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.
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"), ...)
- 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
- 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. For
glm.fitonly the third option is supported. (See
familyfor details of family functions.)
- an optional data frame, list or environment (or object
as.data.frameto a data frame) containing the variables in the model. If not found in
data, the variables are taken from
environment(formula), typically the environment from which
- an optional vector of
prior weightsto be used in the fitting process. Should be
NULLor a numeric vector.
- an optional vector specifying a subset of observations to be used in the fitting process.
- a function which indicates what should happen
when the data contain
NAs. The default is set by the
options, and is
na.failif that is unset. The
na.omit. Another possible value is
NULL, no action. Value
na.excludecan be useful.
- starting values for the parameters in the linear predictor.
- starting values for the linear predictor.
- starting values for the vector of means.
- 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 more
offsetterms can be included in the formula instead or as well, and if more than one is specified their sum is used. See
- a list of parameters for controlling the fitting
glm.fitthis is passed to
- a logical value indicating whether model frame should be included as a component of the returned value.
- the method to be used in fitting the model. The default
"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
- x, y
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.
xis a design matrix of dimension
n * p, and
yis a vector of observations of length
- an optional list. See the
- logical. Should an intercept be included in the null model?
- an object inheriting from class
- character, partial matching allowed. Type of weights to extract from the fitted model object. Can be abbreviated.
glm: arguments to be used to form the default
controlargument if it is not supplied directly.
weights: further arguments passed to or from other methods.
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
families the response can also be specified as a
(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
second. This is the same as
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.
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
is specified, the first in the list will be used. It is often
advisable to supply starting values for a
and also for families with unusual links such as
mustart are evaluated in the same way as variables in
formula, that is first in
data and then in the
For the background to warning messages about
glmreturns an object of class inheriting from
"glm"which inherits from the class
"lm". See later in this section. If a non-standard
methodis used, the object will also inherit from the class (if any) returned by that function.
The generic accessor functions
residualscan be used to extract various useful features of the value returned by
weightsextracts a vector of weights, one for each case in the fit (after subsetting and
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 log-likelihood. 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 log-likelihood plus twice the number of parameters, computed by the
aiccomponent 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
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
1s 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
yvector 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
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
model.frameon the special handling of
- In addition, non-empty fits will have components
effectsrelating 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
weightsdo not just pick out the component of the fit with the same name.
glmmodel was specified by giving a two-column response, the weights returned by
prior.weightsare the total numbers of cases (factored by the supplied case weights) and the component
yof the result is the proportion of successes.
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
glm.fit is supplied as a character string it is
used to search for a function of that name, starting in the
The class of the object return by the fitter (if any) will be
prepended to the class returned by
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.
lm for non-generalized linear models (which SAS
calls GLMs, for
bigglm in package
## 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. 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)) ## for an example of the use of a terms object as a formula demo(glm.vr)