An application of mixed-effects model to evaluate the role of age and size on radial variation in wood specific gravity in teak (Tectona grandis)

Abstract

To test whether radial variation of wood specific gravity (WSG) is controlled by tree age or tree size in teak (Tectona grandis L.f) plantation trees, opposing different-length pith-to-bark strips which represents the differential lateral growth rate was compared using mixed-effects model which considers the heterogeneity of variances and dependency in the data to gain insight into the stochastic processes that govern the wood formation process. Various models were tested in devising an appropriate radial WSG model. Models that accounted for serial correlation in WSG data performed better than the simple structure that assumes zero correlation between measurements. The autoregressive plus random tree effect structure performed better in describing the radial variation pattern. The variability of the data related to random fluctuations during tree development and the wood formation process is modeled by the autoregressive parameter revealing the intrinsic complexity of wood formation. Since they cannot be attributed to observed factors, models should consider temporal or serial correlations when assessing wood quality. The results revealed that tree age is a decisive factor in controlling the WSG of wood, while tree size is statistically less important. Furthermore, the core wood production period varies with the growth rate. It is shown that the core wood area decreased with slow growth. Findings presented here appear to provide the first demonstration of radial variation in WSG with respect to growth rate and age for planted teak growing in Ghana.

Introduction

Teak (Tectona grandis L.f) is a tropical pioneer species growing on more than 260,000 hectares in Ghana [1] and accounts for approximately 70% of the exotic species in the plantation base [2]. Teak in Ghana has been reported to be genetically associated with those in central Laos [3]. Teak is the most commercially important plantation tree species in Ghana, grown in even shorter rotations to keep up with timber demand and reduce restoration costs. However, there is a concern that this practice will produce trees with a large volume of core wood, which will affect usable timber. Lachenbruch et al. defined core wood as wood close to the pith with relatively large radial wood property gradients, and outer wood as wood lying outside the core wood with more uniform wood properties [4]. Core wood is characterized by a low wood specific gravity (WSG), a high microfibril angle, and short fibers, which causes problems in processing and utilization [5]. The need for sustainable plantation resources is amplified by increasing their economic value by providing them as suitable for building, construction, and high-value-added products such as furniture. Therefore, it is important to clarify when and where wood maturation occurs and what is the controlling factor is for xylem maturation. Knowing what, when and where xylem maturation occurs can be critical to the successful development of plantations [6].

Tree growth is a classic spatiotemporal process. The vertical dimension exists due to the primary growth from the apical meristem and the horizontal dimension exists due to secondary growth from the cambium. However, how these processes are spatially and temporally remains unclear. Tree development involves two types of processes at the cellular level: an increase in cell number through division and a change in size through growth, resulting in an increase in height and diameter over time. As part of the tree growth process, there is some interdependence between the cambium sheaths within the trunk [7] since cambium initials are always formed in the preceding growing season [8]. These correlations should be viewed as physiological growth correlations rather than mathematical correlations [9]. This provides the physiological basis for autocorrelation in the tree growth process. Analysis of the variation (variance) over time in the wood properties assessed must allow modeling of the changes in variances and correlation (covariance) with tree age. This modeling process provides a key understanding of the physiological control of wood formation [7]. After a period of tree growth, the young cambium, which forms the core wood, gradually ceases, and the mature cambium which produces the outer wood, is formed thereafter. This has been termed radial shift [10,11,12,13]. Trees undergo physiological and morphological changes through ontogeny. However, the physiological relationships between the life stages of teak trees are poorly understood, although this species is of great ecological and economic interest [14]. The factors known to cause cambium maturity are spatial, temporal, and/or the interaction of both (spatial temporal).

WSG has been described as a function of cell size, shape, proportion, and thickness of cell walls at the anatomical level [15, 16]. A direct relationship between WSG and tree growth or productivity is expected [13, 14]. Thus, the radial pattern of the WSG of a tree offers a record of changes in the biomass allocation during growth and development. This record can tell us whether the WSG is controlled spatially or temporally. Clarifying the spatial and temporal factors in WSG variation can give us the requisite knowledge needed to manipulate the pattern of wood development, particularly in the case of core wood which affects the value and utilization [4].

