CHAPTER 4

SOME PROBLEMS IN TESTING THE SELF-THINNING RULE Introduction

Many self-thinning lines with slopes near y

=

-3/2 have now been reported, and their sheer number is considered strong evidence for the self-thinning rule, or even a self-thinning law (see

references in Hutchings 1983). However, an analysis of size-density data is not simply a matter of regressing log

w

against log N and so demonstrating the self-thinning rule. Some important analytical difficulties must be fully discussed before the large body of evidence is embraced as convincing proof for the rule. The principal problem areas are (1) the data selected to test the

hypothesis, (2) the points used to estimate a thinning line, {3) the . curve fltting methods used, (4) the choice between the log ~-log N

or log B-log N formulations of the rule, and (5) the conclusions drawn from the results. Recommendations for resolving some

difficulties are presented here, and the implications for acceptance of the self-thinning rule are discussed.

Selecting Test Data

Many data sets have been reported to exhibit a linear

relationship between log

w

and log N with a slope near ^y

=

-3/2, but it is usually easy to find evidence for a hypothesis, regardless of whether or not it is generally true. Therefore, the mere

existence of such evidence does not verify the hypothesis. Rigorous verification instead comes from failure of the opposite endeavor, to find evidence that contradicts or falsifies the hypothesis (Popper

1963). The emphasis on compiling corroborative evidence for the thinning rule has diverted attention from data that do not conform.

Violations of the rule have been discussed only for shoot

populations of clonal perennials (Hutchings 1979) and for thinning under very low illumination (Westoby and Howell 1982, Lonsdale and Watkinson 1982), but there are other violations besides these special cases, such as thinning slopes of

y =

-2.59 and -4.5 for tropical trees Shorea robusta and Tectona grandis (O•Neill and DeAngelis 1981),

y =

-1.2 the temperate tree Abies balsamea

(Sprugel 1984), and

y =

-3.2 for seedling populations of the woodland herb Allium ursinum (Ernst 1979).

Information contradicting the self-thinning rule has been missed even in the very sources from which supporting evidence has been drawn. For example, data for one stand of Pinus strobus (Spurr et al. 1957) have been repeatedly cited in self-thinning studies

(Hozumi 1977, 1980, Hara 1984), and a self-thinning slope of

y =

-1.7 has been fit (White 1980). However, the report of Spurr et al.

also presented data for a second stand which gives a thinning slope of

y =

-2.11 (Table A.2). This second, unreported stand

contradicts the self-thinning rule in two ways: the thinning slope is quite different from the predicted value, and both the slope and intercept change from stand to stand. A second example refers to a yield table for Pinus ponderosa (Meyer 1938) reported to give a

..

•

thinning line of log

w =

-1.33 log N + 4.06 {White 1980). However, the log B-log N thinning plot for the complete yield table

{Figure 4.la) shows that 13 thinning lines could be fit since

information is given for 13 values of site index, a general measure of site including soil composition, fertility, slope, aspect, and climate (Bruce and Schumacher 1950). The existence of the twelve unreported thinning lines contradicts two tenets of the

self-thinning rule: the thinning line is not independent of site quality and the thinning intercepts are not species constants~ The data from some yield tables even give different thinning slopes for different site indexes, as shown in Figure 4. lb for Sequoia

sempervirens {Lindquist and Palley 1963).

A second major problem in testing the self-thinning rule arises because the thinning line is an asymptotic constraint approached only as stands become sufficiently crowded. To estimate the slope and position of a thinning line using linear statistics, data points from populations that are not limited by the hypothesized linear constraint must be eliminated. These would include points from young populations that have not yet reached the thinning line, older stands understocked because of poor establishment or

density-independent mortality, and senescent stands. Failure to eliminate such points will bias the thinning line estimates (Mohler et al. 1978), but when the data are confounded by biological

variability and measurement errors, recognition and elimination of

...

N

"'-.. E

0>

'---"' Vl Vl 0

E ₄

0 Cil

E Q) +- (/)

Q 0>

0

_J

3

-2 -1 0

Log₁₀ Density (plants/m2 )

Vl Vl 0

E 0 Cil

E Q) +- (/)

Q 0>

_J 0

5.0

4.5

4.0

3.5

3.0

2.54---,---~-.--~---r--~--~--~--~

-1.4 -1.3 -1.2 -1.1 -1.0

Log10 Density (pI ants/ m²⁾

Figure 4.1. Two forestry yield tables showing variation in thinning line parameters with site index. Dotted lines connect data points and solid lines are PCA thinning lines. (a) shows data for Pinus ponderosa (Meyer 1938). Thinning slopes,

8,

for ten site indexes were near -0.31 (between -0.307 and -0.326), but intercepts ranged from 3.75 to 4.18. (b) shows data for Sequoia sempervirens (Lindquist and Palley 1963). Thinning slopes for six site indexes ranged from -4.15 to -1.81 while intercepts ranged from -1.93 to 2.67. Tables B.l and B.2 give additional information.

spurious points is difficult. Since there is no ~priori estimate of the thinning line position, decisions to eliminate data points must be made~ posteriori (Westoby and Howell 1982}. This is true

even if detailed field notes (Mohler et al. 1978) or mortality curves (Hutchings and Budd 198la) are available to aid the process.

