Growth-dependent tree mortality models based on tree rings

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Growth-dependent tree mortality models based on tree rings Christof Bigler and Harald Bugmann

Abstract: Mortality is a crucial element of population dynamics. However, tree mortality is not well understood, particularly at the individual level. The objectives of this study were to (i) determine growth patterns (growth levels and growth trends) over different time windows that can be used to discriminate between dead and living Norway spruce (Picea abies (L.) Karst.) trees, (ii) optimize the selection of growth variables in logistic mortality models, and (iii) assess the impact of competition on recent growth in linear regression models. The logistic mortality model that we developed for mature stands classified an average of nearly 80% of the 119 trees from one site correctly as being dead or alive. While more than 50% of the variability of recent growth of living trees can be attributed to the influence of competition, this percentage was only 25% for standing dead trees. The predictive power of the logistic mortality model was validated successfully at two additional sites, where 29 of 41 (71%) and 34 of 42 (81%) trees were classified correctly, respectively. This supports the generality of the mortality model for Norway spruce in subalpine forests of the Alps. We conclude that growth trends in addition to the commonly used growth level significantly improve the prediction of growth-dependent tree mortality of Norway spruce. Résumé : La mortalité est un élément crucial de la dynamique de population. Cependant, la mort des arbres n’est pas bien comprise, du moins sur une base individuelle. Les objectifs de cette étude consistaient (i) à déterminer les patrons de croissance (les niveaux de croissance et les tendances de croissance) à l’intérieur de différentes fenêtres de temps qui peuvent être utilisées pour discriminer entre les tiges vivantes et mortes d’épicéa commun (Picea abies (L.) Karst.), (ii) à optimiser la sélection des variables de croissance dans les modèles logistique de mortalité et (iii) à évaluer l’impact de la compétition sur la croissance récente dans les modèles de régression linéaire. Le modèle logistique de mortalité que nous avons développé pour des peuplements matures a en moyenne classé correctement comme étant morts ou en vie près de 80 % des 119 arbres sur un site. Alors que plus de 50 % de la variabilité dans la croissance récente des arbres vivants peut être attribuée à l’influence de la compétition, ce pourcentage tombe à 25 % dans le cas des arbres morts encore debout. Le pouvoir de prédiction du modèle logistique de mortalité a été validé avec succès sur deux sites additionnels où respectivement 29 des 41 (71 %) et 34 des 42 (81 %) des arbres ont été classés correctement. Ces résultats confirment la portée générale du modèle de mortalité pour l’épicéa commun dans les forêts subalpines des Alpes. Nous concluons que la tendance de croissance, en plus du niveau de croissance qui est généralement utilisé, améliore significativement la prédiction de la mortalité liée à la croissance chez l’épicéa commun. [Traduit par la Rédaction]

Bigler and Bugmann

Introduction Tree mortality is a crucial element of population dynamics, and it is important for the maintenance of biological and structural diversity in forested ecosystems (Franklin et al. 1987; McComb and Lindenmayer 1999). A complex of biotic and abiotic factors, acting as mortality agents, lead to tree death over the short to long term (Manion 1981; Waring 1987). Often, these agents are effective only after a substantial time lag, which separates the mortality response from its underlying causes (Pedersen 1999). Also, very different mortality agents may be dominant during different developReceived 27 December 2001 Accepted 28 September 2002. Published on the NRC Research Press Web site at http://cjfr.nrc.ca on 17 January 2003. C. Bigler1 and H. Bugmann. Mountain Forest Ecology, Department of Forest Sciences, Swiss Federal Institute of Technology (ETH), ETH-Zentrum, CH-8092 Zurich, Switzerland. 1

Corresponding author (e-mail: [email protected]).

Can. J. For. Res. 33: 210–221 (2003)

