
0.080
Spawning biomass (t)
SB = (A.Po.W) / (R.F.S)
SB
59,570
157,805
161,244
358,731
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Table 2. Input mean values and 95% confidence intervals for parameters used to generate DEPMbased spawning biomass estimates for Blue Mackerel (from Ward et al. 2015a) , showing
reported spawning biomass and the estimate derived using the mean parameter values.
Species:
Parameter
2
2.2
Blue Mackerel
Variable
Input probability ranges
<95%
Mean
>95%
Spawning area (km )
Egg production (eggs.day-1.m-2)
Female weight (g)
Sex ratio
Batch fecundity (eggs)
Spawning fraction
A
Po
W
R
F
S
17,911
14.6
408.2
0.36
46,468
0.050
17,911
34.6
452.0
0.46
52,182
0.140
17,911
69.1
473.6
0.63
55,053
0.180
Spawning biomass (t)
SB = (A.Po.W) / (R.F.S)
SB
35,100
83,300
83,354
165,000
Derivation of input parameter probability distributions
For the illustrative purposes of analyses presented in this paper, it was not attempted to go back to
the original data to re-calculate the statistical distributions for the input parameters. Instead (other
than for survey area A which is a single fixed value), it was assumed that the probability distributions
for the contributory parameters conformed to either normal or log-normal distributions, with the
given means and 95% confidence intervals in Table 1, or the ranges in Table 2. The survey area was
fixed at the published value. For the other contributory parameters, comparison of the magnitudes
of the -95% CI (left-hand tail) and +95% CI (right-hand tail) was used to select whether the
distributions more closely approximated a normal or lognormal distribution. Left-skewed
distributions (+95% CI > -95% CI) were assumed to be lognormal.
When the 95% confidence intervals are equal, uncertainty in the parameter estimate will conform to
a normal distribution with the given mean and 95% CI. However, the actual input parameter
uncertainty for left-skewed distributions, as determined from survey data, may not correspond
accurately to standard (base-e) lognormal distributions. They may be more strongly skewed to the
left, skewed to the right, or conform to an entirely different probability distribution. The
consequences of mismatch between the assumed lognormal and the actual probability distributions
was evaluated by comparing published (input) and estimated (output) means and CIs or ranges of the
parameter distributions, to evaluate the extent to which particular parameter distributions, and
resulting biomass estimates, might be over- or under-estimated in comparison with the published
values, as a result of the assumption of a lognormal distribution.
All analyses were conducted in a user-friendly dashboard constructed in Microsoft Excel, allowing
managers to explore the analyses. Normal distributions were generated using the randomised
inverse normal function:
NORMINV(RAND(),Mean, StdDev)
Lognormal distributions were generated by applying a log-conversion to the same formula:
EXP(NORMINV(RAND(),LN(Mean), StdDev))
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Standard deviations for normal distributions were approximated from the average of the left and
right 95% confidence intervals by:
AVERAGE(<95%CI, >95%CI) / 1.96 / 2
as each 95% CI = 1.96 * StdDev. Standard deviations for lognormal distributions were approximated
by:
AVERAGE((LN(<95%CI),(LN(>95%CI)) / 1.96 / 2
This does not give separate StdDevs for the left and right tails of the lognormal distribution, and
instead approximates both as an average of the two. This approach was necessitated by the fact that
the NORMINV function only takes a single value for StdDev. It is to be expected that this will
underestimate the width of the right hand tail of the distribution.
Five thousand random samples were drawn from the resulting probability distributions for each of
the input parameters. For a couple of the input parameters with assumed lognormal distributions,
the mean of 5000 samples differed slightly from the published mean value, as a result of the skewing
of the estimated 95% CIs in comparison with published values. A small bias-correction factor,
estimated from the ratio between initial estimated / published mean values, was applied to the 5000
samples, to adjust the mean to the published value. This ensured that the resulting estimated
probability distributions had approximately the same mean (noting that this does vary as a result of
the random re-sampling) as the published values. Bias correction factors were implemented for Jack
Mackerel P0 (0.991) and F (0.988), and for Blue Mackerel P0 (0.983).
