rrvglm {VGAM}

Fitting Reduced-Rank Vector Generalized Linear Models (RR-VGLMs) and Doubly Constrained RR-VGLMs (DRR-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. Doubly constrained RR-VGLMs (DRR-VGLMs) can also be fitted.

Usage

rrvglm(formula, family = stop("'family' is unassigned"),
       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.

Details

In this documentation, M is the number of linear predictors. For RR-VGLMs, 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 mth element is \eta_m = \log(E[Y_m]) for the mth Poisson response. The dimension of \eta is M by definition. 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.

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.

Note

The arguments of rrvglm are in general the same as those of vglm but with some extras in rrvglm.control.

The smart prediction (smartpred) library is packed with the VGAM library.

In an example below, a rank-1 stereotype (reduced-rank multinomial logit) model of Anderson (1984) is fitted to some car data. The reduced-rank regression is performed, adjusting for two covariates. Setting a trivial constraint matrix (diag(M)) for the latent variable variables in x_2 avoids a warning message when it is overwritten by a (common) estimated constraint matrix. It shows that German cars tend to be more expensive than American cars, given a car of fixed weight and width.

If fit <- rrvglm(..., data = mydata) then summary(fit) requires corner constraints and no missing values in mydata. Often the estimated variance-covariance matrix of the parameters is not positive-definite; if this occurs, try refitting the model with a different value for Index.corner.

For constrained quadratic ordination (CQO) see cqo for more details about QRR-VGLMs.

With multiple binary responses, one must use binomialff(multiple.responses = TRUE) to indicate that the response is a matrix with one response per column. Otherwise, it is interpreted as a single binary response variable.

To fit DRR-VGLMs see the arguments H.A and H.C in rrvglm.control. DRR-VGLMs provide structure to the A and C matrices via constraint matrices. So instead of them being general unstructured matrices, one can make many of their elements to have the same value, else be identically equal to 0, for example. This gives greater control over what is modelled as a latent variable, e.g., if one subset of the covariates are physical variables and the remainder are psychological variables then a rank-2 model might have each latent variable a linear combination of each of the types of variables separately.

Incidentally, before I forget, if Corner = TRUE then the @H.A slot indicates that the Index.corner rows of A are estimated. This is a remnant of some internal computations because it is more efficient to estimate the entire A, bar rows str0, and then normalize it. In contrast, optimizing over a subset of A is slow.

Author(s)

Thomas W. Yee

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.

Yee, T. W., Frigau, L. and Ma, C. (2024). Heaping and seeping, GAITD regression and doubly constrained reduced rank vector generalized linear models, in smoking studies. In preparation.

See Also

rrvglm.control, summary.drrvglm, 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 what was formerly in COZIGAM). Methods functions include Coef.rrvglm, calibrate.rrvglm, etc. Data include crashi.

Examples

## Not run: 
# Example 1: RR NB 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 1
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, mydata, pch = "+", col = 'blue', las = 1,
  main = paste0("Var(Y) = mu + ", delta1, " * mu^", delta2))
rrnb2 <- rrvglm(y2 ~ x2 + x3, negbinomial(zero = NULL),
                data = mydata, trace = TRUE)

a21.hat <- (Coef(rrnb2)@A)["loglink(size)", 1]
beta11.hat <- Coef(rrnb2)@B1["(Intercept)", "loglink(mu)"]
beta21.hat <- Coef(rrnb2)@B1["(Intercept)", "loglink(size)"]
(delta1.hat <- exp(a21.hat * beta11.hat - beta21.hat))
(delta2.hat <- 2 - a21.hat)
# delta1.hat:
# exp(a21.hat * predict(rrnb2)[1,1] - predict(rrnb2)[1,2])
summary(rrnb2)

# Obtain a 95 percent CI 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 CI

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

# Plot the abundances and fitted values vs 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 = "red")

# Example 2: stereotype model (RR 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])  # Stdze 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,  # orig.
              Index.corner = c(1, 3),  # Less correlation
              Bestof = 3)
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")

## End(Not run)