Laidig et al : Growth dynamics in early life history of Sebastes jordani 



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Appendix B 



Here we describe the regression model used to relate standard length to otolith radius, which consisted of a series 

 of four linear segments, each describing a different growth stanza. We start with the following equation: 



SL = 



'a + bjx for x<Cj 

 a + b x c x + b 2 (x-ci) for Ci<x<c 2 

 a + bj C! + b 2 (c 2 -c 1 ) + b 3 (x-c 2 ) for c 2 <x<c 3 



( a + bjCj + b 2 (c 2 -Ci) + b 3 (c 3 -c 2 ) + b 4 (x-c 3 ) for c 3 <x 



(1) 



where a is the y-intercept of the first segment, x is the 

 otolith radius, bj, b 2 , b 3 , and b 4 are the slopes of the 

 four segments, and c 4 , c 2 , and c 3 are the points on the 

 x axis corresponding to the intersections. Because not 

 all individuals would be expected to make the transi- 

 tion from one stanza to the next at identical sizes, it 

 is reasonable to smooth the relationship at the segment 

 intersections. We therefore applied the technique 

 described by Bacon and Watts (1971) to smooth the 

 transition between linear segments, assuming that 

 most fish made the transition from one stanza to the 

 next over a small size range. Specifically, we assumed 

 that the probability that an individual fish made the 



transition from one stanza to the next was described 

 by the logistic cumulative distribution function, F(z) = 

 l/(l + exp(-z)), where z = (x-c ; )/g. Note that g is a 

 constant chosen so that 95% of all transitions occur 

 within ± 0.3^m OR of each intersection. (We assumed 

 this rapid rate of transition rather than evaluating the 

 transition rate from the data because preliminary 

 analyses, with g as an estimated parameter, indicated 

 that the density of measurements on the x axis was 

 not sufficient to yield unique solutions. Runs with dif- 

 ferent starting parameters did not converge to the 

 same parameter estimates.) Following Bacon and 

 Watts (1971), we rewrote the model as: 



SL = e + d^x-Cj) + d 2 (x-c 1 )s 1 + d 3 (x-c 2 )s 2 + d 4 (x-c 3 )s 3 



where the terms relate to equation 1 as follows: 



Si = 2*(F((x-Ci)/g) - 1; i=l,2,3; g = 0.1, 



dj = (b, + b 4 )/2, d 2 = (b 2 -b 1 )/2 ) 



d.3 = (b 3 -b 2 )/2, 



d 4 = (b 4 -b 3 )/2, and 



e = a + ((b,+b 4 )/2)c, - ((b,-b 1 )/2)c 1 - ((b 3 -b 2 )/2)c 2 - ((b 4 -b 3 )/2)c 3 . 



Our estimates of the above parameters and their standard errors are in Appendix Table 1. 



(2) 