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mental stages of trees (Franklin et al. 1987; Kozlowski et al. 1991). Usually, growth-independent mortality due to fire, windthrow, and severe epidemics is distinguished from growth-dependent mortality causes such as senescence, competition, climate effects, and weak pathogens (Jenkins and Pallardy 1995; Pedersen 1998b). In this study, we do not attempt to reveal the entire range of causes leading to individual tree mortality, but we attempt to (i) predict the likelihood of tree death based on growthrelated variables and (ii) determine the effect of competition on current growth as the influence of neighboring trees on mortality risk. Improved predictions of tree mortality allow us to understand and more accurately project forest development. This in turn may be extended to estimate the economic and habitat values of forests (Price 1989; Hunter 1999) or to assess the impact of environmental stresses and disturbances on forests using succession models (e.g., Kienast 1991; Bugmann 1997). However, mortality predictions in dynamic succession models such as gap models (Bugmann 2001; Shugart 1984) rely mainly on theoretical considerations and

doi: 10.1139/X02-180

© 2003 NRC Canada


Bigler and Bugmann

are not based on empirical data (Keane et al. 2001; Hawkes 2000). Recently, Wyckoff and Clark (2002) have demonstrated that the implementation of empirically derived mortality functions resulted in a considerable change of the predictions of a succession model. A variety of empirical approaches have been used for describing and modeling tree mortality at different scales. A common approach is followed in stand-based studies focusing on the relationship between stand density and mortality rates of different tree sizes or age-classes (e.g., Glover and Hool 1979; West 1981; Buford and Hafley 1985; Hamilton 1986, 1990). A distinctly different approach is used in individual-based models, which tend to include characteristics such as tree growth rates, size, and competitive status (e.g., Dursky 1997; Dobbertin and Biging 1998; Zhang et al. 1997; Wyckoff and Clark 2000). In individual-based models, tree death is predicted based on a range of variables that can be classified roughly as (i) size-related variables (e.g., diameter at breast height (DBH) or tree height; Buchman et al. 1983; Buford and Hafley 1985), (ii) growth-related variables (e.g., ring widths or basal area increments; Buchman et al. 1983; Kobe 1996; Wyckoff and Clark 2000), (iii) crown-related variables (e.g., leaf area index or crown defoliation; Crow and Hicks 1990; Monserud and Sterba 1999; Dobbertin and Brang 2001), (iv) ratios of crown-related and growth-related variables (e.g., growth efficiency; Coyea and Margolis 1994), and (v) other variables such as age, competition, or social position (Keister 1972; Burgman et al. 1994). Tree ring data (ring widths or basal area increments) have proven to be useful for addressing mortality issues, since tree rings are integrators of biotic and abiotic influences that reflect the entire growth history of a tree (Fritts 1976; Waring and Pitman 1985; Kozlowski et al. 1991; Schweingruber 1996). Furthermore, tree rings have been shown to be an index of mortality risk (Wyckoff and Clark 2000). Long-term low growth rates are a commonly accepted trait of dying trees, as reflected in the low-growth hypothesis (Manion 1981; Kozlowski et al. 1991; Pedersen 1998a). Competition is one of the major biotic factors that can have a long-term negative impact on growth rate (Peet and Christensen 1987; Biging and Dobbertin 1992; de Luis et al. 1998), and it integrates the effects of other factors such as relative water and nutrient availability. Long-term reduced stem growth may be interpreted as a tree’s response to decreasing resources, since under most circumstances, stem growth has a lower priority in resource allocation compared with, for example, allocation to defenses (Waring and Pitman 1985). Changing resource supply can lead to growth trade-offs (Bloom et al. 1985; Loehle 1988), forcing a tree to invest in growth rather than in defense or vice versa (Coley et al. 1985; Herms and Mattson 1992). Trees can be expected to follow either a survival strategy or a growth strategy (also see Loehle 1988), depending on their genetic background (Monserud and Sterba 1999), competitive position, and site conditions. Notably, slow but stationary growth does not necessarily imply a high risk of tree mortality. Higher growth at the expense of investment in defenses may ultimately increase a tree’s probability of dying. In mortality models that are based on diameter growth, the period under consideration is typically 1–5 years prior to

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death (e.g., Buchman et al. 1983; Hamilton 1986). However, recent theoretical and empirical findings have shown that the consideration of longer-term growth rates may be of high importance for many mortality issues (LeBlanc and Foster 1992; Foster and LeBlanc 1993; Jenkins and Pallardy 1995; Filion et al. 1998; LeBlanc 1998; Pedersen 1998b; He and Alfaro 2000; Ogle et al. 2000). Most previous studies have focused on average growth, and little effort has been devoted to investigating the relationship between changes of growth rates and tree mortality, i.e., the relevance of sustained increases or decreases of growth across time (Kaufmann 1996). Trees can adapt to the prevailing environmental conditions, and low growth alone may be an insufficient predictor of tree mortality. Increasing or decreasing growth rates, however, imply an improvement or, more importantly, a deterioration of the conditions as perceived by a tree. We surmise that a combination of growth level and growth trend variables could improve predictions of tree mortality. Such a combination can be interpreted as the interaction of tree adaptation, tree vitality, and changing environmental conditions. In the present article, we provide insight into growth– mortality relationships and the effect of competition on current growth. Data came from a study of Norway spruce (Picea abies (L.) Karst.) mortality at three sites in the European Alps. We generate different stem growth related variables, covering a variety of time windows, as predictors of mortality of P. abies. An ecologically motivated, systematic selection of combinations of these variables is used to derive logistic regression models. We evaluate the ecological significance of this approach and validate the resulting mortality model with respect to its predictive accuracy. Finally, we quantify the impact of competition on recent growth of living and dead trees.

