Estimating The Self-Thinning Boundary Line As A Density-Dependent Stochastic Biomass Frontier

Huiquan Bi




Abstract. The self-thinning rule describes a density-dependent upper boundary of stand biomass for even-aged pure plant stands in a given environment. The econometric approach of stochastic frontier production functions is adopted to estimate the self-thinning boundary line as a density-dependent stochastic biomass frontier using data from even-aged Pinus radiata stands. This method uses all the data points, but recognizes the differences in site occupancy among them. Since no subjective data selection is involved and no information contained in the data is lost, the line can be estimated without subjectivity and more efficiently than the common methods of subjective data selection. As a result, statistical inferences about the estimated self-thinning boundary line can be made objectively and more precisely. In addition, the estimates of site occupancy provide further insight into the dynamics of self-thinning stands.

Key words: Pinus radiata; self-thinning rule; Site occupancy; stochastic biomass frontier; stochastic frontier function.


The self-thinning rule describes a density-dependent upper boundary of stand biomass for even-aged pure plant stands in a given environment by relating mean plant weight to stand density on log scales with a constant slope of -1.5:

log W = C – l.5 log N (1)

where W is mean plant mass, N is stand density and C is a constant (Yoda et al. 1963, White and Harper 1970, White 1980, 1981, 1985, Westoby 1984, Whittington 1984). An equivalent expression of the rule is

log B = C -0.5 log N (2)

where B is the total stand biomass per unit area, the product of Wand N (Westoby and Howell 1981, Westoby 1984).

Since the initial acceptance of the rule, the slope of Eq. 2 has been found to be much more variable than a constant of -0.5 as the rule states (Sprugel 1984, Zeide 1985, von Gadow 1986, Westoby and Howell 1986, Weller 1987a, b). This observed variability has led some to question the fundamental validity and existence of the rule, and eventually to advocate its rejection (Sprugel 1984, Zeide 1985, 1987, von Gadow 1986, Weller 1987a, b, 1990, 1991), while others have been more cautious in doing so (Osawa and Sugita 1989, Lonsdale 1990, Zeide 1991, Sackville Hamilton et al. 1995, Guo and Rundel 1998). The debate on the rule holds a great deal of interest in plant population dynamics and management of even-aged forest stands, because of its theoretical significance (White 1981, Pickard 1983, Long and Smith 1984, Westoby 1984, Yamakura 1985, Voit 1988, Adler 1996) and demonstrated practical implications (Drew and Flewelling 1977, 1979, Long 1985, Hibbs 1987, Tang et al. 1994, 1995, Newton 1997). So far, the deba te has not been adequately resolved.

Much of the debate has centered on the empirical validation of the self-thinning rule, where the lack of objectivity has become a stumbling block. The common methods of estimating the boundary line have been to choose data points that exhibit density-dependent mortality and lie close to an arbitrarily visualized upper boundary before the line is estimated, and to subjectively eliminate data points from populations that are believed to be not undergoing density-dependent mortality. The selected data points are then used to estimate the line through principal component analysis (Mohler et al. 1978, West and Borough 1983, Weller 1987a, Osawa and Sugita 1989, Lonsdale 1990) or reduced major axis regression (Zeide 1991, Osawa and Allen 1993). The subjectivity implies that the same data set can be analyzed by some to provide evidence for rejection, and re-analyzed by others to reach a different conclusion (e.g., Weller 1987a, b, 1990, 1991, Osawa and Sugita 1989, Lonsdale 1990). As pointed out by Weller (1990), th ese methods are arbitrary, subjective, and even circular, but methods to avoid these problems remain undeveloped. Recent works suggest a more objective way of selecting data points (Bi and Turvey 1997) and a more appropriate method of estimating the boundary line (Sackville Hamilton et al. 1995), but the estimation still can not escape from the need for data selection and thus some degree of subjectivity. The lack of an objective method has led to a view that it may even be impossible to validate the self-thinning rule using empirical data (Weller 1990, Zeide 1991). Unless a completely objective method that does not rely on data selection is employed, it is unlikely that the debate will ever be resolved.

Can the self-thinning boundary line be estimated without resorting to data selection? This paper demonstrates how stochastic frontier production functions developed in econometrics for the analysis of maximum potential output can be used to estimate the self-thinning boundary line objectively without data selection, using data from even-aged Pinus radiata stands as an example.