Longitudinal analysis of data using ordinary least squares assumes that the error term is independent [17]. This leads to biased estimates of the model parameters and reduces the efficiency of the estimated parameters, leading to invalid statistical conclusions [18,19,20]. This challenge has increased the popularity of mixed-effects models for analyzing longitudinal data due to their flexible variance–covariance structures that allow for heterogeneity of variance and dependence within observations [21, 22]. The estimated variance–covariance matrix allows valid tests for the significance of mean values [7, 23]. The covariance model is the level of association between consecutive wood increments within a core [24, 25]. However, few studies have evaluated the autocorrelation between errors between successive wood increments for temperate species [26,27,28,29,30], but none for tropical hardwoods. Therefore, since the radial variation of WSG is based on repeated wood increments at different time, it becomes imperative to model the temporal autocorrelation in the data. With the mixed modeling approach, unbiased insights can be gained. Therefore, the objectives of this study were to determine: (1) to develop a model describing the radial variation in teak WSG by determining an appropriate variance–covariance structure; (2) to clarify the effects of age and size on radial variation in teak WSG; and (3) to discuss the broad physiological control in teak WSG using the appropriate correlation structure from the mixed modeling approach. Effective utilization of woody material requires reliable information on the quality distribution (proportion and size of the core wood and outer wood) in a harvested stem. In addition, we subjected our data to Bayesian analysis, which assumes that model parameters come from a broader distribution and do not rely on large sample theory [31]. This was done to solve the small sample problem in this study [32]. As a guide this work and based on the literature [6, 12, 13, 33], our working hypothesis is that age-dependent radial increases should lead to slopes that are steeper in the short radius than in the long radius.

Materials and methods

Three straight-bole 18-year-old T. grandis trees were randomly selected from a private plantation with a spacing of 3 m by 3 m in Dormaa Ahenkro, Bono Region, Ghana. 10-cm-thick discs were sampled from each tree every meter starting at 0.3 m from the ground level. From these discs, two radii being the opposite sides of the tree were selected randomly (see Fig. 1). The discs were eccentric by at least a factor of 1.3. Subsequently, a strip approximately 1–2 cm wide was cut from the disc. The strips were conditioned for several months at 20 °C, 65% relative humidity until achieving 12% dry-base moisture content (MC). Then, the strips were cut into two 1-cm segments (from the pith to the bark) collected from the opposite sides. The air-dried specific gravity (weight at 12% MC / volume at 12% MC), here referred to as WSG, was measured with an electronic densimeter (Alfa-Mirage MD-300S) calibrated to compensate for the water temperature (as water is 1 g/cm³ at 4 °C). Each disc was analyzed as two separate halves, referenced as disc long and disc short, separated by the pith. Each disc was then divided into 10 consistent parts over the radius and the mean value for each fraction was calculated using the relative distance approach [34,35,36,37]. The relative radial distance (RRD) approach has been used to clarify xylem maturation. This was done as it was very difficult to distinguish the annual rings. Using this method, wood strips of different lengths were standardized. The ages at given RRD along a radius were unknown, so we assumed that RRDs along the radius were the same age, for example, at 1% RRD from the pith on both short and long radii the wood should be the same age. To clarify whether the radial WSG pattern is controlled spatially or temporally, the two sides at a height level of the same tree were evaluated using linear mixed models [13]. The different-length radii at the same sampling height of a tree represent growth rate differences in tissues that are uniform in genetic makeup, cambial age, height in the tree, and microenvironment, although aspect and illumination can differ [38]. These differences reflect how trees respond to local environmental conditions and the orientation of the stems for better access to light may lead to eccentric growth [39]. The total number of observations was 360 (3 trees $\times$ 2 radii $\times$ 6 replicates $\times$ 10 RRD levels).

Fig. 1

Schematic diagram of sample preparation and repeated measures data illustration

Statistical analysis

The resulting data constitute spatiotemporal repeated measures on the tree, with RRD as a time factor and aspect as a space factor. Observations from the same unit (tree, disc, and age) tend to be spatially and/or temporally correlated [17, 40]. In the mixed model, the correlation structure of the dependent variable is considered by allowing the parameters to vary randomly from one individual to another around the fixed population mean [41]. The analysis of experimental data with mixed-effects models requires decisions about the specification of the random-effects structure [42]. This implies that in a repeated measures modeling, the best covariance structure describing the correlation among the repeated measures should first be identified. A mixed linear model for expressing the WSG of wood for a specific aspect/radius following [43] is as follows:

$$\begin{gathered} \varvec{Y}_{\varvec{i}} = \, \varvec{X}_{\varvec{i}} \boldsymbol{\beta} \, + \, \varvec{Z}_{\varvec{i}} \varvec{U}_{\varvec{i}} + \, \boldsymbol{\varepsilon}_{\varvec{i}} , \hfill \\ \varvec{U}_{\varvec{i}} \sim \, N \, ({\mathbf{0}}, \, \varvec{G}), \hfill \\ \boldsymbol{\varepsilon}_{i} \sim \, N \, ({\mathbf{0}}, \, \varvec{R}), \hfill \\ \end{gathered}$$

