Introduction

Hinoki cypress (Chamaecyparis obtusa) is one of the most important commercial species in Japan and its afforestation area accounts for 25 % of all plantation forest area in Japan [1]. Historically, hinoki has been used as building materials because it shows remarkable mechanical properties and durability [2]. For instance, it is well known that Horyuji, the world’s oldest wooden architecture, was built more than 1,300 years ago and hinoki was mainly used as its construction member [3]. However, the wood resources have shifted from natural forests to plantation forests recently, and thus the wood properties of hinoki from plantation forests would be different from the past situation. The correct recognition of the wood properties in current hinoki resources should be necessary for the adequate forest management and utilization.

Numerous studies have reported the variation of wood properties in a number of species [4]. However, many of these reports are not quantitative examinations but qualitative examinations, except for the genetic variation which provides various genetic parameters. If the variation patterns of wood properties can be formulated quantitatively, one can interpolate the data or extrapolate the future situations based on the formulae.

There may be no discussed matter with regard to wood quality variation than relationship between growth rate and wood properties [4]. Especially, the effects of growth rate on wood density have been studied intensively. In hinoki cypress, Hirai reported that high growth rate would produce wood having low basic density [5]. Fujiwara et al. [6] examined the variation of basic density of hinoki which was obtained from nine test stands, and reported that there were no significant differences between thinned stands and un-thinned stands. Growth rate affects wood properties at different tree ages, and ignoring this effect could result in growth rate being incorrectly identified as the cause of differing wood properties [7, 8].

Sequences of yearly measurements of wood characteristics are considered as longitudinal (time series) data, and the analysis of longitudinal data is able to evaluate the age effect on variation of these characteristics. In this case, these data are repeated measures made of the same characteristic on the same observational unit (tree, disk, and ring), and such data generally present temporal autocorrelation, heteroscedasticity, and nonstationarity of the mean [9–11]. The mixed-effects analysis technique is frequently used for such grouped data including longitudinal data, repeated measures data, and multilevel data [12, 13]. Correlations among observations made on the same subject or experimental unit can be modeled using random effects, random regression coefficients, and through the specification of a covariance structure. Recent studies have developed models predicting variation in wood properties within a stem or cross section using the mixed-effects analysis for many species, such as pine and spruce [14–21].

Although the general variation pattern of basic density within stem is well known in hinoki, there is little information available about the effects of growth rate on the variation pattern of basic density. The objective of this study is to evaluate the effect of growth rate on variation pattern of basic density within stem. The linear mixed-effects model was used to explain the variation pattern of basic density quantitatively, and the effects of growth rate on its variation were also assessed on the basis of statistical manner.

Materials and methods

Sample materials were obtained from 50-year-old hinoki cypress stands in Tottori University Forest located in Maniwa, Okayama (35°16′ N, 133°36′ E; approximately 540 m elevation). The annual average precipitation and temperature from 2002 to 2012 in the research forest were 1989 mm and 11.6 °C, respectively. Hinoki cypress seedlings were planted in 1962 at an initial plant density of 3333 trees/ha. The test stand consisted of four 10-m square plots, which were two fast growth plots and two slow growth plots. They were adjacent to each other. The information of the test stand is summarized in Table 1. Although the detailed archive of the stand is unclear, the fast growth plots had been thinned several times. On the other hand, the slow growth plots have never been thinned previously. As a result, the stand density between them was quite different.

Table 1 Characteristics of the plots and sample trees in the test stand
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From each plot, four or five, totally nineteen sample trees, were felled in 2012. Height and height to the base of live crown (BLC) of the sample trees were measured. The average height, DBH and BLC of sample trees were 15.9 m, 20.7 cm and 9.1 m, respectively. The height ranged from 10.2 to 18.6 m. DBH ranged from 11.9 to 30.9 cm. Analysis of variance confirmed the significant difference of the DBH among plots (p = 0.006; data not shown).

