110
Miguel A. Zavala
The statistical significance of these patterns was evaluated by calibrating a
top-down probabilistic model for this system. This model was developed for
two main purposes. On the one hand, it permits detection of the environmental factors that have the most significant effect on forest composition. On
the other hand, if the model is successfully tested in a variety of locations, it
could be used to predict species' relative abundance at a given site as a function of local climate and management. Following this idea, a model that considers the total basal area in the plot and the proportion of each species
within the stand was calibrated. The variation in mean holm oak proportion
along a soil moisture gradient can be characterized as a sigmoid response
similar to the one described in a model of logistic regression. However, to my
knowledge no mathematical transformation exists that transforms aU-shaped
function to the binomial distribution assumed in a logistic regression model.
For this reason, it was necessary to derive a probabilistic process that generates a pattern characterized by U-shaped residuals. Briefly, this process considers an index (E) that measures the holm oak proportion ("oakiness") of a
given site and summa,rizes all the underlying processes that affect stand
composition at equilibrium, like soil moisture, management history and altitude. The proportion of holm oak for a given value of this index can be calculated according to a simple response function. In this particular example,
E can be assumed to be normally distributed, with a constant standard deviation and a mean that increases linearly with soil moisture. The resulting
model is then described by four parameters: the slope of the logistic function
that describes the variation in mean holm oak proportion, the standard deviation of the index and the intercept and slope that define the dependence
of the index mean on soil moisture. Given this probabilistic process and the
underlying hypotheses (model parameters) it is then possible to estimate the
values of the parameters that maximize the likelihood of occurrence in relation to our data (see Edwards 1972). The significance of the effect of soil
moisture on holm oak relative abundance can be shown with a statistical test
that compares the likelihood of two alternative models, one of them without
an explicit dependence on soil moisture (ratio likelihood test; Edwards 1972;
1 df, P < 0.001).
The flexibility of this model permits to analyze properly heteroscedastic
data patterns, and to relate stand variability to factors that cannot be described by continuous random variables such as historical management. Topdown models can be used primarily to identify the environmental factors
that determine large-scale species distributions (Landsberg 1986; Chaparro
1996) and for explanatory analyses of landscape structure. Ultimately these
models can be used together with geographic information to generate dynamic spatial maps that render error-bounded predictions on the effect of
different planning strategies on regional forest structure (see Grossman
1991).
Miguel A. Zavala
The statistical significance of these patterns was evaluated by calibrating a
top-down probabilistic model for this system. This model was developed for
two main purposes. On the one hand, it permits detection of the environmental factors that have the most significant effect on forest composition. On
the other hand, if the model is successfully tested in a variety of locations, it
could be used to predict species' relative abundance at a given site as a function of local climate and management. Following this idea, a model that considers the total basal area in the plot and the proportion of each species
within the stand was calibrated. The variation in mean holm oak proportion
along a soil moisture gradient can be characterized as a sigmoid response
similar to the one described in a model of logistic regression. However, to my
knowledge no mathematical transformation exists that transforms aU-shaped
function to the binomial distribution assumed in a logistic regression model.
For this reason, it was necessary to derive a probabilistic process that generates a pattern characterized by U-shaped residuals. Briefly, this process considers an index (E) that measures the holm oak proportion ("oakiness") of a
given site and summa,rizes all the underlying processes that affect stand
composition at equilibrium, like soil moisture, management history and altitude. The proportion of holm oak for a given value of this index can be calculated according to a simple response function. In this particular example,
E can be assumed to be normally distributed, with a constant standard deviation and a mean that increases linearly with soil moisture. The resulting
model is then described by four parameters: the slope of the logistic function
that describes the variation in mean holm oak proportion, the standard deviation of the index and the intercept and slope that define the dependence
of the index mean on soil moisture. Given this probabilistic process and the
underlying hypotheses (model parameters) it is then possible to estimate the
values of the parameters that maximize the likelihood of occurrence in relation to our data (see Edwards 1972). The significance of the effect of soil
moisture on holm oak relative abundance can be shown with a statistical test
that compares the likelihood of two alternative models, one of them without
an explicit dependence on soil moisture (ratio likelihood test; Edwards 1972;
1 df, P < 0.001).
The flexibility of this model permits to analyze properly heteroscedastic
data patterns, and to relate stand variability to factors that cannot be described by continuous random variables such as historical management. Topdown models can be used primarily to identify the environmental factors
that determine large-scale species distributions (Landsberg 1986; Chaparro
1996) and for explanatory analyses of landscape structure. Ultimately these
models can be used together with geographic information to generate dynamic spatial maps that render error-bounded predictions on the effect of
different planning strategies on regional forest structure (see Grossman
1991).