The resulting 5000 values of each contributory parameter were binned into appropriate frequency
bins across the parameter range for plotting of the probability distributions for each parameter.
2.3
Derivation of estimated biomass probability distributions
Biomass probability distributions were the derived from the 5000 random samples drawn from the
probability distributions of the six contributory parameters by application of the DEPM formula. This
will necessarily be skewed to the right as a result of the assumption that some of the input
parameters had right-skewed lognormal distributions, but need not itself be truly lognormal. The
biomass probability distribution can deviate from a simple lognormal distribution, depending on the
way in which the probability distributions of the contributory parameters combine, to result in a
biomass probability distribution that is either more or less skewed than lognormal.
A final bias correction factor was also applied to ensure that the mean of the biomass probability
distribution for each species approximated the published mean value. The published mean biomass
estimate for Jack Mackerel differs slightly from that obtained from simply applying the DEPM formula
to the published mean values of the contributory parameters (see Table 1), perhaps due to rounding
differences in the original calculations. Final spawning biomass bias correction factors were 0.937 for
Jack Mackerel and 0.980 for Blue Mackerel.
2.4
Probability-based advice
The resulting 5000 spawning biomass estimates were binned into 5000 t (Jack Mackerel) or 2000 t
(Blue Mackerel) frequency bins for plotting of the biomass probability distributions. Cumulative
frequencies across these bins were calculated, allowing for the cumulative probability of spawning
biomass being above any chosen value to be estimated. This provides the basis for providing advice
in terms of a preferred level of confidence of biomass being above a chosen value.
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3. Results
3.1
Base-case probability distributions
Results are summarised and displayed in probability distribution display dashboards customised for
each of the species (Appendix A - Dashboards). Displayed results can be recalculated using the F9 key
to generate repeated random re-sampling of the probability distributions, to show how they vary
given the specified means and confidence intervals. Figure 1 shows one realisation of the estimated
probability distributions of the contributory parameters for Jack Mackerel. These distributions
constitute the base-case for Jack Mackerel, for comparison with later exploratory examples using
hypothetical alternative distributions for some of the parameters.
120%
5.0%
4.5%
Spawning area
100%
Egg production
4.0%
3.5%
80%
3.0%
60%
2.5%
2.0%
40%
1.5%
20%
1.0%
0.5%
8%
0.0%
15.0
16.5
18.0
19.5
21.0
22.5
24.0
25.5
27.0
28.5
30.0
31.5
33.0
34.5
36.0
37.5
39.0
40.5
42.0
43.5
45.0
46.5
48.0
49.5
23,500
25,500
27,500
29,500
31,500
33,500
35,500
37,500
39,500
41,500
43,500
45,500
47,500
49,500
51,500
53,500
55,500
57,500
59,500
61,500
63,500
0%
8%
7%
7%
Female weight
6%
6%
5%
5%
4%
4%
3%
3%
Sex ratio
2%
2%
1%
1%
7%
0.526
0.520
0.514
0.508
0.502
0.496
0.490
0.484
0.478
0.472
0.466
0.460
0.454
0.448
0.442
0.430
185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233
0.436
0%
0%
14%
6%
Batch fecundity
12%
5%
10%
4%
8%
3%
6%
2%
4%
1%
Spawning fraction
2%
0%
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.044
0.048
0.052
0.056
0.060
0.064
0.068
0.072
0.076
0.080
0.084
0.088
0.092
0.096
0.100
0.104
0.108
80,000
77,000
74,000
71,000
68,000
65,000
62,000
59,000
56,000
53,000
50,000
47,000
44,000
41,000
38,000
35,000
32,000
29,000
26,000
23,000
20,000
0%
Figure 1. Randomly re-sampled probability distributions for the base-case input parameters: spawning
area (top left), daily egg production (top right), female weight (centre left), sex ratio (centre
right), batch fecundity (bottom left) and spawning fraction (bottom right), used to estimate
spawning biomass for east coast Jack Mackerel.
