\HeaderA{glm}{Fitting Generalized Linear Models}{glm}
\methaliasA{glm.fit}{glm}{glm.fit}
\aliasA{print.glm}{glm}{print.glm}
\aliasA{weights.glm}{glm}{weights.glm}
\keyword{models}{glm}
\keyword{regression}{glm}
\begin{Description}\relax
\code{glm} is used to fit generalized linear models, specified by
giving a symbolic description of the linear predictor and a
description of the error distribution.
\end{Description}
\begin{Usage}
\begin{verbatim}
glm(formula, family = gaussian, data, weights, subset,
    na.action, start = NULL, etastart, mustart,
    offset, control = glm.control(...), model = TRUE,
    method = "glm.fit", x = FALSE, y = TRUE, contrasts = NULL, ...)

glm.fit(x, y, weights = rep(1, nobs),
        start = NULL, etastart = NULL, mustart = NULL,
        offset = rep(0, nobs), family = gaussian(),
        control = glm.control(), intercept = TRUE)

## S3 method for class 'glm':
weights(object, type = c("prior", "working"), ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{formula}] a symbolic description of the model to be fit.
The details of model specification are given below.
\item[\code{family}] a description of the error distribution and link
function to be used in the model. This can be a character string
naming a family function, a family function or the result of a call
to a family function.  (See \code{\LinkA{family}{family}} for details of
family functions.)
\item[\code{data}] an optional data frame, list or environment (or object
coercible by \code{\LinkA{as.data.frame}{as.data.frame}} to a data frame) containing
the variables in the model.  If not found in \code{data}, the
variables are taken from \code{environment(formula)},
typically the environment from which \code{glm} is called.
\item[\code{weights}] an optional vector of weights to be used in the fitting
process.  Should be \code{NULL} or a numeric vector.
\item[\code{subset}] an optional vector specifying a subset of observations
to be used in the fitting process.
\item[\code{na.action}] a function which indicates what should happen
when the data contain \code{NA}s.  The default is set by
the \code{na.action} setting of \code{\LinkA{options}{options}}, and is
\code{\LinkA{na.fail}{na.fail}} if that is unset.  The \dQuote{factory-fresh}
default is \code{\LinkA{na.omit}{na.omit}}.  Another possible value is
\code{NULL}, no action.  Value \code{\LinkA{na.exclude}{na.exclude}} can be useful.
\item[\code{start}] starting values for the parameters in the linear predictor.
\item[\code{etastart}] starting values for the linear predictor.
\item[\code{mustart}] starting values for the vector of means.
\item[\code{offset}] this can be used to specify an \emph{a priori}
known component to be included in the linear predictor
during fitting.  This should be \code{NULL} or a numeric vector of
length either one or equal to the number of cases.
One or more \code{\LinkA{offset}{offset}} terms can be included in the
formula instead or as well, and if both are specified their sum is
used.  See \code{\LinkA{model.offset}{model.offset}}.
\item[\code{control}] a list of parameters for controlling the fitting
process.  See the documentation for \code{\LinkA{glm.control}{glm.control}}
for details.
\item[\code{model}] a logical value indicating whether \emph{model frame}
should be included as a component of the returned value.
\item[\code{method}] the method to be used in fitting the model.
The default method \code{"glm.fit"} uses iteratively reweighted
least squares (IWLS).  The only current alternative is
\code{"model.frame"} which returns the model frame and does no fitting.
\item[\code{x, y}] For \code{glm}:
logical values indicating whether the response
vector and model matrix used in the fitting process
should be returned as components of the returned value.

For \code{glm.fit}: \code{x} is a design matrix of dimension \code{n
      * p}, and \code{y} is a vector of observations of length \code{n}.

