Page 1 - Lovison_alii_2010

Research Article

Received: 16 November 2009, Revised: 14 April 2010, Accepted: 14 July 2010, Published online in Wiley Online Library: 29 December 2010

(wileyonlinelibrary.com) DOI:10.1002/env.1063

Modeling Posidonia oceanica growth data: from

linear to generalized linear mixed models

a
a
b
*
G. Lovison , M. Sciandra , A. Tomasello and S. Calvo b
The statistical analysis of annual growth of Posidonia oceanica is traditionally carried out through Gaussian linear
models applied to untransformed, or log-transformed, data. In this paper, we claim that there are good reasons for
re-considering this established practice, since real data on annual growth often violate the assumptions of Gaussian
linear models, and show that the class of Generalized Linear Models (GLMs) represents a useful alternative for handling
such violations. By analyzing Sicily PosiData-1, a real dataset on P. oceanica growth data gathered in the period
2000–2002 along the coasts of Sicily, we find that in the majority of cases Normality is rejected and the effect of age on
growth is nonlinear. A GLM with Gamma distribution and identity or log link appears to be a satisfactory choice in most
cases. Furthermore, when back-dating techniques are employed, each plant provides a longitudinal set of dependent
data, and a proper statistical analysis should take such dependence into account. We show that the class of Generalized
Linear Mixed Models (GLMM), an extension of GLM’s, provides an effective way to analyze longitudinal P. oceanica
growth data. Again, by using examples taken from Sicily PosiData-1, we show that misleading results can be obtained if
dependence is ignored and that other techniques, like sub-sampling, are not a good option for overcoming the so-called
‘‘pseudo-replications’’ problem. Copyright ß 2010 John Wiley & Sons, Ltd.

Keywords: Posidonia oceanica; annual growth; Generalized Linear Models; Generalized Linear Mixed Models; lepidochro-
nological data

1. INTRODUCTION

Posidonia oceanica represents the key species of the most important and productive ecosystem in subtidal habitats of the Mediterranean Sea
(den Hartog, 1977). It exerts a multifunctional role within coastal systems offering substrate for settlement and food source, as well as
maintaining the chemical and physical characteristics of water and sediments (Procaccini et al., 2003). Moreover, being sensitive to changes
in the environment, P. oceanica is considered a crucial indicator of the quality of coastal marine waters (Pergent-Martini et al., 2005).
A peculiarity of P. oceanica is the presence of persistent morphological features of reiterative modules characterizing its growth, which
lend themselves to back-dating techniques (lepidochronology, internodal length: Pergent, 1990; Duarte et al., 1994), which allow for the
reconstruction of past history of growth variables (annual rhizome elongation and diameter, primary production, number of leaves, flowering
events).
For all these reasons, there is a large body of statistical studies concerning the growth performances of P. oceanica meadows. These studies
are usually carried out through Gaussian linear models applied either to transformed or untransformed data. The aim of this paper is to
underline that there are good reasons for re-considering this established practice, since real data on annual growth of P. oceanica often violate
some (or even all) of the assumptions of Gaussian linear models. In particular such data exhibit marked asymmetry, which makes the
assumption of Normality questionable, and, when related to explanatory variables, show heteroscedasticity and nonlinear relationships,
which make the assumptions of constant variance and linear regression function untenable. We shall show that the class of Generalized
Linear Models (GLMs) is particularly appropriate for handling these departures from the assumptions of the standard Gaussian linear
regression models, still retaining much of their flexibility and interpretation.
Furthermore, when reconstructive data are considered, it is important to look at the annual data available for each shoot as a longitudinal
dataset, and to take into account the dependence of the data; again, GLMs prove to be a flexible tool, since they can be extended to
accommodate such dependence.
After a brief literature review in Section 2, and a presentation of the Sicily PosiData-1 dataset used throughout the paper in Section 3, in
Section 4 we shall discuss the limitations of traditional Gaussian linear models and the potential of GLMs, both theoretically and through
applications to Sicily PosiData-1. In Section 5 we shall show, again through some theoretical discussion and real examples, how the use of

* Correspondence to: G. Lovison, Dipartimento di Scienze Statistiche e Matematiche ‘‘S.Vianelli’’, Universita ` di Palermo, Edificio 13, Viale delle Scienze 90128
Palermo, Italy. E-mail: lovison@unipa.it
a Dipartimento di Scienze Statistiche e Matematiche ‘‘S.Vianelli’’, Universita ` di Palermo, Italy
b Dipartimento di Ecologia, Universita ` di Palermo, Italy

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Environmetrics 2011; 22: 370–382 Copyright ß 2010 John Wiley & Sons, Ltd.