Figure 1
Figure 2
Description of the experimental data used to derive the height–diameter relationships for Robinia pseudoacacia L_
| Sample plot | Parental genotype | Geographic region | Plot size (m2) | Growing space (m2) | Age (years) | Number of measured trees | d (cm) | h (m) |
|---|---|---|---|---|---|---|---|---|
| PP 1 | – | southern | 185.00 | 2.5 | 6 | 48 | 3.6–7.1 | 4.7–6.3 |
| PP 2 | – | southern | 173.25 | 2.5 | 2 | 51 | 3.1–6.0 | 4.7–6.5 |
| PP 3 | – | southern | 202.50 | 2.5 | 3 | 50 | 3.8–4.6 | 6.4–6.5 |
| PP 4 | – | southern | 165.00 | 3.5 | 4 | 67 | 1.5–4.6 | 2.9–4.7 |
| PP 5 | Roszin Varga | northern | 304.00 | 4.5 | 5 | 64 | 2.3–6.7 | 2.8–7.7 |
| PP 6 | Karaisen | northern | 306.04 | 4.5 | 5 | 52 | 2.4–7.6 | 3.9–9.2 |
| PP 7 | Tsarevets | northern | 300.00 | 4.5 | 5 | 70 | 2.2–9.1 | 4.2–9.2 |
Comparison of the mixed-effects height–diameter models by Likelihood Ratio Test
| № | Model | Mixed-effects parameters | df | logLik | Test | Likelihood Ratio | p-value | |
|---|---|---|---|---|---|---|---|---|
| abbreviation | localisation level | |||||||
| 1 | GR.PP.01 | sample plot within geographic region | b0, b1 | 9 | –413.990 | |||
| 2 | PP.01 | sample plot | b0, b1 | 6 | –414.324 | 1 vs. 2 | 0.669 | 0.881 |
| 3 | PP.0 | sample plot | b0 | 4 | –419.808 | 2 vs. 3 | 10.967 | 0.004 |
| 4 | GR.PP.0 | sample plot within geographic region | b0 | 5 | –419.407 | 3 vs. 4 | 0.800 | 0.371 |
| 5 | PP.1 | sample plot | b1 | 4 | –430.327 | 4 vs. 5 | 21.839 | <0.001 |
| 6 | GR.PP.1 | sample plot within geographic region | b1 | 5 | –430.130 | 5 vs. 6 | 0.393 | 0.531 |
| 7 | GR.01 | geographic region | b0, b1 | 6 | –438.471 | 6 vs. 7 | 16.682 | <0.001 |
| 8 | GR.0 | geographic region | b0 | 4 | –440.748 | 7 vs. 8 | 4.553 | 0.103 |
| 9 | GR.1 | geographic region | b1 | 4 | –445.830 | |||
| 10 | simple local | no localisation | – | 3 | –453.059 | 9 vs 10 | 14.457 | <0.001 |
Calibration of the random component of the mixed-effects model with 1-tree height–diameter measurement: accuracy estimates by plots
| Plot | u | Relative errors | MARE% | Bias | ||||
|---|---|---|---|---|---|---|---|---|
| Perc0 | Perc25 | Perc50 | Perc75 | Perc100 | ||||
| PP 1 | –0.789 | –0.719 | –0.150 | –0.077 | –0.003 | 0.166 | 11.829 | –0.363 |
| PP 2 | –0.561 | –0.579 | –0.133 | –0.037 | 0.016 | 0.205 | 11.001 | –0.233 |
| PP 3 | 0.369 | –0.100 | –0.018 | 0.017 | 0.076 | 0.153 | 5.423 | 0.163 |
| PP 4 | –0.981 | –0.350 | –0.182 | –0.067 | 0.063 | 0.391 | 14.612 | –0.228 |
| PP 5 | –0.125 | –0.405 | –0.116 | –0.011 | 0.086 | 0.198 | 10.536 | –0.050 |
| PP 6 | 1.118 | –0.390 | –0.047 | 0.086 | 0.167 | 0.242 | 13.004 | 0.506 |
| PP 7 | 0.968 | –0.673 | –0.025 | 0.068 | 0.118 | 0.238 | 12.661 | 0.403 |
Regression estimates and goodness-of-fit statistics of the mixed-effects and the generalised deterministic height–diameter models
| Mixed-effects model: h = 1.3 + (u + b0)eb1 | ||||||
| ME 0.773 | Model parameters | Model variances | ||||
| b0 | b1 |
| σ2 | |||
| MARE% 10.322 | Estimate | 8.302 | –2.631 | 0.639 | 0.555 | |
| Bias 0.010 | SE | 0.395 | 0.146 | 0.363 | 0.042 | |
| Generalised deterministic model: h = 1.3 + (a0 + a1Spacing)eb1/d | ||||||
| ME 0.724 | Variance function | Model parameters | ||||
| (θ+dη)2 | a0 | a1 | b1 | |||
| MARE% 11.634 | Variance function parameters | Estimate | 7.322 | 0.422 | –2.918 | |
| Bias 0.025 | Θ | H | SE | 0.417 | 0.087 | 0.123 |
| 37.243 | 1.701 | |||||
Local height–diameter models and goodness-of-fit statistics
| Model | Mean Bias (m) | Aggregated MSE (m)* | 90th percentile of ARBi* | |
|---|---|---|---|---|
| abbreviation | formula | |||
| M1 | h = 1.3 + a1d | 0.1515 | 6.9068 | 0.2184 |
| M2 | h = 1.3 + a1d + a2d2 | –0.0024 | 3.8496 | 0.1367 |
| M3 | h = a0da1 | –0.0031 | 4.0965 | 0.1176 |
| M4 | h = 1.3 + exp[a0 + a1/(d + 1)] | –0.0034 | 3.8616 | 0.1100 |
| M5 | h = 1.3 + a0 exp(a1/d) | –0.0005 | 3.7790 | 0.1077 |
| M6 | h = 1.3 + a0da1 | –0.0053 | 4.1714 | 0.1191 |
| M7 | h = 1.3 + a0d/(a1 + d) | 0.1221 | 6.9214 | 0.2412 |