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Fishery Bulletin 89(4). 1991 



tions that hold under most circumstances for large sam- 

 ple sizes (Mendenhall and Scheaffer 1973), -21og e A 

 behaves as a x 2 random variable with s degrees of 

 freedom. Therefore -21og e A may be compared with a 

 critical x 2 value (pertaining to a suitable rejection 

 region) and the null hypothesis either accepted or re- 

 jected. For example, a value of -21og e A of more than 

 3.84 would lead to rejection of the simple model in favor 

 of a model with one extra parameter (df 1) with a re- 

 jection region (significance level) of 0.05 on the x 2 

 distribution. 



Simulations 



Assessment of model performance One hundred 

 simulated data sets were produced and analysed by the 

 models described above. The simulated data sets were 

 produced by simulating values of 61; (for each of the 

 1736 observations comprising the edited data set), 

 using the following equation, 



dli 



[Urf-fli + ei)][l - e-K.'.] + e , 



Looj and e ; were sampled from normal distributions, 

 and K; from a gamma distribution defined by the 

 model 7 maximum-likelihood estimates of their respec- 

 tive f/s and o 2 's. The release-length-measurement 

 error, ej , was sampled from a normal distribution with 

 a mean of and a variance of 5.2428cm 2 . Subroutine 

 GGNML of the International Mathematical and Statis- 

 tical Library (IMSL) was used to generate random nor- 

 mal deviates. IMSL subroutine GGAMR was used to 

 generate gamma deviates. Actual values of 1; and t, 

 from the edited data set were used. 



A second set of simulations was undertaken, assum- 

 ing Looj and Kj to be correlated with a correlation coef- 

 ficient of 0.80. Correlated normal deviates were 

 generated, using IMSL subroutine GGNSM. This was 

 done to test the sensitivity of the models to the assump- 

 tion of independence of L^ and Kj observations. In 

 this set of simulations, K was assumed, for simplicity, 

 to be normally distributed, rather than gamma distrib- 

 uted. With the values of j^ K and o^ 2 encountered in 

 this study, the gamma and normal distributions are vir- 

 tually indistinguishable, and therefore the normal ap- 

 proximation should result in little or no error. This was 

 confirmed in a small number of simulations where 

 analyses of simulated data, produced using normally- 

 distributed Kj values (uncorrelated with L^), gave 

 results virtually identical to simulations where Kj was 

 gamma distributed. 



Analysis of the 100 simulated data sets by each of 

 the models described above provided 100 sets of 

 parameter estimates for each. The means and standard 



errors of these estimates were calculated to (i) derive 

 approximate confidence intervals for the maximum- 

 likelihood estimates produced by the models and (ii) 

 compare model performance, i.e., their ability to 

 estimate known parameter values. 



Testing the assumption of normally-distributed l 2i 



The assumption that, given 1, and t ; , the random 

 variable dlj (or l 2i in the case of models 4 and 7) is nor- 

 mally distributed, is central to all the models described. 

 This assumption is most questionable for model 7, 

 where K variability and release-length-measurement 

 error could have unpredictable effects on the distribu- 

 tion of 1 2 ;. This assumption was tested for 30 com- 

 binations of 1 and t by generating 5000 values of l 2 j 

 for each combination, using the following equation. 



1 2 , = [L^-a+ei)] [l-e-Ki*] + 1 + £i + e,. 



A x 2 goodness-of-fit test (IMSL subroutine GPNOR) 

 with 50 equiprobable categories was then applied to 

 identify possible departures from a normal distribution 

 with mean and variance given by Equations (7) and (8), 

 respectively. Values of 1 of 50, 60, 70, 80, 90, and 100 

 cm were combined with values of t of 2, 4, 6, 8, and 

 10 years to produce the 30 combinations. 



Results 



Residuals analysis 



Using model 1, an analysis of residuals was carried out 

 on the primary data set. An initial fit of model 1 to the 

 1800 observations yielded estimates of L M , K, and 

 o e 2 of 195.614cm, 0.131914/year, and 24.8193cm 2 , 

 respectively. Standardized residuals (R ; = eJo e ) were 

 calculated and plotted against t; (Fig. 1) and against 

 lj (Fig. 2). Examination of Figure 1 reveals an even 

 distribution of standardized residuals about zero and 

 no obvious relationship with time at liberty. However, 

 Figure 2 suggests that the fit model may not be ap- 

 propriate over the entire range of release lengths 

 observed. For release lengths less than 50 cm, there are 

 54 positive residuals but only 8 negative residuals, in- 

 dicating that observed growth was faster than the 

 model would predict for these smaller fish. For release 

 lengths of 50 cm and larger, the pattern of residuals 

 is unremarkable. On this basis, observations with 

 release lengths smaller than 50 cm were excluded from 

 further analyses. 



A refit of model 1 to the amended data set (1738 

 observations) provided estimates of L^,, K, and o e 2 of 

 200.120cm, 0.125836/year, and 23.6157cm 2 , respec- 

 tively. Using these parameter estimates, standardized 



