1. 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

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

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

4. 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

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

6. 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) = corr. (MTL& slope = 0.082)

7. 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)

8. 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

9. Review of Literature
Leslie et al Median Body-Weight at Maturity of Female Rats using Max. Likelihood/Probits
Welch & Foucher Length-at-maturity of Pacific cod using Max. Likelihood/2-Parameter-Special-Sigmoid.
Mollet et al 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.

10. Perspective of Shortfin Mako Maturity Data
Mollet et al 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.

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

12. Earlier version showing binomial data used in calculations

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

14. Quick & Dirty, Using Smallest Mature & Largest Immature
WNA: MTL ~ ( )/2 = 2.98 m;  ~ ( ) = (correct 0.44 m)
SH: MTL ~ ( )/2 = 2.72 m;  ~ ( ) = (correct 0.21 m)

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

16. 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

17. Size-at-maturity of selected female sharks