A 5-cm-thick, knot-free sample disks were cut from each sample tree at height positions of 2, 4 m, and then 4 m intervals until 16 m position. A 3-cm-thick radial strip was cut from each disk and ring width was measured from pith to outward in every five rings. Radial diameter of sample strips ranged from 1.4 to 12.1 cm. The measurements were carried out for two radial directions and ring width was expressed as the mean of both values. After that, the strips were cut into every 5 rings for basic density analysis. Water displace method was used to determine the basic density. The basic density was also determined as the mean of both radial directions. Radial and longitudinal variations of basic density for each plot are presented in Fig. 1.

Fig. 1
figure 1

Radial and longitudinal variations of basic density from different growth rate plots

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Model development

The start model

The mixed model technique [12, 13] was used for modeling the effects of growth rate on the intra-tree variation in basic density. Figure 1 indicates that basic density can be represented by the quadratic function of cambial age, i.e., ring number from pith. On the start model, the fixed effects consisted of population mean and effects of plots determined by growth rate. The random effects consisted of effect of the tree level and vertical stem position level. The model expressed as

$$ \begin{gathered} BD_{hijk} = \left( {\beta_{0} + \beta_{0h} + b_{0i} + b_{0i,j} } \right) + \left( {\beta_{1} + \beta_{1h} + b_{1i} + b_{1i,j} } \right)\,AGE_{k} \\ + \left( {\beta_{2} + \beta_{2h} + b_{2i} + b_{2i,j} } \right)\,AGE_{k}^{2} + \varepsilon_{hijk} , \\ {\boldsymbol{b}}_{i} = \left[ {\begin{array}{*{20}c} {b_{0i} } \\ {b_{1i} } \\ {b_{2i} } \\ \end{array} } \right]\sim N\left( {{\mathbf{0}},\,\,{\boldsymbol{\varPsi}}_{1} } \right),\,\,\,\,\,\,\,{\boldsymbol{b}}_{i,j} = \left[ {\begin{array}{*{20}c} {b_{0i,j} } \\ {b_{1i,j} } \\ {b_{2i,j} } \\ \end{array} } \right]\sim N\left( {{\mathbf{0}},\,\,{\boldsymbol{\varPsi}}_{2} } \right),\,\,\,\,\,\,\,\varepsilon_{hijk} \sim N\left( {0,\,\,\sigma^{2} } \right)\,,\, \\ \end{gathered} $$
(1)

where BD hijk was the basic density of the kth cambial age of the jth vertical stem position in the ith tree in the hth plot; β 0, β 1 and β 2 were the population mean of the basic density; β 0h , β 1h and β 2h were the fixed-effect parameters of hth plot; b i was the tree-level random-effects vector; b i, j was the position-level random-effects vector; ε hijk was the within-group error. The b i were assumed to be independent for different i, the b i, j were assumed to be independent for different i, j and independent of the b i , and the ε hijk were assumed to be independent for different i, j, k and independent of the random effects. The large number of parameters in Eq. (1) makes the optimization of the profiled log-restricted-likelihood quite difficult and unstable [12]. To make the optimization more stable during this model building phase, we simplify Eq. (1) by assuming Ψ 1 and Ψ 2 as diagonal matrices. The models in this article were fitted using the nlme package in R version 3.0.0 [22].

Selecting the fixed-effects structure

First, we evaluate whether the quadratic function of cambial age is adequate to describe the observed data and also test whether the growth rate has significant effect on the intra-tree variation on the basic density. The result of fitting indicated that both first- and second-order terms of age were highly significant (p < 0.001), and thus, the quadratic function of cambial age well describes the radial variation of basic density.

There were no clear effects in the terms of plot (p = 0.087), plot-AGE interaction (p = 0.288), and plot-AGE 2 interaction (p = 0.079). The results indicate that growth rate does not affect the variation pattern of basic density within the stem in hinoki cypress. Consequently, the fixed effect of the Eq. (1) could be simplified to the structure without plot effects.

Determining the variance–covariance structure of random effects

The age quadratic model without plot effects (model 1.1) was examined to determine the variance–covariance structure of random effects. The pair plot for the estimated random effects in the tree level is shown in Fig. 2. There was a weak positive correlation between the AGE and AGE 2 random effects, but no substantial correlation between either of these random effects and the intercept random effects. A blocked diagonal matrix can be used to represent such covariance structure [12], with the intercept random effect corresponding to one block and the AGE and AGE 2 random effects corresponding to another block. There was no remarkable correlation found among the random effects in the vertical stem position level.

