Hoenig et at Estimating survival rate over time for larval fishes 



487 



Here, V(Pi) and ViP-,) are the variances of the pro- 

 portions, found by the usual formula for variance of 

 a binomial, i.e.. 



ViP, 





where A'', is the number of larvae in sample ;', and i 

 can take on the values of 1 and 2. 



Weighted regression 



Equation (3) can be seen to be a logistic model by noting 

 that \oge(Rf) is the logistic transformation. That is, 



letting P, be the proportion in the sample at time t 



rj 



which belongs to group "L" (i.e., ), the logit 



Cl + Ce 

 transformation of the proportion is 



l0git(P,) = l0ge|j^-^| = loge (/?,). 



This allows us to refer to known results for logistic 

 models to determine the optimum weighting scheme. 

 Not all observations on catch composition are of equal 

 value in estimating the relative survival rate. This is 

 because the variance of the ratio of catches for any par- 

 ticular sampling date will be a function of the sample 

 size and the proportions in the population. One method 

 for specifying weights to be used in the regression, 

 which explicitly accounts for this, is: 



weight, = Cei''^ + Cu'l 



When these weights are used in the regression, the 

 resulting estimates are known as minimum logit chi- 

 square estimates. Once the logistic regression model 

 has been fitted, new weights can be computed using 

 the predicted catch composition, rather than the ob- 

 served composition. The regression can then be recom- 

 puted, the weights updated, and the regression recom- 

 puted, until adequate convergence is achieved. This 

 procedure results in maximum likelihood estimates 

 (McCullagh and Nelder 1983). Most standard statistical 

 packages will perform logistic regression so that the 

 user need not specify explicitly the iterative weighting 

 scheme. 



The above weighting scheme is appropriate when lar- 

 vae are sampled randomly, i.e., each larva is sampled 

 independently of all other larvae. This situation is ap- 

 proximated when the expected catch per tow is small 

 (< 1) and tows are random over space. When the ex- 

 pected catches are large, the sampling procedure re- 

 sults in cluster samples. Consequently, the theoretical 

 binomial variance is too small. However, the binomial 



variance becomes increasingly small as the sample size 

 increases and as the proportions approach the extremes 

 (0 or 1), and this is what would be expected for cluster 

 sampling. Thus, weights computed from the binomial 

 variance should be reasonably appropriate. 



Estimation when there are many small samples 



It sometimes occurs that catch rates are extremely low 

 and many small samples are obtained. For example, 

 sampling may be conducted daily with low intensity. 

 In this case, the logistic regression procedure will not 

 work well due to the occurrence of many zero catches. 

 Estimation under the logistic model can still be accom- 

 plished by constructing the likelihood function and solv- 

 ing directly for the difference in mortality rates that 

 maximizes the likelihood function. 



The probability that an individual, randomly selected 

 from the catch at time t, is from group E (early-spawn- 

 ing group) is equal to 



Pt(E) 



C. 



Et 



Cei + Cit 



QEt Neo e-'^fi' 



to Neo e-^f.' + qit Nio e" 



1 + qe 



i^t 



(5) 



where q is the "apparent" initial ratio of abundances 

 (actual initial ratio of abundances if the ratio of catch- 

 ability coefficients is equal to 1) and AZis the difference 

 in instantaneous mortality rates {Ze-Zi). 



The probability that an animal is from group L is the 

 complement of (5): 



P,{L) = 1 



\ + qe 



tiZt 



qe 



t£t 



1 -I- qe^^ 



The likelihood function can then be constructed as the 

 product of the probabilities for each animal in each 

 sample: 



A = 



n 



(=1 



1 



\ + qe 



tiZi, 



n 



qe 



bZt. 



qe 



kZt, 





q«L e 



n (1 + qe'^'') 



(6) 



