Monterey Bay Aquarium, 886 Cannery Row, Monterey, CA 93940, USA Methodology for estimating length-at-maturity with application to elasmobranchs Henry F. Mollet Monterey Bay Aquarium, 886 Cannery Row, Monterey, CA 93940, USA

Introduction N(Z)ormal Cumulative Function (ZCF) Correlation of parameters, Probits, Logits Elasmobranch examples concentrating on shortfin mako with perspective

N(Z)ormal Distribution Function ZDF ((TL - )/)  = mean TL-at-maturity (MTL)  = Stand. deviation (measure of homogeneity) CV = /  Parameters  and  not correlated

N(Z)ormal Cumulative Function ZCF ((TL - )/) Parameters  = MTL and  = stand. dev. (not correlated) Alternate parameter slope = 1/(2)0.5  CV = 25% in example

Logistic (X) vs. ZCF &ZDF (O)

Correlation Normal Cumulative Alternate parameters Mat = ZCF (a + b TL) a = -MTL/ b = 1/ ( b = slope/2) a&b are correlated Logistic Alternate parameters Mat = 1/(1 + exp(a+bTL)) a = -MTL.4. slope b = slope/4 Example corr. (a&b) = 0.999 corr. (MTL& slope = 0.082)

Correlation in VBGF L & k are strongly correlated Differential equation dM/dt =  M2/3 - M  = anabolic parameter (build-up)  = catabolic parameter (break-down) k =  /3 (Symbol for kappa?) M = ( /)3; L  = q ( /) = q ( /3k)

Probit & Logit Fig. 5 from Finney 1964 “Probit Analysis” y = fraction mature for x cm TL ranges (Cannot use raw data) Probit = ZIF (y) + 5 Next step is to use weighted data & working probits Logit procedure is similar Logit = log (y/(1-y)) + 5

Review of Literature Leslie et al. 1945. Median Body-Weight at Maturity of Female Rats using Max. Likelihood/Probits Welch & Foucher 1998. Length-at-maturity of Pacific cod using Max. Likelihood/2-Parameter-Special-Sigmoid. Mollet et al. 2000. Length at maturity of Shortfin mako using Max. Likelihood/Logistic (Common Sigmoid). Best is Normal Cumulative Function (ZCF) in combination with Max. Likelihood loss function (least squares is ok) . That’s what Francis & Ó Maolagáin 2000 used for NZ rig (M. lenticulatus), however, they called it Probit.

Perspective of Shortfin Mako Maturity Data Mollet et al. 2000 15-18 months gestation, 3-year repro-cycle. Based on available data indicating that life-history parameters of different mako populations are similar. However, we were able to substantiate differences for size-at-maturity and mass. Should not affect gestation and repro-cycle.

Mollet, Cliff, Pratt, & Stevens 2000 (Fig Mollet, Cliff, Pratt, & Stevens 2000 (Fig. 4) WNA (n = 61), SH (n = 50 + 32 = 82)

Earlier version showing binomial data used in calculations

Contour plots of loss function (=residual) for size-at-maturity for shortfin mako

Quick & Dirty, Using Smallest Mature & Largest Immature WNA: MTL ~ (3.15+2.8)/2 = 2.98 m; 2.565  ~ (3.15 - 2.80) = 0.35 (correct 0.44 m) SH: MTL ~ (2.80+2.63)/2 = 2.72 m; 2.565  ~ (2.80 - 2.63) = 0.17 (correct 0.21 m)

Contour plots of loss function (=residual) for separate populations from South Africa and Australia

Probit Linear Regression using WNA (eff. n = 24) and SH (eff Probit Linear Regression using WNA (eff. n = 24) and SH (eff. n = 24) maturity data Cannot use raw data; y = fraction mature for 10 cm TL bins Probit = ZIF (y) + 5 Next step is to use weighted data & working probits Logit procedure is similar Logit = log (y/(1-y) + 5

Size-at-maturity of selected female sharks