#### 3.1. Experimental Evidence of a Density-Dependent Height Repression Effect

The density-dependent height repression effect on jack pine and black spruce at approximately 40 years post-establishment within the 3 Nelder plot clusters is shown in

Figure 1: declining mean height with increasing density for trees on identical site qualities and of equal age. Note, due to potential edge effects, inferences were limited to the patterns shown between the nominal initial spacings of 28,621 stems/ha (3rd concentric arc) and 524 stems/ha (23rd concentric arc). Furthermore, using the mean height values of the tallest trees as a surrogate for mean dominant height, the resultant density-dependent height repression effect in terms of site index is illustrated in

Figure 2. Site index was defined as the mean dominant height at a base-age of 50 years breast-height as determined from Ontario-based site index functions developed for jack pine and black spruce by Carmean ([

23] and [

24], respectively). These graphical trends indicated a reduction in productivity with increasing density for both species across all 3 clusters. The pattern of the reduction was nonlinear in nature: site index estimates increased, reaching a plateau and then slightly declined as density decreased. Note, for jack pine for which the data permitted, the repression effect intensified over time as indicated by a gradual shift in the asymptote (peak) towards the lower densities and by the increasing gradient (slope) of the height-density relationship with age (graphic not shown). Graphically assessing the height–density relationship pattern for jack pine at 16, 20 and 40 years post-establishment, and for black spruce at 41 years post-establishment, indicated a consistent pattern of declining productivity with increasing density between 1425 stems/ha and 28,621 stems/ha (e.g.,

Figure 1 and

Figure 2). However, the rate of decline was species-specific and visually more evident for jack pine than for black spruce.

**Figure 1.**
Box and whisker plots graphically illustrating the observed relationship between initial spacing and mean height at approximately 40 year post-establishment by species and plot series. Note, the median value is denoted by the solid square within the open rectangle, first and third quartiles are denoted by the lower and upper horizontal sides of the open rectangle, respectively, and minimum and maximum values are denoted by the end points of the lower and upper whiskers, respectively.

**Figure 1.**
Box and whisker plots graphically illustrating the observed relationship between initial spacing and mean height at approximately 40 year post-establishment by species and plot series. Note, the median value is denoted by the solid square within the open rectangle, first and third quartiles are denoted by the lower and upper horizontal sides of the open rectangle, respectively, and minimum and maximum values are denoted by the end points of the lower and upper whiskers, respectively.

#### 3.2. Quantifying the Density-Dependent Height Repression Effect

As presented in

Table 1, the occurrence of spatial correlation based on time series statistics was quantified by the ratio of the number of plots exhibiting significant (

p ≤ 0.05) partial autocorrelation coefficients to the total number of plots assessed by species, cluster and measurement time. Thus, based on the data from the initial spacing densities of 1425 stems/ha to 28.621 stems/ha, the frequency of spatial correlation as detected by the partial autocorrelation coefficient was relatively low. Specifically, the results indicated significant (

p ≤ 0.05) spatial correlation only at the 1st spatial lag. Only 27% (

n = 22) of all the jack pine relationships assessed and only 33% (

n = 9) of all the black spruce relationships, exhibited spatial correlation effects (

Table 1). For jack pine, the occurrence of significant (

p ≤ 0.05) partial autocorrelation coefficients increased over time: 18%, 29% and 33% of the relationships at age 16, 20 and 40 years, respectively (

Table 1). For both species, the partial autocorrelation coefficients were only significant (

p ≤ 0.05) at the 1st spatial lag suggesting that when present, spatial correlation effects were operating only between first order neighbours (

Table 1).

**Figure 2.**
Box and whisker plots graphically illustrating the observed relationship between initial spacing and site productivity by species and plot series. Note, box and whisker plot denotations are given in

Figure 1.

**Figure 2.**
Box and whisker plots graphically illustrating the observed relationship between initial spacing and site productivity by species and plot series. Note, box and whisker plot denotations are given in

Figure 1.

