vglm: Fitting Vector Generalized Linear Models

Description

vglm fits vector generalized linear models (VGLMs). This very large class of models includes generalized linear models (GLMs) as a special case.

Usage

vglm(formula,
     family = stop("argument 'family' needs to be assigned"),
     data = list(), weights = NULL, subset = NULL,
     na.action, 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(),
     form2 = NULL, qr.arg = TRUE, smart = TRUE, ...)

Arguments

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 often 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 (the weights argument). Also see weightsvglm.

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.

ResSS

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 working 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.

Details

A vector generalized linear model (VGLM) is loosely defined as a statistical model that is a function of $M$ linear predictors and can be estimated by Fisher scoring. 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, $j=1,\ldots,M$, where $M$ is finite. Then one can write $\eta=(\eta_1,\ldots,\eta_M)^T$ as a vector of linear predictors.

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

References

Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: Springer.

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.

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. (2008). The VGAM Package. R News, 8, 28--39.

Examples

Run this code

# Example 1. See help(glm)
(d.AD <- data.frame(treatment = gl(3, 3),
                    outcome = gl(3, 1, 9),
                    counts = c(18,17,15,20,10,20,25,13,12)))
vglm.D93 <- vglm(counts ~ outcome + treatment, poissonff,
                 data = d.AD, trace = TRUE)
summary(vglm.D93)


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


# Example 3. Proportional odds model
fit3 <- vglm(cbind(normal, mild, severe) ~ let, propodds, pneumo)
coef(fit3, matrix = TRUE)
constraints(fit3)
model.matrix(fit3, type = "lm")  # LM model matrix
model.matrix(fit3)               # Larger VGLM (or VLM) matrix


# Example 4. Bivariate logistic model
fit4 <- vglm(cbind(nBnW, nBW, BnW, BW) ~ age, binom2.or, coalminers)
coef(fit4, matrix = TRUE)
depvar(fit4)  # Response are proportions
weights(fit4, type = "prior")


# Example 5. The use of the xij argument (simple case).
# The constraint matrix for 'op' has one column.
nn <- 1000
eyesdat <- round(data.frame(lop = runif(nn),
                            rop = runif(nn),
                             op = runif(nn)), digits = 2)
eyesdat <- transform(eyesdat, eta1 = -1 + 2 * lop,
                              eta2 = -1 + 2 * lop)
eyesdat <- transform(eyesdat,
           leye = rbinom(nn, 1, prob = logitlink(eta1, inv = TRUE)),
           reye = rbinom(nn, 1, prob = logitlink(eta2, inv = TRUE)))
head(eyesdat)
fit5 <- vglm(cbind(leye, reye) ~ op,
             binom2.or(exchangeable = TRUE, zero = 3),
             data = eyesdat, trace = TRUE,
             xij = list(op ~ lop + rop + fill1(lop)),
             form2 = ~  op + lop + rop + fill1(lop))
coef(fit5)
coef(fit5, matrix = TRUE)
constraints(fit5)
fit5@control$xij
head(model.matrix(fit5))


# Example 6. The use of the 'constraints' argument.
as.character(~ bs(year,df=3))  # Get the white spaces right
clist <- list("(Intercept)"      = diag(3),
              "bs(year, df = 3)" = rbind(1, 0, 0))
fit1 <- vglm(r1 ~ bs(year,df=3), gev(zero = NULL),
             data = venice, constraints = clist, trace = TRUE)
coef(fit1, matrix = TRUE)  # Check

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