Material and methods Tree species and study sites Our analysis focuses on Norway spruce, a very widespread, dominant tree species of the subalpine zone in the European Alps. Picea abies is a frost-resistant, shallowrooting species with a broad ecological spectrum (Ellenberg 1996). Three study sites in subalpine forests were selected, located in climatically and geologically different regions of Switzerland (Davos, Bödmeren, and Scatlé) (Table 1). Only mature near-natural or primeval forests with P. abies as the dominant tree species were considered. All three sites are characterized by a similar mortality rate, which can likely be attributed to competition. For Bödmeren and Scatlé, insect outbreaks had resulted in small groups of dead trees, scattered throughout the forest. Details on competition within the samples of the three study sites are given in Table 2. The largest sample of trees was collected in two parallel valleys near Davos, the Dischma and Flüela valleys, located in eastern Switzerland (Table 1). These near-natural forests are dominated by P. abies with small proportions of European larch (Larix decidua Mill.) and Swiss stone pine (Pinus cembra L.). The Davos area is characterized by a continental to suboceanic climate (Landolt et al. 1986). Because of the © 2003 NRC Canada



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Table 1. Site information of the study sites. Davos Coordinates (N, E) Altitude (m a.s.l.) Exposition Slope (%) Geology Annual rainfall (mm) Mean annual temperature (°C) Distance to Dischma Valley (km)

Dischma

Flüela

Bödmeren

Scatlé

46°47′, 09°53′ 1600–2000 SW 60 Silicate (gneiss) 1075 2.4 0

46°49′, 09°52′ 1700–2000 SW 75 Silicate (gneiss) 1075 2.4 3

46°59′, 08°51′ 1500–1600 W 35 Limestone (karst) ≈2500 ≈3–4 80

46°47′, 09°03′ 1500–1700 ENE 60 Silicate (verrucano) 1440 2.2 60

Note: Davos (Dischma and Flüela): Landolt et al. (1986), Krause (1986), Ellenberg (1996), and Bebi (1999), climate station Schatzalp (1868 m above sea level (a.s.l.)); Bödmeren: Bettschart (1994), Hantke (1995), and Kälin and Scagnet (1997), rainfall and temperature estimated for 1400–1650 m a.s.l.; Scatlé: Hartl (1967) and Hillgarter (1971), methods for climate data see Badeck et al. (2001).

Table 2. Competition indices (CIs) and their medians for the three study sites. Median Davos (n = 119)

CI n

4.0

Cl1 = ∑ I ( j) j =1 n

Bödmeren (n = 41) 2.0

Scatlé (n = 42) 4.0

Cl2 = ∑ DBH j

142.0

67.5

116.8

DBH j

57.7

21.1

48.9

5.4

1.5

4.5

2.3

0.4

2.0

j =1 n

Cl3 = ∑ j =1 n

Cl4 = ∑

DISTij DBH2j

2 j =1 DBH i n DBH2j 冫 DBH2i

Cl5 = ∑ j =1

DISTij

Note: I(j), indicator variable for competitor tree j (I(j) = 1 for tree j, I(j) = 0 for other trees); n, number of competitor trees; DBHi, DBH of subject tree i; DBHj, DBH of competitor tree j; DISTij, distance between subject tree i and competitor tree j. References: CI1, Stöckli (1996); CI2 and CI3, Lorimer (1983); CI4 and CI5, Daniels (1976).

very similar conditions in these two valleys, the data from Dischma and Flüela were pooled to form the Davos data set. The two forests in Bödmeren and Scatlé resemble primeval forests, since they are unlikely to have experienced any management during the past few centuries. At the Bödmeren site in central Switzerland, mainly P. abies can be found, at some places together with small amounts of mountain pine (Pinus montana Miller) or birch (Betula spp.) (Bettschart 1994). The suboceanic climate with high rainfall amounts (Table 1) weathered the limestone, leading to a typical karst landscape (Hantke 1995). However, owing to the high daily temperature amplitudes and the low water capacity of the soil, the site is clearly continental in character (Kälin and Scagnet 1997). The second primeval forest is located in Scatlé in eastern Switzerland, growing on the remains of a former landslide. It is composed almost exclusively of P. abies (Hartl 1967; Hillgarter 1971).

