glm
Fitting Generalized Linear Models
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.
 Keywords
 models, regression
Usage
glm(formula, family = gaussian, data, weights, subset, na.action, start = NULL, etastart, mustart, offset, control = list(...), model = TRUE, method = "glm.fit", x = FALSE, y = TRUE, contrasts = NULL, ...)
glm.fit(x, y, weights = rep(1, nobs), start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, nobs), family = gaussian(), control = list(), intercept = TRUE)
"weights"(object, type = c("prior", "working"), ...)
Arguments
 formula
 an object of class
"formula"
(or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under ‘Details’.  family
 a description of the error distribution and link
function to be used in the model. For
glm
this can be a character string naming a family function, a family function or the result of a call to a family function. Forglm.fit
only the third option is supported. (Seefamily
for details of family functions.)  data
 an optional data frame, list or environment (or object
coercible by
as.data.frame
to a data frame) containing the variables in the model. If not found indata
, the variables are taken fromenvironment(formula)
, typically the environment from whichglm
is called.  weights
 an optional vector of ‘prior weights’ to be used
in the fitting process. Should be
NULL
or a numeric vector.  subset
 an optional vector specifying a subset of observations to be used in the fitting process.
 na.action
 a function which indicates what should happen
when the data contain
NA
s. The default is set by thena.action
setting ofoptions
, and isna.fail
if that is unset. The ‘factoryfresh’ default isna.omit
. Another possible value isNULL
, no action. Valuena.exclude
can be useful.  start
 starting values for the parameters in the linear predictor.
 etastart
 starting values for the linear predictor.
 mustart
 starting values for the vector of means.
 offset
 this can be used to specify an a priori known
component to be included in the linear predictor during fitting.
This should be
NULL
or a numeric vector of length equal to the number of cases. One or moreoffset
terms can be included in the formula instead or as well, and if more than one is specified their sum is used. Seemodel.offset
.  control
 a list of parameters for controlling the fitting
process. For
glm.fit
this is passed toglm.control
.  model
 a logical value indicating whether model frame should be included as a component of the returned value.
 method
 the method to be used in fitting the model. The default
method
"glm.fit"
uses iteratively reweighted least squares (IWLS): the alternative"model.frame"
returns the model frame and does no fitting.Usersupplied fitting functions can be supplied either as a function or a character string naming a function, with a function which takes the same arguments as
glm.fit
. If specified as a character string it is looked up from within the stats namespace.  x, y
 For
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
glm.fit
:x
is a design matrix of dimensionn * p
, andy
is a vector of observations of lengthn
.  contrasts
 an optional list. See the
contrasts.arg
ofmodel.matrix.default
.  intercept
 logical. Should an intercept be included in the null model?
 object
 an object inheriting from class
"glm"
.  type
 character, partial matching allowed. Type of weights to extract from the fitted model object. Can be abbreviated.
 ...

For
glm
: arguments to be used to form the defaultcontrol
argument if it is not supplied directly.For
weights
: further arguments passed to or from other methods.
Details
A typical predictor has the form response ~ terms
where
response
is the (numeric) response vector and terms
is a
series of terms which specifies a linear predictor for
response
. For binomial
and quasibinomial
families the response can also be specified as a factor
(when the first level denotes failure and all others success) or as a
twocolumn matrix with the columns giving the numbers of successes and
failures. A terms specification of the form first + second
indicates all the terms in first
together with all the terms in
second
with any duplicates removed.
A specification of the form first:second
indicates the the set
of terms obtained by taking the interactions of all terms in
first
with all terms in second
. The specification
first*second
indicates the cross of first
and
second
. This is the same as first + second +
first:second
.
The terms in the formula will be reordered so that main effects come
first, followed by the interactions, all secondorder, all thirdorder
and so on: to avoid this pass a terms
object as the formula.
NonNULL
weights
can be used to indicate that different
observations have different dispersions (with the values in
weights
being inversely proportional to the dispersions); or
equivalently, when the elements of weights
are positive
integers $w_i$, that each response $y_i$ is the mean of
$w_i$ unitweight observations. For a binomial GLM prior weights
are used to give the number of trials when the response is the
proportion of successes: they would rarely be used for a Poisson GLM.
glm.fit
is the workhorse function: it is not normally called
directly but can be more efficient where the response vector, design
matrix and family have already been calculated.
If more than one of etastart
, start
and mustart
is specified, the first in the list will be used. It is often
advisable to supply starting values for a quasi
family,
and also for families with unusual links such as gaussian("log")
.
All of weights
, subset
, offset
, etastart
and mustart
are evaluated in the same way as variables in
formula
, that is first in data
and then in the
environment of formula
.
For the background to warning messages about ‘fitted probabilities numerically 0 or 1 occurred’ for binomial GLMs, see Venables & Ripley (2002, pp.\ifelse{latex}{\out{~}}{ } 1978).