The probability distributions illustrated in Figure 1 were used to generate the probability distribution
for Jack Mackerel spawning biomass in Figure 2, by application of the DEPM equation to the 5000
input parameter samples. Cumulative probabilities for this distribution have been shaded to show
the biomass ranges below the 90% probability level (dark green), 75% level (dark green and light
green) and 50% (mean) probability level (dark green, light green and yellow). Figure 3 and Figure 4
show the comparable estimated probability distributions of the contributory parameters for Blue
Mackerel, and the resulting spawning biomass probability distribution shaded to show the biomass
ranges below the 90%, 75% and 50% (mean) probability levels.
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Figure 2. One realisation of the probability distribution of east coast Jack Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters
shown in Figure 1. Colours indicate the biomass estimates falling below the >90% (dark
green), >75% (light green), >50% (yellow) and < 50% (red) probability ranges.
Spawning area
100%
80%
60%
40%
20%
17,000
20,000
23,000
26,000
29,000
32,000
35,000
38,000
41,000
44,000
47,000
50,000
53,000
56,000
59,000
62,000
65,000
0%
5%
4%
4%
3%
3%
2%
2%
1%
1%
0%
Egg production
17.0
19.8
22.6
25.4
28.2
31.0
33.8
36.6
39.4
42.2
45.0
47.8
50.6
53.4
56.2
59.0
61.8
64.6
120%
7%
10%
Female weight
6%
8%
Sex ratio
5%
4%
6%
3%
4%
2%
2%
1%
498
492
486
480
474
468
462
456
450
444
438
432
426
420
0.350
0.365
0.380
0.395
0.410
0.425
0.440
0.455
0.470
0.485
0.500
0.515
0.530
0.545
0.560
0.575
0.590
0%
0%
6%
10%
Batch fecundity
8%
5%
Spawning fraction
4%
6%
3%
4%
2%
2%
1%
0%
0.070
0.078
0.086
0.094
0.102
0.110
0.118
0.126
0.134
0.142
0.150
0.158
0.166
0.174
0.182
0.190
0.198
0.206
47,600
48,200
48,800
49,400
50,000
50,600
51,200
51,800
52,400
53,000
53,600
54,200
54,800
55,400
56,000
56,600
0%
Figure 3. Randomly re-sampled probability distributions for the base-case input parameters: spawning
area (top left), daily egg production (top right), female weight (centre left), sex ratio (centre
right), batch fecundity (bottom left) and spawning fraction (bottom right), used to estimate
spawning biomass for east coast Blue Mackerel.
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Figure 4. One realisation of the probability distribution of east coast Blue Mackerel spawning biomass
estimates (t) derived using the base-case probability distributions for the input parameters
shown in Figure 3. Colours indicate the biomass estimates falling below the >90% (dark
green), >75% (light green), >50% (yellow) and < 50% (red) probability ranges.
3.2
Comparison of published and estimated probability distributions
Figure 5 and Figure 6 show comparisons of the published probability distributions with those
estimated here (mean and 95% CIs or ranges) for Jack Mackerel and Blue Mackerel respectively.
Figure 5. One realisation of the comparison between published and estimated probability
distributions (mean and 95% confidence intervals) for the contributory parameters and
resulting spawning biomass distribution for east coast Jack Mackerel. Also shown is the
biomass distribution that would result from a lognormal fit to the mean and estimated SD.
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Figure 6. One realisation of the comparison between published and estimated probability
distributions (mean and 95% confidence intervals) for the contributory parameters and
resulting spawning biomass distribution for east coast Blue Mackerel. Also shown is the
biomass distribution that would result from a lognormal fit to the mean and estimated SD.
The means of the estimated probability distributions correspond closely to the published mean
values, this having been assured by the use of bias correction factors to achieve this. For Jack
Mackerel, the upper and lower 95% confidence intervals are also close to the published values for P0,
W, R and S, and for the resulting spawning biomass probability distribution. Slight under-estimation
of the range for these contributory variables contributes to slight under-estimation of the biomass
range, particularly for the upper 95% CI. However, it is clear that the published distribution for batch
fecundity F is not a simple lognormal distribution, and the assumed lognormal substantially underestimates the upper 95% CI. This should result in over-estimation of the upper 95% CI on spawning
biomass but, when combined with the other parameters has not had a strong effect.