Field sampling and laboratory analyses In summer 2000, pairs of standing dead and living P. abies trees were sampled that were similar with respect to DBH, competition, and microsite conditions. Only trees with a DBH greater than 10 cm were considered. Trees were not included if they had been noticeably affected by strong pathogens, avalanches, wind, fire, or human influences. Dead and living trees of most of the pairs were found some metres away from each other; the maximum distance was 40 m. The following selection criteria for the pairs were applied: (i) their DBH must not differ by more than 30% (this difference was less than 15% for most pairs), (ii) both trees must be surrounded by a similar number of trees of comparable DBH, and (iii) both trees must grow on similar microsites (rockiness, trough, slope, aspect). This sampling strategy corresponds to a matched-pairs case-control study, which is widely used in epidemiology (Breslow and Day 1989; Woodward 1999). In Davos, 60 pairs of living and dead trees were examined, 23 pairs in Bödmeren, and 22 pairs in Scatlé, resulting in a total sample size of 210 trees. The Davos data set was used to fit and validate the models; the Bödmeren and Scatlé data sets were used exclusively to validate the models. From each tree, two cores at breast height (1.3 m) were taken with an increment borer. The DBH of each cored tree was measured as well as the distance to and the DBH of neighboring trees larger than 10 cm DBH within a radius of 5 m. The latter data were used for calculating tree competition indices (see below). A frequency distribution of DBH for the three sites is shown in Fig. 1. The sampling strategy used in this study was feasible only because of the relatively good preservation of the dead trees. Some reliably cross-dated standing dead trees had been dead for 30–40 years but were not greatly decayed. However, about 10% of the trees selected initially could not be cored at all because of advanced wood decay. Growth increments were measured with a Lintab 3 measuring system (F. Rinn S.A., Heidelberg, Germany) and the TSAP tree ring program (Rinn 1996). The last tree rings of dead trees were often close to the measurement limit, given by the minimal number of xylem cell rows. Four trees were excluded during the measuring process because of low wood quality of the cores (one tree from the Davos site, one tree from the Bödmeren site, and two trees from the Scatlé site). © 2003 NRC Canada



Bigler and Bugmann Fig. 1. Frequency distribution of diameter at breast height (DBH) of sampled dead and living trees at the sites Davos (n = 119), Bödmeren (n = 41), and Scatlé (n = 42).

At the Bödmeren site, four dead trees with abrupt growth reductions were not considered in the analysis, since they were sampled 20 m away from a group of trees that had been killed by bark beetles some years ago. In the field, we could not find any distinct signs of bark beetles on the stems of the cored trees, but the laboratory analysis suggested that strong pathogen pressure was the main mortality factor for these trees. To obtain a consistent dendrochronological database without any missing rings, single cores of dead and living trees were cross-dated using the averaged time series of 6–12 well-growing, living trees as a reference at each site (Fritts 1976). The two time series of tree ring widths from each tree were averaged. Death year determination was useful for interpreting the data and comparing competition–growth relationships between dead and living trees. The most recent tree ring of most of the cores was bounded by parts of the bark or by a smooth layer of cambial tissue, thus indicating that no tree rings had eroded. A frequency distribution of the year of tree death is shown in Fig. 2. Ring widths were converted to basal area increments (BAI, cm2/year), assuming a circular outline of stem cross sections (cf. Visser 1995). The BAI is commonly assumed to be a more meaningful indicator of tree growth than ring width, since BAI can remain high despite an apparent decline in ring width (West 1980). Derivation of growth-related variables A range of growth variables was derived from BAI, aimed at optimizing the predictive power of the mortality model. The resulting growth variables were classified as (i) growth level variables and (ii) growth trend variables. We define a growth level variable as the average of growth increments over a certain period of time. Growth level variables can be characterized as the “classical” way to describe growth patterns of dead and living trees (e.g., Jenkins and Pallardy 1995; Pedersen 1998b). In tree mortality studies, growth level variables are commonly referred to as “recent growth” (e.g., Wyckoff and Clark 2000). To derive a first set of growth level variables, we calculated the averages of BAI (cm2/year) over the tree’s last 3, 5, or 7