Stochastic frontier functions

The economic concept of production can be understood generally as the process of transforming a set of inputs into a set of outputs, a process that resembles the growth of plant stands which takes water and nutrients to produce biomass through photosynthesis. In economics, a frontier function represents an unobservable upper boundary of maximum attainable output for any given inputs (Farrell 1957). The econometric approach to estimating stochastic frontiers was first proposed by Aigner et al. (1977), Meeusen and van den Broeck (1977), and Battese and Corra (1977). Over the past two decades, it has been used extensively in the analysis of production efficiency in economics and management science and has received a number of comprehensive reviews (Forsund et al. 1980, Schmidt 1986, Bauer 1990, Greene 1997). The model specification commonly used for a stochastic frontier function has been that of Aigner et al. (1977):

[Y.sub.i] = A[X.sup.[[beta].sub.i]][[X.sup.[[beta].sub.2]].sub.2]…[[X.sup.[[bet a].sub.k]].sub.k][e.sup.[v.sub.i]][e.sup.-[v.sub.i]] (3)

where [Y.sub.i] is the ith observed value of the dependent variable which is called output in econometrics, X’s are inputs, i.e., independent variables, A and [beta]’s are parameters, [e.sup.[v.sub.i]] and [e.sup.-[v.sub.i]] are two error components. The distribution of [v.sub.i] is assumed to be normal with zero mean and constant variance [[[sigma].sup.2].sub.v], and [u.sub.i] is assumed to be the absolute values of a normally distributed variable with zero mean and constant variance [[[sigma].sup.2].sub.[mu]] such that 0 [leq] [u.sub.i] [leq] [infty] and 0 [leq] [e.sup.-[u.sub.i]] [leq] 1. Taking the logarithm, the model becomes

[y.sub.i] = [alpha] + [X.sub.i][beta] + [[varepsilon].sub.i] (4)

where [y.sub.i] = log [Y.sub.i], [alpha] = log A, [X.sub.i] is the ith vector of log-transformed independent variables, and [beta] is a vector of parameters. The error term, [[varepsilon].sub.i] = [v.sub.i] – [u.sub.i], is a compound random variable with two components and each is assumed to be independently and identically distributed across observations. Referring to Greene (1993, 1997), the distribution of [u.sub.i] is assumed to be half normal, with E([u.sub.i]) = ([sqrt{2/[pi]}])[[sigma].sub.u] and Var([u.sub.i]) = (1 – 2/[pi])[[[sigma].sup.2].sub.u]. The distribution of the compound error term, [[varepsilon].sub.i], is asymmetric and nonnormal with a density function

[[lgroup][frac{2}{[pi]}][rgroup].sup.1/2] [Phi][lgroup]-[frac{[[varepsilon].sub.i][lambda]}{[sigma]}][rgroup][l group][frac{1}{[sigma]}][rgroup][phi][lgroup][frac{[[varepsilon].sub. i]}{[sigma]}][rgroup] (5)

where [sigma] = [sqrt{[[[sigma].sup.2].sub.v] + [[[sigma].sup.2].sub.u]}], [lambda] = [[sigma].sub.u]/[[sigma].sub.v] and [Phi] and [phi] are the standard normal distribution and density functions. The mean and variance of [[varepsilon].sub.i] are E([[varepsilon].sub.i]) = -E([u.sub.i]) = -([sqrt{2/[phi]}])[[sigma].sub.u] and V([[varepsilon].sub.i]) = V(u) + V(v) = (1 – 2/[pi])[[[sigma].sup.2].sub.u] + [[[sigma].sup.2].sub.v]. The parameters in the model are commonly estimated by maximum likelihood methods. For a

stochastic frontier function with the above specification, the log-likelihood function is

log L([alpha], [beta], [sigma], [lambda]) = -n ln [sigma] – C

+ [[[sum].sup.n].sub.i=1] {ln[1 – [Phi][lgroup][frac{[[varepsilon].sub.i][lambda]}{[sigma]}][rgroup]] – [frac{1}{2}][[lgroup][frac{[[varepsilon].sub.i]}{[sigma]}][rgroup].su p.2]} (6)

where [[varepsilon].sub.i] = [y.sub.i] – [alpha] – [X.sub.i][beta], C is a constant and n is the number of data points. The normal equations derived from the log-likelihood function and further details on estimation of the model can be found in Aigner et al. (1977). The parameters [alpha] and [beta] possess asymptotic normal distributions with large samples, a property useful for making statistical inferences about the parameters. Once the normal equations are solved by a nonlinear search algorithm, the unobservable production frontier or maximum output line is given by

