Table 1. The fixed-effects ingrowth model for Scots pine estimated with the zeroinfl function in R package pscl. The function estimates the dispersion parameter in form log(1/α). G is the stand basal area, VT and CT are the indicator variables for sub-xeric and xeric or poorer forest types. Log(T) and log(A) are used as offsets, where A = 100 m^{2} is the area of plot and T = 5 yrs is the length of the growth period. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Count model (negative binomial with log link) | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –6.82 | 0.41 | <2e–16 |

G_{pine} | –0.131 | 0.112 | 0.023 |

0.783 | 0.308 | 0.011 | |

VT | 0.633 | 0.250 | 0.011 |

CT | 1.64 | 0.34 | 1.2e–06 |

ln(1/α) | –1.16 | 0.209 | |

Zero-inflation model coefficients (binomial with logit link) | |||

Intercept | –13.6 | 3.5 | 0.0001 |

ln(G_{pine} + 0.01) | –0.307 | 0.083 | 0.00021 |

√G | 6.1 | 1.5 | 5.9E–05 |

G | –0.58 | 0.16 | 0.0003 |

VT | –1.21 | 0.36 | 0.00075 |

α = 3.19; Log-likelihood –905.2 on 11 degrees of freedom. |

Table 2. The fixed-effects ingrowth model for Norway spruce. OMT is an indicator variable for herb-rich or better site. TS is the temperature sum. The other symbols are as in Table 1. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Count model (negative binomial with log link) | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –6.61 | 0.54 | <2e–16 |

log(G_{spruce} + 0.01) | 0.148 | 0.029 | 2.9e–07 |

max(G_{spruce} – 13.0) | –0.0207 | 0.0109 | 0.056 |

TS | 0.00126 | 0.00048 | 0.0082 |

log(1/α) | –0.83 | 0.11 | |

Zero-inflation model coefficients (binomial with logit link) | |||

Intercept | –1.75 | 0.46 | 8.6e–05 |

log(G_{spruce} + 0.01) | –0.359 | 0.073 | 9.0e–07 |

OMT | 1.12 | 0.43 | 0.0085 |

CT | 1.10 | 0.48 | 0.021 |

α = 2.28; Log-likelihood –22987 on 9 degrees of freedom. |

Table 3. The fixed-effects ingrowth model for birch (combined ingrowth of silver and downy birch). The symbols are as in Table 1. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Count model (negative binomial with log link) | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –3.15 | 0.18 | <2e–16 |

G_{pine} | 0.0923 | 0.0090 | <2e–16 |

G_{birch} | –0.109 | 0.045 | 0.016 |

√G_{birch} | 0.349 | 0.150 | 0.020 |

G | –0.113 | 0.0091 | <2e–16 |

VT | –0.50 | 0.16 | 0.0016 |

CT | –0.87 | 0.29 | 0.0032 |

log(1/α) | –1.013 | 0.092 | |

Zero-inflation model (binomial with logit link) | |||

Intercept | –6.40 | 1.57 | 0.000044 |

log(G_{birch} + 0.01) | –0.565 | 0.17 | 0.00078 |

G | 0.142 | 0.033 | 0.000014 |

VT | 0.822 | 0.51 | 0.11 |

CT | 2.41 | 0.75 | 0.0014 |

α = 2.57; Log-likelihood –2700 on 13 degrees of freedom. |

Table 4. Statistics for the fixed-effects ingrowth models. Row “P(Residual > 0)” gives the proportion of positive residuals. | ||||

Variable | Minimum | Maximum | Mean | sd |

Scots pine | ||||

Ingrowth | 0 | 68 | 0.90 | 4.07 |

Censored ingrowth | 0 | 5 | 0.457 | 1.26 |

Probability of extra zeroes (p) | 0.0009 | 0.97 | 0.68 | 0.28 |

Prediction | 0.018 | 8.10 | 0.890 | 1.38 |

Residual | –8.10 | 64.34 | –0.0016 | 3.89 |

P(Residual > 0) | 0.11 | |||

Pearson residual | –0.55 | 28.79 | –0.0020 | 1.13 |

Censored prediction | 0.017 | 2.39 | 0.47 | 0.52 |

Censored residual | –2.37 | 4.90 | –0.001 | 1.15 |

Censored Pearson residual | –1.05 | 9.91 | –0.007 | 0.93 |

Norway spruce | ||||

Ingrowth | 0 | 47 | 2.61 | 5.3 |

Censored ingrowth | 0 | 5 | 1.47 | 1.95 |

Probability of extra zeroes (p) | 0.04 | 0.74 | 0.22 | 0.20 |

Prediction | 0.27 | 5.0 | 2.60 | 1.29 |

Residual | –4.84 | 43.6 | 0.010 | 5.16 |

P(Residual > 0) | 0.27 | |||

Pearson residual | –0.61 | 12.97 | 0.0010 | 1.08 |

Censored prediction | 0.24 | 2.27 | 1.49 | 0.59 |

Censored residual | –2.25 | 4.71 | –0.022 | 1.84 |

Censored Pearson residual | –1.04 | 4.97 | –0.014 | 0.98 |

Birch (silver birch and downy birch) | ||||

Ingrowth | 0 | 120 | 5.45 | 12.8 |

Censored ingrowth | 0 | 5 | 1.77 | 2.14 |

Probability of extra zeroes (p) | 0.0015 | 0.940 | 0.17 | 0.21 |

Prediction | 0.035 | 23.4 | 5.33 | 4.29 |

Residual | –19.4 | 102.7 | 0.12 | 11.7 |

P(Residual > 0) | 0.25 | |||

Pearson residual | –0.60 | 11.4 | –0.0023 | 1.02 |

Censored prediction | 0.035 | 3.31 | 1.75 | 0.80 |

Censored residual | –3.19 | 4.48 | 0.023 | 1.95 |

Censored Pearson residual | –1.47 | 6.03 | 0.0098 | 1.00 |

Table 5. The mixed-effects ingrowth model for Scots pine. VT is the sub-xeric type, and CT is the xeric type. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Conditional model | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –6.47 | 0.56 | <2e–16 |

