Bottom Trawl Surveys - Proceedings of a Workshop Held at Ottawa ...

to be exact. Monte Carlo simulations with
artificial data consistent with model
assumptions led to survivor estimates
distributed as expected. In practice, there is
no assurance that the assumptions underlying the
model hold. Therefore, methods of detecting
failure of assumptions were examined.
The estimator of survivors in Rivard's
implementation depends on two main assumptions
1. Ri (t) = Ni (t) Ki (t)
where Ki(t) = exp (Ki(t))
2. Var { ln Ri(t) - ln(CINTi(t) +
SINTi(t))} =constant
Departures from these assumptions can be
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examined by analysis of the residuals between ln
Ri(t) and ln (CINT + SINT) plus Ki• The
implementation by Rivard (1980) carries out
analysis of variance by linear regression to
test systematic departures in residuals of
years, year-classes and ages. Should this test
show no significant departure, further analysis
can be carried out to determine whether the
first and last age groups have higher or lower
residual variances.
The existing computer program
highlights outliers of residuals more than two
standard deviations from zero, and gives
year-by-year, age-by-age, and year-class
regression coefficients from the analysis of
variance. These are useful in pinpointing
specific problems.
EXAMPLES
The theory outlined above is general
and gives no indication of how successful the
survivor method can be in practice. Precision
of estimates depends heavily on the accuracy of
the research vessel index of abundance. Two
examples show what can happen in applying the
method using Rivard's (1980) implementation.
The first example used data on
Southern Gulf of St. Lawrence cod from Beacham
(1980).
The computer printout is appended
here. Calibration of K factors used ages 3-8
and years 1970-75. It was assumed that K was
constant for ages 6-12.
Five iterations were required to
estimate the K factors. The importance of
integrated survivors in the population number
estimates is an analogous to the sensitivity of
cohort analysis to "terminal F". Due to high
fishing mortality rates in the early 1970s,
survivors of the 64-68 year-classes at age 6
represent 10% or less of the estimated stock
size at age 6. Later year-classes were exposed
to less cumulative fishing mortality and
therefore their integrated survivors represent
up to 40% of estimated population size in the
calibration years and ages. This corresponds to
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the usual uncertainty of cohort analysis
population estimates for the most recent years.
The Tables entitled "Estimated
Survivors" and "Estimated variances of
survivors" are useful in determining the impact
of particular years and ages on the survivor
estimates and the "weighted survivors" Table
shows the contributions of each age-year
combination to the weighted survivor estimate.
Notice that surveys from as early as 1974
contribute up to 5% to the 1979 survivor
estimates. In this example, estimated
coefficients of variation (c.v.) for survivors
at the end of 1979 vary from 16 to 22%.
The analysis of variance of residuals
indicates a year effect which would occur
spuriously with less than 10% probability.
Examination of year coefficients shows a 1971
coefficient with a nominally significant "t"
statistic of -2.56 indicating lower than usual
availability of cod to the survey. The
remaining coefficients are not large in absolute
value. 1971 data has very little effect on 1979
survivors so that no serious violations of
assumptions are indicated by the analysis of
variance.
Examination of residuals shows no
serious outliers in 1977-79 but there are
positive residuals of 0.24 to 0.49 for ages 4-7
in 1979. These influence the "Estimated
Survivors" Table, causing the 1979 estimates for
these year-classes to be higher than those for
1977-78. While the residuals are not large
enough to be statistically significant, the
presence of an availability change to the survey
gear in 1979 cannot be ruled out. If it were
present, the survivors for 1979 would be
overestimated.
The second example uses data from
O'Boyle (1980) for haddock in NAFO division 4X.
The survivor method is applied only to u.s. fall
groundfish survey data. The computer listing is
appended.
The most striking departure from the
previous example is the extremely high
calculated coefficient of variation (over 150%)
of survivor estimates for 1978. The reason is
clearly the highly variable research vessel
abundance index which shows some negative
cross-year mortality estimates. The residual
variation is so high (2.43 on a logarithmic
scale) that a large change in availability to
the survey in 1977 (year effect 0.702 on a
logarithmic scale) is far from being
statistically significant.
The results of this example suggest
that this survey abundance index is inadequate
for quantitative estimation of survivors.
Despite this, qualitatively, the results are
similar to those of O'Boyle which utilize other
abundance indices as well.