[[hat{y}].sub.i] = [hat{[alpha]}] + [X.sub.i][hat{[beta]}] (7)

where [hat{[alpha]}] and [hat{[beta]}] are obtained employing all data points. Since [[varepsilon].sub.i] not [u.sub.i], is estimated by the residual [[hat{[varepsilon]}].sub.i] = [y.sub.i] – [[hat{y}].sub.i] = [y.sub.i] – [hat{[alpha]}] – [X.sub.i][hat{[beta]}], [u.sub.i] must be estimated indirectly (see Jondrow et al. 1982), using

E[[u.sub.i][[sigma].sub.i]] = [[frac{[sigma][lambda]}{1 + [[lambda].sup.2]}]][-[frac{[[varepsilon].sub.i][lambda]}{[sigma]}] + [phi][lgroup]-[frac{[[varepsilon].sub.i][lambda]}{[sigma]}][rgroup]/[ Phi][lgroup]-[frac{[[varepsilon].sub.i][lambda]}{[sigma]}][rgroup]]. (8)

The expectation of [e.sup.-[u.sub.i]], E([e.sup.-[u.sub.i]][[varepsilon].sub.i]), is given by Battese and Coelli (1988, 1992):

E[[e.sup.-[u.sub.i]][[varepsilon].sub.i]] = [[Phi][lgroup][frac{[[u.sup.*].sub.i]}{[[sigma].sub.*]}] – [[sigma].sub.*][rgroup]/[Phi][lgroup][frac{[[u.sup.*].sub.i]}{[[sigma ].sub.*]}][rgroup]]exp[lgroup]-[[u.sup.*].sub.i] + [frac{1}{2}][[[sigma].sup.2].sub.*][rgroup] (9)

where [[u.sup.*].sub.i] = -[varepsilon][[lambda].sup.2]/(1 + [[lambda].sup.2]) and [[[sigma].sup.2].sub.*] = [[[sigma].sup.2].sub.*]/(1 + [[lambda].sup.2]). The expectations of both [u.sub.i] and [e.sup.-[u.sub.i] have been used as a measure of production efficiency relative to the estimated production frontier or the maximum output line.

Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier

The estimation of the self-thinning boundary line fits well into the framework of stochastic frontier functions. Stand biomass at a given stand density in even-aged stands, expressed as a proportion of the maximum attainable biomass at that stand density, reflects the degree of site occupancy, i.e., the extent to which the stands have occupied the growing space and utilized the available resources within the given environment for growth. Before full site occupancy, stands do not have to sacrifice individuals for further growth through competition induced mortality. When the site is fully occupied, stands have accumulated the maximum attainable biomass at that stand density and any further growth will incur mortality. So the maximum attainable biomass at any stand density represents the biomass frontier, and the self-thinning boundary line represents a biomass frontier function depicting the density-dependent upper boundary of stand biomass. However, the theoretical maximum attainable biomass at full site occ upancy is not directly observable (Zeide 1991). The biomass frontier may also be affected by external factors such as soil and climatic variations, insect attacks, diseases, or other changes in the environment specific to each stand and time. The effects of these external factors on the biomass frontier are by and large random across observations and independent from the effects of density-dependent growth and mortality on site occupancy. So any data point of stand biomass and stand density would naturally reflect two component effects: site occupancy due to density-dependent growth and mortality within individual stands, and the effects of external factors that take place at random on the biomass frontier. These two components are compounded and not directly observable, but they can be estimated by formulating the self-thinning boundary line as a density-dependent stochastic biomass frontier function:

B = C[N.sup.[beta]] [e.sup.v] [e.sup.-u] (10)

where B, C, and N are as previously defined, [beta] is a parameter, 0 [leq] u [leq] [infty] is assumed to be half normal, v represents the random effects of external factors on the biomass frontier and is assumed to have a normal distribution with zero mean. Site occupancy is represented by 0 [leq] [e.sup.-u] [leq] 1, with 1 indicating full site occupancy. On logarithmic scales, the equation becomes a density-dependent stochastic biomass frontier function with the same specification as that of Aigner et al. (1977):

log B = [alpha] + [beta] log N + [varepsilon] (11)

where [alpha] = log C, and the error term [varepsilon] is a compound random variable [varepsilon] = v — u. This density-dependent stochastic biomass frontier function was fitted to the data described below using FRONTIER 4.1 of Coelli (1996). Site occupancy was estimated as the conditional expectation of [e.sup.-u] given [varepsilon] as shown in Eq. 9. To be consistent with the previous work of Bi and Turvey (1997), common logarithm was used. Natural logarithm will not change the results for the estimated slope.


