718 



Fishery Bulletin 92(4), 1994 



%BIAS = 



_ Eboot 



E, 



bfsl 



(14) 



E, 



hest 



where E boot is the average of estimates from two thou- 

 sand bootstrap runs and E best is the best estimate 

 from the model fit to the original data. We used the 

 correction factor y y -E boot -E best to remove consistent 

 bias (Efron, 1982). The corrected estimate of log re- 

 cruitment in fishing season y, for example, was P Y ~J y , 

 where B was the original biased log-scale recruit- 

 ment estimate. 



Uncorrected log-scale recruitment estimates were 

 biased by amounts ranging from -7% to 7% and arith- 

 metic scale biomass estimates by amounts ranging 

 from -15% to 27% (Fig. 2). Consistent bias was ex- 

 aggerated when uncorrected recruitments were 

 transformed to arithmetic scale (—40% to 43%, Fig. 3). 



Bias from log-transformed recruitment 

 estimates 



Arithmetic scale recruitments in each fishing sea- 

 son (B ) were calculated 



B 0y =e\^-^ 



(15) 



where VAR is a variance estimated by bootstrapping. 

 The term VARifi )/2 adjusts for bias due to transfor- 

 mation of log- normally distributed random variables 



(Beauchamp and Olson, 1973). Bias due to log trans- 

 formation is in addition to consistent bias estimated 

 by "iy = E b „„r^besf T ne correction for bias due to log 

 transformation increased northern anchovy recruit- 

 ment estimates by 1% to 22% (average 9%). 



Corrections for bias due to log transformation make 

 arithmetic recruitment estimates for northern an- 

 chovy easier to interpret because the amount of bias 

 varies among uncorrected recruitment estimates as 

 a function of their variance. Many stock assessment 

 models (Deriso et al., 1985; Methot, 1990) estimate 

 recruitments as log-scale parameters but corrections 

 for bias in arithmetic-scale recruitment estimates are 

 not made. We recommend that bootstrap or other 

 variance estimates be used to correct arithmetic scale 

 recruitment estimates for bias where appropriate. 



Retrospective bias 



We evaluated potential for retrospective bias in the 

 SMPAR model by comparing our best biomass esti- 

 mates to estimates from runs that omitted data for 

 recent years. Bias corrections for retrospective analy- 

 sis were based on fifty bootstrap iterations. Results 

 indicated a negligible amount of retrospective bias. 



Estimates and comparisons 



Biomass estimates for northern anchovy age 1 and 

 older from the SMPAR (Table 4) and stock synthesis 



1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 



FISHING SEASON 



Figure 2 



Biomass estimates for northern anchovy, Engraulis mordax (age 1+ on 

 15 February, in thousands of metric tons), during the 1963 to 1991 fish- 

 ing seasons from the SMPAR model and the stock synthesis model used 

 by Lo and Methot (1989). Estimates from the SMPAR model are shown 

 with and without correction for bias. 