Fig. 2
figure 2

Scatter plot of the estimated random effects in tree level from model 1.1

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Several models with different structures for the variance–covariance matrices of the estimated random effects were fitted and compared using the log-likelihood ratio test (LRT), the Akaike’s information criterion (AIC), and the Schwarz’s Bayesian information criterion (BIC). According to the fit statistics presented on Table 2, model 1.2 with the blocked diagonal matrix in the tree level was the best of the variance–covariance structure of random effects. Hence, the variance–covariance structure could be represented as

$$ {\text{Var}}\,({\boldsymbol{b}}_{i} ) = {\boldsymbol{\varPsi}}_{1} = \left[ {\begin{array}{*{20}c} {\sigma_{00} } & 0 & 0 \\ 0 & {\sigma_{11} } & {\sigma_{12} } \\ 0 & {\sigma_{21} } & {\sigma_{22} } \\ \end{array} } \right],\,\,\,\,{\text{Var}}\,({\boldsymbol{b}}_{i,j} ) = {\boldsymbol{\varPsi}}_{2} \, = \left[ {\begin{array}{*{20}c} {\sigma_{00} } & 0 & 0 \\ 0 & {\sigma_{11} } & 0 \\ 0 & 0 & {\sigma_{22} } \\ \end{array} } \right] $$
(2)
Table 2 Comparisons of the model performance with different variance–covariance structures for the random effects
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Determining the structure of the within-group error

The within-group error, ε hijk , were assumed to be independent for different i, j, k and independent of the random effects in model 1.2. The plots of residuals against the fitted values and other candidate variance covariates are useful for investigating within-group heteroscedasticity [12]. In this case, the cambial age is a natural candidate for the variance covariate. Figure 3 shows the plots of residuals versus AGE, indicating that the residuals decrease with AGE. Thus, we proceed by specifying the variance structure of the within-group error to account for heteroscedasticity. We use a conditional error variance [12, 23], where we assume

$$ {\text{Var}}\left( {\varepsilon_{ijk} \,|\,{\boldsymbol{b}}_{i,j} ,\,{\boldsymbol{b}}_{ij} } \right) = \sigma^{2} G^{2} \left( {\mu_{ijk} ,\,\upsilon_{ijk} ,\,\delta } \right), $$
(3)

where is μ ijk  = E [y ijk | b i,j , b ij ], υ ijk is a vector of variance covariates, δ is a vector of variance parameters and G(.) is the variance function. A number of variance function can be used in the nlme package, the following two variance structures were tested in this study. The first is the power model which is given as

$$ {\text{Var}}\left( {\varepsilon_{ijk} } \right) = \sigma^{2} \left| {\upsilon_{ijk} } \right|^{2\delta } ,\,\,\,\,G\left( {\upsilon_{ijk} ,\,\,\delta } \right) = \left| {\upsilon_{ijk} } \right|^{\delta } $$
(4)

The second is the exponential model which can be represented as

$$ {\text{Var}}\left( {\varepsilon_{ijk} } \right) = \sigma^{2} \exp \left( {2\delta \upsilon_{ijk} } \right),\,\,\,\,G\left( {\upsilon_{ijk} ,\,\,\delta } \right) = \exp \left( {\delta \upsilon_{ijk} } \right) $$
(5)

The parameter δ is unrestricted, thus, the both variance structures can model a case where the variance increases or decreases with the variance covariate. There were significant increases in the log-restricted-likelihood, as evidenced by the large value of the LRT, indicating that addition of the variance function to the model significantly improves on model 1.2 (Table 3). Based on AIC and BIC, power function of age (model 1.2.1) will be used to represent variance structure of the within-group error.

