# Info

Figure 10.9 Polynomial regression equation describing the structure of salinity (practical salinity units) in the Thau lagoon (Mediterranean Sea), along its main geographic axis, on 25 October 1988.

5 — Partial linear regression

There are situations where two or more complementary sets of hypotheses may be invoked to explain the variation of an ecological variable. For example, the abundance of a species could vary as a function of biotic and abiotic factors. Regression modelling may be used to study one set of factors, or the other, or the two together. Partial regression is a way of estimating how much of the variation of the response variable can be attributed exclusively to one set of factors, once the effect of the other set has been taken into account. The purpose may be to measure the amount of variation (R2) that can be attributed exclusively to one or the other set of explanatory variables, or else to estimate the vector of fitted values corresponding to the exclusive effect of one set of variables. When the purpose of the study is simply to assess the unique contribution of each explanatory variable, there is no need for partial regression analysis since the multiple linear regression coefficients, which are partial regression coefficients, already provide that information.

Consider three data sets. Vector y is the response variable and matrices X and W contain the explanatory variables. Assume that one wishes to model the relationship between y and X, while controlling for the effects of the variables in matrix W, called the matrix of covariables. There are two ways of doing this, i.e. a long and a short way. The long way provides the justification for the short. One proceeds as follows:

• First, compute multiple regressions of y and of each variable in X against matrix W. Calculate the residuals of all these regressions. For vector y for instance, the residuals,

Table 10.5 Data collected at 20 sites in the Thau lagoon on 25 October 1988. There are two response variables (Bna and Ma), three environmental variables (NH4, phaeopigments, and bacterial production), and three spatial variables (the X and Y geographic coordinates measured with respect to arbitrary axes and centred on their respective means, plus the geographic variable X2). The variables are further described in the text. The code names of these variables in the present Section are y, xi to X3, and wi to W3, respectively.

Table 10.5 Data collected at 20 sites in the Thau lagoon on 25 October 1988. There are two response variables (Bna and Ma), three environmental variables (NH4, phaeopigments, and bacterial production), and three spatial variables (the X and Y geographic coordinates measured with respect to arbitrary axes and centred on their respective means, plus the geographic variable X2). The variables are further described in the text. The code names of these variables in the present Section are y, xi to X3, and wi to W3, respectively.

Site

Bna

Ma