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Fishery Bulletin 90(2), 1992 



were iteratively varied, and steps 4 and 5 were re- 

 peated, until the minimum residual sum of squares 

 (RSS) was achieved. The RSS was defined as 



I I (Fu - h)' 



(4) 



where Fjj and Fjj are the observed and estimated 

 frequency within the ith length interval and the jth 

 age-class. 



Mortality simulations 



Mortality simulations examined the performance of 

 IMMAGE and three bias-correction approaches: length 

 elimination, division by capture probabilities, and age 

 elimination. Each of the 1000 repetitions of a simula- 

 tion began by generating a biased length-frequency 

 sample, an unbiased ageing sample, and a selection 

 sample. For all simulations, except those examining 

 IMMAGE, mortality estimation began with an attempt 

 to derive an unbiased age distribution. If length elim- 

 ination was used, this was accomplished in two stages. 

 First, an unbiased length-frequency distribution was 

 estimated by eliminating all length categories with a 

 capture probability of <0.95 [based on capture prob- 

 abilities defined by Equation (3), length-classes 3-18 

 were retained; Fig. 2a]; a probability of 0.95 was used 

 instead of 1.00 because it is better defined. Second, the 

 age-frequency distribution was estimated from the un- 

 biased length-frequency distribution by using the age- 

 ing sample and a procedure known as age-slicing 

 (Mesnil and Shepherd 1990). To do this, lengths from 

 the ageing sample were regressed on the ages, and the 

 fitted linear regression equation was evaluated to 

 determine the length corresponding to each age bound- 

 ary (i.e., 0.5, 1.5, 2.5 days, and so on). Age-frequencies 

 were then estimated by summing length-frequencies 

 between age boundaries. 



If division by capture probabilities was used, the age- 

 frequency distribution was estimated by first dividing 

 the observed length-frequency distribution by esti- 

 mates of capture probability, then converting the 

 length-frequencies to age-frequencies using age-slicing. 

 If age elimination was used, age-classes with a capture 

 probability of <0.95 at the mean length also were 

 eliminated from the age-frequency distribution (age- 

 classes 0-5 days were retained; Fig. 2a). For all three 

 cases, instantaneous mortality rate was then estimated 

 as the slope of an unweighted linear regression to the 

 natural logarithm of numbers-at-age (Ricker 1975). 



For simulations examining IMMAGE, mortality esti- 

 mation proceeded as follows. First, initial values of M 

 and No were obtained from the ageing sample by fit- 

 ting a straight line to the logarithm of numbers-at- 



age. Second, values of It and Var(lt) were estimated 

 from the ageing sample by fitting straight lines to 

 length-at-age and Var(lt)-at-age, using linear regres- 

 sion, and then evaluating the fitted regression equa- 

 tions at each t. Third, capture probabilities were 

 estimated from the selection sample. Fourth, the un- 

 biased age distribution was estimated as Nt=No e '^^. 

 Fifth, the unbiased length distribution of each age-class 

 was estimated as Nt times a normal probability distri- 

 bution with a mean equal to It and a standard devia- 

 tion equal to the square root of Var(lt). The unbiased 

 length distribution for the population was then esti- 

 mated by summing the age-specific length distributions 

 over all age-classes. Sixth, the biased (observed) length 

 distribution was estimated by multiplying the unbiased 

 population length distribution by the estimated capture 

 probabilities. Seventh, Nq and M were iteratively 

 varied, and steps 4-6 repeated, untO the minimum RSS 

 was achieved. The RSS was defined as 



(Fj - Fj)2 



(5) 



where Fj and Fj are the observed and estimated fre- 

 quencies within the jth length interval. 



The IMMAGE application used in the simulations 

 (i.e., one that assumes linear growth and constant mor- 

 tality) is available from the authors as a stand-alone 

 program (IMMAGE, written in Microsoft QuickBasic) 

 designed to run on IBM-compatible microcomputers. 



Results and discussion 



Growth 



The type of size selection examined in the simulations 

 (i.e., a decrease in the probability of capture with in- 

 creasing larval length) complicates the estimation of 

 growth and mortality in slightly different ways. For 

 growth estimation, the primary effect is that the 

 largest larvae in each age-class are undersampled 

 relative to the smallest larvae, and the mean lengths- 

 at-age are therefore underestimated (Fig. 2b). Since 

 the bias progressively increases with age, plots of mean 

 length against age are curvilinear and falsely indicate 

 a declining growth rate (Fig. 3). Such curvilinear or 

 even asymptotic growth patterns are often reported 

 in studies of wild-caught larvae (Bailey 1982, Laroche 

 et al. 1982, Lough et al. 1982, Thorrold 1988, Warlen 

 1988). Although there may be biological reasons to ex- 

 pect such a pattern, especially for species with pro- 

 nounced ontogenetic changes in body form, length- 

 selective sampling may be a contributing factor. 



IMMAGE estimates of the slope (1.499 ±0.005; x 

 1000 replicates ±2 SE) and intercept (10.002 + 0.012) 



