HORWOOD: BIAS IN ALLEN'S RECRUITMENT RATE METHOD 



on when the catches are removed from the popu- 

 lation.) 



(iv) Aging of Catch Biased 



If aging is biased such that each age is wrongly 

 allocated to another specific age then the matrix A 

 may look like Table 2(a), which would indicate that 

 all animals age i were called i + 1 for i = 1 to 4. 

 If the true age of first full recruitment, k, is 3 and 

 this was used in Equation (4) an underestimate 

 would result as described above. This may occur if 

 A" is obtained independently of the catch data. If 

 however the catch data are used to estimate k this 

 will also be aged incorrectly with the result that the 

 estimate of r is correct. In the example k' = A and 

 this value used in Equation (4) yields the correct 

 answer. 



Table 2.— Matrix A of allocated age against true age. 



(a) 



(b) 



Often an age-length key is used to age the catch. 

 This implies that an animal of true age i will be 

 allocated to age j with a probability distribution 

 centered upon i. Table 2(b) shows matrix A in such 

 a case. If, in this example, k = 3, the spreading of 

 some of the catch at age 3 into age 4 means a bias 

 will result from the omission of this group. Conse- 

 quently the k used in Equation (4) needs to be 4 to 

 avoid a negative bias. The value of k used needs to 

 be such as to ensure that all partly recruited ages 

 are counted and if age-length keys are used, it may 

 have to be substantially higher than the true age of 

 first full recruitment. 



(v) Stochastic Results 



In this section we consider the biases introduced 

 by variability occurring in the system. 



Variance in Catch at Age 



Variance in catch at age in the first year was 

 modelled with either a Poisson or Normal distribu- 

 tion as previously described. For the second year 

 the age distribuion was found given these stochas- 

 tic catches and an expected catch in year 2 was 

 obtained given F2. From the expected Cj 2 a second 

 stochastic set of catches was obtained. This was 



repeated for 50 pairs of years with Ui = 0.2(5 - i) 

 for i from 6 to 10 and 1.0 for older ages. 



For a range of A and F2 the average value of r, 

 calculated from Equation (4), was accurate to within 

 a few percent either way of the true value of r. This 

 is in agreement with the findings of Allen (1981). 

 However, even for small values of F^ and F2, and 

 hence low catches, there was no evidence of a gen- 

 eral bias; this is contrary to the findings of Allen 

 (1981). 



Variance of r 



From the population model the variance of r can 

 be calculated for given Fi and F2. For each of the 

 50 simulations described above the mean and vari- 

 ance of r was calculated. The coefficients of vari- 

 ation agreed well with those described by Allen 

 (1981, table 1) ranging from 0.15 for a second year 

 catch of 2,500 to 0.76 for a catch of 75. 



A theoretical variance is derived below. 



Let us define 0, = Z,/{F«(1.0 - exp(-Z,))} 

 and e = 01 /02. 



Let X = the age of first partial recruitment, remem- 

 ber t/, 1 = [/; 9, and CT, is the total catch in year t. 

 Then from Equation (6) it can be seen that r can be 

 written as 



r2 = (I C,2 - 0exp(-Zi)I C,,)/CT, 



1=1 r=l 



Note that ^(^,2) = N,^2Wd2 i = x + 1 to 20 



= ^i Ci_ii 



where ^, = 0(t/,_i exp(-Zi) + (1.0 - [/,_i) 

 exp(-M)) UilUi_i, and writing the catch from the 

 fully recruited plus group at time 2 in terms of that 

 at time 1 we get 



■^(^^21,2) = ^21 Qo,l + ^22 ^21,1) 



where ^22 = ^ exp(-Zi). 



No prior information is known about A',! and so we 

 must assume that £'(C,,i ) = C^i . If C, j is assumed 

 to be a Poisson variable with parameter (Cjj) the 

 sample catch then C,2 can be assumed to be a 

 compound-Poisson variable with a relationship that 

 can be approximated by 



121 



