Growth model of the reared sea urchin Paracentrotus ... - SciViews

d. Results
Theoretical considerations: use of quantile regression instead
of least-square regression
The most widespread method to fit a curve is the use of a least-square
method with one of the many minimization algorithms available [simplex,
(quasi-)Newton, etc.; Sen & Srivastava, 1990; Draper & Smith, 1998;
Nocedal & Wright, 1999]. The algorithm finds the combination of values
for the various parameters in the model (the solution) that leads to a
minimal value for the objective function, which is here the sum of the
square of the residuals (that is, the sum of squared distances between
observed values for the dependent variable and values predicted by the
model at the same levels for the independent variables).
Least-square regression has many advantages over other methods. In
particular, when partial first (gradient matrix) and second (Hessian matrix)
derivatives of the function are calculable for each parameter, convergence
through a solution is accelerated and can be verified (at least for a local
solution, Nocedal & Wright, 1999). In the counterpart, that regression
supposes that the fluctuation around the model (called the error term) is
additive, independent, normally distributed and with a constant standard
deviation (heteroscedasticity). It is also very sensitive to outliers because it
uses the squared residuals. Those constraints, even if not strictly met every
time, particularly in many nonlinear phenomena like growth, appear to be
of minor importance for many authors. Indeed, outliers are eliminated, or
weighing methods are applied to limit their impact.
Yet, there are two arguments against the least-square regression used in
the framework of growth models, particularly when individuals' growth is
influenced by the presence of conspecifics or of other species (indeed a
general case to be verified in each study, except when a single individual is
grown alone in a cage or an aquarium!): first it is sensitive to the
Part IV: A growth model with intraspecific competition
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