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Official Journal of the Japan Wood Research Society

Table 3 Developed model for the radial variation of wood properties and AIC values of each model

From: Relationship between the xylem maturation process based on radial variations in wood properties and radial growth increments of stems in a fast-growing tree species, Liriodendron tulipifera

Model

Equation

Random effects

AIC

Wood fiber length

Vessel element length

Vessel frequency

Vessel diameter

Wood fiber diameter

Wood fiber wall thickness

Air-dry density

MOE

MOR

Compressive strength

b-1

\(\mathrm{WP}_{ij}=\left({b}_{0}+{Tree}_{0j}\right)\mathrm{CA}_{ij}+{b}_{1}+{e}_{ij}\)

Slope

 − 42.601

 − 202.72

590.04

407.34

292.43

119.05

 − 181.55

129.18

367.59

283.83

b-2

\(\mathrm{WP}_{ij}={b}_{0}\mathrm{CA}_{ij}+{b}_{1}+{{Tree}_{1j}+e}_{ij}\)

y-intercept

 − 39.643

 − 194.19

589.34

411.29

251.08

85.408

 − 198.38

132.31

360.97

278.70

c-1

\(\mathrm{WP}_{ij}=\left({c}_{0}+{Tree}_{0j}\right)\mathrm{ln}\left(\mathrm{CA}_{ij}\right)+{c}_{1}+{e}_{ij}\)

Slope

 − 55.725

 − 213.43

562.33

388.96

 − 

 − 

 − 193.37

122.75

359.30

278.85

c-2

\(\mathrm{WP}_{ij}={c}_{0}\mathrm{ln}\left(\mathrm{CA}_{ij}\right)+{c}_{1}+{Tree}_{1j}+{e}_{ij}\)

y value at 1st annual ring from pith

 − 54.718

 − 

559.09

388.93

244.59

67.019

 − 199.62

126.48

356.64

275.82

d-1

\(\mathrm{WP}_{ij}=\left({d}_{0}+{Tree}_{0j}\right){\mathrm{CA}_{ij}}^{2}+{d}_{1}\mathrm{CA}_{ij}+{d}_{2}+{e}_{ij}\)

Opening quadratic parabolas

 − 42.334

 − 

573.03

403.80

330.40

165.56

 − 152.97

131.17

374.51

290.87

d-2

\(\mathrm{WP}_{ij}={d}_{0}{\mathrm{CA}_{ij}}^{2}+\left({d}_{1}+{Tree}_{1j}\right)\mathrm{CA}_{ij}+{d}_{2}+{e}_{ij}\)

Slope of the tangent line at the y-intercept

 − 42.471

 − 195.05

571.63

402.35

299.73

128.39

 − 167.53

132.70

369.72

289.05

d-3

\(\mathrm{WP}_{ij}={d}_{0}{\mathrm{CA}_{ij}}^{2}+{d}_{1}\mathrm{CA}_{ij}+\left({d}_{2}+{Tree}_{2j}\right)+{e}_{ij}\)

y-intercept

 − 41.656

 − 

566.65

402.12

254.35

 − 

 − 183.35

137.92

 − 

284.33

  1. MOE, modulus of elasticity; MOR, modulus of rupture; WPij, wood properties at the ith cambial age of the jth individual tree; CAij, ith cambial age of the jth individual tree; b0, b1, c0, c1, d0, d1, and d2, fixed-effect parameters; Tree0j, Tree1j, and Tree2j, random-effect parameters at the jth individual tree level; eij, residuals; − , the model failed to converge. The bold value represents the minimum AIC value in each wood property