The results from the second part of the spatial autocorrelation detection procedure, which employed regression analysis to determine the significant association among residual errors, indicated that the density-dependent effects on height development were frequent: significant (

p ≤ 0.05) linear relationships with negative slope estimates in 54 of the 83 jack pine relationships (65%), and in 24 of the 27 black spruce relationships (89%). In regards to jack pine, the occurrence of density-dependent height repression increased over time: significant (

p ≤ 0.05) linear relationships with negative slope estimates detected in 50%, 57% and 89% of the relationships at ages 16, 20 and 40, respectively. As presented in

Table 1, the occurrence of spatial correlation based on regression analysis was quantified by the ratio of the number of plots exhibiting significant (

p ≤ 0.05) correlation among adjacent regression residuals (

${\widehat{p}}_{1}$) to the total number of plots assessed by species, cluster and measurement time. Over all the clusters and measurement times, the results indicated that the frequency of significant residual regression relationships was low for jack pine (11%) and totally absence in regards to black spruce (

Table 1). For the significant jack pine relationships, the magnitude of

${\widehat{p}}_{1}$ was considerably less than unity: mean (min/max) absolute

${\widehat{p}}_{1}$ of 0.571 (0.462/0.733). These results indicate that the evidence of significant spatial autocorrelation effects was infrequent for jack pine and completely lacking for black spruce.

**Table 1.**
Plot-level detection of spatial autocorrelation by species and measurement age.

**Table 1.**
Plot-level detection of spatial autocorrelation by species and measurement age.
Species | Age | Nelder Plot Series | Spatial Autocorrelation Detection via Time Series Statistics ^{a} | Spatial Autocorrelation Detection via Regression Statistics ^{b} |
---|

| | | Frequency of significant partial autocorrelation coefficients (n_{s}/n_{t}) | Spatial lag | Frequency of significant density-dependent height repression effects (n’_{s}/n_{t}) | Frequency of significant spatial autocorrelation (n’’_{s}/n’_{s}) |

Jack | 16 | Dunmore | 1/10 | 1st | 3/10 | 1/3 |

Pine | Willison | 3/9 | 1st | 6/9 | 0/6 |

| Terry | 1/9 | 1st | 5/9 | 0/5 |

| 20 | Dunmore | 1/10 | 1st | 3/10 | 1/3 |

| Willison | 2/9 | 1st | 7/9 | 0/7 |

| Terry | 5/9 | 1st | 6/9 | 1/6 |

| 40 | Dunmore | 3/10 | 1st | 8/10 | 0/8 |

| Willison | 2/9 | 1st | 8/9 | 2/8 |

| Terry | 4/8 | 1st | 8/8 | 1/8 |

Black | 41 | Dunmore | 3/10 | 1st | 8/10 | 0/8 |

Spruce | Willison | 3/9 | 1st | 8/9 | 0/8 |

| Terry | 3/8 | 1st | 8/8 | 0/8 |

Therefore, based on the results of these spatial correlation detection analyses which indicated that low levels of spatial autocorrelation were present in only a minority of the plots irrespective of species, it was concluded that there was insufficient evidence to negate the use the OLS parameterization methods in quantifying the height-density relationships. Thus assuming that spatial correlation effects were minimal and inconsequential, a common OLS regression analytical framework was used to estimate the density-dependent height repression parameter: ${\text{\beta}}_{1}$ from the ${H}_{d}={\widehat{\beta}}_{0}+{\widehat{\beta}}_{1}{N}_{e}$ relationship.

For each of the resultant significant relationships, the presence of potential outliers and influential observations were determined via the use of predictor variable—raw residual graphs in association with residual statistics (

i.e., studentized deleted residuals and Cook’s distance measures, respectively, where the probability level for exclusion was set at 0.01 for both measures [

25]). Although no influential observations were detected, outliers were present in 19 of the relationships. These outliers were removed and the models reparameterized and reassessed for compliance with regression assumptions, including the reassessment of the presence of spatial autocorrelation effects.

Table A1 provides the parameter estimates and associated regression statistics for the resultant 54 jack pine and 24 black spruce significant (

p ≤ 0.05) relationships. The associated residual analyses indicated that the models satisfactorily described the

H_{d}-N_{e} relationships. The regression equations explained a moderate level of variation: mean (minimum/maximum) coefficient of determination (

r^{2}) values of 0.512 (0.279/0.760), 0.560 (0.312/0.854), 0.683 (0.387/0.889) for jack pine at ages 16, 20 and 40, respectively, and 0.602 (0.373/0.919) for black spruce at age 41. The magnitude of error was acceptable as reflected by moderate levels of the standard error of the estimate: mean (minimum/maximum) of 0.320 m (0.190/0.455), 0.439 m (0.218/0.736), 0.866 m (0.434/1.270) for jack pine at age 16, 20 and 40, respectively, and 0.834 m (0.369/1.399) for black spruce at age 41. Overall, the regression relationships were in general compliance with the constant variance and normality assumptions underlying OLS parameterization (e.g., horizontal band and invariant pattern of raw residuals when plotted against the predictor variable).

