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Fishery Bulletin 88(3), 1990 



age. It is also determined that larvae can be classified 

 as either fast or slow growing on the basis of the widths 

 of the growth increments. Thus, we can determine the 

 proportion of the larvae that are fast growing. If we 

 sample the same population at time t + l,we can again 

 separate the larvae into fast- and slow-growing groups 

 on the basis of the width of the otolith at time t (not 

 at time t + 1). 



Alternatively, suppose that a sample of larvae at time 

 t has two cohorts, where a cohort is defined to be all 

 larvae hatching in a given week. We can follow the 

 relative abundance of the two cohorts in the catch over 

 time by taking repeated samples from the population. 



We will let the subscripts E and L refer to the two 

 cohorts (e.g., E for early- and L for late-spawned). Let 

 the number of larvae in the population's two cohorts 

 at time t be A^^^, and Ni, . Suppose that the size of each 

 cohort declines exponentially over time such that, for 

 the ;th cohort (; 6 {E,L}), 



iV„ = A,„exp(-Z,0 



where Z, is the instantaneous mortality rate (time^ ■) 

 and Nn) is the initial abundance of the (' th gi'oup. Also 

 suppose that the expected catch of animals from group 

 i at time t (C,(), for a standard unit of effort, is pro- 

 portional to the abundance, i.e., 



C,( = q,t N„ 



where 7,, is the time- and group-specific catchability 

 coefficient. Then the ratio of expected catches, R, , is 



Rt = 



Cei to A^£„ exp(-Z£0 



(1) 



We assume that the ratio of the catchability coeffi- 

 cients (qLflqEt) is constant over the course of the 

 study. Since the ratio of initial abundances (A^o/Afu) 

 is also a constant, equation (1) can be rewritten 



R, = q 



exp(-Z^O 

 exp(-Z£0 



qexp{(ZE - Zi)t] 



(2) 



where g is a nuisance parameter that subsumes the 

 catchability coefficients and initial abundances. Tak- 

 ing logarithms of (2) results in a linear relationship with 

 respect to time: 



\ogAR,) = loge(9) + (Z,: - Z,)t. 



(3) 



Thus, regressing the logarithm of the observed ratio 

 of abundances (Rt) against time results in a linear 

 relationship with slope equal to an estimate of the dif- 



ference in the instantaneous mortality rates. (The 

 proper weighting to use in a weighted regression is 

 discussed below.) Note that it is sometimes necessary 

 to add a small constant to the numerator and denom- 

 inator to avoid dividing by or taking the logarithm 

 of 0. 



Exponentiating the slope estimated by (3) provides 

 an estimate of the ratio of the finite survival rates: 



SJSe = e'^'"/'.- 



where Si = e~^i; S^ = e'^t:. 



By the Taylor's series (delta) method, the asymptotic 

 variance of the estimated survival ratio can be approx- 

 imated by (Seber 1982): 



V{SJSe) = e<- •^■'"'"■' V(slope). 



Diagnostics 



Under the assumptions given above (constant ratio of 

 catchabilities over time and constant difference in in- 

 stantaneous mortality rates), a plot of the logarithm 

 of the ratio of catches versus time would be expected 

 to be linear (assuming the sample sizes are reasonably 

 large). A departure from linearity suggests violation 

 of one or both of the assumptions. This provides a 

 diagnostic procedure to check on the assumptions. If 

 catchabilities vary in a nonsystematic fashion, the fit 

 of the regressions would be low and would not be in- 

 fluenced by increasing sample size. 



Two-sample estimator 



A special case is when only two samples have been ob- 

 tained. Then, exponentiating the slope of the line 

 described by equation (3) reduces to the change-in-ratio 

 estimator of relative survival described by Paulik and 

 Robson (1969). Thus, 



Se 



R2 _ Cl2_^^E1 

 Rl Cix Ce2 



(4) 



where the ~ symbol indicates estimated quantities. If 

 the proportion of early spawners in samples 1 and 2 

 are denoted by P| and P2, respectively, then an esti- 

 mate of the variance can be found by the Taylor's series 

 method to be (Seber 1982, p. 382) 



V(SJSe) = lP,a-P,)]-' 



{(l-P,)^Po-r(P,) 



+ (1-P,)^P,^ V{P.,)}. 



