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Modeling lumber bending stiffness and strength in naturalblack spruce stands using stand and tree characteristics

Chuangmin Liu 1, S.Y. Zhang a,*, Alain Cloutier b,2, Tadeusz Rycabel b,3

a Forintek Canada Corp., 319 rue Franquet, Quebec City, Qué bec, Canada G1P 4R4b Centre de recherche sur le bois, Dé partement des sciences du bois et de la forê t, Université Laval, Qué bec, Qué bec, Canada G1K 7P4

Received 2 October 2006; received in revised form 25 January 2007; accepted 25 January 2007

Abstract

Static bending modulus of rupture (MOR) and modulus of elasticity (MOE) were measured on lumberfrom trees in the natural and mature black spruce stands grown in Eastern Canada. A sample of total 157 trees from the 90–100-year-old black spruce natural stands covering a range of sites

and growing conditions was used for the model development (n = 102) and validation (n = 55). A stepwise regression method was employed to

identify the best variables for predicting MOE and MOR using stand/tree characteristics and wood properties. Then, regression equations with

different explanatory variables were developed to predict lumber bending stiffness and strength. Based on the results of model validation from the

independent dataset, the regression models developed were able to predict the lumber bending MOE and MOR satisfactorily, especially for small-

and middle-sized trees. The results (equation parameter estimates and predictions) obtained in this study, along with those for plantation-grown

black spruce in Eastern Canada, will be highly useful in predicting lumber bending static stiffness and strength for both natural and managed black

spruce stands.

# 2007 Elsevier B.V. All rights reserved.

Keywords: Black spruce; Tree characteristics; Bending strength and stiffness; Regression model; Picea mariana; Stand characteristics

1. Introduction

Black spruce (Picea mariana (Mill.) B.S.P.) is the most

important reforestation and commercial species in eastern

Canada. In Canada, black spruce is mainly used for lumber and

high quality pulp and paper manufacturing. Black spruce is

highly valued for machine stress-rated (MSR) lumber produc-

tion thanks to its strong mechanical properties. Lumber bending

strength or modulus of rupture (MOR) and stiffness or modulus

of elasticity (MOE) are among the most important product

properties used to determine the end uses and MSR grade yield

of black spruce (Bruchert et al., 2000; Zhang et al., 2002).Lumber MOE and MOR can be estimated through various non-

destructive testing (NDT) methods (Ross et al., 1997; Wang

et al., 2002b).

It is well-known that both external tree size (e.g., DBH),

stem taper and internal wood characteristics (e.g., knottiness

and wood density) affect the lumber mechanical properties such

as static bending MOE and MOR (Haartveit and Flate, 2002;

Zhang and Chauret, 2001; Zhang et al., 2002). The relationship

of those lumber properties with stand and tree characteristics

(e.g., stand density, DBH, stem taper, total tree height, crown

size, branchiness) would therefore be useful for estimating the

lumber bending properties using inventory data. Use of easily

measured variables on standing trees is preferred if this does notunduly compromise equation accuracy.

Among tree characteristics, DBH is easily measured and is

the most common parameter determined in forest inventory. In

fact, many inventories record DBH only. Tree height is often

recorded for forest inventory. Tree DBH and height are the most

important variables in determining the yield and quality of

lumber due to their effects on the volume and grade of lumber

(Aubry et al., 1998; Fahey, 1980; Houllier et al., 1995; Zhang

and Chauret, 2001). For softwoods, knottiness is an important

internal wood characteristic affecting strength and appearance

www.elsevier.com/locate/forecoForest Ecology and Management 242 (2007) 648–655

* Corresponding author. Tel.: +1 604 222 5741; fax: +1 604 222 5690.

E-mail addresses: [email protected] (C. Liu),

[email protected] (S.Y. Zhang), [email protected]

(A. Cloutier), [email protected] (T. Rycabel).1 Tel.: +1 418 659 2647.2 Tel.: +1 418 656 5851; fax: +1 418 656 2091.3 Tel.: +1 418 656 2438; fax: +1 418 656 2091.