The frequency and magnitude of the density-dependent height repression effect increased over time as evident by the temporal increase in the number of significant relationships for jack pine (14, 16 and 24 at 16, 20 and 40 years post-establishment, respectively) and increasing negativity of slope estimates: mean (minimum/maximum) slope values of −0.000039 (−0.000061/−0.000021), −0.000060 (−0.000114/0.000023) and −0.000152 (−0.000224/−0.000224) at 16, 20 and 40 years post-establishment, respectively (

Table A1). For jack pine, where multiple temporal relationships were available, the height repression effect was linear in nature and intensified over time as inferred from the temporal increase in the negativity of the slope values (

Figure 3a;

Table 2). In fact, the intensity of the repression effect as measured by the magnitude of the slope parameter estimate increased almost 5 fold from age 16 to age 40: the mean slope was −0.00003 at age 16

versus its value of −0.00014 at age 40.

Figure 3b presents the height–density relationship for black spruce for the remeasurements taken at age 41 for each of the 3 Nelder plot clusters whereas

Figure 3c illustrates the pooled relationship for each species, approximately 40 years after establishment.

**Figure 3.**
Bivariate scatter plots illustrating the relationship between initial density and mean height for (

**a**) jack pine at approximately 16, 20 and 40 years post-establishment for each Nelder plot series; (

**b**) black spruce at approximately 40 years post-establishment for each Nelder plot series; and (

**c**) jack pine and black spruce at approximately 40 years post-establishment for all the Nelder plots combined. The whiskers defined the mean ± standard deviation (SD) values and the dotted line is the estimated height-density relationship for the height repression coefficients given in

Table 2.

**Figure 3.**
Bivariate scatter plots illustrating the relationship between initial density and mean height for (

**a**) jack pine at approximately 16, 20 and 40 years post-establishment for each Nelder plot series; (

**b**) black spruce at approximately 40 years post-establishment for each Nelder plot series; and (

**c**) jack pine and black spruce at approximately 40 years post-establishment for all the Nelder plots combined. The whiskers defined the mean ± standard deviation (SD) values and the dotted line is the estimated height-density relationship for the height repression coefficients given in

Table 2.

**Table 2.**
Resultant mean dominant height-density relationship by species, age and plot series inclusive of the density-dependent height repression coefficient estimates and their 95% confidence limits.

**Table 2.**
Resultant mean dominant height-density relationship by species, age and plot series inclusive of the density-dependent height repression coefficient estimates and their 95% confidence limits.
Species | Age | Plot Series | n | Intercept | Density-Dependent Height Repression Coefficient ^{a} |
---|

| | | | | 95% CI-L | ${\widehat{\beta}}_{1}^{*}$ | 95% CI-U |

Jack Pine | 16 | Dunmore | 3 | 6.634 | −0.000130 | −0.000036 | 0.000058 |

Willison | 6 | 7.028 | −0.000080 | −0.000038 | 0.000004 |

Terry | 5 | 6.019 | −0.000059 | −0.000026 | 0.000007 |

Combined | 14 | 6.563 | −0.000050 | −0.000031 | −0.000012 |

20 | Dunmore | 3 | 9.215 | −0.000216 | −0.000060 | 0.000096 |

Willison | 7 | 9.993 | −0.000094 | −0.000047 | 0.000000 |

Terry | 6 | 8.840 | −0.000077 | −0.000036 | 0.000005 |

Combined | 16 | 9.384 | −0.000067 | −0.000042 | −0.000018 |

40 | Dunmore | 8 | 17.007 | −0.000256 | −0.000139 | −0.000021 |

Willison | 8 | 17.032 | −0.000232 | −0.000124 | −0.000017 |

Terry | 8 | 16.385 | −0.000286 | −0.000154 | −0.000023 |

Combined | 24 | 16.789 | −0.000200 | −0.000137 | −0.000078 |

Black Spruce | 41 | Dunmore | 8 | 11.300 | −0.000202 | −0.000108 | −0.000015 |

Willison | 8 | 12.367 | −0.000198 | −0.000106 | −0.000014 |

Terry | 8 | 10.624 | −0.000185 | −0.000099 | −0.000014 |

Combined | 24 | 11.429 | −0.000150 | −0.000104 | −0.000059 |