213 Fig. 2. Frequency distribution of year of death. Only reliably dated trees that died between 1967 and 1999 are shown (Davos data set, n = 33).

years (BAI3, BAI5, BAI7), with 1999 being the last year for the living trees. For the second set of growth level variables, the variables of the first group were transformed with the natural logarithm (log BAI3, log BAI5, log BAI7). The logtransformed function was used to lower the weight given to very high values of BAI. We define a growth trend variable as the change of growth increments over a certain period of time. The motivation for this variable class resulted from two observations: (i) some trees died after a fast growth collapse, although growth level was still relatively high, and (ii) some trees were able to survive long periods of low, but stationary growth. The slopes of local linear regressions fitted over the last 5, 10, 15,..., 40 years of BAI (locreg5,..., locreg40) were used to characterize the growth trend, with BAI as the dependent variable. The fit of the regression lines was verified visually and was found to be satisfactory in most cases. Fitting of logistic mortality models The prediction of a binary response, such as the tree status “dead” or “alive” from continuous growth variables (independent variables), implies a logistic regression model (Hosmer and Lemeshow 1989; McCullagh and Nelder 1989; Collett 1991). A distinct advantage of logistic regression models is that there are well-established procedures for model fitting and model testing (Collett 1991). The general expression for logistic regression is as follows: [1]

Pr(Y = 1| X) =

eX␤ 1 + eX␤

In our study, Pr(Y = 1|X) is the survival probability of an individual tree expressed as a function of a matrix X of independent variables; Y is the dependent variable (Y = 1 if the tree is alive, Y = 0 if the tree is dead) and ␤ is a vector containing the regression coefficients. All analyses were performed using the “R” software (Ihaka and Gentleman 1996). A series of models was fitted using the data from the Davos site only, with 59 dead trees and 60 living trees. The data from the other two sites (Table 1) were reserved for validation purposes. A two-stage, stepwise variable selection strategy was applied, and models were fitted by maximum log-likelihood © 2003 NRC Canada



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estimation using the Akaike information criterion (AIC).2 The significance level of deviance differences served as the criterion for variable selection.3 The reduction of deviance in a stepwise process can be assumed to be χ2 distributed, even for ungrouped data (McCullagh and Nelder 1989). The starting point for each stepwise regression was the null model, where only the intercept was included as an independent variable. In a first stage of analysis, the most significant variable in each of the three variable groups was determined using a stepwise selection process (eq. 1). Thus, a total of 14 growth variables from the three groups were screened (three nontransformed growth level variables, three logtransformed growth level variables, and eight growth trend variables). The DBH was not considered as a further independent variable, since pairs of dead and living trees with very similar DBH had been sampled. In a second stage, the best linear combination using the three variables selected at the first stage was sought via another logistic regression model. The three selected variables were added stepwise to the null model (eq. 1). The resulting main model, which included significant variables only, was used for all further analyses. The simultaneous inclusion of a nontransformed and a log-transformed growth level variable was disallowed. Interpretation of the mortality model Since similar proportions of dead and living trees in each DBH class were sampled, the derived mortality models cannot be used to calculate the population-level mortality risk but only the relative mortality risk of an individual tree given the observed values of the independent variables (Woodward 1999). This restriction has implications for both application and interpretation of the models. The application of our mortality model to a population sample is appropriate only when odds ratios, an approximation of the relative mortality risk between two trees, are used (Hosmer and Lemeshow 1989; Woodward 1999). The odds can be derived from eq. 1 with odds = p$ /(1 − p$) = eX␤ where p$ is the estimated survival probability Pr(Y = 1|X). The odds that a given tree r will die relative to the odds that another tree s will die can be expressed as the odds ratio: [2]

 k  $ = exp  ∑ β$ i(xri − x si) Ψ  i = 0 

where k is the number of independent variables, β$ i is the estimated coefficient of variable i in eq. 1, and (xri – xsi) is the difference of variable i between tree r and tree s. If all variables except i are held constant in eq. 2, a 95% confidence interval can be approximated by exp[β$ i(xri − x si) ± 1.96 (xri − x si)SE$ (β i)], where SE$ (β i) is the estimated standard error of βi (Woodward 1999).

