R: Latent Variable Plot for QO models

lvplot.qrrvglm {VGAM} | R Documentation

## Latent Variable Plot for QO models

### Description

Produces an ordination diagram (latent variable plot) for quadratic ordination (QO) models. For rank-1 models, the x-axis is the first ordination/constrained/canonical axis. For rank-2 models, the x- and y-axis are the first and second ordination axes respectively.

### Usage

    lvplot.qrrvglm(object, varI.latvar = FALSE, refResponse = NULL,
        add = FALSE, show.plot = TRUE,
        rug = TRUE, y = FALSE, type = c("fitted.values", "predictors"),
        xlab = paste("Latent Variable", if (Rank == 1) "" else " 1", sep = ""),
        ylab = if (Rank == 1) switch(type, predictors = "Predictors",
        fitted.values = "Fitted values") else "Latent Variable 2",
        pcex = par()$cex, pcol = par()$col, pch = par()$pch,
        llty = par()$lty, lcol = par()$col, llwd = par()$lwd,
        label.arg = FALSE, adj.arg = -0.1,
        ellipse = 0.95, Absolute = FALSE,
        elty = par()$lty, ecol = par()$col, elwd = par()$lwd, egrid = 200,
        chull.arg = FALSE, clty = 2, ccol = par()$col, clwd = par()$lwd,
        cpch = "   ",
        C = FALSE, OriginC = c("origin", "mean"),
        Clty = par()$lty, Ccol = par()$col, Clwd = par()$lwd,
        Ccex = par()$cex, Cadj.arg = -0.1, stretchC = 1,
        sites = FALSE, spch = NULL, scol = par()$col, scex = par()$cex,
        sfont = par()$font, check.ok = TRUE, jitter.y = FALSE, ...)

### Arguments

`object` | A CQO object.
`varI.latvar` | Logical that is fed into `Coef.qrrvglm`.
`refResponse` | Integer or character that is fed into `Coef.qrrvglm`.
`add` | Logical. Add to an existing plot? If `FALSE`, a new plot is made.
`show.plot` | Logical. Plot it?
`rug` | Logical. If `TRUE`, a rug plot is plotted at the foot of the plot (applies to rank-1 models only). These values are jittered to expose ties.
`y` | Logical. If `TRUE`, the responses will be plotted (applies only to rank-1 models and if `type = "fitted.values"`.)
`type` | Either `"fitted.values"` or `"predictors"`, specifies whether the y-axis is on the response or eta-scales respectively.
`xlab` | Caption for the x-axis. See `par`.
`ylab` | Caption for the y-axis. See `par`.
`pcex` | Character expansion of the points. Here, for rank-1 models, points are the response y data. For rank-2 models, points are the optimums. See the `cex` argument in `par`.
`pcol` | Color of the points. See the `col` argument in `par`.
`pch` | Either an integer specifying a symbol or a single character to be used as the default in plotting points. See `par`. The `pch` argument can be of length M, the number of species.
`llty` | Line type. Rank-1 models only. See the `lty` argument of `par`.
`lcol` | Line color. Rank-1 models only. See the `col` argument of `par`.
`llwd` | Line width. Rank-1 models only. See the `lwd` argument of `par`.
`label.arg` | Logical. Label the optimums and C? (applies only to rank-2 models only).
`adj.arg` | Justification of text strings for labelling the optimums (applies only to rank-2 models only). See the `adj` argument of `par`.
`ellipse` | Numerical, of length 0 or 1 (applies only to rank-2 models only). If `Absolute` is `TRUE` then `ellipse` should be assigned a value that is used for the elliptical contouring. If `Absolute` is `FALSE` then `ellipse` should be assigned a value between 0 and 1, for example, setting `ellipse = 0.9` means an ellipse with contour = 90% of the maximum will be plotted about each optimum. If `ellipse` is a negative value, then the function checks that the model is an equal-tolerances model and `varI.latvar = FALSE`, and if so, plots circles with radius `-ellipse`. For example, setting `ellipse = -1` will result in circular contours that have unit radius (in latent variable units). If `ellipse` is `NULL` or `FALSE` then no ellipse is drawn around the optimums.
`Absolute` | Logical. If `TRUE`, the contours corresponding to `ellipse` are on an absolute scale. If `FALSE`, the contours corresponding to `ellipse` are on a relative scale.
`elty` | Line type of the ellipses. See the `lty` argument of `par`.
`ecol` | Line color of the ellipses. See the `col` argument of `par`.
`elwd` | Line width of the ellipses. See the `lwd` argument of `par`.
`egrid` | Numerical. Line resolution of the ellipses. Choosing a larger value will result in smoother ellipses. Useful when ellipses are large.
`chull.arg` | Logical. Add a convex hull around the site scores?
`clty` | Line type of the convex hull. See the `lty` argument of `par`.
`ccol` | Line color of the convex hull. See the `col` argument of `par`.
`clwd` | Line width of the convex hull. See the `lwd` argument of `par`.
`cpch` | Character to be plotted at the intersection points of the convex hull. Having white spaces means that site labels are not obscured there. See the `pch` argument of `par`.
`C` | Logical. Add C (represented by arrows emanating from `OriginC`) to the plot?
`OriginC` | Character or numeric. Where the arrows representing C emanate from. If character, it must be one of the choices given. By default the first is chosen. The value `"origin"` means `c(0,0)`. The value `"mean"` means the sample mean of the latent variables (centroid). Alternatively, the user may specify a numerical vector of length 2.
`Clty` | Line type of the arrows representing C. See the `lty` argument of `par`.
`Ccol` | Line color of the arrows representing C. See the `col` argument of `par`.
`Clwd` | Line width of the arrows representing C. See the `lwd` argument of `par`.
`Ccex` | Numeric. Character expansion of the labelling of C. See the `cex` argument of `par`.
`Cadj.arg` | Justification of text strings when labelling C. See the `adj` argument of `par`.
`stretchC` | Numerical. Stretching factor for C. Instead of using C, `stretchC *` C is used.
`sites` | Logical. Add the site scores (aka latent variable values, nu's) to the plot? (applies only to rank-2 models only).
`spch` | Plotting character of the site scores. The default value of `NULL` means the row labels of the data frame are used. They often are the site numbers. See the `pch` argument of `par`.
`scol` | Color of the site scores. See the `col` argument of `par`.
`scex` | Character expansion of the site scores. See the `cex` argument of `par`.
`sfont` | Font used for the site scores. See the `font` argument of `par`.