(1)

where Y_i = vector of observations, β = p $\times$ 1 vector of fixed effects with incidence matrix X; p = number of levels for fixed effects. U_i = q $\times$ 1 dimensional vector of random effects with corresponding design matrix Z_i; q = number of levels for random effects. ε_i = n $\times$ 1 vector of residual effects. R = variance–covariance matrix of errors. G = random-effects covariance matrix.

The expected value and variances are E[Y_i] = Xβ, var [U_i] = G = ${\sigma }_{b}^{2}A$ and var[e] = R = ${\sigma }_{e}^{2}I$ for A the numerator relationship matrix and I an identity matrix.

Model development

The longitudinal data (time series) of WSG were analyzed using linear models to determine the effects of the differential growth rate on radial trends. In model development, it is imperative to find a parsimonious model that specifies the covariance structure of the data and to obtain unbiased statistical tests of fixed effects. Hence, our major focus is on modeling the covariance structure following [44].

Conceptually, the design is a split plot in which trees and aspects are the main plot treatments and radial positions (cambial age) the minor plot treatments. The effect of RRD and the differential growth rate on the WSG was examined using a mixed-model approach with repeated measures [45, 46] as follows:

$$Y_{ijk} = \, \mu \, + \, \alpha_{i} + \, d_{ij} + \, \tau_{k} + \left( {\alpha \tau } \right)_{ik} + \, e_{ijk} ,$$

(2)

where Y_ijk is the SG measured in RRD k on the jth tree assigned to the ith radius (growth category); μ is the overall mean effect; α_i, τ_k, (ατ)_ik are parameters corresponding to the fixed effects of radii i, RRD k and the interaction of i radii and RRD k, respectively; d_ij is the random effect of the jth tree within ith radii; d_ij is normally distributed random variable with mean zero and σ²_d corresponding to tree j in radii i, and e_ijk is a normally distributed random variable with mean zero and variance σ²_e, independent of d_ij, corresponding to tree j in radii i at RRD k. We used the MIXED procedure of the SAS software [47] to fit the model with a restricted maximum likelihood (REML) method. Degrees of freedom were determined using the between–within method. The homogeneity and normality of the residuals were checked to ensure that assumptions were met. We determined the statistical significance (α = 0.05) of fixed effects with the F-test. The variables components of random effects and their standard errors were expressed as a percentage of the total variation of all random effects. As a result of small tree replication, the results of the Z-test must be considered indicative only [45]. Moreover, a quadratic evolution of WSG for each radius, with tree-specific intercepts and with correlated errors within radii was modeled as follows:

$${Y_{ij}} = ({\beta _{{A_0}}} + {\mkern 1mu} {b_i}){\mkern 1mu} + {\beta _{{A_1}}}t{\mkern 1mu} + {\beta _{{A_2}}}{t^2} + {\mkern 1mu} {\varepsilon _{ij}}\left( t \right)\;{\rm{in\, radius }}\,1\,{\rm{ or}}\;({\beta _{{B_0}}} + {\mkern 1mu} {b_i}){\mkern 1mu} + {\beta _{{B_1}}}t{\mkern 1mu} + {\beta _{{B_2}}}{t^2} + {\mkern 1mu} {\varepsilon _{ij}}\left( t \right)\;{\rm{in \, radius }}\,2,$$

(3)

where t is the RRD = 0.1, 0.2, …, 1; ε_ij is the random error term associated with the jth rrd of the ith radii, and it is assumed that ε_i ~ N(0, Σ), where the bold character of ε_i denotes the vector of {ε_ij} specific to the ith radii, and Σ is a symmetric positive definite variance–covariance matrix. Graphical techniques were used to check for normality and homoscedasticity of the residuals. We used PROC BGLIMM of the SAS software [47] to generate the model parameter estimates with their associated errors and the random-effects variance based on Hamiltonian Monte Carlo algorithm. The model ran for 500 burn-in iterations and 50⁵ sampling iterations, saving every 100th iterations for a total of 5000 generated posterior distributions of the model parameters. Along with the WSG estimates computed from the fitted model and applied to new data, a prediction uncertainty interval was also constructed, represented here by the range between 2.5 and 97.5% of the estimated distribution density.