Validation of the mortality model Prediction accuracy is a useful criterion for comparing logistic models of tree mortality (Avila and Burkhart 1992; Dursky 1997; Dobbertin and Brang 2001). Comparing the observed with the predicted state of each individual tree allows us to develop a semimechanistic understanding of mortality processes at the scale of the individual. This is a distinct advantage over population-based mortality models, where measured and predicted mortality rates can only be compared at the stand scale, e.g., with respect to DBH classes or some other aggregated variable. A goodness-of-fit evaluation of a logistic model requires that the variables be grouped, i.e., each independent variable has to be divided into intervals (Collett 1991). This was not possible in our case because of the relatively small size of the data sets. Therefore, goodness-of-fit could not be evaluated by applying a χ2 test to the deviance. Instead, goodnessof-fit was tested via validation (Hosmer and Lemeshow 1989) and checks of model diagnostics (Collett 1991). Validation is especially important as an assessment tool when the fitted model is used to predict the behavior of independent subjects. Two validation methods were applied to determine the predictive accuracy of the model. Note that the validation methods to be described are appropriate only because the models were fitted and validated with similar proportions of dead and living trees. For both methods, a deterministic approach was used, where the threshold for the response in the logistic model was fixed at 0.5. Thus, a tree was predicted to be dead when Pr(Y = 1|X) ≤ 0.5 and to be alive when Pr(Y = 1|X) > 0.5. The two methods were as follows. (1) An internal validation was performed where only data from the Davos site were used. A model was fitted with a random sample containing 50% of the trees from Davos and was then applied to the remaining 50% of the data to predict the status (dead or alive) of these trees. This cross-validation with random subsampling, i.e., sampling of a fitting data set and validating the model with the remaining data, was done with R = 200 resamplings (Fielding and Bell 1997; Manly 1997; Chernick 1999). (2) An external validation was performed by fitting a model for the Davos site using the full data set and testing this model against the data sets from the Bödmeren and Scatlé sites (cf. Fielding and Bell 1997). This second validation method allowed us to assess the extent to which the model can be generalized to Norway spruce populations growing under different environmental conditions (cf. Table 1). Relationship between growth and competitive status We used a regression modeling approach to estimate the proportion of variability in recent tree growth that is attributable to competition. Distance-independent and distancedependent, size-related competition indices (CIs) were

2

The AIC (Venables and Ripley 1999) is used to compare models with a different number of independent variables. The statistical computing software “R” defines AIC = –2(maximized log likelihood) + 2p, where p is the number of estimated parameters. Generally, the model with the smallest AIC is selected. 3 The deviance D is calculated in the fitting process of generalized linear models (e.g., logistic regression models) and is derived from the maximized likelihood L, which is a measure of how likely a particular model is, given the observed data (Collett 1991): D = − 2{log L$ − log L$ F}. It is a measure of the difference between a particular model of interest (likelihood L) and the full model (likelihood LF), which fits the data perfectly. The deviance difference is the difference between the deviances of two models. © 2003 NRC Canada



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Fig. 3. Individual growth curves and medians of dead trees (n = 59) and living trees (n = 60) of the Davos data set.

Fig. 4. Comparisons of two sampled pairs of a living tree and a dead tree from the Davos site.

computed for all three sites (Table 2). Only the Davos site was used for relating recent tree growth to competitive status. To select the most significant CI, only living trees from the Davos site were considered. Two different procedures were applied in the process of model selection. First, a stepwise procedure with the AIC as the selection criterion was used, relating the five CIs from Table 2 linearly to log BAI3: [3]

log BAI3 = b0 + b1CI

To avoid heteroscedasticity, we used the log transform of BAI3 in the model. Second, a separate linear model (eq. 3) was developed for each of the five CIs in Table 2. The adjusted R2 was used as a criterion for model comparison (Zar 1999). The final variable selected was used to calculate a linear model for reliably dated dead trees only that had died after 1985.

