vglm {VGAM}R Documentation

Fitting Vector Generalized Linear Models

Description

vglm is used to fit vector generalized linear models (VGLMs). This is a large class of models that includes generalized linear models (GLMs) as special cases.

Usage

vglm(formula, family, data = list(), weights = NULL, subset = NULL, 
     na.action = na.fail, etastart = NULL, mustart = NULL, 
     coefstart = NULL, control = vglm.control(...), offset = NULL, 
     method = "vglm.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE, 
     contrasts = NULL, constraints = NULL, extra = list(), 
     qr.arg = FALSE, smart = TRUE, ...)

Arguments

In the following, M is the number of linear predictors.

formula a symbolic description of the model to be fit. The RHS of the formula is applied to each linear predictor. Different variables in each linear predictor can be chosen by specifying constraint matrices.
family a function of class "vglmff" (see vglmff-class) describing what statistical model is to be fitted. This is called a ``VGAM family function''. See CommonVGAMffArguments for general information about many types of arguments found in this type of function.
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 vglm is called.
weights an optional vector or matrix of (prior) weights to be used in the fitting process. If weights is a matrix, then it must be in matrix-band form, whereby the first M columns of the matrix are the diagonals, followed by the upper-diagonal band, followed by the band above that, etc. In this case, there can be up to M(M+1) columns, with the last column corresponding to the (1,M) elements of the weight matrices.
subset an optional logical vector specifying a subset of observations to be used in the fitting process.
na.action a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The ``factory-fresh'' default is na.omit.
etastart starting values for the linear predictors. It is a M-column matrix. If M=1 then it may be a vector.
mustart starting values for the fitted values. It can be a vector or a matrix. Some family functions do not make use of this argument.
coefstart starting values for the coefficient vector.
control a list of parameters for controlling the fitting process. See vglm.control for details.
offset a vector or M-column matrix of offset values. These are a priori known and are added to the linear predictors during fitting.
method the method to be used in fitting the model. The default (and presently only) method vglm.fit uses iteratively reweighted least squares (IRLS).
model a logical value indicating whether the model frame should be assigned in the model slot.
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.
contrasts an optional list. See the contrasts.arg of model.matrix.default.
constraints an optional list of constraint matrices. The components of the list must be named with the term it corresponds to (and it must match in character format exactly). Each constraint matrix must have M rows, and be of full-column rank. By default, constraint matrices are the M by M identity matrix unless arguments in the family function itself override these values. If constraints is used it must contain all the terms; an incomplete list is not accepted.
extra an optional list with any extra information that might be needed by the VGAM family function.
qr.arg logical value indicating whether the slot qr, which returns the QR decomposition of the VLM model matrix, is returned on the object.
smart logical value indicating whether smart prediction (smartpred) will be used.
... further arguments passed into vglm.control.

Details

A vector generalized linear model (VGLM) is loosely defined as a statistical model that is a function of M linear predictors. The central formula is given by

eta_j = beta_j^T x

where x is a vector of explanatory variables (sometimes just a 1 for an intercept), and beta_j is a vector of regression coefficients to be estimated. Here, j=1,...,M where M is finite. Then one can write eta=(eta_1,...,eta_M)^T as a vector of linear predictors.

Most users will find vglm similar in flavour to glm. The function vglm.fit actually does the work.

Value

An object of class "vglm", which has the following slots. Some of these may not be assigned to save space, and will be recreated if necessary later.

extra the list extra at the end of fitting.
family the family function (of class "vglmff").
iter the number of IRLS iterations used.
predictors a M-column matrix of linear predictors.
assign a named list which matches the columns and the (LM) model matrix terms.
call the matched call.
coefficients a named vector of coefficients.
constraints a named list of constraint matrices used in the fitting.
contrasts the contrasts used (if any).
control list of control parameter used in the fitting.
criterion list of convergence criterion evaluated at the final IRLS iteration.
df.residual the residual degrees of freedom.
df.total the total degrees of freedom.
dispersion the scaling parameter.
effects the effects.
fitted.values the fitted values, as a matrix. This is usually the mean but may be quantiles, or the location parameter, e.g., in the Cauchy model.
misc a list to hold miscellaneous parameters.
model the model frame.
na.action a list holding information about missing values.
offset if non-zero, a M-column matrix of offsets.
post a list where post-analysis results may be put.
preplot used by plotvgam, the plotting parameters may be put here.
prior.weights initially supplied weights.
qr the QR decomposition used in the fitting.
R the R matrix in the QR decomposition used in the fitting.
rank numerical rank of the fitted model.
residuals the working residuals at the final IRLS iteration.
rss residual sum of squares at the final IRLS iteration with the adjusted dependent vectors and weight matrices.
smart.prediction a list of data-dependent parameters (if any) that are used by smart prediction.
terms the terms object used.
weights the weight matrices at the final IRLS iteration. This is in matrix-band form.
x the model matrix (linear model LM, not VGLM).
xlevels the levels of the factors, if any, used in fitting.
y the response, in matrix form.


