Livingston et ai.: Estimating the annual proportion of nonspawning adult Macruronus novaezelandiae 
103 
The migration ratio, x, cannot be estimated very 
precisely because both trawl surveys will be subject 
to measurement error and because an unknown num- 
ber of fish will have died naturally or have been 
caught between December and May. But to obtain 
the best estimate of x, the numbers of adult fish on 
the Southern Plateau in December and May were 
estimated from the total numbers of fish in the sur- 
veys and the proportion that were in the adult age 
group. The age distributions were estimated from the 
length-frequency distributions and the age-length 
keys were calculated from the samples collected for 
age determination. 
In determining the number of fish that moved out 
of the survey area before May, it is necessary to ac- 
count for fish that died between December and May. 
Five months of natural mortality was applied to the 
number of fish observed in the December surveys to 
estimate the number of fish that would be expected 
in the May surveys. The catch taken on the South- 
ern Plateau between December and May in these two 
years was 10,595 t in 1992 and 8,339 t in 1993. Be- 
cause the estimated size of the stock was 860,000 t 
in May 1992 and 1.3 million t in May 1993 (Cordue 5 ), 
fishing mortality was considered to be negligible. 
The discrepancy between the number of fish ex- 
pected in the May survey and the number observed 
was the maximum number of fish that could be con- 
sidered to have left the Southern Plateau to spawn. 
This number was used in Equation 2 to obtain an 
estimate x of the migration ratio x. An estimate of 
the total proportion of fish that will spawn (p) w r as 
calculated by using x and an estimate of p. of p + in 
Equation 3. Standard errors of these numbers were 
calculated by a resampling procedure that included 
uncertainty regarding the total number of fish in the 
December and May surveys. 
Procedure for estimating the total 
proportion of adult fish spawning 
The estimation procedure for the proportion of adult 
spawning fish was as follows: 
1 The total number of fish on the Southern Plateau 
in December, N 1 was selected from a normal dis- 
tribution with mean equal to the estimated value 
for this survey and with standard deviation equal 
to the standard error of this estimate. This number 
of fish was then distributed over the length-fre- 
quency distribution (assumed to be known exactly). 
5 Cordue, P. 1996. Natl. Inst. Water and Atmospheric Res. P.O. 
Box 14-901 Kilbirnie, Wellington, New Zealand. Personal 
commun. 
2 An age-length key (including ageing error) was 
generated by sampling with replacement from the 
fish in the age-length sample from the December 
survey and was applied to the December length- 
frequency distribution to estimate the number of 
adult fish in December, n r The number of adult 
fish expected to be alive in May was calculated 
by applying the natural mortality M to n v as 
n x e- M5112 . 
3 The same procedures described in 1 and 2 above 
were applied to the May surveys to obtain the total 
number of fish on the Southern Plateau in May 
(N 2 ) and the number of adult fish in May (n 2 ) 
4 The number of fish apparently missing (n m ) was 
estimated as 
5 Taken as a fraction of the number n 2 on the South- 
ern Plateau in May, x (the migration ratio) was 
estimated by 
n 2 
6 p + was calculated by using the simulated 
age-length key and the histological sample as de- 
scribed above. Hence p was estimated as 
- _ P + + x 
This process was repeated 1,000 times. The standard 
errors of each of the values was calculated from the 
standard deviation of the distribution of the 1,000 
values. 
Results 
Trawl surveys 
The four surveys were successfully completed with a 
combined total of 495 stations sampled. Gear param- 
eters were within the range necessary for survey 
standardization (Hurst et al. 6 ), thereby permitting 
the direct comparison of survey results used for data 
analysis (Chatterton and Hanchet, 1994; Schofield 
and Livingston, 1994, a and b; Ingerson et al., 1995). 
6 Hurst, R. J., N. Bagley, T. Chatterton, S. Hanchet, K. A. 
Schofield, and M. Vignaux. 1992. Standardisation of hoki/ 
middle depth time series trawl surveys. NIWA Greta Point 
Internal Report 194, NIWA Greta Point library, Wellington, New 
Zealand, 87 p. 