Most of the adult biology parameters for Blue Mackerel represent ranges, rather than 95% CIs
(Figure 6). The published upper and lower ranges for W, F or S do not conform to normal or
lognormal distributions and appear to be strongly skewed to the right. It is unlikely that the 95% CIs
for the individual surveys from which these ranges were drawn were skewed to the right. For the
purposes of illustrative analyses in this paper, it was assumed that these parameters showed normal
distributions. There is close correspondence between the published and estimated ranges for P0, the
one parameter for which 95% CIs were available, and quite close correspondence for R. The
assumption of normal distributions for the apparently right-skewed W, F and S results in the
estimated distributions over-estimating the upper range for these parameters. However, in
combination, the estimated parameter distributions only result in a slight over-estimation of the
lower and upper bounds for spawning biomass.
For both Jack Mackerel and Blue Mackerel, comparative estimated spawning biomass distributions
derived from the published mean and 95% CI values slightly under-estimate the upper 95% CI,
confirming that spawning biomass probabilities are not true log-normal distributions, are more
strongly skewed to the left, and have longer right-hand tails.
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Nonetheless, the resulting spawning biomass distributions have approximately the same means and
correspond quite closely to the published distributions, and are certainly adequate for the illustrative
purposes of this paper.
3.3
Probability-based biomass estimates
The probability distributions shown for Jack Mackerel in Figure 2 and for Blue Mackerel in Figure 4
can provide the basis for providing advice on estimated biomass at alternative probability levels.
Advice on biomass estimates derived from stock assessments is often provided in terms of the mean
value, usually with the associated standard deviation or 95% confidence intervals. Management
action is usually based on this mean value. However, this is not always the case. Under the
Commonwealth Harvest Strategy Policy, the probability of biomass being above the limit reference
point must be at least 90%. Management advice can be provided at any confidence level, depending
on the perceived risk of managing to a particular level of confidence for a biomass estimate.
The mean biomass estimate corresponds to a 50% probability that biomass is higher or lower than
that value. The lower 95% confidence interval corresponds to a 97.5% probability that biomass is
higher than that value (the 95% CI spanning the range from 2.5% to 97.5%). Provided the shape of
the probability distribution is known, cumulative probabilities at any point along the distribution
indicate the probability that biomass is larger than the biomass value at that point.
Figure 7 and the associated table show the Jack Mackerel biomass values with 50%, 75% and 90%
probability that biomass is higher than that biomass. The 50% (mean) biomass value (157,811 t) is
close to the published value of 157,805 t, although varies slightly around this as the distribution is
randomly re-sampled. If the biomass probability distribution in Figure 2 is an accurate reflection of
the variance in estimated biomass, then there is a 75% probability that biomass is larger than
~135,000 t and a 90% probability that biomass is larger than ~115,000 t (to the nearest 5000 t).
Figure 7. Jack Mackerel biomass estimates with a 50%, 75% and 90% probability that that biomass is
higher than each value.
Figure 8 and the associated table show the comparable Blue Mackerel biomass values with 50%, 75%
and 90% probabilities that biomass is higher than that biomass. The 50% (mean) biomass value
(83,312 t) is close to the published value of 83,300 t (although varies around this as the distribution is
randomly re-sampled). If the biomass probability distribution in Figure 4 is an accurate reflection of
the variance in biomass, then there is a 75% probability that biomass is larger than ~70,000 t and a
90% probability that biomass is larger than ~62,000 t (to the nearest 2000 t).
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Figure 8. Blue Mackerel biomass estimates with a 50%, 75% and 90% probability that that biomass is
higher than each value.
When it preferred to be precautionary in giving advice, perhaps as a result of particular uncertainty
in some of the contributory parameters used in the DEPM equation, a higher probability level than
50% can be used as the basis for that advice, providing higher certainty that biomass is at least as
large as the advised level. Alternately, advice can be given for a range of probabilities, and managers
can choose the probability level they would prefer to manage to, depending on the risk at each level.
