98 



Fishery Bulletin 101(1) 



The following values were used for tiie biological parameters: 



w„=li, (1.19) 



/_=l_e-o.i, (1.20 1 



and M = 0.2 and .s = 1, where the a for M and s are subscripted. 



Appendix II: extensions 



The recruitment-environmental submodel that we used to 

 analyze the snapper stock is simple, and other submodels 

 may improve the fit to the data and the explanatory ability of 

 the environmental time series. A dome-shaped relationship 

 has been observed between abundance of tuna larvae and 

 SST (Forsbergh, 1989), indicating that a quadratic or higher- 

 order polynomial submodel may be more appropriate. 



and Walters (1992, p. 285-287) could be used to integrate 

 spawner-recruitment models and environmental time se- 

 ries into the stock assessment model. 



R, =/'(S,)exp(/3/, -i-f, -I- a), 



(II.4) 



where /"(S,l = the function for the stock-recruitment rela- 

 tionship and 

 S, = the spawning biomass at time t. 



The equation for the Ricker (1954) and Beverton and 

 Holt ( 1957) models would be 



i?( = S^ exp(a - ftS,) exp(/3/^ -(- f^ -t- a). Ricker (II. 5) 



R, = -2-^exp(/}/, +£,+ a), Beverton-Holt (II.6) 

 b + S, 



R, = exp(a -I- PJ, + /}.,/,- + ... PJl' + f, ). 



fll.l) 



Regime-switching models (Granger, 1993) that have the 

 ability to favor two levels of values may be more appropri- 

 ate for species that are hypothesized to experience two 

 environmental regimes. 



where a and b are parameters of the stock recruitment 

 models. 



Appendix III: the hypothesis test problem 

 for the environmental model 



X, = hub -lb- 



1 + exp 



.ln(19)A_k. 



+ /i^exp(f,), (II. 2) 



where lb = the lower bound of the model parameter (low 

 regime); 

 ub = the upper bound of the model parameter ( high 



regime); 

 l^g = the environmental time series value that 



gives a SC^f influence: and 

 the environmental ti: 

 gives a 95% influence. 



lyg = the environmental time series value that 



If Zq, is only slightly higher than /,„, the model will have 

 two regimes. Therefore, the model can be simplified by set- 

 ting Ic,- as a small fixed value above I^g, allowing for the 

 use of a regime-shifting model that requires estimation of 

 only the lower bound, upper bound, and the value of the 

 environmental time series at the point of change. 



The method can be easily extended to include multiple 

 environmental factors. 



X, = lu exp 



e, +XA^''+" 



(11.3) 



where / indexes the environmental factor 



The method we have used assumes that recruitment 

 is independent of spawner biomass (i.e. we penalize the 

 deviation from a mean recruitment modified by the rela- 

 tionship with the environmental time series). Maunder 

 < 1998a) suggested applying the method to stock-recruit- 

 ment relationships, and the models described by lliiborn 



Let, 

 where 



Pi 



N 



is the sample size and n^ is the number from category 7 

 in the sample. The negative log-likelihood (ignoring con- 

 stant) is 



-lnL = N^-p,\n(Pj). 



X'- =2(lnLj-lnL|,) = 2N 



^p,ln(p,,)-^p^ln(po,,) 



V J 



therefore X' 's proportional to A^. LnLi (estimate (5) has 

 one more parameter than LnL^ (/j=0) and therefore will 

 be at least slightly larger (two sets if independent random 

 numbers always have a nonzero correlation). Therefore, 

 there will be some value of N for which ;f->3.84. Now, con- 

 sider a simple example where 



Pj = W— and .V, = nexpijil^ + t)). 



with the penalty - In Prior (f | ct) = V 



2a' 



and CT is a constant. Consider two models: 1) f ^ = and 2) 

 estimate f. For model 1, as A' increases x~ increases in 

 proportion to N, as explained above, because the penalty 

 term is constant. However, for model 2, as N gets large, the 