ln(G_{pine} + 0.01) | 0.323 | 0.074 | 1.4E–05 |

√G | –1.04 | 0.13 | 6.4E–15 |

VT | 0.73 | 0.23 | 0.0018 |

CT | 1.23 | 0.37 | 0.00082 |

Zero-inflation model | |||

Intercept | –5.40 | 0.88 | 8.6e–10 |

α = 0.135; σ = 3.06; Log-likelihood = –820.6 on 7 Df. |

Table 6. The mixed-effects ingrowth model for Norway spruce. TS is the temperature sum and CT is the xeric type. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Count model | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –7.58 | 0.69 | <2e–16 |

ln(G_{spruce} + 0.01) | 0.287 | 0.032 | <2e–16 |

max(G_{spruce} – 13.0) | –0.0532 | 0.0126 | 2.4e–05 |

LS | 0.000716 | 0.00060 | 0.235 |

Zero-inflation model | |||

Intercept | –3.74 | 0.27 | <2e–16 |

ln(G_{spruce} + 0.01) | –0.32 | 0.12 | 0.0096 |

CT | 1.31 | 1.02 | 0.20 |

α = 0.0071; σ = 1.79; Log-likelihood = –2156.2. |

Table 7. The mixed-effects ingrowth model for birch. | |||

Predictor | Estimate | Std. Error | Pr(>|z|) |

Conditional model | |||

log(T) | 1 | ||

log(A) | 1 | ||

Intercept | –4.27 | 0.20 | <2e–16 |

G | –0.127 | 0.010 | <2e–16 |

log(G_{birch} + 0.01) | 0.221 | 0.030 | 1.6e–13 |

G_{pine} | 0.0815 | 0.0107 | 2.7e–14 |

CT | –0.21 | 0.19 | 0.26 |

Zero-inflation model | |||

Intercept | –5.03 | 0.86 | 4.2e–09 |

G | 0.0929 | 0.033 | 0.004 |

α = 0.101; σ = 2.056; Log-likelihood = –2615.2. |

Table 8. Statistics for the mixed-effects ingrowth models. Notations (11) and (12) refer to Eqs. 11 and 12, respectively. | ||||

Variable | min | max | mean | sd |

Scots pine | ||||

Probability of extra zeroes (p) | 0.0045 | 0.0045 | 0.0045 | 0 |

Prediction | 0.0001 | 0.767 | 0.039 | 0.072 |

Prediction | 0.010 | 82.91 | 4.21 | 7.84 |

Residual (11) | –0.6668 | 67.87 | 0.8567 | 4.05 |

P(Residual (11) > 0) | 0.16 | |||

Residual (12) | –72.12 | 54.48 | –3.318 | 7.60 |

P(Residual(12) > 0) | 0.055 | |||

Pearson residual (11) | –8.01E–05 | 0.260 | 0.002 | 0.012 |

Pearson residual (12) | –0.0087 | 0.252 | –0.007 | 0.012 |

Censored prediction | 0.0075 | 1.86 | 0.390 | 0.335 |

Censored residual | –1.80 | 4.83 | 0.068 | 1.17 |

Censored Pearson residual | –0.84 | 6.80 | 0.019 | 0.94 |

Norway spruce | ||||

Probability of extra zeroes (p) | 0.0068 | 0.277 | 0.042 | 0.060 |

Prediction | 0.090 | 1.37 | 0.695 | 0.382 |

Prediction | 0.45 | 6.83 | 3.47 | 1.90 |

Residual (11) | –1.30 | 46.27 | 1.91 | 5.25 |

P(Residual(11) > 0) | 0.43 | |||

Residual (12) | –6.47 | 43.37 | –0.86 | 5.20 |

P(Residual(12) > 0) | 0.24 | |||

Pearson residual (11) | –0.040 | 6.560 | 0.12 | 0.39 |

Pearson residual (12) | –0.20 | 6.41 | –0.038 | 0.39 |

Censored prediction | 0.33 | 2.07 | 1.34 | 0.54 |

Censored residual | –2.02 | 4.64 | 0.13 | 1.85 |

Censored Pearson residual | –0.99 | 4.81 | 0.070 | 1.07 |

Birch (silver birch and downy birch) | ||||

Probability of extra zeroes (p) | 0.0074 | 0.45 | 0.055 | 0.050 |

Prediction (11) | 0.0070 | 5.50 | 1.18 | 1.09 |

Prediction (12) | 0.058 | 45.5 | 9.79 | 9.00 |

Residual (11) | –5.23 | 117.2 | 4.27 | 12.4 |

P(Residual(11) > 0) | 0.43 | |||

Residual (12) | –43.3 | 96.7 | –4.35 | 12.9 |

P(Residual(12) > 0) | 0.18 | |||

Pearson residual (11) | –0.014 | 2.30 | 0.057 | 0.17 |

Pearson residual (12) | –0.12 | 2.20 | –0.041 | 0.17 |

Censored prediction | 0.052 | 3.27 | 1.63 | 0.75 |

Censored residual | –3.22 | 4.57 | 0.142 | 1.98 |

Censored Pearson residual | –1.55 | 4.81 | 0.080 | 1.07 |