BIAS AND VARIANCE IN ALLEN'S RECRUITMENT RATE METHOD 



J. W. HORWOOD' 



ABSTRACT 



The method of estimation of the recruitment rate of a population based upon the methodology of Allen 

 is reviewed, and a simpler formulation is presented. The estimator is evaluated for bias and variability. 

 If the recruitment pattern at age is constant with time, the technique shows no bias provided the age 

 of first full recruitment is not underestimated. Use of age-length keys will tend to spread partially recruited 

 ages upwards so the age of first full recruitment should accommodate this. An approximate analytical 

 formula for the variance of the estimated recruitment rate is given, and this shows that variance decreases 

 with increasing second year catches. 



If recruitment to a fishery changes such that it occurs at an earlier age with time, then high values 

 of recruitment rate are given and vice versa. This could be interpreted erroneously as an increasing or 

 decreasing population growth rate, if it was assumed that the pattern of new recruitments at age had 

 been constant. It is also found that a fluctuating recruitment pattern will give a negatively biased rate. 

 These last considerations suggest that the method should not be used unless a constant recruitment pat- 

 tern can be established. For a series of years of data other techniques should be used. 



Allen (1966, 1968) introduced a technique for the 

 estimation of the proportion of new recruits to total 

 recruits in an exploited stock. This statistic is par- 

 ticularly useful since it is necessary for the simula- 

 tion of stock dynamics and hence it is used in the 

 estimation of stock size. As proposed by Allen the 

 statistic is obtained directly from the catch of the 

 previous 2 years, and this is in contrast to estimates 

 obtained by virtual population analysis, which re- 

 quires several years of data for a comparable esti- 

 mate to converge to a satisfactory answer. Allen 

 termed this parameter r^ but here it will be de- 

 noted by r. The only data needed to calculate this 

 recruitment rate are the proportions of catch at age 

 for 2 consecutive years, and a knowledge of the age 

 of first full recruitment. Allen (1973) used this 

 method to calculate the recruitment rates of fin 

 whales in the Antarctic and so constructed a stock 

 and recruit relationship and estimated stock sizes. 

 The use of Allen's method of obtaining recruitment 

 rates has been advocated by Ricker (1975) and 

 Gulland (1977: Chapter 1 by Ricker, Chapter 4 by 

 Gulland, Chapter 14 by Allen and Chapman). The 

 method has been used extensively in stock assess- 

 ments by the International Whaling Commission but 

 often has given average values that were thought 

 to be unreasonably low. This can be seen for sei 

 whales in Ohsumi (1978) and for minke whales in 



'Ministry- of Agriculture, Fisheries and Food Directorate of Fish- 

 eries Research, Fisheries Laboratory, Lowestoft, Suffolk NR33 

 OHT, England. 



Chapman (1983). In addition, the rates have been 

 very variable even with moderately high catches 

 (Ohsumi 1978; Allen 1982). Some properties of the 

 estimate have been considered. Ricker (1975) inves- 

 tigated the effect of changes in some of the popu- 

 lation parameters and found a negative bias if the 

 first age of full recruitment was underestimated, 

 and Allen (1981) looked at the sensitivity of the 

 method to variable catches at age and concluded that 

 although variability of the estimate was high for 

 catches less than a few hundred, the bias was small 

 even for low catches. This still left unresolved the 

 question of why the estimated recruitment rates 

 were often so low and why they were more variable 

 than Allen's (1981) simulations predicted. Ohsumi's 

 results gave coefficients of variation over 200%, 

 whereas Allen's simulations gave about 75% if 

 catches were low. 



This study investigates the behavior of the esti- 

 mated recruitment rate from Allen's model. The 

 estimator is fully introduced in order to demonstrate 

 that the existence of a free and selectable param- 

 eter (Allen's T) is erroneous; once established the 

 estimator then reduces to a simple form. A typical 

 age-structured model is introduced and the effects 

 on the estimated recruitment rate of changes from 

 year to year of demographic parameters of the 

 model are investigated. The effect of sampling 

 variability in the number of animals caught at age 

 is reviewed in relation to bias and variance of the 

 estimate, and an expression for the approximate 

 analytical variance of the recruitment rate is pre- 



Manuscript accepted October 1986. 

 FISHERY BULLETIN: VOL. 85, NO. 1. 1987. 



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