Improving the Precision of Forestry Experimentation with Covariance Analysis

INTRODUCTION

Since sustainable management of tropical forest ecosystem has become internationally an important issue and the environmental certification concept has been created, new silvicultural techniques have earlier been tested by Ivorian researchers in order to satisfy industrial product demands without damaging the environment (Zobi, 2002). One of those silvicultural techniques experimented by researchers is the method of thinning (Balboa-Murias et al., 2006; Utsugi et al., 2006). Closed to forest economy, this method consists of eliminating trees of no commercial species in order to increase the increment rate of the whole group of the commercial ones.

Thus, the statement saying that the method of thinning stimulates the dynamic of trees growth was proved by researchers through inventories in time and space (Chave, 1999; Blanco et al., 2006; Jaakkola et al., 2006; Kanninen et al., 2004). Since 1978, Sodefor (Société de Dévelopement des Forêts) has decided to confirm this statement by experimentation. For this raison, it created three permanent experimental designs. The main purpose of this experimentation is to check up whether the elimination of no commercial species (secondary species) can have a positive effect on commercial species (principal species) growth.

This study also aims at examining whether a statistical method (covariance analysis) (Dahdouh-Guebas and Koedam, 2006; Headley et al., 2005), can be used to solve the inadequacy of the experimental design which results from a skewness on the assessment of a silvicultural treatment effect.

MATERIALS AND METHODS

Experimental material and data-gathering: This study has been conducted from 1978 to 1998 on the experimental design installed in the classified forest of Mopri. This forest which covers an area of 33000 ha is located between 5° 40' and 5° 55' of Northern-latitude and 4° 52' and 5° 02' of Western-longitude (Vennetier and Laclavere, 1983). It is a semi-deciduous rainforest with an average annual rainfall of 1138 mm. The soils are ferralitic and the relief presents a weak slope, going from West to East (Roose, 1981).

The experimental design is a square of 3000 m sidelong or a total area of 900 ha. But the test is carried out in the central square of the design and its area is 400 ha. This one was subdivided into 25 parcels of 16 ha each. Each parcel comprises of two distinct parts which are the buffer zone and the central zone. The central zone of 4 ha is composed of 4 quadrants of 1 ha each. Among the 25 parcels of the experimental design, 15 were chosen randomly, thinned out and constitute the treated quadrants. The 10 others parcels which have not been treated constitute the untreated quadrants. Only the biggest trees among the secondary species were eliminated.

The observations and measurements were made in the quadrants. Only trees which diameter at 1.30 m above ground is superior or equal to 10 cm were considered. They were classified according to their economical importance: principal species and secondary species. The diameter of trees of principal species was measured. Their coordinates (x, y), their scientific and commercial names were determined. Contrary to the principal trees, the precision on the diameter of the secondary trees was approximate: they were merely gathered by range of diameter.

Measurements were done every 2 years. The proportion of basal area eliminated corresponds to the rate of thinning. This rate expresses the thinning intensity and represents the degree of stress undergone by the forest stand. They distinguish two types of thinning which are T₁ and T₂. T₁ stands for the average thinning (rate from 25-35%) and T₂ stands for the highest thinning (rate from 36- 50%). For the untreated quadrants, the total basal area before thinning (G_i) is equal to the total basal area after thinning (G_r).

Principle of covariance analysis: The principle of covariance analysis is based on the comparison of several regression models (Aznar and Guijarro, 2007; Inglot and Ledwina, 2006; Pines et al., 1992; Dagnélie, 1998; Mead, 1988). The goal of a regression model is to explain a quantitative variable (increase rate in basal area of principal species-R) by the means of qualitative variables (thinning and site) and quantitative variable (initial total basal area-G_i). The models that covariance analysis compares are of three types:

Model 1 explains the R by the means of qualitative variables treatment and site. It is a model of two-ways analysis of variance. Mathematically, it is formulated as follows:

Model 2 explains R by the means of two qualitative variables treatment and site and a quantitative variable G_i. It is a model of covariance analysis with only one covariable. It is formulated as follows:

Model 3 explains R by the means of the qualitative variable site and the quantitative variable G_i. Mathematically, it is formulated as follows:

RESULTS

A classical analysis of variance of data concerning increase in basal area in the treated and untreated quadrants was done.

The F test points out a highly significant effect of the site and a very highly significant effect of the silvicultural treatment on the increase rate (R) (Table 1). The intensity of the treatments (T₁ and T₂) allows two independent comparisons (treated versus untreated and T₁ versus T₂) and to use the method of contrasts.

It was noticed that the difference observed between the silvicultural treatments is mainly due to the difference between the mean value of R (ā_i) in the treated quadrants (1.05 m² ha^-1 year^-1) and that of the untreated ones (0.53 m² ha^-1 year^-1). It was also observed that the T₂ is lightly superior to the T₁.

