LO ET. AL.: ESTIMATES OF LARVAL MORTALITY 



significance (a = 0.05) when the difference is 

 zero. The power is symmetrical about D = 0; 

 thus, only partial figiu'es were given in Table 7. 



For example, if the true difference was 0.5 and 

 one of the mortality coefficients was 2.0, with a 

 probability of 0.86, a sample size of 30 positive 

 tows from each of two populations will detect a 

 significant difference in their mortality coeffi- 

 cients. The probability would be only 0.76 if one 

 of the mortality coefficients was 2.5. In genei-al, 

 to achieve the same probability of detecting a 

 given difference between mortality coefficients, 

 a larger sample size is required for a larger p. To 

 detect a significant difference with a probability 

 of 0.96, when the true difference is 0.5 and one of 

 the mortahty coefficients is 1.0, 30 positive tows 

 are required from each population. If p = 2.5, 

 however, 60 positive tows are required to detect 

 the same difference with a probability of 0.95. If 

 the difference is gi'eater than 1 , at most 20 posi- 

 tive tows from each population would be suffi- 

 cient. 



The two methods serve different purposes. 

 The cv method provides a 95% confidence inter- 

 val for the difference. The Power Method as- 

 signs a probability to the detection of a differ- 

 ence, but provides no information on the magni- 

 tude of the difference. 



Estimates of p with Various Growth 

 Rates 



Mortality is defined as the decline of produc- 

 tion with larval age. Thus an overestimate of 

 larval age, predicted from an underestimate of 

 growth rate, will underestimate mortality rate. 

 Similarly, an overestimate of gi'owth rate will 

 result in an overestimate of mortality rate. 



The mortality coefficient O) was fixed at 1.5. 

 Data from February, region 7, temperature 

 15°C, were used to construct five populations, 

 corresponding to five combinations of growth 

 coefficients for yolk-sac and feeding larvae 

 (Table 8). Each population was surveyed 50 

 times with a sample size of 50 plankton tows. 

 The estimated mortality coefficient ib) was cal- 

 culated by assuming standard growth coeffi- 

 cients for February, region 7, temperature 15°C 

 (Table 8). When the population growth coeffi- 

 cients (a,„) were underestimated by the stan- 

 dard coefficients, the estimated mortahty coeffi- 

 cient (b) was less than p = 1.5; conversely when 

 growth was overestimated, the mortality coeffi- 

 cient was also overestimated. 



Because the yolk-sac stage is short, relative to 

 the feeding stage, we can reasonably assume 

 that the gi'owth coefficient for feeding larvae 

 (a,„) has the largest effect on the estimated mor- 

 tality coefficient (b). When the estimated mortal- 

 ity coefficient is plotted against a,„ (Fig. 9), it is 

 apparent that the bias in estimating mortality 

 rate, caused by errors in the assumed growth 

 rate, is asymmetrical: greater when actual 

 growth is slower than assumed growth and 

 smaller when actual gi'owth is faster than as- 

 sumed. When the actual gi'owth was half the 

 assumed rate, the mortality coefficient was over- 

 estimated by 80%; when the actual growth was 

 double the assumed rate, the mortality coeffi- 

 cient was underestimated by only 16% (Table 8). 



The coefficient, a„,, determines the instan- 

 taneous gi'owth rate (IGR) at age t as the IGR = 

 a„, ln(L„/Lo) exp[-a,„(f - ^o)] where L» 

 is the maximum fish length, and Lo is the min- 

 imum fish length for t > 6.28 days (Fig. 6). 

 Large value of a,„ implies that the IGR is large 

 for the small value of age t, and the IGR de- 

 creases rapidly as the fish ages. Because both 

 the IGR and the instantaneous mortality rate 

 (IMR = p/0 are two different nonlinear func- 

 tions of age (t), the relationship between these 

 two coefficients (a„, and p) is also nonlinear and 

 thus the bias is asymmetric. 



Table 8. — Five sets of coefficients for two-step Gompertz 

 growtfn curves (Fig. 6) used to simulate five populations. 

 Also listed are the standard coefficients used in tfie analy- 

 sis of survey data for region 7 in February with a tempera- 

 ture of IS^C. The estimated mortality coefficient (b) is 

 listed as average of 50 computer runs. The true mortality 

 coefficient (P) was 1 .5. 



DISCUSSION AND CONCLUSIONS 



The simulation model and its methodology have 

 general applicability to larval fish of many 

 species, although these results apply directly to 

 estimates of northern anchovy larval mortality 

 rates derived from CalCOFI surveys. Results 

 may differ because of differences in the param- 



411 