`check.ok` | Logical. Whether a check is performed to see that `noRRR = ~ 1` was used. It doesn't make sense to have a latent variable plot unless this is so.
`jitter.y` | Logical. If `y` is plotted, jitter it first? This may be useful for counts and proportions.
`...` | Arguments passed into the `plot` function when setting up the entire plot. Useful arguments here include `xlim` and `ylim`.

### Details

This function only works for rank-1 and rank-2 QRR-VGLMs with argument `noRRR = ~ 1`.

For unequal-tolerances models, the latent variable axes can be rotated so that at least one of the tolerance matrices is diagonal; see `Coef.qrrvglm` for details.

Arguments beginning with “`p`” correspond to the points e.g., `pcex` and `pcol` correspond to the size and color of the points. Such “`p`” arguments should be vectors of length 1, or n, the number of sites. For the rank-2 model, arguments beginning with “`p`” correspond to the optimums.

### Value

Returns a matrix of latent variables (site scores) regardless of whether a plot was produced or not.

### Warning

Interpretation of a latent variable plot (CQO diagram) is potentially very misleading in terms of distances if (i) the tolerance matrices of the species are unequal and (ii) the contours of these tolerance matrices are not included in the ordination diagram.

### Note

A species which does not have an optimum will not have an ellipse drawn even if requested, i.e., if its tolerance matrix is not positive-definite.

Plotting C gives a visual display of the weights (loadings) of each of the variables used in the linear combination defining each latent variable.

The arguments `elty`, `ecol` and `elwd`, may be replaced in the future by `llty`, `lcol` and `llwd`, respectively.

For rank-1 models, a similar function to this one is `perspqrrvglm`. It plots the fitted values on a more fine grid rather than at the actual site scores here. The result is a collection of smooth bell-shaped curves. However, it has the weakness that the plot is more divorced from the data; the user thinks it is the truth without an appreciation of the statistical variability in the estimates.

In the example below, the data comes from an equal-tolerances model. The species' tolerance matrices are all the identity matrix, and the optimums are at (0,0), (1,1) and (-2,0) for species 1, 2, 3 respectively.

### Author(s)

Thomas W. Yee

### References

Yee, T. W. (2004). A new technique for maximum-likelihood canonical Gaussian ordination. Ecological Monographs, 74, 685–701.

### See Also

`lvplot`, `perspqrrvglm`, `Coef.qrrvglm`, `par`, `cqo`.

### Examples

    set.seed(123); nn <- 200
    cdata <- data.frame(x2 = rnorm(nn),  # Has mean 0 (needed when I.tol=TRUE)
                        x3 = rnorm(nn),  # Has mean 0 (needed when I.tol=TRUE)
                        x4 = rnorm(nn))  # Has mean 0 (needed when I.tol=TRUE)
    cdata <- transform(cdata, latvar1 =  x2 + x3 - 2*x4,
                              latvar2 = -x2 + x3 + 0*x4)
    # Nb. latvar2 is weakly correlated with latvar1
    cdata <- transform(cdata,
                lambda1 = exp(6 - 0.5 * (latvar1-0)^2 - 0.5 * (latvar2-0)^2),
                lambda2 = exp(5 - 0.5 * (latvar1-1)^2 - 0.5 * (latvar2-1)^2),
                lambda3 = exp(5 - 0.5 * (latvar1+2)^2 - 0.5 * (latvar2-0)^2))
    cdata <- transform(cdata,
                spp1 = rpois(nn, lambda1),
                spp2 = rpois(nn, lambda2),
                spp3 = rpois(nn, lambda3))
    set.seed(111)
    ## Not run:
    p2 <- cqo(cbind(spp1, spp2, spp3) ~ x2 + x3 + x4, poissonff,
              data = cdata, Rank = 2, I.tolerances = TRUE,
              Crow1positive = c(TRUE, FALSE))  # deviance = 505.81
    if (deviance(p2) > 506) stop("suboptimal fit obtained")
    sort(deviance(p2, history = TRUE))  # A history of all the iterations
    Coef(p2)

    ## End(Not run)

    ## Not run:
    lvplot(p2, sites = TRUE, spch = "*", scol = "darkgreen", scex = 1.5,
           chull = TRUE, label = TRUE, Absolute = TRUE, ellipse = 140,
           adj = -0.5, pcol = "blue", pcex = 1.3, las = 1, Ccol = "orange",
           C = TRUE, Cadj = c(-0.3, -0.3, 1), Clwd = 2, Ccex = 1.4,
           main = paste("Contours at Abundance = 140 with",
                      "convex hull of the site scores"))
    ## End(Not run)
    ## Not run:
    var(latvar(p2))  # A diagonal matrix, i.e., uncorrelated latent vars
    var(latvar(p2, varI.latvar = TRUE))  # Identity matrix
    Tol(p2)[, , 1:2]  # Identity matrix
    Tol(p2, varI.latvar = TRUE)[, , 1:2]  # A diagonal matrix

    ## End(Not run)

[Package VGAM version 1.1-3 Index]