% File src/library/stats/man/formula.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2022 R Core Team
% Distributed under GPL 2 or later

\name{formula}
\alias{formula}
\alias{formula.default}
\alias{formula.formula}
\alias{formula.terms}
\alias{formula.data.frame}
\alias{DF2formula}
\alias{as.formula}
\alias{print.formula}
\alias{[.formula}

\title{Model Formulae}
\usage{
formula(x, \dots)
DF2formula(x, env = parent.frame())
as.formula(object, env = parent.frame())

\method{print}{formula}(x, showEnv = !identical(e, .GlobalEnv), \dots)
}
\description{
  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"}.
}
\arguments{
  \item{x, object}{\R object, for \code{DF2formula()} a \code{\link{data.frame}}.}
  \item{\dots}{further arguments passed to or from other methods.}
  \item{env}{the environment to associate with the result, if not
    already a formula.}
  \item{showEnv}{logical indicating if the environment should be printed
    as well.}
}
\details{
  The models fitted by, e.g., the \code{\link{lm}} and \code{\link{glm}}
  functions are specified in a compact symbolic form.
  The \code{\link{~}} operator is basic in the formation of such models.
  An expression of the form \code{y ~ 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.
 \itemize{
  \item The \code{*} operator denotes factor crossing: \code{a*b} is
  interpreted as \code{a + b + a:b}.
  \item The \code{^}
  operator indicates crossing to the specified degree.  For example
  \code{(a+b+c)^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.
  \item 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}.
  \item The \code{/} operator provides a shorthand, so that
  \code{a / b} is equivalent to \code{a + b \%in\% a}.
  \item The \code{-} operator removes the specified terms, hence
  \code{(a+b+c)^2 - a:b} is identical to \code{a + b + c + b:c + a:c}.
  It can also used to remove the intercept term: when fitting a linear
  model \code{y ~ x - 1} specifies a line through the origin.
  A model with no intercept can be also specified as \code{y ~ x + 0}
  or \code{y ~ 0 + x}.
 }

  While formulae usually involve just variable and factor
  names, they can also involve arithmetic expressions.
  The formula \code{log(y) ~ 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{\link{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 ~ 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
  containing calls to function \code{\link{offset}}.  These calls
  indicate terms with coefficient fixed to one.  Some model-fitting
  functions recognize other \dQuote{specials} like \code{strata} and
  \code{cluster} (typically by setting the \code{specials} argument of
  \code{\link{terms.formula}}).  Because specials are nothing more than
  syntax, disambiguation using \code{\link{::}} or similar is nonsense
  (even if packages happen to define so-named functions) and the result
  is often a model that the user does not intend.  For example, the
  formula \code{y ~ stats::offset(w) + x} does \emph{not} specify a
  model in which the coefficient of \code{w} is fixed to one.

  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{\link{terms.formula}}.  In the context of
  \code{\link{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 \code{formula} method for data frames.  When there's
  \code{"terms"} attribute with a formula, e.g., for a
  \code{\link{model.frame}()}, that formula is returned.  If you'd like the
  previous (\R \eqn{\le} 3.5.x) behavior, use the auxiliary
  \code{DF2formula()} which does not consider a \code{"terms"} attribute.
  Otherwise, 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.
}
\note{
  In \R versions up to 3.6.0, \code{\link{character}} \code{x} of length
  more than one were parsed as separate lines of \R code and the first
  complete expression was evaluated into a formula when possible.  This
  silently truncates such vectors of characters inefficiently and to some
  extent inconsistently as this behaviour had been undocumented.  For this
  reason, such use has been deprecated.  If you must work via character
  \code{x}, do use a string, i.e., a character vector of length one.

  E.g., \code{eval(call("~", quote(foo + bar)))} has been an order of magnitude
  more efficient % 20 times faster
  than \code{formula(c("~", "foo + bar"))}.

  Further, character \dQuote{expressions} needing an \code{\link{eval}()}
  to return a formula are now deprecated.
}
\section{Environments}{
  A formula object has an associated environment, and
  this environment (rather than the parent
  environment) is used by \code{\link{model.frame}} to evaluate variables
  that are not found in the supplied \code{data} argument.

  Formulas created with the \code{~} operator use the
  environment in which they were created.  Formulas created with
  \code{as.formula} will use the \code{env} argument for their
  environment.
}
\value{
  All the functions above produce an object of class \code{"formula"}
  which contains a symbolic model formula.
}
\references{
  \bibshow{R:Chambers+Hastie:1992:c2}
}
\seealso{
  \code{\link{~}}, \code{\link{I}}, \code{\link{offset}}.

  For formula manipulation: \code{\link{update.formula}},
  \code{\link{terms.formula}}, and \code{\link{all.vars}}.
  For typical use: \code{\link{lm}}, \code{\link{glm}}, and
  \code{\link{coplot}}.
  For formula construction: \code{\link{reformulate}}.
}
\examples{
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 <- paste0("x", 1:25)
(fmla <- as.formula(paste("y ~ ", paste(xnam, collapse= "+"))))
## Equivalent with reformulate():
fmla2 <- reformulate(xnam, response = "y")
stopifnot(identical(fmla, fmla2))
}
\keyword{models}
