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

are inside this interval. Additionally, a τ = 0.5 median curve visualizes
possible asymmetry and its change with time. When a representative
sample of the whole size distribution at each time is available, typically
with n > 50, points representing unconditional quantiles can be superposed
on the graph to show how well quantile regressions match them, as we did
in Figs. 28B, 31 and 34A. There is no residuals analysis for this kind of
quantile regression, at least in the way it is conceived for least-square
regression.
Nonlinear quantile regression using interior point algorithm proposed
by Koenker & Park (1996) is compatible with many growth models. For
the cases where representative samples are measured at each time interval,
we have proposed a modified method to use with so-called 'envelope
models', that is, models of the form Y = f(t, τ) as eq. 35. These models
calculate a 3D-surface enveloping data (Fig. 33). Curves for all quantiles
0 < τ < 1 are calculated at once. They contain most of the information in
the initial dataset, including individual variability. They are particularly
useful when individual variability in itself expresses some aspects of the
phenomenon studied, like growth in the presence of an intraspecific
competition. We have developed basic tools to fit and diagnose them
(namely, basic graphical residuals analysis, Fig. 34). Tests of significance
of the fitting can be elaborated as extensions of some tests for
unconditional size-distributions, like χ 2 or Kolmogorov-Smirnov tests or
by other means (Koenker & Machado, 1999). Confidence intervals for the
parameters do not yet exist. Tests for comparing various models fitted on
the same dataset, or to compare a single model fitted on various datasets
remain also to be developed. However, one difficulty in conceptualizing
such tests for envelope models is that one of the "independent variables",
τ, is estimated according to the dependent variable y (eq. 36) and is thus
not really independent.
Flexible, unconstrained envelope models can be incredibly complex,
with dozens of parameters. They are impossible to fit in practice.
Part IV: A growth model with intraspecific competition
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