Fig. 3
figure 3

Plot of residuals versus cambial age for the model 1.2 having homoscedastic within-group errors

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Table 3 Comparisons of the model performance with different within-group error variance structures
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Since the age trend of basic density could be considered as time series data, we need to pay attention to the temporal autocorrelation. Serial correlation structures are used to model dependence in time series data [24]. The empirical autocorrelation function provides a useful tool for investigating serial correlation in time series data. A plot of the estimated autocorrelation coefficients against lags for model 1.2.1 indicates that autocorrelations were significant even at cambial age lag 3 and 4 (data not shown). The following second-order moving average MA(2) was the best of the candidate correlation structures based on the statistics presented in Table 4.

$$ \varepsilon_{t} = \sum\limits_{j = 1}^{2} {\theta_{j} a_{t - j} + a_{t} } , $$
(6)

where ε t is the current within-subject error term, θ j are the moving-average parameters (j = 1, 2) and a t is a homoscedastic noise term centered at 0 (E[a t ] = 0). The estimated normalized autocorrelation structure for model 1.2.1.4 residuals was

$$ \begin{gathered} \hat{\rho } = \left[ {\hat{\rho }(1),\,\,\hat{\rho }(2),\,\,\hat{\rho }(3),\,\,\hat{\rho }(4),\,\,\hat{\rho }(5),\,\,\hat{\rho }(6),\hat{\rho }(7),\hat{\rho }(8)} \right]^{\text{T}} \\ = \left[ { - 0.198,\,\, - 0.087,\,\, - 0.112,\,\, - 0.194,\,\, - 0.098,\,\, - 0.042,\, - 0.097} \right]^{\text{T}} , \\ \end{gathered} $$
(7)

where \( \hat{\rho }(l) \) is the empirical autocorrelation calculated at lag l and the estimated parameters for the MA(2) model were θ 1 = 0.348, and θ 2 = 0.256.

Table 4 Comparisons of the model performance with different within-groups correlation structures
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We selected the MA(2) model that is more adequate to represent the autocorrelation at small lags even though the second-order autoregressive model AR(2) model and the MA(2) model indicated almost the same AIC, BIC and log-restricted-likelihood. This is because the empirical autocorrelation at larger lags tends to be less reliable due to the estimation by fewer residuals pairs [12]. In our data, there were differences of the ring number from pith among the disks. For example, a total ring number of a disk was 42 while that of another was only 37, such disproportion of the data also might be possibly the cause of the autocorrelation at larger lags.

Final model

We have tested the models having various mean and covariance structures, and have finally obtained the following model to describe the intra-tree variation in basic density.

$$ \begin{gathered} BD_{ijk} = \left( {\beta_{0} + b_{0i} + b_{0i,j} } \right) + \left( {\beta_{1} + b_{1i} + b_{1i,j} } \right)\,AGE_{k} + \left( {\beta_{2} + b_{2i} + b_{2i,j} } \right)\,AGE_{k}^{2} + \varepsilon_{ijk} , \\ {\boldsymbol{b}}_{i} \sim N\left( {{\mathbf{0}},\,\,\left( {\begin{array}{*{20}c} {\sigma_{00} } & 0 & 0 \\ 0 & {\sigma_{11} } & {\sigma_{12} } \\ 0 & {\sigma_{21} } & {\sigma_{22} } \\ \end{array} } \right)} \right)\,\,,\,\,{\boldsymbol{b}}_{i,j} \sim N\left( {{\mathbf{0}},\,\,\left( {\begin{array}{*{20}c} {\sigma_{00} } & 0 & 0 \\ 0 & {\sigma_{11} } & 0 \\ 0 & 0 & {\sigma_{22} } \\ \end{array} } \right)} \right), \\ \varepsilon_{ijk} \sim N\left( {0,\,\,\,\,\sigma^{2} G_{ijk}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {\upsilon_{ijk} ,\,\,\delta } \right)\,\,H_{ijk} \left( {\varphi ,\,\,\theta } \right)\,\,G_{ijk}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {\upsilon_{ijk} ,\,\,\delta } \right)} \right), \\ G_{ijk} \left( {\upsilon_{ijk} ,\,\,\delta } \right) = \left| {AGE} \right|^{\delta } ,\,\,\,\,\,\,H_{ijk} \left( {\varphi ,\,\,\theta } \right) = {\text{ARMA}}\left( {0,\,\,2} \right), \\ \end{gathered} $$
(8)

where Η ijk (ϕ, θ) is the serial correlation function and ARMA is mixed autoregressive-moving average model. The remaining elements of the model have been described previously. Parameter estimates, corresponding standard errors and p values for fixed effects of model 1.2.1.4 [Eq. (8)] are given in Table 5.

