r.regression.multi calculates a multiple linear regression from raster maps, according to the formula
Y = b0 + sum(bi*Xi) + E
where
X = {X1, X2, ..., Xm}
m = number of explaining variables
Y = {y1, y2, ..., yn}
Xi = {xi1, xi2, ..., xin}
E = {e1, e2, ..., en}
n = number of observations (cases)
In R notation:
Y ~ sum(bi*Xi)
b0 is the intercept, X0 is set to 1
r.regression.multi is designed for large datasets that can not be processed in R. A p value is therefore
not provided, because even very small, meaningless effects will become significant with a large number of
cells. Instead it is recommended to judge by the estimator b, the amount of variance explained (R squared
for a given variable) and the gain in AIC (AIC without a given variable minus AIC global must be
positive) whether the inclusion of a given explaining variable in the model is justified.
Theglobalmodel
The b coefficients (b0 is offset), R squared or coefficient of determination (Rsq) and F are identical to
the ones obtained from R-stats’s lm() function and R-stats’s anova() function. The AIC value is identical
to the one obtained from R-stats’s stepAIC() function (in case of backwards stepping, identical to the
Start value). The AIC value corrected for the number of explaining variables and the BIC (Bayesian
Information Criterion) value follow the logic of AIC.
Theexplainingvariables
R squared for each explaining variable represents the additional amount of explained variance when
including this variable compared to when excluding this variable, that is, this amount of variance is
explained by the current explaining variable after taking into consideration all the other explaining
variables.
The F score for each explaining variable allows testing if the inclusion of this variable significantly
increases the explaining power of the model, relative to the global model excluding this explaining
variable. That means that the F value for a given explaining variable is only identical to the F value
of the R-function summary.aov if the given explaining variable is the last variable in the R-formula.
While R successively includes one variable after another in the order specified by the formula and at
each step calculates the F value expressing the gain by including the current variable in addition to the
previous variables, r.regression.multi calculates the F-value expressing the gain by including the
current variable in addition to all other variables, not only the previous variables.
The AIC value is identical to the one obtained from the R-function stepAIC() when excluding this variable
from the full model. The AIC value corrected for the number of explaining variables and the BIC value
(Bayesian Information Criterion) value follow the logic of AIC. BIC is identical to the R-function
stepAIC with k = log(n). AICc is not available through the R-function stepAIC.
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