With such ^~ posteriori manipulations, no thinning analysis can be done in a strictly objective way. Figure 4.2 presents three data sets that illustrate these problems. Each plot shows how the slope, intercept, r2, significance level, and confidence interval all

change with the points used to fit the thinning line (Table 4.1).

The results are very sensitive to certain points, yet there is no objective way to decide whether or not to include those points.

The sensitivity of thinning line parameters to the choice of data points also has important statistical implications. The uncertainties about including or excluding some points should be counted in forming confidence intervals and performing tests of significance. However, existing statistical methods do not take such uncertainties into account, so estimated r 2 values are too high and confidence intervals are too narrow.

Some data sets show more that one region of linear behavior and so present the analyst with still another subjective decision: Which linear region is relevant to the self-thinning rule? Figure 4.3 presents a yield table for Populus deltoides (Williamson 1913) that illustrates this problem. White (1980) fit the thinning line

log

w =

-1.8 log N + 3.08 through the data for ages 7 to 15, while

~ N

...__,_ E

0>

----_(/)

Ill 0

E 0

·-m

-

⁰ 0 .s:;

(/)

t£ 0 --'

4.0

---..

3.8

3.6

3.4·

3.2

'~:

0 ' ^D

0

-1.5 -1.0 -o.5 o.o ·o.5

Log,. Density (plants/m²⁾

...__,_ 'E 2.8

~

., Ill

E

^2.6

0

m

-; 2.4 .s:; 0 (/) _g'

..

2.2 --'

.c

0 ¹⁰

0

0

2 . 0 + - - - . - - - , - - - . . . , . . - - - . . J

2.0 2.5 3.0 3.5

Log,. Density (pI ant s/m²⁾

Ill Ill 0

g

^2.5

·-m

-

⁰ 0 .s:;

(/) 5! 2.0 0>

0 --'

3.0 3.5

Log,. Density (plants/m²⁾

Figure 4.2. Three examples of the sensitivity of self-thinning line parameters to the points chosen for analysis. In each plot, several thinning lines (capital letters) are fitted to different combinations of points (numbers) as indicated in Table 4.1. Data in (a), (b), and (c) are for

Populus tremuloides (Pollard 1971, 1972), Triticum sp. (Puckridge and Donald 1967, White and Harper 1970), and Tagetes patula (Ford 1975).

0 0

4.0 (X) 0

Table 4.1. Three Examples of the Sensitivity of the Fitted Self- thinning Line to the Points Chosen for Analysis.

Points

Figurea Lineb Includedc r2 pd

4.5a A l-2 1.00

B 1-3 0.94 0.16

c

l-4 0.94* 0.030

0 2-4 0.99* 0.010 4.5b A l-10 0.63* 0.0061

B 1-3,5-10 0.86* 0.0003

c

5-10 0.91* 0.0030

D 6-10 0.98* 0.0008 E 1-3,5-9 0.80* 0.0027 F 1-3,5-8 0.66* 0.027 4.5c A 1-10 0.34 0.075 B l-8 0.06 0.54

c

3-10 0.48 0.056

D 4,6,9, 10 0.84 0.079

PCA Thinning Linee

Slope Intercept

95% CI

a

-0.19 3.73

-0.25 3.66

-0.26 [-0.48,-0.07] 3.65 -0.53 [-0.64,-0.42] 3.72 -0.20 [-0.33,-0.08] 3.32 -0.24 [-0.32,-0.15] 3.43 -0.31 [-0.46,-0.18] 3.70 -0.36 [-0.45,-0.28] 3.84 -0.16 [-0.25,-0.08] 3.26 -0.13 [-0.24,-0.02] 3.17

-0.41 4.36

+0.24 2.22

-0.70 5.38

-0.35 4.03

aspecies names and references are given in the legend of Figure 4.2.

bThese letters label the fitted lines in Figure 4.2.

CThe numbers of data points in Figure 4.2 used to fit the thinning line.

dThe statistical significance of the log B-log N correlation.

eThe method of fitting the thinning line by principal component analysis is discussed in Chapter 5.

*Significant at the 95% confidence level (P ~ 0.05).

...-...

N 4.2

"-.._

E

0>

'--"'

(/) 4.0

(/)

0

E --

0

m

3.8

E

Q)

+- (/)

52 3.6

0>

0

~

3.4~--~---~---~---~

-2.0 -1.5 -1.0 -0.5