Value
 coefficients
 a named vector of coefficients
 residuals
 the 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
NA
.  fitted.values
 the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.
 rank
 the numeric rank of the fitted linear model.
 family
 the
family
object used.  linear.predictors
 the linear fit on link scale.
 deviance
 up to a constant, minus twice the maximized loglikelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.
 aic
 A version of Akaike's An Information Criterion,
minus twice the maximized loglikelihood plus twice the number of
parameters, computed by the
aic
component of the family. For binomial and Poison families the dispersion is fixed at one and the number of parameters is the number of coefficients. For gaussian, Gamma and inverse gaussian families the dispersion is estimated from the residual deviance, and the number of parameters is the number of coefficients plus one. For a gaussian family the MLE of the dispersion is used so this is a valid value of AIC, but for Gamma and inverse gaussian families it is not. For families fitted by quasilikelihood the value isNA
.  null.deviance
 The deviance for the null model, comparable with
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.  iter
 the number of iterations of IWLS used.
 weights
 the working weights, that is the weights in the final iteration of the IWLS fit.
 prior.weights
 the weights initially supplied, a vector of
1
s if none were.  df.residual
 the residual degrees of freedom.
 df.null
 the residual degrees of freedom for the null model.
 y
 if requested (the default) the
y
vector used. (It is a vector even for a binomial model.)  x
 if requested, the model matrix.
 model
 if requested (the default), the model frame.
 converged
 logical. Was the IWLS algorithm judged to have converged?
 boundary
 logical. Is the fitted value on the boundary of the attainable values?
 call
 the matched call.
 formula
 the formula supplied.
 terms
 the
terms
object used.  data
 the
data argument
.  offset
 the offset vector used.
 control
 the value of the
control
argument used.  method
 the name of the fitter function used, currently always
"glm.fit"
.  contrasts
 (where relevant) the contrasts used.
 xlevels
 (where relevant) a record of the levels of the factors used in fitting.
 na.action
 (where relevant) information returned by
model.frame
on the special handling ofNA
s. In addition, nonempty fits will have components
glm
returns an object of class inheriting from "glm"
which inherits from the class "lm"
. See later in this section.
If a nonstandard method
is used, the object will also inherit
from the class (if any) returned by that function.The function summary
(i.e., summary.glm
) can
be used to obtain or print a summary of the results and the function
anova
(i.e., anova.glm
)
to produce an analysis of variance table.The generic accessor functions coefficients
,
effects
, fitted.values
and residuals
can be used to
extract various useful features of the value returned by glm
.weights
extracts a vector of weights, one for each case in the
fit (after subsetting and na.action
).An object of class "glm"
is a list containing at least the
following components:qr
, R
and effects
relating to the final weighted linear fit.Objects of class "glm"
are normally of class c("glm",
"lm")
, that is inherit from class "lm"
, and welldesigned
methods for class "lm"
will be applied to the weighted linear
model at the final iteration of IWLS. However, care is needed, as
extractor functions for class "glm"
such as
residuals
and weights
do not just pick out
the component of the fit with the same name.If a binomial
glm
model was specified by giving a
twocolumn response, the weights returned by prior.weights
are
the total numbers of cases (factored by the supplied case weights) and
the component y
of the result is the proportion of successes.
Fitting functions
The argument method
serves two purposes. One is to allow the
model frame to be recreated with no fitting. The other is to allow
the default fitting function glm.fit
to be replaced by a
function which takes the same arguments and uses a different fitting
algorithm. If glm.fit
is supplied as a character string it is
used to search for a function of that name, starting in the
stats namespace. The class of the object return by the fitter (if any) will be
prepended to the class returned by glm
.
References
Dobson, A. J. (1990) An Introduction to Generalized Linear Models. London: Chapman and Hall.
Hastie, T. J. and Pregibon, D. (1992) Generalized linear models. Chapter 6 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.
McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. New York: Springer.
See Also
anova.glm
, summary.glm
, etc. for
glm
methods,
and the generic functions anova
, summary
,
effects
, fitted.values
,
and residuals
.
lm
for nongeneralized linear models (which SAS
calls GLMs, for ‘general’ linear models).
loglin
and loglm
(package
\href{https://CRAN.Rproject.org/package=#1}{\pkg{#1}}MASSMASS) for fitting loglinear models (which binomial and
Poisson GLMs are) to contingency tables.
bigglm
in package \href{https://CRAN.Rproject.org/package=#1}{\pkg{#1}}biglmbiglm for an alternative
way to fit GLMs to large datasets (especially those with many cases).
esoph
, infert
and
predict.glm
have examples of fitting binomial glms.
Examples
library(stats)
## 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)
utils::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. 3002)
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))
## Not run:
# ## for an example of the use of a terms object as a formula
# demo(glm.vr)
# ## End(Not run)