Results and discussion

WSG mean and variability

Table 1 gives the descriptive statistics of the WSG values for each radius. Figure 2 shows the WSG distribution across the radial direction for each growth category. The pooled mean WSG value was 0.666, ranging from 0.610 to 0.715 between trees with a coefficient of variation (CV) of 11.23%. WSG increased with height from 0.673 at 0.3 m to 0.704 at 5.3 m. This translates to an increase in WSG of about 11.6% from bottom to top. However, at 2.3 m height level, WSG was strikingly low (0.625). With radial variation, WSG increased from pith to bark across all height levels, averaging 0.603 to 0.687 from pith to cambium representing about 14% increase in WSG with time. Spatially, the radial dimension showed higher variation than the axial dimension in WSG. The results presented here are similar to those described in the literature [48]. Our mean WSG value was higher than those reported [49,50,51] but similar to those reported [52, 53].

Table 1 Descriptive statistics of the WSG between the two radii

Fig. 2

Contour plot visualizing the WSG variation with RRD per radius

Table 2 gives the Pearson correlation between WSG measured at different radial positions. REML covariance and correlation are printed above and below the bold highlighted diagonal, respectively, of the matrix in Table 3. The correlations between WSG at RRD 1 and later times are in the first column of the matrix. Correlation generally decreased over time. Covariance and correlations between adjacent WSG measurements on the same strip are more correlated at later time periods than at earlier times, for example, the covariance and correlation between adjacent WSGs of RRD 9 and 10 are 0.0049 and 0.8609, respectively, which are much larger than their corresponding covariance and correlation between adjacent WSG measurements of 0.0021 and 0.6932 at RRD 1 and 2. The correlations of the WSG formed in the early ages of the tree (RRDs 1 and 2) decreased rapidly with the WSG formed later. This indicates the phase where the trees are growing faster (high annual increment). The low correlation of RRD 1 with later ages suggests that early growth is probably not a good predictor of later WSG. Additionally, the low correlations of RRD 1 with later ages suggest that whatever causes these early differences, they do not persist and are not scaled up as the trees grow. The temporal positive-to-negative and negative-to-positive shifts in the correlations may indicate intrinsic differences in the underlying wood formation processes during the young and adult growth stages. The correlations increased substantially after RRD 4. The strength of correlation increases from RRD 5, implying that the WSG changes only slightly after RRD 5. From RRD 5, the tree can be said to have entered the mature phase. The stability of the correlation implies that no meaningful change in genotypic ranking is expected to occur after RRD 5. The results indicated that the most rapid change within the trajectories occurred before RRD 5. This is approximately the termination point of the juvenile core wood as the tree starts to produce high WSG. The correlation matrix can be seen as the pattern of cambial activity during tree growth. The diagonals are the estimated variances of the errors associated with each RRD measurements. Note that they increase marginally from 0.0026 at RRD 2 to 0.0062 at RRD 6. After RRD 6, the variances stabilized (Fig. 3). This trend is consistent with most growth data, and so the covariance structure models that we consider for these data should accommodate heterogeneous variance with time, however, it is reasonable to assume homogenous variances across time. The phenomenon may be due to the initial fast growth of the tree at a younger age resulting in wider volume increments (growth rings), low correlations, and high variation. At later stages in the tree’s life, high WSG is produced, slowing down growth resulting in smaller wood increments and hence high correlation and low variation. Adult slow growth produces relatively uniform wood in its wood anatomy while young fast growth produces more complex wood because of the changing anatomical patterns from year to year. Cambial aging can be seen as a positive process because it brings order out of chaos.

