760 



Fishery Bulletin 88(4), 1990 



Appendix A 



Derivation of Confidence Bands 



The predicted size after one molt can be obtained by 

 regressing tiie size at recapture on the size at tagging 

 for those animals which molted once (text equation 1). 

 Thus, 



predicted recapture size = a + bX 



(A.l) 



where a and h are parameter estimates and X is the 

 size at tagging. Estimates of the variances and covar- 

 iances (i.e., V(a), V(6), Cov{aJ))) are easily obtained 

 in the standard way. 



The predicted size after two molts is obtained from 

 A.l as 



Y = predicted size after two months 



= a + b{a + bX) = a + ab + 6-X. 



An approximate (asymptotic) estimate of the variance 

 of the size after two months, V(Y), is found by the 

 Taylor's series or delta method to be 



V(Y) = 



dY 



V(-0 + {^^Y V{b) 

 db 



. 2 i"i (" 



da db 



Cov {ii,li) 



where the derivatives are evaluated at the parameter 

 estimates. Thus. 



V(Y) = (1 + b)- Via) + id + 2bx)- Y(b) 



2(1 + 6)(o + 26x) Co\'(a.b). 



An approximate 95% confidence band is thus obtained 

 as 



Y ± 2 \/V(Y). 



Appendix B Estimating molt 

 increments for a quadratic model 



Over the size range of animals we studied, the relation- 

 ship between post- and pre-molt size appeared linear. 

 However, when a wide range of pre-molt sizes is con- 

 sidered, it is common to find a curvilinear relationship 

 which may be modeled satisfactorily by a quadratic 

 equation. The non-linear regression model (equation 3) 

 in the text can be generalized to allow estimation for 

 the quadratic model. 



Let the size at tagging be denoted by X, and assume 

 the size after one molt is given by 



size after one molt = « + 6X -i- cX-. (B.l) 



Then the size after two molts is given by 



size after 2 molts = n + b(ii + liX + cX-) 



+ c(a + bX + cX~)~. (B.2) 



As before, let Y be the size at recapture (for animals 

 molting once or twice), and define Z to be an indicator 

 variable for whether an animal molted once or twice, 

 i.e., let 



Z = 



0, animal molted once 



1, animal molted twice 



Then equations (B.l) and (B.2) can be combined in a 

 single non-lineai' regression as 



Y = a + {ab + r(-r)Z + bX + b(b - 1 + 2r;r)ZX 



-I- c-X -I- c(b + 2ac + b- - 1)ZX- 



+ 2bc-ZX^ + (■■'ZX-' + e (B.3) 



where e is the random error term. Equation (B.3) 

 reduces to (B.l) when Z = and to (B.2) when Z = 1. 



