FISHERY BULLETIN: VOL. 85, NO, 1 



size as this average. The ratio of this catch to the 

 average population is the net recruitment rate that 

 we would wish to use; these values varied from 0.04 

 (/cg = 6) to 0.05 {k2 = 14). This confirmed that the 

 distortion of the age structure and population size 

 by the change in selection had very little effect and 

 that a value of r of 0.095, a net rate of 0.046, would 

 be needed to calculate a stabilizing catch. The Table 

 3 averages are much smaller and we must conclude 

 that if there is no trend in recruitment but a fluc- 

 tuation of more than 1 year then the average esti- 

 mated rates will be largely but undeterminably 

 negatively biased, even if k is not underestimated. 

 A 50-yr simulation confirmed this to be true. 



As can be gleaned from the above, selection plays 

 an important role in determining r. However, this 

 technique treats overlapping pairs of years as being 

 independent and implies a selection pattern for a 

 pair of years, say 1980 and 1981 and a different one 

 for years 1981 and 1982; these assumptions may be 

 inconsistent. The difference may be small or large 

 but there is no criterion for acceptability. Some cur- 

 rent techniques take arrays of catch-at-age data and 

 obtain best fits to the overall pattern (Beddington 

 and Cooke 1981; Pope and Shepherd 1982), and Pope 

 and Shepherd reduced consideration to two param- 

 eters. What is clear is that selection and recruitment 

 or fishing rates are confounded, and these latter 

 techniques make the assumptions clearly and would 

 be expected to replace analyses of pairs of years. 



CONCLUSIONS 



If the recruitment pattern to the exploited popula- 

 tion is constant then the following conclusions may 

 be stated. 



1. The "T" of Allen's technique is shown to be 

 necessarily unity and this gives rise to Equa- 

 tion (4) for the estimation of recruitment rate. 



2. If the age of first full recruitment is selected 

 correctly then calculated recruitment rates are 

 unbiased for changing fishing efforts or for an 

 increasing or decreasing population. 



3. If the age of first full recruitment is overesti- 

 mated then an unbiased recruitment rate is 

 found. If it is underestimated then a negative 

 bias ensues. Inspection of Equation (4) however 

 would caution use of an assumed high value of 

 k, such that a and (i were near unity, and this 

 is reflected in the higher variances given by the 

 approximate variance formula. 



4. Aging bias and the use of age-length keys may 

 spread the partially recruited age groups into 



allocated higher ages. The age of first full re- 

 cruitment should be high enough to encompass 

 this spreading. 



5. No bias was detected in recruitment rates from 

 a series of stochastic simulations although Allen 

 (1981) found a small negative bias with low 

 catches. As found by Allen (1981) coefficients 

 of variation of the recruitment rates are high. 



6. Equation (8) provides an approximate formula 

 for the variance of the recruitment values given 

 a fixed effort in the pairs of years. To use this 

 the recruitment pattern needs to be estimated 

 from the data as described by Allen (1966). 



If the recruitment pattern is not constant, serious 

 biases follow: 



7. If there is a trend to earlier recruitment over 

 a period of years high recruitment values will 

 be seen and vice versa. These are likely to be 

 interpreted as true increases or decreases. 



8. If the recruitment pattern fluctuates about a 

 mean then the net or gross average recruitment 

 rate will be negatively biased, the bias increas- 

 ing with the amplitude of the fluctuations. It is 

 likely that many of the very low rates found by 

 the International Whaling Commission are due 

 to this feature. 



9. These last two points indicate that for the tech- 

 nique to be useful it is necessary to establish 

 that the recruitment pattern has been constant. 

 This is likely to prove difficult and consequent- 

 ly much of the value of this simple method is 

 lost. 



10. For groups of years of data, alternative tech- 

 niques should be investigated. 



11. Finally it appears that the Allen recruitment 

 rate, as calculated in this study or through 

 Allen's original equations with T = 1, should 

 be used with great care. It is subject to large 

 and undeterminable biases and large variances. 

 Where possible other techniques should be used. 



ACKNOWLEDGMENT 



Thanks are extended to T. Featherstone, a stu- 

 dent of Brunei University, for assistance in this 

 study, and to D. C. Chapman and J. M. Breiwick 

 for criticisms of the manuscript. 



LITERATURE CITED 



Allen, K. R. 



1966. Some methods for estimating exploited populations. J. 



124 



