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MODELING POSIDONIA OCEANICA GROWTH DATA

statistical methods more advanced than those provided by traditional linear models becomes even more crucial when dealing with dependent
data like those arising from reconstructive techniques. In particular, we shall discuss the potential of Generalized Linear Mixed Models
(GLMM) as a tool for dealing with such dependence.
All statistical and graphical analyses have been carried out using R (R Development Core Team, 2008), a public domain statistical
environment freely downloadable from the URL www.R-project.org.

2. A BRIEF REVIEW OF THE LITERATURE
Several pieces of evidence coming from different areas of the Mediterranean Sea indicate a general decline of the P. oceanica meadows
in recent years, mainly induced by increasing human pressures, which calls for the implementation of monitoring programs to provide
the necessary knowledge about their status and evolution in space and time. Assessment of the status of P. oceanica meadows is based on
the monitoring of a set of variables sensitive to changes in environmental conditions (Balestri et al., 2003).
Backdating measures are considered a powerful tool for investigating P. oceanica meadows dynamics of growth in space and time (Duarte
et al., 1994). These measures are based on the analysis of some persistent morphological features of reiterative modules characterizing
seagrass growth (Guidetti, 2001), and allow the determination of seagrass shoots and rhizomes age (Pergent, 1990, Duarte et al., 1994).
Currently, about 60% of Mediterranean marine laboratories, focusing their research on seagrass ecosystems, use back-dating measures to
monitor P. oceanica meadows, because they provide information about: (a) temporal evolution of above and below ground production; (b)
rate of sedimentation; (c) importance of sexual reproduction; (d) dynamics of the meadow, and (e) reaction to environmental factors
(Pergent-Martini et al., 2005). Back-dating techniques have been applied to investigate the effect on P. oceanica rhizomes growth of climate
change (Marba ´ and Duarte, 1997), deterioration of marine coastal environment (Guidetti and Fabiano, 2000; Balestri et al., 2004; Tomasello
et al., 2007), sedimentation (Boudouresque et al., 1984), water transparency (Guidetti, 2001), water temperature and hydrodynamic regime
(La Loggia et al., 2004), and sexual reproduction (Balestri and Vallerini, 2003; Calvo et al., 2006).
As for the statistical methods that have been used in the literature to analyze P. oceanica growth data, Table 1 provides a synoptic
summary, reporting the reference, the response variable/s analyzed, the statistical models and methods, and the design employed and, in
the last three columns, whether or not Normality tests, heteroscedasticity tests, and data transformations have been carried out.
Inspection of the table confirms the predominant role of the Gaussian linear model, and of techniques based on it, the absence of checks of
Normality, the presence of heteroscedasticity tests in a minority of cases and the resort, again in a few cases, to the logarithmic transformation
of the data. As far as the treatment of time dependence is concerned, except Calvo et al. (2006) no paper takes explicitly into account the
longitudinal nature of growth data obtained from back-dating techniques. Most of the papers using such data treat them as replicates over
time of independent samples (what we have named ‘‘cross-sections repeated over time’’ in Table 1). Only a few papers try to deal with the
problem by ‘‘sub-sampling’’, i.e., by random sampling a year from each shoot, in order to obtain independent replicates.

3. THE SICILY POSIDATA-1 DATASET

The dataset used in the applications throughout this paper was collected between 2000 and 2002 along the coasts of Sicily (see Figure 1).
More specifically, 17 sites were sampled in the survey. At each site, a total of 66 to 180 orthotropic (vertical) shoots, attached to plagiotropic
(horizontal) rhizome portions, were randomly collected by SCUBA diving at three stations (see Table 2), except for the Marzamemi site in
which rhizomes were sampled from only two stations. Depth at each station is approximately constant, whilst stations within sites have
depths varying between 6 and 27 m.
Samples were studied using lepidochronological analysis (Pergent, 1990). This technique, based on the observation of the cyclic variations
in the sheath thickness along the rhizomes of P. oceanica, allowed us to isolate and date rhizome segments corresponding to a 1-year period of
growth (lepidochronological year). Each lepidochronological year was dated starting from the rhizome apex (the sampling year) downward
and then back-dating each cycle with its corresponding rhizome segment. This procedure was performed for the entire cycles sequence until
meeting the rhizome segment connected to the horizontal axis, which represents the year of shoot birth. Finally, the length of each rhizome
segment was measured and its age was determined. In particular, the age of a rhizome segment is its distance in years from the horizontal axis
(Pergent and Pergent-Martini, 1990). Thus, for each rhizome the life history of growth performances (mm/year) was reconstructed.
In the Sicily PosiData-1 dataset we have series of different lengths varying between 1 and 48 lepidochronological years, but 75% of the
longitudinal series consist of 11 lepidochronological years or less (see Table 3). In all 16 938 lepidochronological years were observed
representing the total number of observations in the dataset.

4. CROSS-SECTIONAL ANALYSIS OF P. OCEANICA GROWTH DATA

Let us begin by considering the simple, cross-sectional setting in which no use is made of the longitudinal series obtained from back-dating
techniques, and the growth performance is referred only to a specific year (typically, the last available) or to an average taken over a limited
number of years (typically, the last two or three).

4.1. From linear to Generalized Linear Models
As pointed out in Section 2, it is a common practice in the literature to analyze P. oceanica annual growth data by using the well known
classical Gaussian linear models on the original, or on properly transformed, data. In order to show why it is suitable to re-consider the use of 371

Environmetrics 2011; 22: 370–382 Copyright ß 2010 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/environmetrics