Fitting Generalized Linear Models
Description
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.
Usage
1 2 3 4 5 6 7 8 9 10 11 12  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)
## S3 method for class 'glm'
weights(object, type = c("prior", "working"), ...)

Arguments
formula 
an object of class 
family 
a description of the error distribution and link
function to be used in the model. For 
data 
an optional data frame, list or environment (or object
coercible by 
weights 
an optional vector of ‘prior weights’ to be used
in the fitting process. Should be 
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 
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 
control 
a list of parameters for controlling the fitting
process. For 
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 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 
x, y 
For For 
contrasts 
an optional list. See the 
intercept 
logical. Should an intercept be included in the null model? 
object 
an object inheriting from class 
type 
character, partial matching allowed. Type of weights to extract from the fitted model object. Can be abbreviated. 
... 
For For 
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 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.\sspace197–8).
Value
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:
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 
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 
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 
null.deviance 
The deviance for the null model, comparable with

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

df.residual 
the residual degrees of freedom. 
df.null 
the residual degrees of freedom for the null model. 
y 
if requested (the default) the 
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 
data 
the 
offset 
the offset vector used. 
control 
the value of the 
method 
the name of the fitter function used, currently always

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

In addition, nonempty fits will have 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
.
Author(s)
The original R implementation of glm
was written by Simon
Davies working for Ross Ihaka at the University of Auckland, but has
since been extensively rewritten by members of the R Core team.
The design was inspired by the S function of the same name described in Hastie & Pregibon (1992).
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
\CRANpkgMASS) for fitting loglinear models (which binomial and
Poisson GLMs are) to contingency tables.
bigglm
in package \CRANpkgbiglm 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  ## 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)

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