374 



Fishery Bulletin 90(2). 1992 



Dividing the observed length-frequency distribution 

 by estimates of capture probability was as ineffective 

 because the mortality estimates (-0.035 + 0.030) had 

 a highly significant negative bias of 83% (Fig. 4c). The 

 negative bias and the strong negative skew in the fre- 

 quency distribution are due to the infrequent genera- 

 tion of large larvae. When such larvae occur in a 

 sample, their relative abundance is greatly magnified 

 by their extremely small capture probabilities. The 

 overestimation of large length-classes results in a cor- 

 responding overestimation of old age-classes. Because 

 the overestimated age-frequencies are at the extreme 

 of the age range, they exert considerable influence on 

 the slope of the mortality regression and thereby result 

 in the underestimation of M. 



Elimination of the biased age-frequencies nearly 

 eliminates this problem and results in mortality esti- 

 mates (0.192 ± 0.002) with a significant but small nega- 

 tive bias of 4% and a considerable reduction in variance 

 (Fig. 4d). The apparently greater effectiveness of age 

 elimination compared with length elimination is a func- 

 tion of the variance in length-at-age within the larval 

 fish population. For example, when length elimination 

 is applied to the simulated population, essentially all 

 of the undersampled lengths are removed. However, 

 this creates a new bias in the age distribution, because 

 some age-classes experience greater elimination than 

 others (Fig. 2a). Clearly, if no variation exists in length- 

 at-age, length elimination will be identical to age 

 elimination. To our knowledge, the use of age elimina- 

 tion has not been reported in the literature. 



Mortality estimates produced by IMMAGE (0.201 

 + 0.002) are unbiased and, along with age elimination, 

 have the smallest variance (Fig. 4e). This superior per- 

 formance is achieved because, unlike length or age 

 elimination, IMMAGE uses all of the sampled length- 

 frequency data and, unlike correction, uses capture 

 probability multiplicatively and therefore avoids magni- 

 fying the sampling error. 



Practical application of IMMAGE 



The application of IMMAGE using the Monte Carlo 

 simulation has been chosen because of its simplicity. 

 Although linear growth, constant mortality, and mono- 

 tonically increasing size selection may not always be 

 suitable for a particular application, the IMMAGE 

 procedure is extremely adaptable in the way growth, 

 mortality, and size selection can be specified. For 

 example, growth could be specified as an exponential 

 or a Laird-Gompertz function, and size selection could 

 be specified as a double logistic function (Somerton and 

 Kobayashi 1989) describing extrusion and avoidance 

 simultaneously. Mortality could be specified as either 

 a stage-specific function, where the mortality rates of 



yolksac and feeding larvae differ, or an inverse func- 

 tion of age (Lo 1986). More importantly perhaps, mor- 

 tality could also be specified as a function of length. 



Although mortality rates of larvae likely decline with 

 length for many species (Pepin 1991), length-dependent 

 mortality rates are difficult to estimate because such 

 mortality induces a progressive bias in mean length- 

 at-age (if the largest larvae in an age-class survive 

 better than the smallest, the apparent growth rate is 

 positively biased; Methot and Kramer 1979). This 

 problem can be circumvented with IMMAGE by 

 estimating the size-selected length-frequency distribu- 

 tion using a length-based population model (Somerton 

 and Kobayashi 1990) which mimics the growth and sur- 

 vival of individual members of an age-class over time. 

 Using such an approach, the likelihood of size-depen- 

 dent mortality could be tested against constant or age- 

 dependent mortality based on goodness-of-fit to the 

 observed length-frequencies. 



Several variations on the application of IMMAGE 

 described herein may be more appropriate in other 

 cases. First, the estimation of growth and mortality 

 parameters could be accomplished simultaneously 

 rather than separately by allowing the mortality pro- 

 cess model to include growth parameters as variables 

 rather than as known quantities. Second, the objective 

 function used for parameter estimation could be speci- 

 fied as a likelihood function rather than a sum-of- 

 squares function. This would be especially appropriate 

 in cases where the errors about the observed length- 

 frequencies are not normally distributed. Third, prior 

 estimates of some parameters could be included in the 

 growth and mortality process models rather than 

 estimating all parameters directly from the three 

 samples (i.e., ageing, length-frequency, and selection 

 samples). 



We believe the best way of estimating parameter 

 variances for an IMMAGE application is to use a 

 sample reuse technique known as boot-strapping 

 (Efron and Tibshirani 1991), because all sources of 

 sampling variability can be included. Boot-strapping 

 IMMAGE, however, is computationally intensive and 

 potentially time-consuming. To facilitate variance 

 estimation on slow computers, we have therefore in- 

 cluded in the IMMAGE program an approximate tech- 

 nique that is based on the inverse of the information 

 matrix (Ratkowsky 1983). 



When obtaining larval fish samples free of selection 

 bias is difficult, IMMAGE can still obtain unbiased 

 estimates of growth and mortality parameters. Not 

 only does IMMAGE provide estimates that are more 

 accurate and precise than other approaches, its greater 

 flexibility in form allows estimation of length-depen- 

 dent mortality rates that are perhaps biologically more 

 realistic than the constant rates now estimated. 