3.4
Exploratory alternative scenarios for Jack Mackerel
Generation of alternative spawning biomass distributions, and provision of advice either against the
mean or other probability levels of those alternative scenarios, can be used to provide advice for any
plausible alternative scenario of input parameters. This is the usual approach for dealing with
substantial uncertainty in some of the input parameters to a stock assessment, where sensitivity
tests are run using alternative values of input parameters to generate alternative assessments. The
relative likelihood of these alternative scenarios, and the relative risks of managing under one or
other scenario, can then be reported in a probability matrix, from which managers can choose the
risk-probability scenario under which they would prefer to manage.
Three hypothetical alternate scenarios were explored for Jack Mackerel, to illustrate how alternative
values or increased uncertainty for a couple of the parameters could be evaluated, and the effect on
biomass estimates reported.
Alternative Scenario 1: Extended spawning area
Figure 9 shows a hypothetical extended spawning area probability distribution, in which an
exponential decline in likelihood of spawning occurring has been modelled extending beyond the
reported spawning area of 23,960 km2. This could occur, for example, if eggs are found right to the
edge of the surveyed area, and there is some information from acoustics or exploratory fishing that
fish occurred over an additional ~6000 km2 beyond the edge of the survey area. The modelled
distribution provides for a declining likelihood of spawning occurring out to ~30,000 km2, but with
little likelihood of spawning occurring beyond that. The effect on the probability distribution of
spawning biomass estimates shown in Figure 10 is expected. The probability that spawning occurred
over a larger area results in a proportional increase in the biomass distribution compared to the
base-case in Figure 2, with the mean estimate increasing from ~157,800 t to ~171,400 t (Table 3).
Provided information can be obtained on the presence of fish beyond the survey area, an approach
like this can compensate for incomplete survey coverage within a probabilistic framework.
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25%
Extended Area
20%
15%
10%
5%
64,000
62,000
60,000
58,000
56,000
54,000
52,000
50,000
48,000
46,000
44,000
42,000
40,000
38,000
36,000
34,000
32,000
30,000
28,000
26,000
24,000
0%
Figure 9. Randomly re-sampled probability distribution of spawning area (km2) for east coast Jack
Mackerel, modelled to include an exponentially declining likelihood of spawning occurring
beyond the best estimate of spawning area shown in Figure 1. The maximum value on the xaxis is the total survey area.
Figure 10. One realisation of the probability distribution of east coast Jack Mackerel spawning
biomass estimates (t) derived using the base-case probability distributions for the input
parameters shown in Figure 1, but incorporating the exponentially declining likelihood that
spawning occurred beyond the best estimate of spawning area, as shown in Figure 9.Colours
indicate the biomass estimates falling within the >90% (dark green), >75% (light green), >50%
(yellow) and < 50% (red) probability ranges.
Alternative Scenario 2: Uncertain spawning fraction
Hypothetical high uncertainty in one of the adult biological spawning parameters, such as might
occur if there were difficulties obtaining samples and a range of previous estimates was assumed, as
occurred for Blue Mackerel (Ward et al. 2015a), was explored using the same approach. The StdDev
for Jack Mackerel spawning fraction was doubled from 0.0057 to 0.0115, retaining the mean value of
0.056. The resulting spawning fraction probability distribution (Figure 12) has the same mean, but
substantially wider variance than the base-case (Figure 1). The mean of the resulting biomass
distribution remains close to the published value and the base-case but the wider uncertainty results
in the 75% probability and 90% probability estimates decreasing by 5000 t each to 125,000 t and
105,000 t respectively (Table 3). A precautionary approach under such uncertainty would be to
advise managing, for example, to the lower 75% probability biomass estimates.
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8%
Uncertain spawning fraction
7%
6%
5%
4%
3%
2%
1%
0.106
0.100
0.094
0.088
0.082
0.076
0.070
0.064
0.058
0.052
0.046
0.040
0.034
0.028
0.022
0.016
0.010
0%
Figure 11. Randomly re-sampled probability distribution of increased uncertainty around the basecase estimated spawning fraction for east coast Jack Mackerel shown in Figure 1.
