\Header{family}{Family Objects for Models}
\alias{binomial}{family}
\alias{Gamma}{family}
\alias{gaussian}{family}
\alias{inverse.gaussian}{family}
\alias{poisson}{family}
\alias{print.family}{family}
\alias{quasi}{family}
\alias{quasibinomial}{family}
\alias{quasipoisson}{family}
\keyword{models}{family}
\begin{Description}\relax
Family objects provide a convenient way to specify the details of the
models used by functions such as \code{\Link{glm}}.  See the
documentation for \code{\Link{glm}} for the details on how such model
fitting takes place.\end{Description}
\begin{Usage}
\begin{verbatim}
family(object)

binomial(link = "logit")
gaussian(link ="identity")
Gamma(link = "inverse")
inverse.gaussian(link = "1/mu^2")
poisson(link = "log")
quasi(link = "identity", variance = "constant")
quasibinomial(link = "logit")
quasipoisson(link = "log")

print.family(x, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{link}] a specification for the model link function.
The \code{binomial} family admits the links \code{"logit"},
\code{"probit"}, \code{"log"}, and \code{"cloglog"} (complementary
log-log);
the \code{Gamma} family the links \code{"identity"},
\code{"inverse"}, and \code{"log"};
the \code{poisson} family the links \code{"identity"}, \code{"log"},
and \code{"sqrt"};
the \code{quasi} family the links \code{"logit"}, \code{"probit"},
\code{"cloglog"},  \code{"identity"}, \code{"inverse"},
\code{"log"}, \code{"1/mu$\,\hat{\,}$2"} and \code{"sqrt"}.
The function \code{\Link{power}} can also be used to create a
power link function for the \code{quasi} family.

The other families have only one permissible link function:
\code{"identity"} for the \code{gaussian} family, and
\code{"1/mu$\,\hat{\,}$2"} for the \code{inverse.gaussian} family.
\item[\code{variance}] for all families, other than \code{quasi}, the
variance function is determined by the family.  The \code{quasi}
family will accept the specifications \code{"constant"},
\code{"mu(1-mu)"}, \code{"mu"}, \code{"mu$\,\hat{\,}$2"} and \code{"mu$\,\hat{\,}$3"} for
the variance function.
\item[\code{object}] the function \code{family} accesses the \code{family}
objects which are stored within objects created by modelling
functions (e.g. \code{glm}).
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The \code{quasibinomial} and \code{quasipoisson} families differ from
the \code{binomial} and \code{poisson} families only in that the
dispersion parameter is not fixed at one, so they can ``model''
over-dispersion.  For the binomial case see McCullagh and Nelder
(1989, pp. 124--8).  Although they show that there is (under some
restrictions) a model with
variance proportional to mean as in the quasi-binomial model, note
that \code{glm} does not compute maximum-likelihood estimates in that
model.  The behaviour of S-PLUS is closer to the quasi- variants.\end{Details}
\begin{References}\relax
McCullagh P. and Nelder, J. A. (1989)
\emph{Generalized Linear Models.}
London: Chapman and Hall.

Dobson, A. J. (1983)
\emph{An Introduction to Statistical Modelling.}
London: Chapman and Hall.

Cox, D. R. and  Snell, E. J. (1981).
\emph{Applied Statistics; Principles and Examples.}
London: Chapman and Hall.\end{References}
\begin{SeeAlso}\relax
\code{\Link{glm}}, \code{\Link{power}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
nf <- gaussian()# Normal family
nf
str(nf)# internal STRucture

gf <- Gamma()
gf
str(gf)
gf$linkinv
all(1:10 == gf$linkfun(gf$linkinv(1:10)))# is TRUE
gf$variance(-3:4) #- == (.)^2

## quasipoisson. compare with example(glm)
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
d.AD <- data.frame(treatment, outcome, counts)
glm.qD93 <- glm(counts ~ outcome + treatment, family=quasipoisson())
glm.qD93
anova(glm.qD93, test="F")
summary(glm.qD93)
## for Poisson results use
anova(glm.qD93, dispersion = 1, test="Chisq")
summary(glm.qD93, dispersion = 1)

## tests of quasi
x <- rnorm(100)
y <- rpois(100, exp(1+x))
glm(y ~x, family=quasi(var="mu", link="log"))
# which is the same as
glm(y ~x, family=poisson)
glm(y ~x, family=quasi(var="mu^2", link="log"))
glm(y ~x, family=quasi(var="mu^3", link="log")) # should fail
y <- rbinom(100, 1, plogis(x))
# needs to set a starting value for the next fit
glm(y ~x, family=quasi(var="mu(1-mu)", link="logit"), start=c(0,1))
\end{ExampleCode}
\end{Examples}

