Mertz and Myers: Estimating the predictability of recruitment 



659 



(tfln(m)) =(<7 cl ) 2 +... + (CF c ,-) 2 



(9) 



where o ln( „ ;l is the standard deviation of In N it and 

 o ci is the standard deviation of C,. Correspondingly, 



(C7 Infi ) 2 =(CT cl ) 2 +... + (CT c4 ) 2 . 



(10) 



We designate the correlation coefficient relating log 

 recruitment and log abundance in stage i to be r ni , 

 and the corresponding coefficient relating log recruit- 

 ment and stage-specific mortality to be r cr With Equa- 

 tions 7, 8, 9, and 10, it is easily demonstrated that 



u \nim) 



(11) 



this relation holds for i = 1,2,3; for i = 4, one has a 

 correlation coefficient of 1.0, because we have stipu- 

 lated that abundances are evaluated at the end of a 

 given stage. Corresponding to Equation 11 we have 



(12) 



°infl 



Predictability of recruitment: density 

 dependence 



We prescribe density dependence of the form dis- 

 cussed by Myers and Cadigan ( 1993, a and b), which 

 is the same form as that used in key factor analysis 

 (Varley and Gradwell, 1960; Manly, 1990; Bradford, 

 1992). In this formulation, mortality during the ju- 

 venile stage is increased (decreased) for years in 

 which larval abundance is high (low). Specifically, 



AC 4 = aAlnN 3 +e, 



(13) 



where a gauges the strength of the density-depen- 

 dence and £ (which should not be identified with the 

 e introduced in section 2) represents the portion of 

 juvenile mortality uncorrected with late-larval 

 abundance. It follows that 



2 2/ \ 2 2 



tfc4=« (<Tl n(n 3)) +<X;, 



(14) 



where a E is the standard deviation of £. With this 

 formulation the quantities of interest can be readily 

 calculated. 



The quantities o ln( „,, remain as given in Equation 

 9, for i = 1,2,3; for i = 4, again, o lnl ,„, = q^, which is 

 now given by 



a XnR ) 2 =(l-2a)[(cr cl ) 2 + (a c2 f +(o c3 ) 2 ] (15) 



+(a r4 r. 



The correlation coefficients of interest may also be 

 calculated: 



r ri =-(l-a) 



Mnfl 



d'= 1,2,3) 



(16) 



-a[(cr cl ) 2 +(cr c2 ) 2 +(ct c3 ) 2 ] + (ct c4 ) 2 



< J \nR°c4 



The coefficients r ni are given by 



r m= (l- a) ^ml (» = 1,2,3), 



(17) 



and, again, r ni = 1. 



This treatment may be generalized to any case in 

 which there exists a linear relation, analogous to Equa- 

 tion 13, among the stage-specific mortalities and log 

 abundances. One could easily examine the case where 

 two or more stage-specific mortalities are positively 

 correlated, an effect that would enhance predictability. 

 However, a relationship of this sort will also increase 

 Oj nR , an outcome which is undesirable, as we show in 

 the results section, when calculated values of a lnR are 

 compared with those estimated from fisheries data. 



Predictability of raw recruitment 



Thus far, we have formulated relationships bearing 

 on the predictability, from prerecruit mortalities or 

 abundances, of log- transformed recruitment. It seems 

 intuitively likely that raw recruitment will be con- 

 siderably less predictable, which is unfortunate, be- 

 cause it is the untransformed recruitment which is 

 sought for fisheries management purposes. In this 

 section we undertake a quantitative investigation of 

 the predictability of recruitment. The results that 

 follow do not depend on the presence or absence of 

 density dependence (or other interstage correlations). 

 Let r/ ; be the coefficient for the correlation between 

 R and C,. We wish to find a relationship between this 

 quantity and the coefficient for the correlation between 

 In R and C,, r cr This quantity, r c ' ( , is calculated from 



r ci = (Gci a R >' 



J" J°° p(R,C l )dRdC l -C,R 



. (18) 



The joint probability p(i?,C,) is obtained through 



d(\nR) 

 dR ' (19) 



p(R,C i ) = p(lnR,C i ) 



