% File src/library/stats/man/influence.measures.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2026 R Core Team
% Distributed under GPL 2 or later

\name{influence.measures}
\title{Regression Deletion Diagnostics}

\concept{studentized residuals}
\concept{standardized residuals}
\concept{Cook's distances}
\concept{Covariance ratios}
\concept{\I{DFBETAs}}
\concept{\I{DFFITs}}
\concept{\I{PRESS}}

\alias{influence.measures}
%\alias{print.infl}
%\alias{summary.infl} <- To document: has 'digits' & return()s
\alias{hat}
\alias{hatvalues}
\alias{hatvalues.lm}
\alias{hatvalues.nls}
\alias{rstandard}
\alias{rstandard.lm}
\alias{rstandard.glm}
\alias{rstudent}
\alias{rstudent.lm}
\alias{rstudent.glm}
\alias{dfbeta}
\alias{dfbeta.lm}
\alias{dfbetas}
\alias{dfbetas.lm}
\alias{dfbetas.glm}
\alias{dffits}
\alias{dffits.lm}
\alias{dffits.glm}
\alias{covratio}
\alias{cooks.distance}
\alias{cooks.distance.lm}
\alias{cooks.distance.glm}
\usage{
influence.measures(model, infl = influence(model))

rstandard(model, \dots)
\method{rstandard}{lm}(model, infl = lm.influence(model, do.coef = FALSE),
          sd = sigma(model),
          type = c("sd.1", "predictive"), \dots)
\method{rstandard}{glm}(model, infl = influence(model, do.coef = FALSE),
          type = c("pearson", "deviance"), \dots)

rstudent(model, \dots)
\method{rstudent}{lm}(model, infl = lm.influence(model, do.coef = FALSE),
         res = infl$wt.res, \dots)
\method{rstudent}{glm}(model, infl = influence(model, do.coef = FALSE), \dots)

dffits(model, \dots)
\method{dffits}{lm}(model, infl = lm.influence(model, do.coef = FALSE),
		   res = weighted.residuals(model), \dots)
\method{dffits}{glm}(model, infl = lm.influence(model, do.coef = FALSE),
		   res = weighted.residuals(model), \dots)

dfbeta(model, \dots)
\method{dfbeta}{lm}(model, infl = lm.influence(model, do.coef = TRUE), \dots)

dfbetas(model, \dots)
\method{dfbetas}{lm}(model, infl = lm.influence(model, do.coef = TRUE), \dots)
\method{dfbetas}{glm}(model, infl = lm.influence(model, do.coef = TRUE), \dots)

covratio(model, infl = lm.influence(model, do.coef = FALSE),
         res = weighted.residuals(model))

cooks.distance(model, \dots)
\method{cooks.distance}{lm}(model, infl = lm.influence(model, do.coef = FALSE),
               res = weighted.residuals(model),
               sd = sqrt(deviance(model)/df.residual(model)),
               hat = infl$hat, \dots)
\method{cooks.distance}{glm}(model, infl = influence(model, do.coef = FALSE),
               res = infl$pear.res,
               dispersion = summary(model)$dispersion,
               hat = infl$hat, \dots)

hatvalues(model, \dots)
\method{hatvalues}{lm}(model, infl = lm.influence(model, do.coef = FALSE), \dots)
\method{hatvalues}{nls}(model, \dots)% ../R/nls.R %

hat(x, intercept = TRUE)
}
\arguments{
  \item{model}{an \R object, typically returned by \code{\link{lm}} or
    \code{\link{glm}}.}
  \item{infl}{influence structure as returned by
    \code{\link{lm.influence}} or \code{\link{influence}} (the latter
    only for the \code{glm} method of \code{rstudent} and
    \code{cooks.distance}).}
  \item{res}{(possibly weighted) residuals, with proper default.}
  \item{sd}{standard deviation to use, see default.}
  \item{dispersion}{dispersion (for \code{\link{glm}} objects) to use,
    see default.}
  \item{hat}{hat values \eqn{H_{ii}}{H[i,i]}, see default.}
  \item{type}{type of residuals for \code{rstandard}, with different
    options and meanings for \code{lm} and \code{glm}.  Can be
    abbreviated.}

  \item{x}{the \eqn{X} or design matrix.}
  \item{intercept}{should an intercept column be prepended to \code{x}?}
  \item{\dots}{further arguments passed to or from other methods.}
}
\description{
  This suite of functions can be used to compute some of the regression
  (leave-one-out deletion) diagnostics for linear and generalized linear
  models discussed in \bibcitet{R:Belsley+Kuh+Welsch:1980},
  \bibcitet{R:Cook+Weisberg:1982}, etc.
}
\details{
  The primary high-level function is \code{influence.measures} which produces a
  class \code{"infl"} object tabular display showing the \I{DFBETA}s for
  each model variable, \I{DFFIT}s, covariance ratios, Cook's distances and
  the diagonal elements of the hat matrix.  Cases which are influential
  with respect to any of these measures are marked with an asterisk.

  The functions \code{dfbetas}, \code{dffits},
  \code{covratio} and \code{cooks.distance} provide direct access to the
  corresponding diagnostic quantities.  Functions \code{rstandard} and
  \code{rstudent} give the standardized and Studentized residuals
  respectively. (These re-normalize the residuals to have unit variance,
  using an overall and leave-one-out measure of the error variance
  respectively.)

