Applied Ecophysiology: An Integrative Form to Know How Culture Environment Modulates the Performance of Aquatic Species from an Energetic Point of View - Aquaculture and the Environment: A Shared Destiny - page 183

requirement of regression models, namely that each data point be independent from others
Zar, (1999); Zuur et al . , (2007). Finally, size-at-age variation generally increases with age,
resulting in strong heterogeneity of variance in different ages, another impediment for
ordinary regression analysis Zuur et al . , (2007). These characteristics violate important
regression assumptions, thus producing unreliable models, i.e. models with dubious F , X 2
and p values that cannot be used for prediction purposes.
A methodological approach is fitting a regression model to the weight of each individual at
a known fixed time ( t 2 ) against its weight at an earlier point in time ( t 1 ). This method ensures
linearity of the X-Y function and independence of data points (as long as they are
individually labelled). The resulting linear equation describes a type of relative growth, and
its slope represents the proportionality of the difference between two individuals at t 2
relative to the difference between them at t 1 . In turn, the line's intercept represents the final
weight reached by the smallest individual in the dataset. In this context, comparing the
slopes of different lines gives information on how much inter-individual weight differences
change over experimental time. Concomitantly, comparing different line intercepts informs
on individual growth rates: lines with different intercepts indicate different growth rates,
because two animals with the same initial weight reach different final weights within the
same period. Although this approach allows indirect corroboration of whether individuals
with different final size have different growth rates, it does not permit estimation (mean
value ± standard error) of those parameters in the equation that describe individual growth
over time.
Generalized linear mixed models (GLMM) are a statistical tool that can complement growth
analyses, because they allow modelling of the large variability in individual size observed
within a culture population. GLMM can be applied to non-normal data in which random
effects are present (Zuur et al., 2009). By incorporating components that modify the
structure of variance, mixed models yield more-reliable estimators of model coefficients. In
addition, through certain variance and correlation structures, mixed models may produce
new parameters that estimate size-at-age variability and the time elapsed for two size
measures to be statistically unrelated. GLMM procedures include the validation of models
by visual inspection of residuals Montgomery & Peck, (1992); Draper & Smith, (1998. ),
thereby assuring that regression assumptions are adequately met.
Cephalopod growth has some remarkable characteristics: (i) growth rates are among the
highest in metazoans (the highest in invertebrate metazoans, higher than those of fish and
similar to those of mammals) Calow, (1987); (ii) it lacks an asymptotic growth phase
Moltschaniwskyj, (2004); (iii) it is highly plastic owing to its strong dependence on abiotic
and biotic factors, mainly temperature Pecl et al . , (2004), the amount and quality of food
André et al . , (2008), and sexual maturation Semmens et al., (2004); (iv) it follows a biphasic
pattern (as it often does in captivity), consisting of an initial rapid exponential phase
followed by a second phase, where growth slows down progressively (André et al . , 2009);
and (v) it is highly variable intra-specifically Pecl et al., (2004); Leporati et al., (2007).
Size-at-age variability has been attributed to the lack of a strong association between age
and size of these soft bodied invertebrates, and is well documented both in wild and in
culture populations Leporati et al . , (2007); Leporati et al . , (2008). This great variability,
patent even under controlled temperature and food conditions, has been associated with