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Fishery Bulletin 95(4), 1997 
evaluate the implicit functional form of the growth 
model without having to specify an explicit and per- 
haps invalid nonlinear function; and 2) a polyphasic 
parametric growth function fitted to the size-at-age 
data on the basis of the functional form implied by 
the nonparametric smooth. Polyphasic growth means 
that there is more than one growth phase or cycle in 
postnatal development, suggesting ontogenetic shifts 
in growth rates manifested by at least two growth 
spurts between birth and the onset of adult matu- 
rity (see Gasser et al., 1984). The polyphasic growth 
function used in our study was the Peil and Helwin 
(1981) parameterization comprising a summation of 
logistic functions as follows: 
y t = X{ a '[ 1 + tanh (^ a_ ^ ) )]} + e< ' (1) 
i= 1 
assumed that the polyphasic form (Eq. 1) used here 
also has sound statistical properties. In principle, 
Equation 1 was fitted by heteroscedasticity-robust 
nonlinear least-squares (HRNLS) with a hetero- 
scedasticity-consistent covariance matrix estimator 
(HCCME) to account for growth variability and mea- 
surement error (see Davidson and MacKinnon, 1993). 
In practice, Equation 1 was fitted with RATS (Doan, 
1992), which implements HRNLS with White’s 
HCCME. Otherwise, the generalized method of mo- 
ments (GMM) approach can be used for robust non- 
linear regression estimation (Davidson and Mac- 
Kinnon, 1993). The age-specific growth-rate function 
for the Kemp’s ridley sea turtle was derived analyti- 
cally by taking the first derivative of the fitted Equa- 
tion 1 with the software program MATHEMATICA 
(Wolfram Research, 1993). 
where y t = 
tanh(z) = 
j = 
mean length at age t ; 
(asymptotic mean)/; length in phase i\ 
growth coefficient in phase i; 
age at the inflection point of phase i; 
( e z - e~ z )/(e z + e ~ z ) and 2 = ((!.(/ - 8 ; )); 
number of growth phases; and 
an appropriate random error structure. 
Parameters of the standard logistic function (mono- 
phasic with skewed symmetric inflexion and suggest- 
ing one growth spurt) are well known to have excel- 
lent statistical properties (Ratkowsky, 1990). It was 
Results 
The size and estimated age data for the 70 Kemp’s 
ridley sea turtles presented in Zug et al. (in press) 
are shown in Figure 1A with a locally weighted re- 
gression smoothing known as LOWESS (see Cleve- 
land, 1993) superimposed to reveal the implicit func- 
tional form. The LOWESS procedure can be imple- 
mented by using S software (Becker et al., 1988). The 
nonparametric smooth (Fig. 1A) implies a polyphasic 
function with two sequential growth phases, with the 
first decelerating around 30 cm SCL and the second 
Age estimate (years) 
Figure 1 
(A) Scatterplot of size-at-age estimates for the 69 Kemp’s ridley sea turtles stranded along the Atlantic Bight and Gulf 
coasts of the United States, with an additional estimate of mean hatchling size (age=0 yr). Open circles and solid dot 
are the original data estimates (n= 70) from Zug et al. (1997). Solid dot is the outlier discounted in the parametric 
model (Table 1). The curve in (A) is a LOWESS (locally weighted robust regression) smooth superimposed to highlight 
the underlying size-at-age function without presuming the functional form. (B) Scatterplot of the Atlantic Bight 
subsample estimates (n= 55), with hatchling size included and a LOWESS smooth superimposed. (C) Scatterplot of the 
Gulf of Mexico subsample estimates (n=14), with hatchling size included and a LOWESS smooth superimposed. 