\item[\code{contrasts}] an optional list. See the \code{contrasts.arg}
of \code{model.matrix.default}.
\item[\code{object}] an object inheriting from class \code{"glm"}.
\item[\code{type}] character, partial matching allowed.  Type of weights to
extract from the fitted model object.
\item[\code{intercept}] logical. Should an intercept be included in the
\emph{null} model?
\item[\code{...}] further arguments passed to or from other methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
A typical predictor has the form \code{response \textasciitilde{} terms} where
\code{response} is the (numeric) response vector and \code{terms} is a
series of terms which specifies a linear predictor for \code{response}.
For \code{binomial} and \code{quasibinomial} families the response can
also be specified as a \code{\LinkA{factor}{factor}} (when the first level
denotes failure and all
others success) or as a two-column matrix with the columns giving the
numbers of successes and failures.  A terms specification of the form
\code{first + second} indicates all the terms in \code{first} together
with all the terms in \code{second} with duplicates removed.  The terms in
the formula will be re-ordered so that main effects come first,
followed by the interactions, all second-order, all third-order and so
on: to avoid this pass a \code{terms} object as the formula.

A specification of the form \code{first:second} indicates the
the set of terms obtained by taking the interactions of
all terms in \code{first} with all terms in \code{second}.
The specification \code{first*second} indicates the \emph{cross}
of \code{first} and \code{second}.
This is the same as \code{first + second + first:second}.

\code{glm.fit} is the workhorse function.

If more than one of \code{etastart}, \code{start} and \code{mustart}
is specified, the first in the list will be used.  It is often
advisable to supply starting values for a \code{\LinkA{quasi}{quasi}} family,
and also for families with unusual links such as \code{gaussian("log")}.

All of \code{weights}, \code{subset}, \code{offset}, \code{etastart}
and \code{mustart} are evaluated in the same way as variables in
\code{formula}, that is first in \code{data} and then in the
environment of \code{formula}.
\end{Details}
\begin{Value}
\code{glm} returns an object of class inheriting from \code{"glm"}
which inherits from the class \code{"lm"}. See later in this section.

The function \code{\LinkA{summary}{summary}} (i.e., \code{\LinkA{summary.glm}{summary.glm}}) can
be used to obtain or print a summary of the results and the function
\code{\LinkA{anova}{anova}} (i.e., \code{\LinkA{anova.glm}{anova.glm}})
to produce an analysis of variance table.

The generic accessor functions \code{\LinkA{coefficients}{coefficients}},
\code{effects}, \code{fitted.values} and \code{residuals} can be used to
extract various useful features of the value returned by \code{glm}.

\code{weights} extracts a vector of weights, one for each case in the
fit (after subsetting and \code{na.action}).

An object of class \code{"glm"} is a list containing at least the
following components:

\begin{ldescription}
\item[\code{coefficients}] a named vector of coefficients
\item[\code{residuals}] the \emph{working} residuals, that is the residuals
in the final iteration of the IWLS fit.  Since cases with zero
weights are omitted, their working residuals are \code{NA}.
\item[\code{fitted.values}] the fitted mean values, obtained by transforming
the linear predictors by the inverse of the link function.
\item[\code{rank}] the numeric rank of the fitted linear model.
\item[\code{family}] the \code{\LinkA{family}{family}} object used.
\item[\code{linear.predictors}] the linear fit on link scale.
\item[\code{deviance}] up to a constant, minus twice the maximized
log-likelihood.  Where sensible, the constant is chosen so that a
saturated model has deviance zero.
\item[\code{aic}] Akaike's \emph{An Information Criterion}, minus twice the
maximized log-likelihood plus twice the number of coefficients (so
assuming that the dispersion is known).
\item[\code{null.deviance}] The deviance for the null model, comparable with
\code{deviance}. The null model will include the offset, and an
intercept if there is one in the model
\item[\code{iter}] the number of iterations of IWLS used.
\item[\code{weights}] the \emph{working} weights, that is the weights
in the final iteration of the IWLS fit.
\item[\code{prior.weights}] the case weights initially supplied.
\item[\code{df.residual}] the residual degrees of freedom.
\item[\code{df.null}] the residual degrees of freedom for the null model.
\item[\code{y}] the \code{y} vector used. (It is a vector even for a binomial
model.)
\item[\code{converged}] logical. Was the IWLS algorithm judged to have converged?
\item[\code{boundary}] logical. Is the fitted value on the boundary of the
attainable values?
\item[\code{call}] the matched call.
\item[\code{formula}] the formula supplied.
\item[\code{terms}] the \code{\LinkA{terms}{terms}} object used.
\item[\code{data}] the \code{data argument}.
\item[\code{offset}] the offset vector used.
\item[\code{control}] the value of the \code{control} argument used.
\item[\code{method}] the name of the fitter function used, currently always
\code{"glm.fit"}.
\item[\code{contrasts}] (where relevant) the contrasts used.
\item[\code{xlevels}] (where relevant) a record of the levels of the factors
used in fitting.
\end{ldescription}