0378-1127/$ – see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.foreco.2007.01.077

mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.foreco.2007.01.077http://dx.doi.org/10.1016/j.foreco.2007.01.077mailto:[email protected]:[email protected]:[email protected]:[email protected]

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set to 0.15 and 0.05, respectively. Therefore, all parameters left

in the final regression models after stepwise selection were

significant at the 0.05 level. Because all sample trees were from

six stands with different site quality, the effect of site was

introduced as a class level into the final regression models and

tested for significance using analysis of covariance (ANCOVA).

The models were evaluated based on the multiple coefficient

of determination ( R2), the root mean square error (RMSE), the

mean absolute error (MAE) and error index (EI) of the

predictions, and bias. The R2, RMSE, MAE, RE, and bias were

computed as follows:

R2 ¼ 1

PðY j ˆ Y jÞ

2PðY j ¯ Y Þ

2 (3)

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPm j¼1 ðY j

ˆ Y jÞ2

m

s (4)

MAE ¼Pm

j¼1 jY j ˆ Y jj

m(5)

EI ¼ jY j ˆ Y jj (6)

bias ð%Þ ¼ð1=mÞ

Pm j¼1ðY j

ˆ Y jÞ

¯ Y 100 (7)

where Y j and ˆ Y j are the observed response variable value of tree

j and predicted response variable value of tree j for each

regression model, respectively; ¯ Y is the mean of observed

response variable value for the m trees; j = 1, 2, . . ., m; and

m is the number of trees. The RMSE, MAE, and EI provide an

indication of model fit due to no cancellation between positiveand negative values. Positive bias represents underprediction by

the model and negative bias represents overprediction by the

model. For model comparison, those statistical criteria were

evaluated based on the test data set. For each regression model,

the normality of distribution of residuals (observed–predicted),

heteroscedasticity, and multicollinearity were checked by using

the Shapiro–Wilk test (Shapiro and Wilk, 1965), visually by

plotting the residuals against the estimated lumber volume, and

using the maximum variance inflation factor (MVIF), respec-

tively. SAS (SAS Institute Inc., 1999) was used for all compu-

tations.

3. Results

3.1. Model development

Table 3 presents the results of the stepwise linear regression

equations using the fit data set. All parameters in the equations

after stepwise selection were statistically significant at the 0.05

probability level. In the two models, maximum VIFs in the

range of 1.432–3.068 indicated that multicollinearity was not

severe. MOE was best predicted by wood density, DBH, and

crown length, and this model explained 65% of the total

variance in MOE (model 1). In the MOR model (model 2), only

two variables (DBH and wood density) were selected as

predictors by stepwise regression.

Although wood density is an important factor for bending

strength and stiffness, the variable is much more difficult to

measure than tree characteristics. Since the objective of this

study was to predict the MOE and MOR using the input data

from forest inventory, we further conducted the stepwise

regression analysis without wood density variable. The resultsare listed in Table 3. From Table 3, when wood density was

excluded from model 1, stem taper and stand density were

added into model 1a at the 0.05 probability level. Based on

statistical criteria such as R2, RMSE and MAE, it was expected

that the model 1a seemed to have lower prediction precision

than the model 1. In the MOR model (model 2a), DBH became

the best single predictor of MOR. Approximately 58% of the

variance in MOR was explained by this model (Table 3).

The parameter estimates in the above four models were

obtained by ordinary least squares (OLS) method. Like the

study by Lei et al. (2005), the SUR method was used to estimate

the parameters in the system of two equations (i.e., models 1band 2b). The results are presented in Table 3. From Table 3, the

model 1b had comparable prediction precision to the model 1a

in terms of R2, RMSE, and MAE. In the MOR model, the

endogenous variable MOE was by far the best single predictor

of MOR based on the t -statistics and its p-value of less than

0.0001, followed by DBH.