  Note that for \emph{multivariate} \code{lm()} models (of class
  \code{"mlm"}), these functions return 3d arrays instead of matrices,
  or matrices instead of vectors.

  Values for generalized linear models are approximations, as described
  in \bibcitet{R:Williams:1987} (except that Cook's distances are scaled as
  \eqn{F} rather than as chi-square values).  The approximations can be
  poor when some cases have large influence.

  The optional \code{infl}, \code{res} and \code{sd} arguments are there
  to encourage the use of these direct access functions, in situations
  where, e.g., the underlying basic influence measures (from
  \code{\link{lm.influence}} or the generic \code{\link{influence}}) are
  already available.

  Note that cases with \code{weights == 0} are \emph{dropped} from all
  these functions, but that if a linear model has been fitted with
  \code{na.action = na.exclude}, suitable values are filled in for the
  cases excluded during fitting.

  For linear models, \code{rstandard(*, type = "predictive")} provides
  leave-one-out cross validation residuals, and the \dQuote{PRESS}
  statistic (\I{\bold{PRE}dictive \bold{S}um of \bold{S}quares}, the same as
  the CV score) of model \code{model} is \preformatted{   PRESS <- sum(rstandard(model, type="pred")^2)}

  The function \code{hat()} exists mainly for S (version 2)
  compatibility; we recommend using \code{hatvalues()} instead.
}
\note{
  For \code{hatvalues}, \code{dfbeta}, and \code{dfbetas}, the method
  for linear models also works for generalized linear models. These were
  updated for R 4.6 to use \emph{working} residuals and weights; it is
  assumed that the model has methods suitable for
  \code{\link{weighted.residuals}()}.

  The \code{\link{nls}} method for \code{hatvalues()} is based on theory in
  \bibcitet{R:StLaurent+Cook:1992}.
}
\author{
  Several R Core Team members and John Fox, originally in his \CRANpkg{car}
  package; \code{hatvalues.nls} by R Dev Day participants, see \PR{18897}.
}
\references{
  \bibshow{*,
    R:Fox:2000,
    R:Fox:2002,
    R:Fox+Weisberg:2019}
}
\seealso{
  \code{\link{influence}} (containing \code{\link{lm.influence}}).

  \sQuote{\link{plotmath}} for the use of \code{hat} in plot annotation.
}
\examples{
require(graphics)

## Analysis of the life-cycle savings data
## given in Belsley, Kuh and Welsch.
lm.SR <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data = LifeCycleSavings)

inflm.SR <- influence.measures(lm.SR)
which(apply(inflm.SR$is.inf, 1, any))
# which observations 'are' influential
summary(inflm.SR) # only these
\donttest{inflm.SR          # all}
plot(rstudent(lm.SR) ~ hatvalues(lm.SR)) # recommended by some
plot(lm.SR, which = 5) # an enhanced version of that via plot(<lm>)

## The 'infl' argument is not needed, but avoids recomputation:
rs <- rstandard(lm.SR)
iflSR <- influence(lm.SR)
all.equal(rs, rstandard(lm.SR, infl = iflSR), tolerance = 1e-10)
## to "see" the larger values:
1000 * round(dfbetas(lm.SR, infl = iflSR), 3)
cat("PRESS :"); (PRESS <- sum( rstandard(lm.SR, type = "predictive")^2 ))
stopifnot(all.equal(PRESS, sum( (residuals(lm.SR) / (1 - iflSR$hat))^2)))

## Show that "PRE-residuals"  ==  L.O.O. Crossvalidation (CV) errors:
X <- model.matrix(lm.SR)
y <- model.response(model.frame(lm.SR))
## Leave-one-out CV least-squares prediction errors (relatively fast)
rCV <- vapply(seq_len(nrow(X)), function(i)
              y[i] - X[i,] \%*\% .lm.fit(X[-i,], y[-i])$coefficients,
              numeric(1))
## are the same as the *faster* rstandard(*, "pred") :
stopifnot(all.equal(rCV, unname(rstandard(lm.SR, type = "predictive"))))


## Huber's data [Atkinson 1985]
xh <- c(-4:0, 10)
yh <- c(2.48, .73, -.04, -1.44, -1.32, 0)
lmH <- lm(yh ~ xh)
\donttest{summary(lmH)}
im <- influence.measures(lmH)
\donttest{ im }
is.inf <- apply(im$is.inf, 1, any)
plot(xh,yh, main = "Huber's data: L.S. line and influential obs.")
abline(lmH); points(xh[is.inf], yh[is.inf], pch = 20, col = 2)

## Irwin's data [Williams 1987]
xi <- 1:5
yi <- c(0,2,14,19,30)    # number of mice responding to dose xi
mi <- rep(40, 5)         # number of mice exposed
glmI <- glm(cbind(yi, mi -yi) ~ xi, family = binomial)
\donttest{summary(glmI)}
signif(cooks.distance(glmI), 3)   # ~= Ci in Table 3, p.184
imI <- influence.measures(glmI)
\donttest{ imI }
stopifnot(all.equal(imI$infmat[,"cook.d"],
          cooks.distance(glmI)))
}
\keyword{regression}
