VGAM (version 1.1-6)

# rrvglm: Fitting Reduced-Rank Vector Generalized Linear Models (RR-VGLMs)

## Description

A reduced-rank vector generalized linear model (RR-VGLM) is fitted. RR-VGLMs are VGLMs but some of the constraint matrices are estimated. In this documentation, $$M$$ is the number of linear predictors.

## Usage

rrvglm(formula, family = stop("argument 'family' needs to be assigned"),
data = list(), weights = NULL, subset = NULL,
na.action = na.fail, etastart = NULL, mustart = NULL,
coefstart = NULL, control = rrvglm.control(...), offset = NULL,
method = "rrvglm.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE,
contrasts = NULL, constraints = NULL, extra = NULL,
qr.arg = FALSE, smart = TRUE, ...)

## Arguments

formula, family, weights

See vglm.

data

an optional data frame containing the variables in the model. By default the variables are taken from environment(formula), typically the environment from which rrvglm is called.

subset, na.action

See vglm.

etastart, mustart, coefstart

See vglm.

control

a list of parameters for controlling the fitting process. See rrvglm.control for details.

offset, model, contrasts

See vglm.

method

the method to be used in fitting the model. The default (and presently only) method rrvglm.fit uses iteratively reweighted least squares (IRLS).

x.arg, y.arg

logical values indicating whether the model matrix and response vector/matrix used in the fitting process should be assigned in the x and y slots. Note the model matrix is the LM model matrix; to get the VGLM model matrix type model.matrix(vglmfit) where vglmfit is a vglm object.

constraints

See vglm.

extra, smart, qr.arg

See vglm.

further arguments passed into rrvglm.control.

## Value

An object of class "rrvglm", which has the the same slots as a "vglm" object. The only difference is that the some of the constraint matrices are estimates rather than known. But VGAM stores the models the same internally. The slots of "vglm" objects are described in vglm-class.

## Details

The central formula is given by $$\eta = B_1^T x_1 + A \nu$$ where $$x_1$$ is a vector (usually just a 1 for an intercept), $$x_2$$ is another vector of explanatory variables, and $$\nu = C^T x_2$$ is an $$R$$-vector of latent variables. Here, $$\eta$$ is a vector of linear predictors, e.g., the $$m$$th element is $$\eta_m = \log(E[Y_m])$$ for the $$m$$th Poisson response. The matrices $$B_1$$, $$A$$ and $$C$$ are estimated from the data, i.e., contain the regression coefficients. For ecologists, the central formula represents a constrained linear ordination (CLO) since it is linear in the latent variables. It means that the response is a monotonically increasing or decreasing function of the latent variables.

For identifiability it is common to enforce corner constraints on $$A$$: by default, the top $$R$$ by $$R$$ submatrix is fixed to be the order-$$R$$ identity matrix and the remainder of $$A$$ is estimated.

The underlying algorithm of RR-VGLMs is iteratively reweighted least squares (IRLS) with an optimizing algorithm applied within each IRLS iteration (e.g., alternating algorithm).

In theory, any VGAM family function that works for vglm and vgam should work for rrvglm too. The function that actually does the work is rrvglm.fit; it is vglm.fit with some extra code.

## References

Yee, T. W. and Hastie, T. J. (2003). Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15--41.

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

Anderson, J. A. (1984). Regression and ordered categorical variables. Journal of the Royal Statistical Society, Series B, Methodological, 46, 1--30.

Yee, T. W. (2014). Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889--902.

rrvglm.control, lvplot.rrvglm (same as biplot.rrvglm), rrvglm-class, grc, cqo, vglmff-class, vglm, vglm-class, smartpred, rrvglm.fit. Special family functions include negbinomial zipoisson and zinegbinomial. (see Yee (2014) and COZIGAM). Methods functions include Coef.rrvglm, calibrate.rrvglm, summary.rrvglm, etc. Data include crashi.

## Examples

Run this code
# NOT RUN {
# Example 1: RR negative binomial with Var(Y) = mu + delta1 * mu^delta2
nn <- 1000       # Number of observations
delta1 <- 3.0    # Specify this
delta2 <- 1.5    # Specify this; should be greater than unity
a21 <- 2 - delta2
mydata <- data.frame(x2 = runif(nn), x3 = runif(nn))
mydata <- transform(mydata, mu = exp(2 + 3 * x2 + 0 * x3))
mydata <- transform(mydata,
y2 = rnbinom(nn, mu = mu, size = (1/delta1)*mu^a21))
plot(y2 ~ x2, data = mydata, pch = "+", col = 'blue', las = 1,
main = paste("Var(Y) = mu + ", delta1, " * mu^", delta2, sep = ""))
rrnb2 <- rrvglm(y2 ~ x2 + x3, negbinomial(zero = NULL),
data = mydata, trace = TRUE)

(delta1.hat <- exp(a21.hat * beta11.hat - beta21.hat))
(delta2.hat <- 2 - a21.hat)
# exp(a21.hat * predict(rrnb2)[1,1] - predict(rrnb2)[1,2])  # delta1.hat
summary(rrnb2)

# Obtain a 95 percent confidence interval for delta2:
se.a21.hat <- sqrt(vcov(rrnb2)["I(latvar.mat)", "I(latvar.mat)"])
ci.a21 <- a21.hat +  c(-1, 1) * 1.96 * se.a21.hat
(ci.delta2 <- 2 - rev(ci.a21))  # The 95 percent confidence interval

Confint.rrnb(rrnb2)  # Quick way to get it

# Plot the abundances and fitted values against the latent variable
plot(y2 ~ latvar(rrnb2), data = mydata, col = "blue",
xlab = "Latent variable", las = 1)
ooo <- order(latvar(rrnb2))
lines(fitted(rrnb2)[ooo] ~ latvar(rrnb2)[ooo], col = "orange")

# Example 2: stereotype model (reduced-rank multinomial logit model)
data(car.all)
scar <- subset(car.all,
is.element(Country, c("Germany", "USA", "Japan", "Korea")))
fcols <- c(13,14,18:20,22:26,29:31,33,34,36)  # These are factors
scar[, -fcols] <- scale(scar[, -fcols])  # Standardize all numerical vars
ones <- matrix(1, 3, 1)
clist <- list("(Intercept)" = diag(3), Width = ones, Weight = ones,
Disp. = diag(3), Tank = diag(3), Price = diag(3),
Frt.Leg.Room = diag(3))
set.seed(111)
fit <- rrvglm(Country ~ Width + Weight + Disp. + Tank +
Price + Frt.Leg.Room,
multinomial, data = scar, Rank = 2, trace = TRUE,
constraints = clist, noRRR = ~ 1 + Width + Weight,
Uncor = TRUE, Corner = FALSE, Bestof = 2)
fit@misc\$deviance  # A history of the fits
Coef(fit)
biplot(fit, chull = TRUE, scores = TRUE, clty = 2, Ccex = 2,
ccol = "blue", scol = "orange", Ccol = "darkgreen", Clwd = 2,
main = "1=Germany, 2=Japan, 3=Korea, 4=USA")
# }


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