Table 5 The estimated fixed effects parameters of basic density with the quadratic function of cambial age
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A final assessment of the adequacy of model 1.2.1.4 [eq. (8)] is given by the plot of the augmented prediction [12, 19] for a chosen tree (Fig. 4). Observed values of basic density, prediction by estimated fixed-effects parameters of model 1.2.1.4, which random effects were excluded, and that of containing random effects. The predictions containing the random effects follow the observed values closely, indicating that the final model explains the density variation data well.

Fig. 4
figure 4

Population prediction (setting all random effects to 0), subjects specified prediction (containing random effects) from the final model 1.2.1.4 and observed values of basic density versus cambial age. The solid lines, the dots line and filled circles indicate the population prediction, subjects specified prediction and observed values, respectively

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Discussion

This study applied the linear mixed model to evaluate the effects of growth rate on the intra-tree variation in the basic density. The basic density decreased from pith to outward and the unique pattern was found in any height position (Fig. 1). The results were consistent with the previous reports in hinoki cypress [25, 26]. From Fig. 4, the final model with the quadratic function of cambial age [Eq. (8)] explained the variation of basic density successfully. The density variation pattern found in hinoki is different from those of general coniferous tree, such as pine and larch, where the wood density increases with age [4, 7]. Decrease of wood density with age can be also found in sugi (Cryptomeria japonica) [26, 27].

There were no significant effects in the terms of plot, plot-AGE interaction, and plot-AGE 2 interaction. The results indicate that growth rate does not affect the variation pattern of basic density within the stem in hinoki cypress. There were few studies about the effects of growth rate on the variation pattern of basic density in hinoki cypress in spite of its importance for forest management and utilization. Fujiwara et al. [6] reported the similar results, but Hirai’s report [5] is inconsistent with our result. Growth rate affects wood density at different tree ages, and the inconsistency of these results would be due to the ignorance of age effects [4, 8]. The modeling approach employed in this study can assess the effects of growth rate on the variation of wood density quantitatively considering the age effects. The effect of growth rate on wood density varies greatly among species [4], and it can be confirmed that hinoki shows the similar tendency as the hard pines, Douglas-fir, and larch species.

The defining characteristics of mixed-effects models are that they are applied to data where the observations are grouped according to one or more levels of experimental units and that they incorporate both fixed-effects terms and random-effects term [12]. Moreover, they present an inherent flexibility that allows for development of a unique variance–covariance structure alleviating the problems of nonconstant variance and autocorrelation among the repeated measurements. The final model has two levels of mixed effects with random effects at the tree and vertical position levels. The random-effects estimates were found to be larger at the tree level than the vertical position level (data not shown). This means that the basic density values at each height position are relatively consistent, but the variation among trees is more noticeable, due to the unique patterns of basic density corresponding to height level.

In this study, the variation of wood density within stem was modeled using the linear mixed-effects model. The methodology presented in this paper can easily extend for other wood properties [16, 18–20, 28]. In this case, it should be noted that the age trend of each traits is quite different. For instance, tensile strength in hinoki increases from pith to outward [29], and thus, the fitting model function should be different from that of wood density. We used the polynomial model that is linear in the parameters. By increasing the order of a polynomial model, one can get increasingly accurate approximations to the true regression function within the observed range of the data [12]. These empirical models are based only on the observed relationship between the response and the covariates and do not include any theoretical considerations about the underlying mechanism producing the data. As the next step, it would be useful to apply the nonlinear model that is based on a model for the mechanism producing the response, and that also provides more reliable predictions for the response variable outside the observed range of the data [16, 19, 20].