NOTE Chaloupka and Zug: A polyphasic growth function for Lepidochelys kempii 
851 
at >60 cm SCL. It is proposed that a polyphasic 
growth model might be a better mathematical de- 
scription of growth than the monophasic (monotonic) 
von Bertalanffy (Caillouet et al., 1995b; Schmid, 
1995; Zug et al., 1997) or the monophasic (non- 
monotonic) Gompertz functions (Caillouet et al., 
1986) proposed for this species. A similar polyphasic 
growth function comprising two phases is also evi- 
dent for the Atlantic Bight subsample (Fig. IB) and 
is suggested for the Gulf subsample (Fig. 1C) despite 
a very sparse data field in the latter case. 
Figure 1 also highlights the considerable variabil- 
ity (heterogeneity) inherent in sea turtle growth and 
why heteroscedasticity-robust estimation procedures 
(e.g. HCCME, GMM) should be used to derive re- 
gression parameter estimates for growth model fits. 
There is also a major outlier in Figure 1A indicated 
by a solid dot — this value was discounted in the ex- 
plicit parametric model fit because no parametric 
model could be as robust in respect to this outlier as 
the nonparametric smooth displayed in Figure 1A. 
Growth variability in sea turtle studies is a complex 
function of demographic (sex, maturity status) and 
geographic factors as well as a function of the time- 
dependent nature of the implicit sampling design 
(confounding year and cohorts effects) and instru- 
mental measurement error. For instance, Caillouet 
et al. (1986) have shown conclusive evidence of so- 
matic growth variability due to cohort (year-class) 
effects for captive reared Kemp’s ridley sea turtles. 
The small sample size, mixed cross-sectional sam- 
pling design, and insufficient data on demographic 
and geographic covariates precluded any reliable 
estimate of these additional sources of growth record 
variability in the current study. 
The parametric growth curve proposed here to 
match the nonparametric smooth (Fig. 1A) for the 
Kemp’s ridley data comprises separate logistic 
growth functions for each of the two inferred growth 
phases integrated into a single explicit polyphasic 
function — Equation 1. The statistical fit of this func- 
tion to the growth data (Fig. 1A) is shown in Table 1. 
The growth model with robust estimation and with 
elimination of the extreme outlier (see Fig. 1A) fit- 
ted the data well with significant parameter esti- 
mates even allowing for family-wise error-rate ad- 
justment, small parameter estimate standard errors, 
and no aberrant residual behavior (see Judge et al., 
1985, or Ratkowsky, 1990, for a discussion of nonlin- 
ear regression fitting and goodness-of-fit criteria). 
Despite the good fit, significant growth variability, 
probably due to instrumental measurement error and 
confounding of year and cohort sampling effects, was 
not accounted for by the model (residual variance: 
g 2 =29.1). 
Table 1 
Parameter estimates for the polyphasic logistic growth 
function (Eq. 1) fitted to the Kemp’s ridley sea turtle size- 
at-age growth data in Zug et al. (1997). See Equation 1 for 
definitions of parameters. 
Asymptotic 
Parameter 
Estimate 
standard error 
t-ratio 
Inference 
«i 
13.6467 
2.7463 
4.97 
P < 0.001 
Pi 
0.7901 
0.2989 
2.64 
P < 0.008 
»i 
1.1169 
0.4303 
2.59 
P < 0.009 
a 2 
17.6595 
3.9288 
4.49 
P< 0.001 
P 2 
0.3059 
0.1274 
2.40 
P < 0.016 
S 2 
7.6361 
0.5407 
14.12 
P< 0.001 
The expected polyphasic size-at-age function is 
shown in Figure 2 A (age=skeletochronological age 
estimate) and presented numerically in Table 2 for 
comparative purposes. The explicit size-at-age 
growth function (Fig. 2A) was then differentiated 
with respect to estimated age by an analytical solu- 
tion to Equation 1 to derive the age-specific growth 
rate function (Fig. 2B). The expected age-specific 
growth rate function (Fig. 2B) displays an initial 
posthatchling growth rate >5 cm SCL/year, increas- 
ing to 11 cm SCL/year >1 year of age (A 1 ) or 13 cm 
SCL, slowing to 2 cm SCL/year by 3-4 years of age 
(ca. 27 cm SCL), marking the end of the first growth 
phase (i.e. 20^=27.3 in Table 1; mid-curve asymptote 
in Figs. 1A and 2A). The growth rate then rises to 
6 cm SCL/year near 8 years of age (5 2 ) or to 46 cm 
SCL before declining slowly to negligible growth ap- 
proaching adulthood >15 years of age at a size >62 
cm SCL, marking the end of the second growth phase 
(i.e. 2(a 1 +a 2 )=62.6 in Table 1; upper asymptote in 
Figs. 1A and 2A). 
Discussion 
Monophasic von Bertalanffy growth functions have 
been proposed for the Kemp’s ridley sea turtle by 
Caillouet et al. (1995b), Schmid (1995), Zug et al., 
(1997), and others (Marquez, 1994, and references 
therein). With the von Bertalanffy growth function, 
however, a monotonic decreasing growth-rate func- 
tion is implied and hence no growth spurt at any age 
or size. The statistical validity of that function fitted 
to a limited data span and of the Fabens mark- 
recapture analogue used by Schmid (1995) and Zug 
et al. (1997) has been reviewed critically by 
Chaloupka and Musick (1997). It is questionable 
whether a monophasic von Bertalanffy function fits 
the mean growth profile for the complete postnatal 