Table 2 Observed WSG correlation between relative radial distance groups (Pearson correlation)

Table 3 REML covariance and correlation estimates for unstructured covariance structure for WSG repeated measures data

Fig. 3

The variance function in the WSG data

As shown in Table 4, the inverse of the covariance matrix for the unstructured model yielded larger (absolute) values along the first off-diagonal bold highlighted band. This gives us an indication of a first-order structure, justifying the use of the first-order autoregressive correlation structure. First-order structures model conditional independence, since any pair of measurements made on the same strip that are more than a time interval apart are conditionally independent [54]. The covariance and correlation matrices displayed as a function of lag for simple, compound symmetric, autoregressive (1), autoregressive (1) plus random tree effect, Toeplitz and antedependence structures are summarized in Table 5. Figure 4 shows the correlation function of lag for unstructured, autoregressive (1), compound symmetric and autoregressive (1) plus random effect correlation structures. We expect that traits separated by developmental changes will be uncorrelated; however, for dissociation to occur, the correlations need not be zero [55], justifying the importance of the first-order autoregressive structure. The correlation structure can be viewed as the physiological correlations in wood formation.

Table 4 Estimated concentration matrix or inverse of the covariance matrix for the unstructured covariance model

Table 5 REML covariance and correlation estimates for unstructured covariance structure for WSG repeated measures data

Fig. 4

Correlograms of the six correlation structures

Comparison of fits of covariance structures

To evaluate the impact of accounting for serial correlation (within-tree dependence), models with and without incorporating a function to model the error structure were compared. The results of fitting the seven covariance structures to the data are shown in Table 6. The likelihood ratio statistic (− 2Inλ_N) is the difference between the—2L_ML of the current and reference model (simple). The P values indicate that there is significant evidence to conclude that the six other covariance structures performed better than the simple ones. Thus, we may conclude that the correlation structure in the data is needed. Heterogeneous models of the compound symmetric and first-order autoregressive models were tested. They did not perform better than the homogenous models (data not shown). The heterogeneous Toeplitz model did not converge in the SAS system.

Table 6 Summary results for fitting different covariance structures to WSG data

With the specific goal of a parse modeling of the covariance structure, we use the BIC criterion. AIC and BIC values for the six covariance structures are shown in Table 7. ‘Unstructured’, it has the largest AIC, but Toeplitz ranks second in both AIC and BIC. AR(1) + RE ‘autoregressive with random tree effect’ is first in BIC and third in AIC. The discrepancy between AIC and BIC for the UN structure reflects the penalty for the large number of parameters in the UN covariance matrix [55]. Overparameterization can lead to uninterpretable and complex models. Based on examining of the correlation estimates in Tables 3 and 7 and the relative values of BIC, we conclude that AR(1) + RE ‘autoregressive with random tree effect’, is the best choice of covariance structure describing the radial patten of WSG variation. The random tree effect which takes care of spatial variation, significantly improved the model’s performance. The scaled residual and the Q-Q plot of the residuals appeared normal and well behaved (Fig. 5).

Table 7 Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC) for seven covariance structures

Fig. 5

Profiles of the scaled residuals (a) and Q–Q plot of the residuals (b) from the AR(1) + RE correlation structure

More importantly, the parameters of the AR(1) + RE are of interest for the purpose of the study. The variance quantifies the spatial variation, the variance is the temporal variation, and the first-order autocorrelation coefficient explains the physiological correlation between actual (t) and previous WSG (t-1) since cambial initials were always formed in the previous season [8]. The degree to which the fixed effects (differential growth rate and RRD) do not adequately describe the WSG is also measured by the first-order autocorrelation parameter.

The mixed-effects model was able to capture the variation at the different levels considered, that is between-tree variation, within-tree (residual) variation and autocorrelation function, which justified its use as shown in Table 8. The estimated first-order autoregressive process AR(1) was high with a value of 0.8475. The total variance was 0.005127, which was corrected by autocorrelation of the residuals. Therefore, it is possible to express the variability of each level as a proportion of the total variance: the spatial variation (growth category level variation) was not significant and accounted for about 23% of the total radial variation of the WSG showing that the effects of differential radial growth rate on the WSG were low. This spatial variation is the variation caused by the differential cambium production on the two opposite sides of the same tree. Temporal variation (pith-to-bark) was significant explaining about 77% of the total variation in radial WSG. Overall, there is enough variability at both levels to justify the application of mixed-effects models.