This slot information is repeated at vglm-class.

Note

This function can fit a wide variety of statistical models. Some of these are harder to fit than others because of inherent numerical difficulties associated with some of them. Successful model fitting benefits from cumulative experience. Varying the values of arguments in the VGAM family function itself is a good first step if difficulties arise, especially if initial values can be inputted. A second, more general step, is to vary the values of arguments in vglm.control. A third step is to make use of arguments such as etastart, coefstart and mustart.

Some VGAM family functions end in "ff" to avoid interference with other functions, e.g., binomialff, poissonff, gaussianff, gammaff. This is because VGAM family functions are incompatible with glm (and also gam in the gam library and gam in the mgcv library).

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

The theory behind the scaling parameter is currently being made more rigorous, but it it should give the same value as the scale parameter for GLMs.

In Example 5 below, the xij argument to illustrate covariates that are specific to a linear predictor. Here, lop/rop are the ocular pressures of the left/right eye (artificial data). Variables leye and reye might be the presence/absence of a particular disease on the LHS/RHS eye respectively. See fill for more details and examples.

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. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

vglm.control, vglm-class, vglmff-class, smartpred, vglm.fit, fill, rrvglm, vgam. Methods functions include coef.vlm, predict.vglm, summary.vglm, etc.

Examples

# Example 1. Dobson (1990) Page 93: Randomized Controlled Trial :
counts = c(18,17,15,20,10,20,25,13,12)
outcome = gl(3,1,9)
treatment = gl(3,3)
print(d.AD <- data.frame(treatment, outcome, counts))
vglm.D93 = vglm(counts ~ outcome + treatment, family=poissonff)
summary(vglm.D93)

# Example 2. Multinomial logit model
data(pneumo)
pneumo = transform(pneumo, let=log(exposure.time))
vglm(cbind(normal, mild, severe) ~ let, multinomial, pneumo)

# Example 3. Proportional odds model
fit = vglm(cbind(normal,mild,severe) ~ let, cumulative(par=TRUE), pneumo)
coef(fit, matrix=TRUE) 
constraints(fit) 
fit@x # LM model matrix
model.matrix(fit) # Larger VGLM model matrix

# Example 4. Bivariate logistic model 
data(coalminers)
fit = vglm(cbind(nBnW, nBW, BnW, BW) ~ age, binom2.or, coalminers, trace=TRUE)
coef(fit, matrix=TRUE)
fit@y

# Example 5. The use of the xij argument
n = 1000
eyes = data.frame(lop = runif(n), rop = runif(n)) 
eyes = transform(eyes, 
                 leye = ifelse(runif(n) < logit(-1+2*lop, inverse=TRUE), 1, 0),
                 reye = ifelse(runif(n) < logit(-1+2*rop, inverse=TRUE), 1, 0))
fit = vglm(cbind(leye,reye) ~ lop + rop + fill(lop),
           binom2.or(exchangeable=TRUE, zero=3),
           xij = op ~ lop + rop + fill(lop), data=eyes)
coef(fit)
coef(fit, matrix=TRUE)
coef(fit, matrix=TRUE, compress=FALSE)

# Here's one method to handle the xij argument with a term that
# produces more than one column in the model matrix. 
POLY3 = function(x, ...) {
    # A cubic 
    poly(c(x,...), 3)[1:length(x),]
}

fit = vglm(cbind(leye,reye) ~ POLY3(lop,rop) + POLY3(rop,lop) + fill(POLY3(lop,rop)),
           binom2.or(exchangeable=TRUE, zero=3),  data=eyes,
           xij = POLY3(op) ~ POLY3(lop,rop) + POLY3(rop,lop) + 
                             fill(POLY3(lop,rop)))
coef(fit)
coef(fit, matrix=TRUE)
coef(fit, matrix=TRUE, compress=FALSE)
predict(fit)[1:4,]

[Package VGAM version 0.7-7 Index]