\name{glm}
\title{Fitting Generalized Linear Models}
\alias{glm}
\alias{glm.fit}
\alias{weights.glm}
\alias{print.glm}
\concept{regression}
\concept{logistic}
\concept{log-linear}
\concept{loglinear}
\description{
  \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.
}
\usage{
glm(formula, family = gaussian, data, weights, subset,
    na.action, start = NULL, etastart, mustart,
    offset, control = glm.control(\dots), model = TRUE,
    method = "glm.fit", x = FALSE, y = TRUE, contrasts = NULL, \dots)

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)

\method{weights}{glm}(object, type = c("prior", "working"), \dots)
}
\arguments{
  \item{formula}{a symbolic description of the model to be fit.
    The details of model specification are given below.}

  \item{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{\link{family}} for details of
    family functions.)}

  \item{data}{an optional data frame, list or environment (or object
    coercible by \code{\link{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{weights}{an optional vector of weights to be used in the fitting
    process.  Should be \code{NULL} or a numeric vector.}

  \item{subset}{an optional vector specifying a subset of observations
    to be used in the fitting process.}

  \item{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{\link{options}}, and is
    \code{\link{na.fail}} if that is unset.  The \dQuote{factory-fresh}
    default is \code{\link{na.omit}}.  Another possible value is
    \code{NULL}, no action.  Value \code{\link{na.exclude}} can be useful.}

  \item{start}{starting values for the parameters in the linear predictor.}

  \item{etastart}{starting values for the linear predictor.}

  \item{mustart}{starting values for the vector of means.}

  \item{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{\link{offset}} terms can be included in the
    formula instead or as well, and if both are specified their sum is
    used.  See \code{\link{model.offset}}.}

  \item{control}{a list of parameters for controlling the fitting
    process.  See the documentation for \code{\link{glm.control}}
    for details.}

  \item{model}{a logical value indicating whether \emph{model frame}
    should be included as a component of the returned value.}

  \item{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{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{contrasts}{an optional list. See the \code{contrasts.arg}
    of \code{model.matrix.default}.}

  \item{object}{an object inheriting from class \code{"glm"}.}
  \item{type}{character, partial matching allowed.  Type of weights to
    extract from the fitted model object.}

  \item{intercept}{logical. Should an intercept be included in the
    \emph{null} model?}

  \item{\dots}{further arguments passed to or from other methods.}
}
\details{
  A typical predictor has the form \code{response ~ 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{\link{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{\link{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}.
}
\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{\link{summary}} (i.e., \code{\link{summary.glm}}) can
  be used to obtain or print a summary of the results and the function
  \code{\link{anova}} (i.e., \code{\link{anova.glm}})
  to produce an analysis of variance table.

  The generic accessor functions \code{\link{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:

  \item{coefficients}{a named vector of coefficients}
  \item{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{fitted.values}{the fitted mean values, obtained by transforming
    the linear predictors by the inverse of the link function.}
  \item{rank}{the numeric rank of the fitted linear model.}
  \item{family}{the \code{\link{family}} object used.}
  \item{linear.predictors}{the linear fit on link scale.}
  \item{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{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{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.  Note that this will be
    incorrect if the link function depends on the data other than
    through the fitted mean: specify a zero offset to force a correct
    calculation.}
  \item{iter}{the number of iterations of IWLS used.}
  \item{weights}{the \emph{working} weights, that is the weights
    in the final iteration of the IWLS fit.}
  \item{prior.weights}{the case weights initially supplied.}
  \item{df.residual}{the residual degrees of freedom.}
  \item{df.null}{the residual degrees of freedom for the null model.}
  \item{y}{the \code{y} vector used. (It is a vector even for a binomial
    model.)}
  \item{converged}{logical. Was the IWLS algorithm judged to have converged?}
  \item{boundary}{logical. Is the fitted value on the boundary of the
    attainable values?}
  \item{call}{the matched call.}
  \item{formula}{the formula supplied.}
  \item{terms}{the \code{\link{terms}} object used.}
  \item{data}{the \code{data argument}.}
  \item{offset}{the offset vector used.}
  \item{control}{the value of the \code{control} argument used.}
  \item{method}{the name of the fitter function used, currently always
    \code{"glm.fit"}.}
  \item{contrasts}{(where relevant) the contrasts used.}
  \item{xlevels}{(where relevant) a record of the levels of the factors
    used in fitting.}

  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{\link{residuals}} and \code{weights} do \bold{not} just pick out
  the component of the fit with the same name.

  If a \code{\link{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.
}
\seealso{
  \code{\link{anova.glm}}, \code{\link{summary.glm}}, etc. for
  \code{glm} methods,
  and the generic functions \code{\link{anova}}, \code{\link{summary}},
  \code{\link{effects}}, \code{\link{fitted.values}},
  and \code{\link{residuals}}. Further, \code{\link{lm}} for
  non-generalized \emph{linear} models.

  \code{\link[datasets]{esoph}}, \code{\link[datasets]{infert}} and
  \code{\link{predict.glm}} have examples of fitting binomial glms.
}
\author{
  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).
}
\references{
  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.
}

\examples{
## 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))

\dontrun{
## for an example of the use of a terms object as a formula
demo(glm.vr)
}}
\keyword{models}
\keyword{regression}
