Returns Important Matrices etc. of a QO Object
This methods function returns important matrices etc. of a QO object.
Coef.qrrvglm(object, varI.latvar = FALSE, refResponse = NULL, ...)
object |
A CQO object.
The former has class |
varI.latvar |
Logical indicating whether to scale the site scores (latent variables)
to have variance-covariance matrix equal to the rank-R identity
matrix. All models have uncorrelated site scores (latent variables),
and this option stretches or shrinks the ordination axes if |
refResponse |
Integer or character. Specifies the reference response or reference species. By default, the reference species is found by searching sequentially starting from the first species until a positive-definite tolerance matrix is found. Then this tolerance matrix is transformed to the identity matrix. Then the sites scores (latent variables) are made uncorrelated. See below for further details. |
... |
Currently unused. |
If I.tolerances=TRUE or eq.tolerances=TRUE (and its
estimated tolerance matrix is positive-definite) then all species'
tolerances are unity by transformation or by definition, and the spread
of the site scores can be compared to them. Vice versa, if one wishes
to compare the tolerances with the sites score variability then setting
varI.latvar=TRUE is more appropriate.
For rank-2 QRR-VGLMs, one of the species can be chosen so that the
angle of its major axis and minor axis is zero, i.e., parallel to
the ordination axes. This means the effect on the latent vars is
independent on that species, and that its tolerance matrix is diagonal.
The argument refResponse allows one to choose which is the reference
species, which must have a positive-definite tolerance matrix, i.e.,
is bell-shaped. If refResponse is not specified, then the code will
try to choose some reference species starting from the first species.
Although the refResponse argument could possibly be offered as
an option when fitting the model, it is currently available after
fitting the model, e.g., in the functions Coef.qrrvglm and
lvplot.qrrvglm.
The A, B1, C, T, D matrices/arrays
are returned, along with other slots.
The returned object has class "Coef.qrrvglm"
(see Coef.qrrvglm-class).
Consider an equal-tolerances Poisson/binomial CQO model with noRRR = ~ 1.
For R=1 it has about 2*S+p2 parameters.
For R=2 it has about 3*S+2*p_2 parameters.
Here, S is the number of species, and p2=p-1 is
the number of environmental variables making up the latent variable.
For an unequal-tolerances Poisson/binomial CQO model with
noRRR = ~ 1, it has about 3*S-1+p2 parameters
for R=1, and about 6*S -3 +2*p2 parameters
for R=2.
Since the total number of data points is n*S, where
n is the number of sites, it pays to divide the number
of data points by the number of parameters to get some idea
about how much information the parameters contain.
Thomas W. Yee
Yee, T. W. (2004). A new technique for maximum-likelihood canonical Gaussian ordination. Ecological Monographs, 74, 685–701.
Yee, T. W. (2006). Constrained additive ordination. Ecology, 87, 203–213.
cqo,
Coef.qrrvglm-class,
print.Coef.qrrvglm,
lvplot.qrrvglm.
set.seed(123) x2 <- rnorm(n <- 100) x3 <- rnorm(n) x4 <- rnorm(n) latvar1 <- 0 + x3 - 2*x4 lambda1 <- exp(3 - 0.5 * ( latvar1-0)^2) lambda2 <- exp(2 - 0.5 * ( latvar1-1)^2) lambda3 <- exp(2 - 0.5 * ((latvar1+4)/2)^2) # Unequal tolerances y1 <- rpois(n, lambda1) y2 <- rpois(n, lambda2) y3 <- rpois(n, lambda3) set.seed(111) # vvv p1 <- cqo(cbind(y1, y2, y3) ~ x2 + x3 + x4, poissonff, trace = FALSE) ## Not run: lvplot(p1, y = TRUE, lcol = 1:3, pch = 1:3, pcol = 1:3) # vvv Coef(p1) # vvv print(Coef(p1), digits=3)
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