Table 16 Estimated parameters and goodness of fit of the H–D models for N. pumilio fitted to the complete database

Simple H–D model

\({H}_{ij}=1.3+\frac{DB{H}_{ij}^{2}}{{\left(a+b\times {DBH}_{ij}\right)}^{2}}+{\varepsilon }_{ij}\)

 a

2.827768

0.133800

3.48

2.75

0.63

3.48

 b

0.205948

0.003295

Simple NLME model

\({H}_{ij}=1.3+\frac{DB{H}_{ij}^{2}}{{\left(\left(a+{u}_{i}\right)+b\times DB{H}_{ij}\right)}^{2}}+{\varepsilon }_{ij}\)

\({u}_{i}N\left(0,D\right)\)

\({\varepsilon }_{i}N\left[0,{R}_{i}\right]\)

 a

2.8369380

0.185100

3.40

2.66

0.64

3.45

 b

0.2061832

0.003319

\({\sigma }_{u}^{2}\)

0.09093102

0.301548

\({\sigma }^{2}\)

1.18632864

1.089187

\(\delta\)

0.2925926

Generalized H–D model

\({H}_{ij}=1.3+\frac{DB{H}_{ij}^{2}}{{\left(a0+a1\times DBHma{x}_{i}+\left(b1\times D{q}_{i}+b2\times NlogN{P}_{i}\right)\times {DBH}_{ij}\right)}^{2}}+{\varepsilon }_{ij}\)

 a0

4.202

6.924e-01

3.60

2.81

0.60

3.61

 a1

-7.122e-03

3.539e-03

 b1

1.860e-03

8.561e-05

 b2

2.923e-02

2.715e-03

Generalized NLME H–D model

\({H}_{ij}=1.3+\frac{{DBH}_{ij}^{2}}{{\left(a0+{u}_{i}+a1\times DBHma{x}_{i}+\left(b1\times D{q}_{i}+b2\times NlogN{P}_{i}\right)\times {DBH}_{ij}\right)}^{2}}+{\varepsilon }_{ij}\)

\({u}_{i}N\left(0,D\right)\)

\({\varepsilon }_{i}N\left[0,{R}_{i}\right]\)

 a0

2.935442

1.3530663

3.42

2.66

0.64

3.49

 a1

0.000124

0.0069137

 b1

0.001779

0.0001023

 b2

0.031011

0.0030591

\({\sigma }_{u}^{2}\)

0.190482

0.4364421

\({\sigma }^{2}\)

1.151112

1.0728989

\(\delta\)

0.297914