Table 8 Parameter estimates in the covariance and correlation matrices for the AR(1) + RE structure

The first-order autocorrelation parameter measures the influence of WSG in the previous year on WSG for the current growing season. This parameter was estimated at 0.8475, so observations of WSG from the same strip but one RRD apart are highly correlated by 0.882, observations 2 RRD apart by 0.782 and observations 3 RRD apart by 0.697. The significant first-order autocorrelation parameter implies that the WSG of the wood formed in the previous year (t-1) influences the growth in the current year (t) and changes over time, which seems to be characteristic of deciduous trees [56]. This means that if a given year’s growing conditions favored thickening and hardening of the cell wall, it is very like that the WSG would be similar in the next growing year, most likely due to stored photosynthates and carbohydrates the cambial meristems use for the next year [57]. Furthermore, the significant autocorrelation coefficient implies that errors within-tree cannot be assumed to be independent and that precautions must be taken to remove autocorrelations before hypothesis testing to ensure that normality assumptions are met and that statistical tests are unbiased.

Effects of differential growth rate on WSG

Type 3 fixed effects were used to assess the statistical significance of fixed effects as shown in Table 9. The main effect of growth category (radii), which reflects size effect, was not statistically significant (p > 0.05). The lack of statistical significance between the WSG values along the long and short radii indicates that within a tree, factors other than lateral growth rate can drive the shift from low WSG (core wood) to high WSG (outer wood). The radii-by-rrd interaction was not significant (p > 0.05). This indicates that there is a similar response in the WSG in both direction and magnitude on opposite sides of the same tree. At an assumed age, the WSG values show no difference from long radius to short radius. The rrd main effect representing presumed age, is highly statistically significant (p < 0.0001). This illustrates a high temporal variation of WSG. The shift from low WSG to high WSG, which is analogous to xylem maturity, is likely driven by cambium age rather than distance (size) in the selected teak trees. WSG increases with tree age until it reaches an asymptote. This suggests that the changes follow a fixed trajectory on both the long and short radius.

Table 9 Results of the fixed hypothesis test with AR(1) + RE structure (ANOVA F-tests)

The long-radius WSG mean values were slightly larger than the short-radius mean values, although they were not statistically significant as shown in Table 10. WSG comparisons of presumed age groups along the two radii also showed no significant differences. The WSG formed at the same time are similar regardless of the side of the tree on which cambium is produced slowly or rapidly. This finding is consistent with the characterization of WSG changes per unit time rather than per unit size, as reported by Williamson and Wiemann [13].

Table 10 Between-subject comparison with AR(1) + RE structure. Differences in mean WSG between long radius (growing rapidly) and short radius (growing slowly) are almost same over time (RRD)

Modeling polynomial change over time

Data from each aspect were combined to represent the specific growth rate, and polynomial modeling of the fixed effects of radius and presumed age was applied using the AR(1) + RE covariance structure. Correlation between errors from consecutive assumed ages on the same core was included in the model. The third-degree polynomials showed no significant evidence of third-degree age terms (p = 0.827, type 1 tests of fixed effects tests). The quadratic term was significant, implying that there is a quadratic age effect on the radial WSG in the selected teak. The results are shown in Table 11. The scaled residuals and the Q-Q plot of the residuals of the quadratic model are shown in Fig. 6. They appeared normal and well behaved.

Table 11 Final nonlinear mixed-effects model. β_A0 -β_B2 correspond to the fixed effect regression model (intercept, linear, quadratic trend), the bi-parameter is the corresponding random effects

Fig. 6

Profiles of the scaled residuals (a) and Q–Q plot of the residuals (b) of the quadratic model with AR(1) + RE correlation structure

The fitted polynomial equations are as follows:

R1: WSG = 0.5691 + 0.3652 RRD – 0.3112 RRD².

R2: WSG = 0.5831 + 0.3114 RRD – 0.3051 RRD².