Results of the Shapiro–Wilk test indicated that the normality

assumption for each regression model was held with a p-value

greater than 0.05. The plots of residuals (observed–predicted)

confirmed that the residuals for all the developed regression

models were evenly distributed over the predicted product

Table 3

Results for the linear regression equations for MOE and MOR

Model number Fitted regression equation Fit data set Test data set

R2 RMSE MAE R2 RMSE MAE

1 MOE = 2705.36 176.34DBH + 175.31CL + 21808WD 0.65 1213.20 882.38 0.55 1109.07 1058.652 MOR = 21.18 1.6273DBH + 128.02WD 0.68 8.46 6.28 0.53 9.49 7.68

1a MOE = 17,065 222.38DBH + 261.78CL 3094.96ST + 0.41699SD 0.55 1366.31 1035.35 0.43 1117.36 1363.112a MOR = 100.78 2.4412DBH 0.58 9.60 7.33 0.50 10.24 9.48

1b MOE = 17031.74 222.49DBH + 262.38CL 3083.06ST 0.43011SD 0.55 1366.31 1035.07 0.43 1116.99 1362.362b MOR = 25.98 1.3242DBH + 0.00455MOE 0.79 6.82 5.30 0.72 7.23 5.84

Note. CL, crown length; DBH, diameter at breast height; SD, standdensity; ST, stem taper; WD, wood density; R2, multiple coefficient of determination; RMSE, root

mean square error; MAE, mean absolute error.

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recovery and had no obvious trend. Results of testing the effect

of site using ANCOVA analysis showed that the site effect in

each of six models was not significant at the probability level of

0.05. It indicated that the six equations developed here could be

used across all the sites in the study area. However, MOE and

MOR data from the 55 sample trees collected as the

independent data set were needed to further validate the

accuracy of all the developed models.

3.2. Model evaluation

With the test data (55 observations), MOE and MOR were

predicted using the six models under study. The root mean

square error (RMSE) and mean absolute prediction error

(MAE) for the test data set are also summarized in Table 3. For

each model, the difference in R2, RMSE, and MAE values

between the fit data set and test data set were relatively small.

This indicated that those models had an ability to estimate

MOE/MOR when using new inputs in the test data set. Model 1

had lower RMSE and MAE values than model 1a for the testdata set (Table 3). It appears that, on average, model 1 including

wood density predicted the MOE slightly more accurately than

did models 1a and 1b. For MOR models, it was evident that

model 2b in which MOE was included increased predictive

ability when compared to models 2 and 2a in terms of RMSE,

and MAE (Table 3).

To evaluate the model performance across tree sizes, the

computed prediction bias were averaged over 2-cm diameter

classes and listed in Table 4. It was evident that all MOE models

had relatively large biases for large-sized (24-cm DBH class)trees. It appears that all models estimated the MOE more

accurately for small- and medium-sized trees (10–22 cm DBHclasses). For MOE, the overall biases of models 1, 1a, and 1b

were 3.22%, 3.40%, and 3.41%, respectively. This indicated that

the three models produced underestimations for MOE. From

Table 4, we have noticed that the MOR models performed better

and had smaller biases for medium-sized trees (12–22 cm DBH

classes). On the other hand, they gave relatively larger MOE

values for the 10 cm DBH class and the 24 cm DBH class or

above in the test data set (Table 4), especially models 2 and 2a.

To assess the statistical significance of the prediction error

index (EI) for the different models, we performed paired t tests,

which can be used to compare the difference (pairing by tree)

between the error indices for any two models for each response

variable (Zhang et al., 2003). The normality assumption for the

prediction error index was assessed using the Shapiro–Wilk test

(Shapiro and Wilk, 1965) before we applied the paired t test.