The data set for this work was used by Bi and Turvey (1997) to demonstrate a method of selecting data points for fitting the self-thinning boundary line. The data were from experiments established in the 1950s in three P. radiata tree farms owned by A.P.M. Forests Proprietary Limited in the Gippsland region of Victoria, Australia. The experiments were designed to test the effect of different thinning regimes on stand growth across the tree farms. Twelve unthinned control plots that underwent self-thinning provided a total of 121 measurements for estimating the self-thinning boundary line. The initial planting densities of these plots were unknown. When plots were measured at the age of 10–12 yr, stand density ranged from 1230 to 2090 trees/ha. Plot area varied from 0.0512 ha to 0.0958 ha. These plots were generally measured every two years and the last measurement was taken between ages 26 and 35 yr. At each measurement, dbh and height of individual trees were obtained. The reduction in the number of trees was recorded in three categories: died from unidentified causes, removed because of Sirex noctilio attack, and removed for unknown reasons.

The biomass equations developed by Baker et al. (1984) for P. radiata in Gippsland, Victoria were based on a sample of 62 unpruned 9–28-yr-old trees growing in closed-canopy stands with density ranging from 610 to 2163 trees/ha. The equations predict the mass of needles and live branches from diameter and that of stem from both diameter and tree height. The total dry mass of each tree was calculated as the sum of the mass of these components estimated using these equations. The sum of individual tree biomass gave the total stand biomass. Some trees were older than 28 years, beyond the age range covered by the empirical biomass equations. However, the equations are the best available at present for P. radiata in the region.


The maximum likelihood estimates of the self-thinning boundary line as a density-dependent stochastic biomass frontier (Fig. 1) are

log B = 7.13 – 0.53 log N. (12)

The standard errors of the estimates of [alpha] and [beta] are 0.236 and 0.078. The t ratio for the difference between the estimated slope and -0.5 is 0.38. So we may conclude that the slope is not significantly different from -0.5, the stated slope of the self-thinning rule, at 95% confidence level. The estimated intercept is slightly, but not significantly, greater than the intercept of 6.99 for stemwood only determined visually by Drew and Flewelling (1977) for 54 P. radiata stands in New Zealand. This comparison is made after converting their relationship between mean tree volume and stand density to one that is comparable with the results in this paper using the average basic density of stemwood of P. radiata in New Zealand (Cown et al. 1991).

Since the compound error term of Eq. 11, [varepsilon] = v – u, has an asymmetric and nonnormal distribution with a genative mean, most residuals are negative, therefore most observations lie below the estimated boundary line (Fig. 1). However, a negative mean does not imply that all observations taken from the distribution are negative. A small number of residuals can still be positive, and the corresponding observations would certainly lie above the estimated boundary line. When a stand is approaching full site occupancy, i.e., u [rightarrow] 0 and [e.sup.-u] [rightarrow] 1, and the random effects of external factors represented by v have a positive impact on the biomass frontier, a positive residual can be expected. Examples can be seen in Fig. 1 where observations from two stands approaching full site occupancy, at log stand density between 3.1 and 3.2, lie above the estimated self-thinning boundary line. Experimental records show that superphosphate was applied at establishment at the level of 251 kg/ha and 188 kg/ha, respectively, to the two stands (Bi 1989). Other stands were not fertilized, apart from the one with the leftmost trajectory in Fig. 1 which had a substantial early reduction in stand density. The topmost observations above the estimated boundary line come from the most heavily fertilized stand. This external factor would certainly have a positive effect on the biomass frontier and result in positive residuals for these observations. The magnitude of the positive residuals is related to the variance of v, which has a normal distribution with zero mean.

The variance of v, the random external effects on the biomass frontier, would also relate directly to the variation in the growth trajectory followed, by individual stands if the rate of growth and mortality were the same for stands with the same degree of site occupancy. It is reasonable to expect a relatively large variance of v for forest stands because of the time frame and spatial variation over the course of their growth. Stands at the same site occupancy can not be expected to have the same overall rate of growth and mortality due to local variation in competition and mortality within each stand (Kenkel 1988, Kenkel et al. 1989, 1997, Adler 1996). Any difference will result in further variation in the growth trajectory of individual stands. Variations in growth trajectory such as those in Fig. 1 are common for even-aged forest stands (e.g., Drew and Flewelling 1977, 1979, Osawa and Sugita 1989) rather than an exception. Even for herbaceous plants in glasshouse experiment, the growth trajectory can sti ll be quite variable (Westoby and Howell 1986).

