Cadrin and Vaughan: Virtual population analysis estimates for Atlantic menhaden stock assessment 
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converged estimates for the additive properties of 
mean square difference. Sample sizes for historical 
differences were low, because of the limited number 
of historical stock assessments, but the following 
retrospective analyses have greater sample size for 
estimating RMS difference (rc>30). 
Retrospective comparisons 
Retrospective analysis was performed in two stages 
to investigate consistency of both elements of the 
estimation rule: 1) estimation of fully recruited F by 
ad hoc methods and 2) estimation of partial recruit- 
ment to the fishery at ages 0 and 1 by separable 
VPA (SVPA; Pope and Shepherd, 1982). Both ana- 
lytical stages assumed that menhaden were fully 
recruited to the fishery at age 2 and that M was 0.45 
for all ages, over the entire time period. 
Fully recruited F was approximated by using three 
alternative ad hoc methods for the first element of 
the analysis. Conventional catch curves (Beverton 
and Holt, 1957; Ricker, 1975; Gulland, 1983) and 
modified catch curves (Chapman and Robson, 1960; 
Robson and Chapman, 1961) were used to estimate 
mortality of the age-5 cohort over the four terminal 
years of the catch record (i.e. ages 2-5). These two 
catch-curve methods assumed that F in the current 
year was similar to F experienced by that cohort over 
the previous three years. The third ad hoc method, 
log catch ratios (Ricker, 1975; Gulland, 1983), derived 
fully recruited F from the negative log ratio of age- 
s' 1 ' abundance in the terminal year to age-2 + abun- 
dance in the previous year and assumed that F in 
the last year was similar to F in the previous year. 
All three ad hoc methods assume that menhaden are 
fully recruited and equally available to the fishery 
at age-2 + , which was confirmed through inspection 
of back-calculated F from the 1995 VPA. 
The second element of the assessment, estimation 
of partial recruitment, was performed by using SVPA 
on a fixed number of years. For example, a retro- 
spective series of 5-year SVPA’s was produced with 
the following algorithm. 
Step 1 SVPA was run on an initial time series of 
catch-at-age data (e.g. 1955-60) with the 
appropriate estimate of fully recruited F 
in the terminal year. 
Step 2 Catch data in the starting year (e.g. 1955) 
were deleted, and catch data from a 
new terminal year (e.g. 1961) were ap- 
pended. 
Step 3 SVPA was rerun on the revised time series 
with the appropriate estimate of fully re- 
cruited F in the new terminal year. 
Steps 2 and 3 were repeated until 1995 was the ter- 
minal year. 
Significance of retrospective bias was tested with 
a conventional (-ratio test (H 0 : mean difference=0). 
Normality was tested by using the Shapiro and Wilk 
( 1965) method. Results of (-ratio tests were confirmed 
by using non parametric chi-square and sign tests. Dis- 
persion of retrospective estimates from converged es- 
timates was compared among alternative assess- 
ment rules by using RMS of retrospective differences. 
Retrospective estimates of fully recruited F in terminal 
years were compared with back-calculated estimates 
of fully recruited F from the 1995 VPA, as the aver- 
age of ages 2-5 (weighted by abundance), to derive 
retrospective differences: 
Note that there is no k subscript, as there were in 
the formulae for historical comparisons, because all 
retrospective estimates of fully recruited F were for 
terminal years (i.e. &=0). The relative retrospective 
difference (A F t JF t 1995 ) was also calculated to remove 
the magnitude of the estimate from estimates of gen- 
eral inconsistency. 
Back-calculated estimates of fully recruited F from 
the 1995 VPA were used in terminal years to com- 
pare retrospective inconsistency of SVPA settings 
without including retrospective inconsistency from 
ad hoc estimates of terminal F. The fixed number of 
years in each series of retrospective SVPA’s was varied 
from three to ten years by using the algorithm described 
for the five-year example above. Retrospective consis- 
tency was compared among the eight series of retro- 
spective SVPA’s according to RMS difference of age- 
1 F estimates. Full recruitment of age- 2 and oldest 
age (6 + ) menhaden was confirmed through inspec- 
tion of back-calculated F at age from the 1995 VPA 
and was not adjusted for retrospective comparison. 
Final SVPA runs were performed with seven years 
of catch-at-age and catch-curve estimates of termi- 
nal F to emulate more realistic inconsistency and 
describe the general magnitude and direction of ret- 
rospective differences. Retrospective estimates of R 
were derived directly from SVPA terminal estimates 
of age-1 abundance. SSB was estimated from termi- 
nal SVPA estimates of age-3 + abundance and esti- 
mated weight at age of spawners. Percent MSP 
was calculated according to egg production per recruit 
(Vaughan, 1990; AMAC, 1992). Retrospective dif- 
ferences and relative differences of management vari- 
able estimates were log transformed [e.g. loge ( R+ con- 
stant)] to test bias, and geometric mean square was 
used to estimate mean square difference because dif- 
ferences had skewed distributions. 