Results Fitting and interpretation of the mortality model Many dead trees at the three study sites showed growth decreases prior to death that lasted up to several decades (Fig. 3). Notably, long periods of relatively low, but stationary growth did not necessarily imply an increased mortality

probability (e.g., see the period from 1770 to 1820 of the living tree DAB203 in Fig. 4a). Rather, strongly negative growth trends appear to imply a high mortality probability (e.g., see the dead tree DAB033 in Fig. 4b). In the first stage of the stepwise regression analysis for the Davos data, BAI3 and log BAI3 best reflected the status of the trees within the groups of growth level variables (Table 3). Within the growth trend variables group, locreg25 had the highest significance and the lowest AIC (Table 3). In the second stage, the above three variables were included in a stepwise logistic regression. In the resulting main model, only the growth level variable log BAI3 and the growth trend variable locreg25 were significant (Table 4). The internal validation procedure, where a model was fitted with 50% of the Davos data set and applied to the remaining 50% of the data set (described below in more detail for the main model), was used here additionally as a tool to verify the variable selection. Averaged misclassification rates for all combinations of one growth trend variable (locreg) with one log-transformed growth level variable (log BAI) are given in Table 5. These results confirmed the selection of the main model that was developed in the two-stage stepwise regression. Most of the dead trees had very low BAI during the last 3 years, whereas the living trees were characterized by larger © 2003 NRC Canada



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Table 3. Coefficients, p values, and AIC values of logistic regressions with one independent variable for the Davos data set (n = 119).

BAI3 BAI5 BAI7 log BAI3 log BAI5 log BAI7 locreg5 locreg10 locreg15 locreg20 locreg25 locreg30 locreg35 locreg40

Coefficient

p >|z|

0.224 0.164 0.134 0.962 0.851 0.793 1.153 1.731 2.290 3.859 5.332 5.326 5.327 6.488

4.76 2.09 4.30 1.51 7.32 2.11 1.36 7.66 1.98 2.45 1.06 1.86 4.17 3.20

× × × × × × × × × × × × × ×

AIC 10–5*** 10–4*** 10–4*** 10–6*** 10–6*** 10–5*** 10–3** 10–4*** 10–3** 10–4*** 10–4*** 10–4*** 10–4*** 10–4***

137.0 145.5 149.9 136.0 142.2 145.8 152.9 151.5 154.3 145.3 138.1 142.7 142.4 139.5

Note: Values for the intercepts are not shown. **, p < 0.01; ***, p < 0.001.

Table 4. Mortality model fitted for the Davos data set (n = 119).

Coefficient SE z p >|z|

Intercept

log BAI3

locreg25

–0.568 0.316 –1.800 7.18 × 10–2

0.898 0.220 4.082 4.46 × 10–5***

4.507 1.333 3.381 7.22 × 10–4***

Note: The coefficients, standard errors, and z and p values (***, p < 0.001) of the logistic regression with two independent variables are shown.

growth increments (Fig. 5a). The distribution of the growth trend is shifted towards positive or slightly negative values for the living trees, while most dead trees had negative growth trends over the last 25 years prior to death (Fig. 5b). A scatterplot of log BAI3 versus locreg25 (Fig. 6) further illustrates the discriminatory power of the growth trend variable locreg25 for dead trees with intermediate growth level (–0.5 < log BAI3 < 3) but a clearly negative growth trend (locreg25 < –0.2). Growth level as the only independent variable would fail to classify these trees correctly. Furthermore, trees exhibiting a very low growth level (log BAI3 < 0) always showed stationary growth, with locreg25 close to zero. The response surface of the main model (Fig. 7) shows that trees with a distinct negative growth trend and an intermediate growth level are subject to an increased risk of dying. Trees with a low growth level (log BAI3 < 0) combined with stationary growth over the last 25 years (cf. Fig. 6) had a similar risk profile. Note that trees with a low growth level (log BAI3 < 0) combined with a distinctly positive or negative growth trend do not exist in our sample (cf. Fig. 6). According to our model, trees with a high past growth level (log BAI3 > 2) and a positive or slightly negative growth trend over the past 25 years (locreg25 > –0.2) are now more likely to be alive than dead. The use of the odds ratio for determining the relative mortality risk is exemplified here for two trees. Let us assume

Table 5. Average percentage of misclassified trees for the internal validation of the Davos data set (n = 119, R = 200 simulations) using various combinations of growth level variables (log BAI) and growth trend variables (locreg).

locreg5 locreg10 locreg15 locreg20 locreg25 locreg30 locreg35 locreg40

log BAI3

log BAI5

log BAI7

21.58 21.82 23.77 21.48 20.58* 21.55 22.59 22.47

21.53 22.75 24.99 23.06 21.84 22.46 24.16 24.44

22.67 22.13 26.36 24.19 21.49 22.88 24.9 25.56

*Main model.

