# Returns Important Matrices etc. of a QO Object This methods function returns important matrices etc. of a QO object. ```r Coef.qrrvglm(object, varI.latvar = FALSE, refResponse = NULL, ...) ``` ## Arguments - `object`: A CQO object. The former has class `"qrrvglm"`. - `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 `TRUE`. See below for further details. - `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. ## Details 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`. ## Returns 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`). ## References 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. ## Author(s) Thomas W. Yee ## Note 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. ## See Also `cqo`, `Coef.qrrvglm-class`, `print.Coef.qrrvglm`, `lvplot.qrrvglm`. ## Examples ```r 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) ```

Coef.qrrvglm function