Zobel and Sprague defined the core wood zone as the area of rapid property changes near the pith and the outer wood as more uniform towards the bark [5]. The pith-to-bark profiles for each growth category yielded different but identical curves, particularly different starting points (intercepts) and slopes in the outer wood zone. The difference in the parameter β₀, which is the starting point (first-formed WSG) of the mean pith-to-bark profile was 0.014, corresponding to an approximately 2.5% increase in WSG on the short radius. The inflection point (shift from low to high WSG) was estimated from the first derivative of the quadratic polynomial. The maximum estimated of the quadratic function of the core wood part x₀ = − β₁/(2β₂) in WSG for R1 were 0.6 and 0.5 for R2 since this is the presumed age when WSG values tend to reach an asymptote (WSG levels at this point). These correspond to approximately 11 years and 9 years in R1 and R2, respectively. The longer radius may have had a higher rate of cell production and therefore required more time to complete the wood formation process. The differential growth rate can affect the age of the transition from core wood to outer wood, with rapid growth increasing the proportion of core wood. Our working hypothesis seems to have been confirmed. The slope gradient on the slow-growing side is steeper and flatter on the fast-growing side. The shorter radius and long radius follow a fixed trajectory. The low WSG is larger in the fast-growing core, indicating that fast growth appears to be associated with a longer period of core wood content. This case suggests that maturation might be controlled by cambium age. This suggests that xylem maturation (production of higher WSG) begins after reaching a certain cambium age. The higher growth rate can result in cells having less time to expand as the cambium produces new cells faster, resulting in the long radius having more core wood (less WSG) than the short radius [4]. It is evident that slowing young growth can reduce the low WSG (core wood) area. During the young phase, the low WSG averaged 0.619 with a CV of 3.7% on the long core and 0.619 with a CV of 2.6% on the short core. After maturity, the overall mean was 0.668 (CV: 1.2%) in the long core and 0.657 (CV: 0.99%) in the short core, a difference of 0.011. The overall mean WSG was 0.644 in the long core with a CV of 4.7% and 0.642 with a CV of 3.5%. The slow growing side has more consistent wood with a small WSG difference between young stage and adult stage WSG. The WSG increases by about 13% over the lifetime of the tree, which translates into an increase in modulus of rupture by about 245 kg/cm² [58, 59]. This pattern could be driven by the greater need for hydraulic function before maturity [60, 61] and for mechanical strength after maturity to achieve stiffness [10, 62,63,64]. During the early growing seasons, the total stem cross-section is small, and a large proportion of void volume is required, producing large-diameter and thin-walled fibers [65, 66]. As a stem increases in size, the percentage void volume required is reduced and WSG can increase to achieve maximum mechanical efficiency observed that as the cambium ages [65], the cell diameter decreases and the cell wall thickness increases. The same amount of material spread over a smaller area increases the WSG. From a physiological point of view, since the tree is taller, the water tension is much higher and the cell walls must therefore be thicker to withstand the pressure. Such effects may explain the steady increase in WSG observed in the outer forest region. In both cores, WSG in the outer wood differed significantly from the core wood. On the R1, t(8) = − 4.22, p = 0.0029; and R2 t(4.406) = − 4.09, p = 0.0123.

From Table 12, the WSG formed during youth differed significantly between the fast-growing side and the slow-growing side. WSG was significantly higher on the slow-growing side during the young phase of tree growth. After maturity, the WSG differences are the same in the two cores. This phenomenon is supported by the correlation structure. WSG exhibits compensatory the growth as growth in WSG of the first-formed wood decays to the WSG of later formed wood [67] and can be represented by this equation WSG(t) = 0.9452e^−0.099t. The decreasing temporal correlation structure over time can be attributed to the decreasing crown effect on later formed wood, differential gene expression, radial morphogen (i.e., auxin) gradient, and changing microenvironmental conditions under which the trees evolved. The temporal correlation structure provides statistical evidence that young cambium differs from adult cambium in terms of physiology and epigenetics. We expect that there would be differences in early radial growth rate and WSG, but both sides of the same tree can achieve nearly similar WSG after maturity. This leads to a reduction of the variation in WSG in the later growth stages. These changes have been associated with changes in the physiological system and gene pattern and activity [68, 69]. Genes do not function simultaneously and vary with age, i.e., a qualitative change [68] since genes on both sides of the same tree may not function optimally at the same time. By changes in gene activity is meant that active genes can be more or less active (gradual change with age), i.e., a quantitative change [68] since there is a different activity of genes at different ages. Maturation is related to genetic changes [68].

