Mertz and Myers: Estimating the predictability of recruitment 



661 



has a median value of about 0.75 (Fig. 2). Of neces- 

 sity, the discussion of the importance of measurement 

 error cannot be precise. We examine the effect of o E 

 in the interval 0.3 to 0.5, a range about midway be- 

 tween 0.08 and 0.75, on the variance of M estimates. 

 If the error in the log-transformed survey abun- 

 dances is characterized by o E = 0.3 (corresponding to 

 a CV of approximately 30% in the untransformed 

 abundances), then for the range of mortalities in 

 Bradford's regression, M ~ 10 - * to 0.4d _1 , Equation 

 25 shows that estimation error accounts for 309;- ( up- 

 per end of range) to 100% (lower end of range) of the 

 variance in M. In other words, the true variance in 

 M amounts to between 0% (lower end of range) and 

 70% (upper end of range) of the variance of M. If o f = 

 0.5, then the true variance represents 09c (lower end 

 of range) to 20% of the estimated variance of M. Of 

 more interest is the range of mortalities for major 

 fish species, entered in Table 1 of Bradford (1992), 

 M =10" 2 to 10- 1 d- 1 . For this range, with o E = 0.3, we 

 find that the true variance constitutes about one-half 

 of the estimated variance of M. If o E = 0.5, then the 

 true variance is estimated to make no contribution 

 to the estimated variance. On the basis of these num- 



bers, but somewhat arbitrarily, we assume, for the 

 range M = 10~ 2 to 10 _1 d _1 , that the true variance 

 represents 25% of the estimated variance of M, so 

 that Var(M) = 0.04 M 2 or 



a m - 0.2M , 



(26) 



where o m is the standard deviation of M. 



Finally, we wish to utilize Equation 26 to obtain a 

 relationship between the interannual variability in 

 cumulative mortality in a given stage and the mean 

 cumulative mortality. Since the M values in 

 Bradford's data base are largely stage averages, the 

 cumulative mortality is just C = M t s , where t s is the 

 stage duration. It also follows that the standard de- 

 viation of cumulative mortality a c , is given by a c = 

 O m t s . Applying these relations to Equation 26, we 

 arrive at o c = 0.2 C , where we have placed a bar over 

 the C to indicate that we are relating the interannual 

 variability of C (represented by a c ) to its mean value 

 ( C ). We can be more specific, since Bradford's re- 

 gression applies across stages, and make the stan- 

 dard deviation and mean specific to each stage i: 



0.2C, 



(27) 



The coefficient in Equation 27 is only half as large 

 as that in Bradford's regression (i.e. the square root 

 of the factor 0.15 which appears in Equation 1). This 

 adjustment of slope, arising from correction for esti- 

 mation error, could be too severe (Bradford and Ca- 

 bana, in press; Bradford 1 ); nevertheless, we take 

 Equation 27 at face value, use it to predict o inJi , and 

 compare the derived values to data. In the discus- 

 sion we comment on the influence of the slope param- 

 eter in Equation 27 on the predictability calculations. 



Predictability of recruitment: no density 

 dependence 



In Table 1 we present the estimates of the correlation 

 coefficients derived from Equations 11 and 12; in the 

 final column the calculated o lnfl , from Equation 10, 

 appears. If we had used relation ( Equation 1 ) in the 

 calculation of c lnR , without adjusting for measurement 

 error, then the calculated values of a lnR would be one 

 and a half times as large. It is evident that (Fig. 3) o ]nR 

 is overestimated for cod, anchovies, and plaice. Myers 

 and Cadigan ( 1993, a and b) have shown that density- 

 dependent juvenile mortality can be expected to ap- 

 preciably attenuate larval variability in cod and plaice. 



1 Bradford, M. Dept. 1994. Fisheries and Oceans, West Van- 

 couver Laboratory, 4160 Marine Dr., West Vancouver. B.C. V7V 

 1NG, Canada. Personal commun. 