Figure 12. One realisation of the probability distribution of east coast Jack Mackerel spawning
biomass estimates (t) derived using the base-case probability distributions for the input
parameters shown in Figure 1, but incorporating increased uncertainty around spawning
fraction, as shown in Figure 11. Colours indicate the biomass estimates falling within the
>90% (dark green), >75% (light green), >50% (yellow) and < 50% (red) probability ranges.
Alternative Scenario 3: Increased spawning fraction
The effect of a change in the mean value of one of the contributory parameters was explored by
assuming a 50% increase in Jack Mackerel spawning fraction, increasing the mean value from 0.056
to 0.067, but maintaining the StdDev at 0.0057. This could be a scenario where difficulties were
experienced in collecting adequate samples of fish to provide reliable estimates of adult biological
parameters, and there are differing alternative values available from previous surveys, with no clarity
regarding which might be the more correct value for the current survey. Alternative values can be
explored and the more precautionary scenario chosen as the basis for advice. Figure 13 shows the
resulting probability distribution for the increased spawning fraction, similar in variance to the base
case Figure 1, but with the higher assumed mean value. The resulting spawning biomass distribution
is shown in Figure 14. The assumption of increased spawning fraction results in a decrease in
estimated mean biomass from the base case value of ~157,800 t to ~132,100 t. Advice based on this
scenario would be more precautionary than the mean estimate from the base case, but less so than
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reporting against the 75% probability for the base case. Expectedly, the 75% and 90% probability
spawning biomass estimates under this scenario decrease further, to 110,000 t and 90,000 t
respectively (Table 3).
14%
Increased spawning fraction
12%
10%
8%
6%
4%
2%
0.106
0.100
0.094
0.088
0.082
0.076
0.070
0.064
0.058
0.052
0.046
0.040
0.034
0.028
0.022
0.016
0.010
0%
Figure 13. Randomly re-sampled probability distribution of an increase in the estimated spawning
fraction for east coast Jack Mackerel, compared with the base-case in Figure 1.
Figure 14. One realisation of the probability distribution of east coast Jack Mackerel spawning
biomass estimates (t) derived using the base-case probability distributions for the input
parameters shown in Figure 1, but using the increase in spawning fraction shown in Figure 13.
Colours indicate the biomass estimates falling within the >90% (dark green), >75% (light
green), >50% (yellow) and < 50% (red) probability ranges.
Table 3. Maximum estimated spawning biomass within the >50% (mean), >75% and >90% probability
ranges under the Fixed area base case, Extended spawning area, Uncertain spawning Fraction
and Increased spawning fraction scenarios.
Scenario
Fixed area
Extended area
Uncertain spawning fraction
Increased spawning fraction
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SB P > 75%
SB P > 90%
157,848
171,404
157,925
132,112
130,000
140,000
125,000
110,000
110,000
115,000
105,000
90,000
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The results of the base case and alternative scenarios (Table 3), are plotted in Figure 15. Compared
to the base case, at each probability level (50%, 75% and 90%), increased spawning area results in an
increase in estimated biomass. Increased uncertainty in spawning fraction provides about the same
mean biomass estimate as the base case, but slightly lower 75% and 90% probability biomass
estimates. Increased spawning fraction results in reduced biomass estimates at all probability levels.
Figure 15. Maximum estimated spawning biomass at the >50% (mean), >75% and >90% probability
ranges under the Fixed area base case (Figure 2), Extended spawning area (Figure 10),
Uncertain spawning Fraction (Figure 12) and Increased spawning fraction (Figure 14)
scenarios (values from Table 3).
4. Discussion and Recommendations
Spawning biomass estimates from DEPM surveys are derived by a mathematically simple process of
raising egg counts by a number of substantial raising factors, to obtain an estimate of spawning
biomass. The total raising factor resulting from the combination of the contributory parameters is
substantial. For example, assuming that the 3,530 Jack Mackerel eggs found by Ward et al. (2015b),
with an average diameter of 0.92 mm (from Ward et al 2015 b, Figure 2), had the same density as
seawater (although they would actually be somewhat lighter, being buoyant), giving an estimated
combined weight of 1.47 g, the final weight raising factor to obtain the reported mean spawning
biomass estimate of 157,805 t would be ~1.1 x 1011.
