F F S S P P M M 0 0 4 4 ORAL PRESENTATIONS - SESSION 2 4th International Workshop on Functional-Structural Plant Models, 7-11 june 2004 –Montpellier, France Edited by C. Godin et al., pp. 65-69 Analysis of longitudinal data applied to plant architecture study C. Véra 1 , Y. Guédon 1 , C. Lavergne 2 , Y. Caraglio 1 1 Unité Mixte de Recherche CIRAD/CNRS/INRA/IRD/Université Montpellier II Botanique et Bioinformatique de l'Architecture des Plantes, TA 40/PS2, 34398 Montpellier Cedex 5, France E-mail: carine.vera@cirad.fr , guedon@cirad.fr , caraglio@cirad.fr 2 Unité Mixte de Recherche CNRS/Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France E-mail : christian.lavergne@inria.fr Introduction Plant growth is modulated by environmental factors such as rainfall, temperature or light. Several methods for modelling cambial growth (tree-ring) response to climate have been proposed in dendrochronology; see Monserud (1986) and Guiot (1986). Those methods belong to the field of time-series analysis and are mainly based on autoregressive moving average (ARMA) models. Climate influence has been modelled using multiple regression and principal components analysis (Fritts and Swetnam, 1989). Our study focuses on the primary growth analysis (shoot elongation) of perennial plants, of forest trees more precisely, according to climatic factors, and for various plant ages. The response variable of the model is the shoot length measured along the stem for successive years and for several individuals. As these plant growth data are collected a posteriori, once the trees have been cut down, the limits between annual shoots have to be identified on the basis of morphological markers (mainly cataphyll scars). Successive annual shoot lengths along the stem have the property of being mutually correlated. This type of data is named, longitudinal data, in statistics. Dendrochronological data are made up of a small number of long series whereas longitudinal data are made up of a larger number of shorter series. Growth is subject to three main sources of variation: a trend (slow varying component), showing a succession of growth phases and mainly expressing ontogeny, climatic fluctuations and individual effects (influence of unobserved factors such as pests and diseases). The proposed method of analysis is based on the multiphasic modelling of those different sources of variation. A Markov chain represents the succession of growth phases. A linear mixed model associated with each growth phase combines both fixed effects (trend and climatic covariates) and individual random effects representing the heterogeneity between individuals within a given phase. The originality of the proposed method lies in the combination of a multiphasic modelling and a linear mixed modelling. Statistical modelling of the data with the family of the multiphasic linear mixed models The statistical modelling approach will be illustrated using a sample of 46 fifteen-year-old sessile oaks (Quercus petraea, (Matt) Liebl, Fagaceae) that grew in the private forest of Louppy-le-château (north east of France). The length of each successive shoot has been measured along the stem. The annual shoot length series are represented in Figure 1. Tree growth can be decomposed into a trend and local fluctuations (Diggle et al., 2002). The trend, whose nature is mainly endogenous, shows that tree growth period is made up of a succession of growth phases. Three different phases delimited by precise boundaries can be identified (Figure 2): during the first one (before 1986), annual growth does not evolve (waiting phase), the second one (from 1987 to 1990) corresponds to a rapid increase of the annual shoot length (establishment phase of the young tree) and the third one (after 1990) coincides with a stabilization of the annual growth. Local fluctuations result from the difference between the data and the trend. They mainly express the influence of environmental factors on growth. Synchronism of individuals (see Figure 1) highlights a year effect due mainly to climatic factors. Moreover, we have checked that the amplitude of the fluctuations is not constant but is proportional to the trend level. Consequently, variance is not constant along the sequences.