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 (BalboaMurias
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) (DahdouhGuebas 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 datagathering: 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 Northernlatitude and
4° 52' and 5° 02' of Westernlongitude (Vennetier and Laclavere,
1983). It is a semideciduous 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 T_{1}
and T_{2}. T_{1} stands for the average thinning (rate
from 2535%) and T_{2} 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
speciesR) by the means of qualitative variables (thinning and site) and
quantitative variable (initial total basal areaG_{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 twoways analysis of variance. Mathematically,
it is formulated as follows:
a_{ij }= m + t_{i}
+ b_{j} + e_{ij} 
(1) 
Where:
a_{ij} 
= 
R of the quadrants j, having received treatment i 
m 
= 
Mean R of all the quadrants 
t_{i} 
= 
Effect of treatment i (treated and untreated) 
b_{j} 
= 
Effect of the quadrant j 
e_{ij} 
= 
Experimental error 
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:
a_{ij }= m + t_{i}
+ b_{j} + α (u_{i}ū) + e_{ij} 
(2) 
Where:
u_{i} 
= 
G_{i} of the quadrants having received treatment i 
ū 
= 
Mean G_{i} of all the quadrants 
α 
= 
Linear coefficient of regression of R (a_{i}) in G_{i}
(u_{i}) 
Model 3 explains R by the means of the qualitative variable site and
the quantitative variable G_{i}. Mathematically, it is formulated
as follows:
a_{ij }= m + b_{j}
+ α (u_{i}ū) + e_{ij} 
(3) 
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 (T_{1} and T_{2}) allows
two independent comparisons (treated versus untreated and T_{1} versus
T_{2}) 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 m^{2} ha^{1
} year^{1}) and that of the untreated ones (0.53 m^{2}
ha^{1 } year^{1}). It was also observed that the T_{2}
is lightly superior to the T_{1}.
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 m^{2} ha^{1}) is significantly different from
that of the untreated one (21.60 m^{2} 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 m^{2} 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 m^{2} 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 m^{2} ha^{1 }year^{1})
and the corrected R values (0.73 and 0.92 m^{2} ha^{1 }year^{1}).
As a result, the productivity increases of 0.2 m^{2} ha^{1
}year^{1 }in the untreated quadrants while it depletes of
0.13 m^{2} 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).
CONCLUSION
Covariance analysis constitutes a set of methods which are all together
related to the analysis of variance and the regression. Its use is justified
in this study where the influence of two classification factors (treatment
and site) on a quantitative variable (rate of increase) is studied.
We proceed like in a traditional analysis of variance, eliminating by
regression the effect of the auxiliary variable observed in the same quadrant.
In contrast to analysis of variance, covariance analysis can be used only
under relatively strict conditions such as the distributions normality,
the variances equality, etc. These application conditions must always
be confirmed. This statistical technique should not be used systematically
and it is always preferable to consult a biometrician during the experimental
plan development.
ACKNOWLEDGMENTS
We thank SODEFOR and CIRADFORET for the installation of permanent experimental
design in Côte d`Ivoire and datagathering. We also thank Drs. Kadio
A., J.P. Pascal, J.G. Bertault, J.F. Dhôte, P. Couteron and D. Chessel
for their effective contribution and their advices in data analysis. We
are very grateful with the French Mission of Cooperation and Cultural
Action (MCAC) in Abidjan and with FORAFRI project which financed this
study.