\HeaderA{formula}{Model Formulae}{formula}
\aliasA{as.formula}{formula}{as.formula}
\methaliasA{formula.data.frame}{formula}{formula.data.frame}
\methaliasA{formula.default}{formula}{formula.default}
\methaliasA{formula.formula}{formula}{formula.formula}
\methaliasA{formula.terms}{formula}{formula.terms}
\aliasA{print.formula}{formula}{print.formula}
\aliasA{[.formula}{formula}{[.formula}
\keyword{models}{formula}
\begin{Description}\relax
The generic function \code{formula} and its specific methods provide a
way of extracting formulae which have been included in other objects.

\code{as.formula} is almost identical, additionally preserving
attributes when \code{object} already inherits from
\code{"formula"}.  The default value of the \code{env} argument is
used only when the formula would otherwise lack an environment.
\end{Description}
\begin{Usage}
\begin{verbatim}
formula(x, ...)
as.formula(object, env = parent.frame())
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, object}] \R{} object.
\item[\code{...}] further arguments passed to or from other methods.
\item[\code{env}] the environment to associate with the result.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The models fit by, e.g., the \code{\LinkA{lm}{lm}} and \code{\LinkA{glm}{glm}} functions
are specified in a compact symbolic form.
The \code{\textasciitilde{}} operator is basic in the formation of such models.
An expression of the form \code{y \textasciitilde{} model} is interpreted
as a specification that the response \code{y} is modelled
by a linear predictor specified symbolically by \code{model}.
Such a model consists of a series of terms separated
by \code{+} operators.
The terms themselves consist of variable and factor
names separated by \code{:} operators.
Such a term is interpreted as the interaction of
all the variables and factors appearing in the term.

In addition to \code{+} and \code{:}, a number of other operators are
useful in model formulae.  The \code{*} operator denotes factor
crossing: \code{a*b} interpreted as \code{a+b+a:b}.  The \code{\textasciicircum{}}
operator indicates crossing to the specified degree.  For example
\code{(a+b+c)\textasciicircum{}2} is identical to \code{(a+b+c)*(a+b+c)} which in turn
expands to a formula containing the main effects for \code{a},
\code{b} and \code{c} together with their second-order interactions.
The \code{\%in\%} operator indicates that the terms on its left are
nested within those on the right.  For example \code{a + b \%in\% a}
expands to the formula \code{a + a:b}.  The \code{-} operator removes
the specified terms, so that \code{(a+b+c)\textasciicircum{}2 - a:b} is identical to
\code{a + b + c + b:c + a:c}.  It can also used to remove the intercept
term: \code{y \textasciitilde{} x - 1} is a line through the origin.  A model with no
intercept can be also specified as \code{y \textasciitilde{} x + 0} or \code{y \textasciitilde{} 0 + x}.

While formulae usually involve just variable and factor
names, they can also involve arithmetic expressions.
The formula \code{log(y) \textasciitilde{} a + log(x)} is quite legal.
When such arithmetic expressions involve
operators which are also used symbolically
in model formulae, there can be confusion between
arithmetic and symbolic operator use.

To avoid this confusion, the function \code{\LinkA{I}{I}()}
can be used to bracket those portions of a model
formula where the operators are used in their
arithmetic sense.  For example, in the formula
\code{y \textasciitilde{} a + I(b+c)}, the term \code{b+c} is to be
interpreted as the sum of \code{b} and \code{c}.

Variable names can be quoted by backticks \code{`like this`} in
formulae, although there is no guarantee that all code using formulae
will accept such non-syntactic names.

Most model-fitting functions accept formulae with right-hand-side
including the function \code{\LinkA{offset}{offset}} to indicate terms with a
fixed coefficient of one.  Some functions accept other
\sQuote{specials} such as \code{strata} or \code{cluster} (see the
\code{specials} argument of \code{\LinkA{terms.formula}{terms.formula})}.

There are two special interpretations of \code{.} in a formula.  The
usual one is in the context of a \code{data} argument of model
fitting functions and means \sQuote{all columns not otherwise in the
formula}: see \code{\LinkA{terms.formula}{terms.formula}}.  In the context of
\code{\LinkA{update.formula}{update.formula}}, \bold{only}, it means \sQuote{what was
previously in this part of the formula}.

When \code{formula} is called on a fitted model object, either a
specific method is used (such as that for class \code{"nls"}) or the
default method.  The default first looks for a \code{"formula"}
component of the object (and evaluates it), then a \code{"terms"}
component, then a \code{formula} parameter of the call (and evaluates
its value) and finally a \code{"formula"} attribute.

There is a method for data frames.  If there is only one column this
forms the RHS with an empty LHS.   For more columns, the first column
is the LHS of the formula and the remaining columns separated by
\code{+} form the RHS.
\end{Details}
\begin{Value}
All the functions above produce an object of class \code{"formula"}
which contains a symbolic model formula.
\end{Value}
\begin{Section}{Environments}
A formula object has an associated environment, and
this environment (rather than the parent
environment) is used by \code{\LinkA{model.frame}{model.frame}} to evaluate variables
that are not found in the supplied \code{data} argument.

Formulas created with the \code{\textasciitilde{}} operator use the
environment in which they were created.  Formulas created with
\code{as.formula} will use the \code{env} argument for their
environment.  Pre-existing formulas extracted with
\code{as.formula} will only have their environment changed if
\code{env} is given explicitly.
\end{Section}
\begin{References}\relax
Chambers, J. M. and Hastie, T. J. (1992)
\emph{Statistical models.}
Chapter 2 of \emph{Statistical Models in S}
eds J. M. Chambers and T. J. Hastie, Wadsworth \& Brooks/Cole.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{I}{I}}, \code{\LinkA{offset}{offset}}.

For formula manipulation: \code{\LinkA{terms}{terms}}, and \code{\LinkA{all.vars}{all.vars}};
for typical use: \code{\LinkA{lm}{lm}}, \code{\LinkA{glm}{glm}}, and
\code{\LinkA{coplot}{coplot}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
class(fo <- y ~ x1*x2) # "formula"
fo
typeof(fo)# R internal : "language"
terms(fo)

environment(fo)
environment(as.formula("y ~ x"))
environment(as.formula("y ~ x", env=new.env()))

## Create a formula for a model with a large number of variables:
xnam <- paste("x", 1:25, sep="")
(fmla <- as.formula(paste("y ~ ", paste(xnam, collapse= "+"))))
\end{ExampleCode}
\end{Examples}