that the first tree has a growth level variable of log BAI3 = 1 and the second tree is characterized by log BAI3 = 1.5. Assuming that both trees have a stationary growth (locreg25 = 0), we find according to eq. 2 and the coefficients in Table 4 that the odds ratio for these two trees is 1.57 with a 95% confidence interval of (1.26, 1.94), i.e., the first tree, which has a lower growth level, is nearly 1.6 times more likely to die compared with the second tree. Validation The internal validation, based on a random split of the Davos data set, revealed that 19.1% of the dead trees were misclassified as being alive and 21.7% of the living trees were predicted to be dead (Table 6). Overall, 20.6% of the trees were misclassified. The use of only one growth variable would, for the same validation procedure, result in an increase of the misclassification rate to 27.3% if log BAI3 was the only independent variable and to 31.6% if locreg25 was the only independent variable. In the external validation (Table 6), 22.2% of the dead trees at the Bödmeren site were misclassified, whereas 34.8% of the living trees were predicted to be dead in the model. At this site, 70.7% of all trees were assigned to the correct group. For Scatlé, 25.0% of the dead trees were misclassified, while 13.6% of the living trees were assigned to the wrong group. Overall, 81.0% of all trees were classified correctly at Scatlé. Relationship between growth and competitive status In the stepwise regression using all five CIs (cf. Table 2), as well as in the comparison of the separate models, CI5 turned out to best reflect recent growth (log BAI3) of living trees with respect to both selection criteria, the AIC for the stepwise regression and the adjusted R2 for the separate models. The adjusted R2 values for the five CIs in the linear models for living trees were 25.9% (CI1), 29.9% (CI2), 23.7% (CI3), 47.7% (CI4), and 51.5% (CI5). For the living trees, 52.3% of the variability of log BAI3 could be attributed to competition using CI5 as an independent variable (Table 7), while for the dead trees that died between 1986 and 1999, this amounted to 24.9% (Table 7). © 2003 NRC Canada



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Fig. 5. Frequency distribution of (a) growth level variable log BAI3 and (b) growth trend variable locreg25 for dead and living trees (Davos data set, n = 119).

Fig. 6. Scatterplot of log BAI3 (growth level) and locreg25 (growth trend) grouped by dead or alive (Davos data set, n = 119). Many dead trees had a low to very low growth level (log BAI3 < 0) combined with a growth trend of around 0 or a growth level of greater than –0.5 in combination with a growth trend of less than –0.2.

Fig. 7. Response surface of logistic mortality model with fitted survival probabilities Pr(Y = 1|X). High values of Pr(Y = 1|X) indicate low mortality probabilities.

Table 6. Validation of the mortality model with log BAI3 and locreg25 as independent variables (0 = dead, 1 = living) for internal validation with random subsampling (data from Davos only, n = 119, R = 200 simulations) and external validation with data from Bödmeren (n = 41) and Scatlé (n = 42).

The relationship between competition and recent growth is shown in Fig. 8.

Discussion Growth levels, growth trends, and tree mortality Norway spruce trees are able to endure low, stationary growth for many decades, in our case up to 150 years (cf. Kaufmann 1996). Most of the dead Norway spruce trees in our sample were characterized by a continuously decreasing growth trend during the last decades prior to death (Figs. 3 and 4). This growth pattern is likely attributable to competition or to decreasing resources in general. Some trees in our sample showed strong, stairstep-like growth reductions, which often resulted in rapid death. Intense pathogen attacks

Internal Davos External Bödmeren Scatlé

Predicted = 0

Predicted = 1

Observed = 0 Observed = 1

23.85±2.50 6.63±2.63

5.62±2.77 23.91±2.39

Observed Observed Observed Observed

14 8 15 3

4 15 5 19

= = = =

0 1 0 1

Note: Numbers of observed and predicted (mean ± SD) trees are shown.

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Can. J. For. Res. Vol. 33, 2003 Table 7. Linear regression models relating recent growth (log BAI3) and competition (CI5) of living trees and trees that died between 1986 and 1999 (Davos data set). Coefficient (p) Model Living trees (n = 60) Trees that died between 1986 and 1999 (n = 19)

Intercept

R2 (%)

CI5 –16

2.436 (