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, singular.ok = 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, singular.ok = TRUE)
# S3 method for glm
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
glmthis can be a character string naming a family function, a family function or the result of a call to a family function. Forglm.fitonly the third option is supported. (Seefamilyfor details of family functions.)- data
an optional data frame, list or environment (or object coercible by
as.data.frameto a data frame) containing the variables in the model. If not found indata, the variables are taken fromenvironment(formula), typically the environment from whichglmis called.- weights
an optional vector of ‘prior weights’ to be used in the fitting process. Should be
NULLor 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
NAs. The default is set by thena.actionsetting ofoptions, and isna.failif that is unset. The ‘factory-fresh’ default isna.omit. Another possible value isNULL, no action. Valuena.excludecan 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
NULLor a numeric vector of length equal to the number of cases. One or moreoffsetterms 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.fitthis 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.User-supplied 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:xis a design matrix of dimensionn * p, andyis a vector of observations of lengthn.- singular.ok
logical; if
FALSEa singular fit is an error.- contrasts
an optional list. See the
contrasts.argofmodel.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 defaultcontrolargument 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
two-column 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 re-ordered so that main effects come
first, followed by the interactions, all second-order, all third-order
and so on: to avoid this pass a terms object as the formula.
Non-NULL 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
\(w_i\) unit-weight 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.197--8).
Value
glm returns an object of class inheriting from "glm"
which inherits from the class "lm". See later in this section.
If a non-standard 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:
a named vector of coefficients
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.
the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.
the numeric rank of the fitted linear model.
the family object used.
the linear fit on link scale.
up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.
A version of Akaike's An Information Criterion,
minus twice the maximized log-likelihood plus twice the number of
parameters, computed via 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 quasi-likelihood the value is NA.
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.
the number of iterations of IWLS used.
the working weights, that is the weights in the final iteration of the IWLS fit.
the weights initially supplied, a vector of
1s if none were.
the residual degrees of freedom.
the residual degrees of freedom for the null model.
if requested (the default) the y vector
used. (It is a vector even for a binomial model.)
if requested, the model matrix.
if requested (the default), the model frame.
logical. Was the IWLS algorithm judged to have converged?
logical. Is the fitted value on the boundary of the attainable values?
the matched call.
the formula supplied.
the terms object used.
the data argument.
the offset vector used.
the value of the control argument used.
the name of the fitter function used (when provided as a
character string to glm()) or the fitter
function (when provided as that).
(where relevant) the contrasts used.
(where relevant) a record of the levels of the factors used in fitting.
(where relevant) information returned by
model.frame on the special handling of NAs.
In addition, non-empty 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 well-designed 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 two-column 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 non-generalized linear models (which SAS
calls GLMs, for ‘general’ linear models).
loglin and loglm (package
MASS) for fitting log-linear models (which binomial and
Poisson GLMs are) to contingency tables.
bigglm in package biglm 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)
# NOT RUN {
## 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)
# }
# NOT RUN {
summary(glm.D93)
# }
# NOT RUN {
## Computing AIC [in many ways]:
(A0 <- AIC(glm.D93))
(ll <- logLik(glm.D93))
A1 <- -2*c(ll) + 2*attr(ll, "df")
A2 <- glm.D93$family$aic(counts, mu=fitted(glm.D93), wt=1) +
2 * length(coef(glm.D93))
stopifnot(exprs = {
all.equal(A0, A1)
all.equal(A1, A2)
all.equal(A1, glm.D93$aic)
})
# }
# NOT RUN {
## 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)
# }
# NOT RUN {
# A Gamma example, from McCullagh & Nelder (1989, pp. 300-2)
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))
## Aliased ("S"ingular) -> 1 NA coefficient
(fS <- glm(lot2 ~ log(u) + log(u^2), data = clotting, family = Gamma))
tools::assertError(update(fS, singular.ok=FALSE), verbose=interactive())
## -> .. "singular fit encountered"
# }
# NOT RUN {
## for an example of the use of a terms object as a formula
demo(glm.vr)
# }