Many of the contributory parameters to DEPM spawning biomass estimates are recognised as being
uncertain, as well as seasonally and inter-annually variable, particularly adult biological parameters
such as sex-ratio on the spawning grounds, batch fecundity and spawning fraction. Apparently
moderate changes (biologically speaking) in these parameters will have substantial effects on
resulting spawning biomass estimates, both in terms of the mean value and the uncertainty around
that value. Where it is intended to use the DEPM-derived spawning biomass estimates as absolute
estimates of biomass for the purposes of determining recommended biological catches (as is the
cases under the SPF harvest strategy), it becomes particularly important to ensure that the estimates
for these parameters are reliable. Given the uncertainty around mean estimates for these
parameters, and the cascade effect that this will have on uncertainty around the spawning biomass
estimate, it is advisable that this uncertainty be fully characterised and reported.
Ward et al. (2015b) recognise the effect that ranges in values of contributory parameters have on
biomass estimates, and provide a figure showing the change in estimated mean spawning biomass as
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values of other individual parameters are varied across their known maximum ranges (presumably
keeping other parameters constant at their mean values in each case) (Figure 16). However, while
these figures illustrate the possible effect that each parameter can have across its entire range,
without incorporating the probability that each parameter will take on certain values, they do not
constitute influence plots. There should be the highest probability of the mean value occurring, with
very low probability of either the minimum or maximum values occurring.
Figure 16. Sensitivity analysis of the effects of individual parameters on estimates of spawning
biomass of Jack Mackerel, showing estimated mean (red arrows), minimum and maximum
(black arrows) values (from Ward et al. 2015b)
The methods shown in this paper provide an adequate approach to post-hoc reconstruction of
probability distributions around contributory parameters and biomass estimates. Advice can then be
provided in terms of the probability of a range of alternative estimated biomass values, or of
different biomass estimates derived from alternative assumptions regarding uncertain input
parameters. Consultation with managers and Management Advisory Committees could advise the
Scientific Panel regarding which alternative, precautionary, probability levels would be considered
appropriate, or should be reported on, when cases of particular uncertainty regarding input
parameters arise.
However, the correct approach would be to determine and provide these probability distributions as
part of the original analysis. Where these are normal or true lognormal distributions, it should be
possible to mathematically derive deterministic probability distributions for the input parameters.
The biomass probability distribution can then be derived by Monte-Carlo re-sampling of the input
parameter distributions, as done in this paper. It is likely, however, that some of the parameter
distributions may not conform to normal or true lognormal distributions (as shown in Figure 5 and
Figure 6), and may take on a variety of statistical or empirical distributions. The problem then
becomes one of integrating multiple different probability distributions to obtain a statistically reliable
posterior distribution of spawning biomass. This is appropriately dealt with by application of
Bayesian methods to the analysis of probability distributions, and use of Markov Chain Monte-Carlo
(MCMC) estimation to determine the posterior probability distribution of spawning biomass.
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5. References
Ward T.M., P. Burch, L.J. Mcleay and A.R. Ivey (2011) Use of the daily egg production method for stock
assessment of sardine, Sardinops sagax; lessons learned over a decade of application off Southern
Australia. Reviews in Fisheries Science, 19(1): 1–20.
Ward T.M., G. Grammer, A. Ivey, J. Carrol, J. Keane, J. Stewart and L. Litherland (2015a) Egg distribution,
reproductive parameters and spawning biomass of Blue Mackerel, Australian Sardine and Tailor off
the east coast during late winter and early spring. Final Report, FRDC Project No. 2014-033, 86 pp.
Ward T.M., O. Burnell, A. Ivey, J. Carrol, J. Keane, J. Lyle and S. Sexton (2015b) Summer spawning patterns and
preliminary DEPM survey of Jack mackerel and Australian sardine off the east coast. Final Report,
FRDC Project No. 2013-053, 64 pp.
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6. Appendix A - Dashboards
Probability distribution display dashboard for Jack Mackerel
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Probability distribution display dashboard for Blue Mackerel
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