In addition, non-empty fits will have components \code{qr}, \code{R}
and \code{effects} relating to the final weighted linear fit.

Objects of class \code{"glm"} are normally of class \code{c("glm",
    "lm")}, that is inherit from class \code{"lm"}, and well-designed
methods for class \code{"lm"} will be applied to the weighted linear
model at the final iteration of IWLS.  However, care is needed, as
extractor functions for class \code{"glm"} such as
\code{\LinkA{residuals}{residuals}} and \code{weights} do \bold{not} just pick out
the component of the fit with the same name.

If a \code{\LinkA{binomial}{binomial}} \code{glm} model is specified by giving a
two-column response, the weights returned by \code{prior.weights} are
the total numbers of cases (factored by the supplied case weights) and
the component \code{y} of the result is the proportion of successes.
\end{Value}
\begin{Author}\relax
The original \R{} implementation of \code{glm} was written by Simon
Davies working for Ross Ihaka at the University of Auckland, but has
since been extensively re-written by members of the R Core team.

The design was inspired by the S function of the same name described
in Hastie \& Pregibon (1992).
\end{Author}
\begin{References}\relax
Dobson, A. J. (1990)
\emph{An Introduction to Generalized Linear Models.}
London: Chapman and Hall.

Hastie, T. J. and Pregibon, D. (1992)
\emph{Generalized linear models.}
Chapter 6 of \emph{Statistical Models in S}
eds J. M. Chambers and T. J. Hastie, Wadsworth \& Brooks/Cole.

McCullagh P. and Nelder, J. A. (1989)
\emph{Generalized Linear Models.}
London: Chapman and Hall.

Venables, W. N. and Ripley, B. D. (2002)
\emph{Modern Applied Statistics with S.}
New York: Springer.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{anova.glm}{anova.glm}}, \code{\LinkA{summary.glm}{summary.glm}}, etc. for
\code{glm} methods,
and the generic functions \code{\LinkA{anova}{anova}}, \code{\LinkA{summary}{summary}},
\code{\LinkA{effects}{effects}}, \code{\LinkA{fitted.values}{fitted.values}},
and \code{\LinkA{residuals}{residuals}}. Further, \code{\LinkA{lm}{lm}} for
non-generalized \emph{linear} models.

\code{\LinkA{esoph}{esoph}}, \code{\LinkA{infert}{infert}} and
\code{\LinkA{predict.glm}{predict.glm}} have examples of fitting binomial glms.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## 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))
glm.D93 <- glm(counts ~ outcome + treatment, family=poisson())
anova(glm.D93)
summary(glm.D93)

## an example with offsets from Venables & Ripley (2002, p.189)
data(anorexia, package="MASS")

anorex.1 <- glm(Postwt ~ Prewt + Treat + offset(Prewt),
                family = gaussian, data = anorexia)
summary(anorex.1)

# A Gamma example, from McCullagh & Nelder (1989, pp. 300-2)
clotting <- data.frame(
    u = c(5,10,15,20,30,40,60,80,100),
    lot1 = c(118,58,42,35,27,25,21,19,18),
    lot2 = c(69,35,26,21,18,16,13,12,12))
summary(glm(lot1 ~ log(u), data=clotting, family=Gamma))
summary(glm(lot2 ~ log(u), data=clotting, family=Gamma))

## Not run: 
## for an example of the use of a terms object as a formula
demo(glm.vr)
## End(Not run)\end{ExampleCode}
\end{Examples}