Results indicated that the normality assumption held for the

prediction error index from each regression model. Table 5

presents the results of these paired t tests for each possible pair

of models. A negative paired t statistic implies that modelrowhas a comparatively better fit than modelcolumn, because the

difference between the error indices of two models is calculated

as modelrow modelcolumn, where row and column refer to theposition of the fitted models being compared in Table 5. In this

context, Table 5 shows that models 1a and 1b for MOE had

significantly higher error indices than model 1. For example,

the comparison between models 1 and 1a resulted in t = 3.02( p = 0.0040). Clearly, for the MOE, the model 1a had

insignificant difference in error index than the model 1b atthe probability level of 0.05 (Table 5). For the MOR, the

difference between the error indices for any two models was

significant at the probability level of 0.05.

4. Discussion

Three models (models 1, 1a, and 1b) to predict MOE and

three models (models 2, 2a, and 2b) to predict MOR were

developed using stepwise regression analysis. Explanatory

variables were stand and tree characteristics, and wood

properties such as wood density. Wood density and DBH were

selected to predict MOR (model 2) and crown length along withthe two variables were selected to predict MOE (model 1) by

stepwise regression. The signs of the regression coefficients

were as expected as for all variables in the two models. Wood

density was positively correlated with MOE and MOR. Some

studies (Via et al., 2003; Aubry et al., 1998; Oja et al., 2000,

2001) reported that density was a positive contributor to the

MOE and MOR of softwood species. Log DBH contributed

negatively to both static bending and stiffness of natural black

Table 4

The bias (%) in each DBH class and overall bias (%) for model validation using the test data set

DBH class Number of trees MOE MOR

Model 1 Model 1a Model 1b Model 2 Model 2a Model 2b

10 5 5.41 7.02 7.02 16.12 15.01 11.7412 8 7.49 9.19 9.18 8.10 10.79 2.41

14 10 1.59 0.40 0.39 5.54 2.55 3.7516 6 0.08 3.81 3.81 7.35 0.96 6.5918 3 5.26 8.11 8.15 0.94 0.79 2.1820 6 1.79 4.17 4.15 3.65 0.35 1.1822 7 8.48 4.11 4.13 10.63 8.43 0.10

24 3 12.41 13.78 13.80 12.86 19.35 1.66

26 2 10.47 9.53 9.53 20.09 28.98 7.79

28 2 14.17 9.25 9.27 29.65 26.27 20.20

30 3 4.51 12.15 12.15 19.19 43.14 19.42

Total 55 3.22 3.40 3.41 1.49 4.10 1.41

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spruce trees like the findings in other tree species such as

Norway spruce (Haartveit and Flate, 2002) and four softwood

species (Wang et al., 2002a). Crown length positively

influenced static bending strength and stiffness. From

Table 2, crown width seemed to be closely related to both

MOE and MOR. Becauseit had a strong linear relationship with

DBH ( R2 = 0.51), crown length was not included in the modelsby stepwise regression, through efforts to minimize multi-

collinearity effects by excluding sets of highly correlated

variables (Freund and Littell, 2000).

MOE and MOR are important indicators for the quality of

wood products. The use of MOE and MOR models allowed

industries to estimate thelumber quality of individual trees based

on stand and tree characteristics from forest inventory. However,

forest inventory very rarely collects the wood density data,

although wood density had impacts on bending strength and

stiffness. Models 1a and 2a were useful in a practical sense when

wood density was excluded from the two models. Based on the

model 1a, the impact of either stand density or stem taper wasstatistically significant at the probability level of 0.05. Large

values of stemtaper could affect MOE negatively (Haartveit and

Flate, 2002). Stand density had a positive effect on MOE due to

its effects on growth rate, wood density and branchiness.

The system of two equations (i.e., models 1b and 2b) were

obtained by SUR method. The selected variables in the two

equations and values of R2 were similar to those in the study by

Lei et al. (2005), although the estimated parameters wereobviously different due to different types of black spruce stands

(i.e., plantations in the study of Lei et al. (2005) and natural

stands in this study). MOE was found to be the most important

predictor for bending strength like other studies (e.g., Castera

et al., 1996; Divos and Tanaka, 1997; Green et al., 2001, 2004;

Lei et al., 2005).