However, the high value of the coefficient of variation (100x√0.16÷ 0.84 = 47.6%) leads to some doubt about the validity of these results. Indeed, this high value of the coefficient of variation shows that some significant sources of variation between both groups of quadrants were not taken into account during the experimentation.

Therefore, it is interesting to take into consideration the characteristics of the site. The effect of those characteristics remains very significant, even though the site effect is evaluated basing on the treatment effect. For that, the mean value of G_i (ū_i) was considered because it is directly accessible. Treated quadrants ū_i (31.77 m² ha^-1) is significantly different from that of the untreated one (21.60 m² ha^-1).

Table 1:	Analysis of variance result for sites and treatments effect on the increase rate (R)
df = degree of freedom, *** = Very highly significant (p≤0.001), ** = Highly significant (p≤0.01)

Table 2:	Decomposition of the sum of squares for the 3 models of regression
df = degree of freedom

Table 3:	Correction of R values using covariance analysis

The ū_i value was taken into consideration through covariance analysis. The purpose of this statistical method is to eliminate the influence of ū_i value from the mean value of R (α_i). It allows the comparison of both groups of quadrants regarding that ū_i value is identical. The obtained result shows that models (2) and (3) are equivalent because SS₍₂₎ = SS₍₃₎ and df₍₂₎ = df₍₃₎. However, in addition to all the sources of variation considered in model (3), the model (2) expressed the effect of the silvicultural treatment (t_i) which is consequently null. Thus, the effect of the silvicultural treatment is not significant (5%) when ū_i value is regarded as identical in both types of quadrants (i.e., treated and untreated) (Table 2).

Regarding ū_i value, ā_i values were revaluated by using the technique of mean correction that is associated to covariance analysis method. This correction or adjustment of ā_i values aims at determining values that probably would be observed if both types of quadrants had the same ū_i value. That explains the introduction of the quantity α (u_i -ū) in the mathematical expression of models (2) and (3). For any quadrant j, this quantity represents the presumably linear relationship between G_i (ū_i) and R (ā_i). We suppose that ū_i is the mean G_i of all the quadrants that have received treatment i. For this treatment i, the R value is corrected by subtracting the quantity α (u_i -ū) from ā_i. The value of the coefficient of regression α is 0.032 and the mean value of G_i for all the quadrants (ū) is 27.68 m² ha^-1.

The positive value of the coefficient of regression α indicates that G_i trends to increase R values in the quadrants. When G_i value decreases, R value increases. The difference between the corrected mean (0.73 and 0.92 m² ha^-1 year^-1) is not significant, but R value of untreated quadrants remains lower than that of the treated quadrants (Table 3).

The introduction of variable G_i allows to estimate the gain of precision by comparing model (1) and model (2). Table 1 gives an estimate value of the error variance that is 11.68 ÷ 75 = 0.156 for model (1) and 10.94÷ 75 = 0.148 for model (2). The relationship (0.156÷0.148 = 1.054) between these two estimates shows that using covariance analysis have the same effect on the experimentation precision like the multiplication of the replicates number by 1.054.

DISCUSSION

In tropical forests, biological material precedes most of the time the experimental plan and the installation of experimental design. It is therefore difficult, even impossible, for the experimenter to constitute homogeneous blocks in a strict sense of the experimental statistics. The permanent design of Mopri is a good example that illustrates that reality.

Although silvicultural treatments were attributed randomly, both groups of quadrants (treated and untreated) differ from their initial total basal area. This is due to the diversity of environmental factors which effects are not sometimes considered by the experimenter. In the case of Mopri, with covariance analysis, it was discovered that the variable initial basal area that was neglected at the beginning of the experimentation appeared very important. Although the precision increase is low (1.054), there are a significant difference between the observed R values (0.53 and 1.05 m² ha^-1 year^-1) and the corrected R values (0.73 and 0.92 m² ha^-1 year^-1). As a result, the productivity increases of 0.2 m² ha^-1 year^-1 in the untreated quadrants while it depletes of 0.13 m² ha^-1 year^-1 in the treated quadrants.

These corrections are particularly significant on the scale of tropical forests which areas represent generally several thousands of hectares.

This example shows that in forestry, with the method of covariance analysis, it is possible to take into consideration some important variables omitted during a study. This statistical technique should however not be used abusively, because significance tests issues are lightly important than those of experimental organization and control (Atkinson and Donev, 1992). The best solution for the forest experimenter should be always consulting a biometrician. In that case, the biometrician could advise the experimenter to give up the test if its implementation is not going to be successful. Indeed, what is the importance of performing an experiment which has only few chances to highlight interesting differences? The biometrician should indicate the way in which the experimentation must be performed in order to obtain the most effective control of the environmental heterogeneity. According to the number of replicates, he should calculate the expected precision (Foster, 2001; Guimarães and Guimarães, 2006; Snedecor and Cochran, 1989).