DeMartini et al.: Validated morphological metric for lobster size at maturity 



33 



Appendix 



Method for estimation of maturation 

 with pleopod metrics 



To model the allometry we used the power function 

 Y=(5X' 1 ' and assumed multiplicative error. The logarithmic 

 transformation of this function leads to a linear regres- 

 sion model. Specifically, we defined ln(Y) = f 1 (X)+e 1 

 where f i (X)=a 1 +l\\n{X) and a, =ln((3j), as the allometric 

 relationship for juvenile lobsters and ln(Y) = f 2 (X)+e 2 , 

 where f 2 (X)=a 2 +fi 2 \n(X) and a 2 =ln(d 2 ), as the allometric 

 relationship for adult lobsters. The errors, e l and e 2 , were 

 assumed to be independent and normally distributed 

 with mean and variance <jj 2 and o 2 2 , respectively. 

 We assumed that maturation occurred over a range 

 of tail widths. Dividing the domain of x into four 

 intervals, we defined the probability that a lobster 

 with observed tail width x was mature (m) as 



P(m\x)-- 



x-e n 



exp 



' J o +e l -2x ' 



0,-x 



exp 



e l -e Q 



e <x< 



x <e 



x<e x 



x>( 



(i) 



When y=0, Pim\x) increases linearly from to 1 over 

 the interval [8 , 6 l ]. For y>0, the curves are sigmoidal, 

 symmetrical, and the rate that the probability changes 

 with respect to tail width is bell shaped (the sigmoidal 

 curve first accelerates, then decelerates). The point of 

 inflection, {6^+0^12, is the tail width at which 50% of 

 the lobsters are expected to be mature. For both species, 

 we assumed that yaO. 



Defining the allometry model and the probability 

 of maturity as above, we expressed the model relat- 

 ing pleopod length to tail width as ln( Y)=/ 1 (X)(1- 

 P(m\x))+f 2 iX)Pim\x)+e, where f are independent normal 

 variates with mean and covariance V m . 



Assuming (.t,,y ( ) i=l, ... ,n independent paired obser- 

 vations and cP- = o^=o 2 2 , V !H =Io 2 +M, where / is the (n x 

 n) identity matrix, M is the diagonal matrix M U =A 2 (x t ) 

 P(m,l.v,) (1-P(m,l.v,)), and AU,)^*,)-/^*,). Hence, we 

 have a weighted least squares problem with weights 



(o'+Mj 



if x l s or x l a 6 X 



ife <x i <e 1 . 



(2) 



To fit the model, we defined a 3 =c< 2 -c< 1 and P 3 =P 2 ~Pi 

 and expressed the model as ln( Y)=f 3 (X)P(m\x)+f 1 (X)+e, 

 where f 3 (X)=a 3 + /3 3 ln(X). To ensure that the curve in 

 the transition range was monotonically increasing (if 

 P 3 >0), 8 was bounded such that o aexp(-cc 3 //3 3 ), and 

 if /3 3 <0, 6 1 was bounded such that 0jsexp(-a 3 //3 3 ). The 

 curve was fitted by using iteratively reweighted least 

 squares. The weights were recomputed at each iteration. 



While fitting the lobster data to the specified model, 

 we observed that one or more of the parameters in- 

 volved in defining the sigmoidal curve departed from 

 linear behavior. Under these circumstances, the con- 

 fidence interval derived by assuming the asymptotic 

 properties of maximum likelihood estimates may be 

 invalid (Ratkowsky, 1983). Therefore, we computed ap- 

 proximate 95% confidence intervals for the point of 

 inflection using the bootstrap method. Specifically, we 

 used case resampling with 1000 bootstrap replications. 

 Confidence intervals were derived by using the studen- 

 tized bootstrap confidence limits (Davison and Hinkley, 

 1997). 