Bi and Turvey (1997) demonstrated a method of selecting data points for fitting the self-thinning boundary line using the same data set of 121 points. The method involves the division of the 121 points into a specified number of intervals. From each interval, the point having the maximum stand biomass is selected to contribute to the fitting process using principal component analysis. In their example, the process was repeated six times with the number of intervals increasing from five to ten. So the self-thinning line was repeatedly estimated using five, six, seven, eight, nine, and ten data points selected from a total of 121. The estimated slope varied from -0.63 to -0.49, whilst the estimated intercept varied from 7.02 to 7.44. Since the number of selected data points was small relative to the total number of observations and only the few selected ones were used in the estimation, results were sensitive to differences in the number of selected points, and which data points were selected. Both could not b e determined without some degree of subjectivity. This problem was overcome in this paper by taking the stochastic frontier approach. The parameter estimates using eight selected data points in Bi and Turvey (1997) were exactly the same as those estimated using the stochastic frontier function. However, the variance of their estimated slope was much greater and so the confidence interval was too wide [-0.99, -0.20] to be useful for any convincing statistical inferences. In comparison, the estimated slope using the stochastic biomass frontier function had a much smaller standard error, so the estimation was far more efficient. The improved efficiency would enable more precise statistical inference about the estimated slope and intercept based on data from a smaller number of plant populations.

The formulation of the self-thinning boundary line as a density-dependent stochastic biomass frontier implies that stands travel along the boundary line only when site occupancy is 1. The estimated site occupancy ranged from 0.57 to 0.98, with most observations above 0.7 (Fig. 1). Mortality started well before stands reached full site occupancy and became much greater when site occupancy exceeded 0.9. Site occupancy generally increased following lighter mortality and decreased following heavier mortality. After the initial mortality, site occupancy of any single stand varied mostly between 0.70 and 0.98 along its thinning trajectory. These results highlight another shortcoming of data selection in addition to its subjectivity. No matter what criteria are used, the selected data points will not have the same degree of site occupancy, but they contribute equally to the parameter estimation. So the estimated parameters may be in part a reflection of the variation in site occupancy among the selected data points rather than an exact depiction of the self-thinning boundary line. This also helps to explain why it is impossible to estimate the self-thinning boundary line from individual thinning trajectories, an approach that has been taken by many in their analysis of the self-thinning rule. Because no single stand grows in a state of full site occupancy throughout the course of biomass accumulation, an obseved thinning trajectory may not contain even a single data point at full site occupancy, let alone the number of observations required for appropriate statistical estimation of the self-thinning boundary line. Estimating the self-thinning boundary line from individual thinning trajectories is just like trying to estimate a distribution from a single observation. This fallacy must be rectified if the self-thinning rule is to be appropriately evaluated in the future.

As is often the case with growth data from repeated measurements of plant stands, the errors may not be independent from each other for observations from the same stand. Some degree of positive temporal autocorrelation may well exist (West et al. 1984, West 1995). As in the case of ordinary least square regression, the standard errors of the parameters in a stochastic frontier function also tend to be underestimated in the presence of positive autocorrelation. However, this does not pose a problem for estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier since the parameter estimates are unbiased and statistical inferences about the estimated slope are more conservative towards agreeing with the self-thinning rule in the presence of positive autocorrelation. The autocovariance structure present in data commonly used for estimating the self-thinning boundary line and the consequent magnitude of underestimation may need to be addressed by further research.


A. P. M. Forests kindly provided the data for this work. We thank Shimin Cai and Anne Chi for technical assistance. Dr. Phil West, Mr. Jack Simpson, and Mr. Charles Mackowski provided helpful comments.

(1.) Research and Development Division, State Forests of New South Wales, P.O. Box 100, Beecroft, New South Wales 2119 Australia.

(2.) Department of Agricultural Economics, University of Sydney, New South Wales 2006 Australia

(3.) Greenfield Resource Options, S Central Avenue, Indooroopilly, Queensland 4068 Australia

(4.) E-mail:


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COPYRIGHT 2000 Ecological Society of America

COPYRIGHT 2000 Gale Group

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