Watt et al. (2006) reported that site had a highly significant

effect on MOE across 21 plots covering a wide environmental

range in New Zealand. Results from ANCOVA in this study

showed that the site effect on MOE and MOR was not

significant at the probability level of 0.05. This probably was

because the range of site quality (i.e., from 5.5 to 12.5 m at baseage of 50 years) was still not large enough in our data set. Onthe

other hand, the models developed here were able to account for

variance in MOE or MOR across sites in the study area due to

the insignificant site effect. In this study, all the trees were

within the narrow age range (90–100 years). Therefore, the tree

age was not included in the models.

Since we were more interested in estimating MOE and MOR

using easily measured variables from forest inventory

information, the scatter plots of observed and predicted

MOE and MOR for models 1a and 2a are illustrated in

Fig. 1. It can be observed that the regression line for the two

regression models was close to a straight line. Hence, these

results show that model 1a and 2a could be applied to estimateMOE and MOR from measurements of DBH, crown length,

stem taper, and stand density for black spruce natural stands in

the study area. However, when wood density is available, it is

preferable to use models 1 and 2 in order to improve the

prediction accuracy. And when observed MOE from various

non-destructive testing methods are available, model 2b should

be used to predict MOR more accurately.

5. Conclusion

Lumber static bending modulus of elasticity (MOE) and

modulus of rupture (MOR) are important quality parameters for

Table 5

Comparison of prediction error indices in the two response variables for the models by paired t test based on the test data set

MOE MOR

Model 1 Model 1a Model 1b Model 2 Model 2a Model 2b

Model 1 – 3.02 (0.0040) 3.02 (0.0040)Model 1a – 1.54 (0.1275)

Model 1b –Model 2 – 2.56 (0.0135) 2.74 (0.0085)Model 2a – 4.22 (0.0001)

Model 2b –

Note. Numbers in parentheses are p values for the tests.

Fig. 1. Observed MOE and MOR against predicted MOE and MOR for (a)

model 1a and (b) model 2a, respectively.


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the lumber industry. In this study, a stepwise regression method

was employed to identify the best variables for estimating the

MOE and MOR based on stand and tree characteristics for

black spruce stands grown in Eastern Canada. When modeling

MOE using wood properties and stand and tree characteristics

(model 1), the best explanatory variables were wood density,

DBH, and crown length. They explained approximately 65% of

the total variance in MOE. When only stand and tree

characteristics (models 1a and 1b) were used, the best

explanatory variables were DBH, crown length, stem taper,

and stand density. The four variables explained approximately

55% of the total variance in MOE. In terms of the error index,

the difference in prediction accuracy between the models 1 and

1a (or 1b) for the test data set was statistically significant at the

probability level of 0.05. MOE as an exogenous variable in the

system (i.e., models 2a and 2b) was successfully used to predict

MOR. However, MOR could also be predicted using DBH

alone or using both DBH and wood density although the

prediction accuracy was significantly different based on the

independent test data set. Two models (models 1a and 1b)involving only stand and tree characteristics can be used for

estimating MOE and MOR. In conclusion, the developed MOE

and MOR models in this study were based on stand and tree

characteristics and/or an MOE model that can be obtained by

static, dynamic bending vibration and dynamic longitudinal

vibration as well as statistical approach (Divos and Tanaka,

1997). This study provides an approach to modeling wood

properties from tree and stand characteristics in natural black

spruce stands.

In this paper, we have focused mainly on the development of

models for lumber static bending stiffness and strength models

of natural black spruce stands. The results of this study arelimited to naturally grown black spruce trees. The density of

wood (lumber) in plantation trees may be different from that in

natural stands. Zhang et al. (2002) reported that lumber bending

stiffness and strength from plantation-grown black spruce are

significantly lower than from natural black spruce stands.

Therefore, the results (equation parameter estimates and

predictions) obtained in this study, along with those for

plantation-grown black spruce (Lei et al., 2005), would be very

useful for estimating lumber static bending stiffness and

strength for both natural and plantation-grown black spruce

stands.

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