Table 12 Between-subject comparison for AR(1) + RE structure and polynomial model for RRD

Application to an independent dataset

The best model was introduced into the Bayesian framework, fitted using the Markov chain Monte Carlo (MCMC) approach, with the exact model structure on the same data. To prevent the posterior distribution from being influenced by the means chosen for the model's parameter values, we used weakly non-informative priors. The parameters estimated by the Bayesian framework are presented in Table 13. A focus was on examining the pattern of WSG variation between the two radii. For Radii 1 and 2, the predicted slopes were 0.00483 and 0.00543, respectively. The difference in slope was − 0.00060 (95% Credible interval: − 0.00385 to 0.00269). This suggests that the radial WSG variation may not be affected by the differential growth rate since the 95% credible interval contains zero. The steeper slope in the short radius may indicate that radial increases in WSG are age dependent, consistent with our working hypothesis. A significant difference between trees was found by the Bayesian model. It has been reported that random tree-level variation is a major source of variation in a population [14, 70,71,72]. The autoregressive correlation parameter was high and significant. Even after accounting for time trend, the typical assumption of a random structure over time is wrong since wood data are typically collected in time series and the wood data contain significant temporal autocorrelation. The independence assumption is satisfied by the residuals of a suitable AR(1) model, which can be put to the test. We should recognize that tree growth is a significant repository of the interactions between various environmental, molecular, cellular, developmental, and physiological elements. Only models that take growth process variability into account may explain these random fluctuations.

Table 13 Posterior summary and intervals

Mechanism of maturation in teak

Previous studies have identified the transition from core wood to outer wood after 8–10 years [49, 73,74,75], which is consistent with our results. However, Bhat et al. (2001) reported that teak in India starts maturing at around 15–25 years [76], and Hidayati et al. (2014) reported that teak from Indonesia starts maturing at 12 years [37]. The differences in the maturation time of teak planted worldwide can be attributed to proximity to the equator and several other factors including genetic control, breed selection, soil conditions and traits considered [6]. In tropical latitudes, changes in day length are large enough to affect plant development around the equinox [77]. The transition from core wood to outer wood in teak is characterized by (1) an increase in fiber length of about 50% [37, 73, 76]; (2) a decrease in microfibril angle of more than 50-fold from the pith to the bark [76] and (3) from large annual rings to narrow rings [51, 74, 76, 78]. These disclosures support the relative stability in the correlation structure of WSG later in tree life (referred to herein as xylem maturity). Aging of the cambium plays an important role in the radial variation of WSG. This leads to systematic variations in the secondary tissues that affect both the anatomical and physical properties of wood. At a young age, short fibers are produced, these cells have a large diameter and a thin wall [65, 66], which explains the initially wide rings, high MFA, low WSG. As the cambium ages, the fibers increase in size, cell diameter decreases, and cell wall thickness increases [65]. This leads to a reduction in ring width and a lower MFA, which also explains the steady increase in WSG observed in the outer wood portion.

Radial increases in WSG have been studied in several tropical pioneering species. It has been linked to the mechanical reinforcement of the trunk that would result from stiffer/stronger wood at the periphery [10, 11, 79]. Tropical pioneers exhibit rapid growth in height and diameter by producing a low WSG as juveniles but require greater stability later in tree development. Given this phenomenon, as de Castro et al. noted [33], one would expect WSG to be spatially rather than temporally controlled. However, ecologically pioneering species are shade intolerant and therefore require electromagnetic energy from light to grow in space, making them more sensitive to time; thus, the development of pioneer species is controlled by time [80]. The age effect indicates an intrinsic control of wood development limiting the younger and often shorter cambium cells in the size of cells they can produce [81].

Conclusion

The influence of the different growth rate on the WSG is studied using a linear mixed modeling technique. Overall, the linear mixed model method fits our data better. It has been found that models that account for serial correlation in the data tend to perform better than models with independent and identically distributed (simple) structure. This should serve as a warning to wood technologists who ignore serial correlations as unworthy. The serial correlation provides important background knowledge to understand a complex system like the tree and its growth process. We have also made suggestions for choosing an appropriate covariance structure. The models presented here could serve as a prototype for wood technologists who wish to clarify the response of trees to different growth rates over time, even in the absence of annual rings, particularly in tropical timber species, and to understand the physiological differences in the tree growth process. It only requires that the data be collected in a repeated measures design, so that the statistical analysis can account for the time series nature of the data, taking into account the variance function and autocorrelation between repeated measures. Analysis of wood properties in time series format can help us to understand the underlying wood formation process and the pattern of variation of wood properties over time. Despite the data limitations in the current analysis, the model we developed provides a novel way to study WSG variation in response to random fluctuations during tree growth. The data collection process in future work could be assisted by the Bayesian model that we present here. Using Bayesian model and posterior summary, we can test our predictions on future observations.