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 = glm.control(...), 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 = glm.control(), intercept = TRUE)
 
## S3 method for class 'glm':
weights(object, type = c("prior", "working"), ...)

formula

The details of model specification are given below. |

family

function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.) |

data

coercible by as.data.frame to 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 glm is called. |

weights

process. Should be NULL or a numeric vector. |

subset

to be used in the fitting process. |

na.action

when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit. Another possible value is NULL, no action. Value na.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

known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length either one or equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if both are specified their sum is used. See model.offset. |

control

process. See the documentation for glm.control for details. |

model

should be included as a component of the returned value. |

method

The default method “glm.fit” uses iteratively reweighted least squares (IWLS). The only current alternative is “model.frame” which returns the model frame and does no fitting. |

x, y

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 dimension n * p, and y is a vector of observations of length n. |

contrasts

of model.matrix.default. |

object	an object inheriting from class “glm”.
type

extract from the fitted model object. |

intercept

null model? |

...	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 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 duplicates removed. 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.

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.

glm.fit is the workhorse function.

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.

glm returns an object of class inheriting from “glm” which inherits from the class “lm”. See later in this section.

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

in the final iteration of the IWLS fit. Since cases with zero weights are omitted, their working residuals are NA. |

fitted.values

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

log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero. |

aic

maximized log-likelihood plus twice the number of coefficients (so assuming that the dispersion is known). |

null.deviance

deviance. The null model will include the offset, and an intercept if there is one in the model |

iter	the number of iterations of IWLS used.
weights

in the final iteration of the IWLS fit. |

prior.weights	the case weights initially supplied.
df.residual	the residual degrees of freedom.
df.null	the residual degrees of freedom for the null model.
y

model.) |

converged	logical. Was the IWLS algorithm judged to have converged?
boundary

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

“glm.fit”. |

contrasts	(where relevant) the contrasts used.
xlevels

used in fitting. |

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 is 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.

The original implementation of glm was written by Simon Davies working for Ross Ihaka at the University of Auckland, but has since been extensively re-written by members of the R Core team.

The design was inspired by the S function of the same name described in Hastie \& Pregibon (1992).

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.

anova.glm, summary.glm, etc. for glm methods, and the generic functions anova, summary, effects, fitted.values, and residuals. Further, lm for non-generalized linear models.

esoph, infert and predict.glm have examples of fitting binomial glms.

## 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)
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))
 
## Not run: 
## for an example of the use of a terms object as a formula
demo(glm.vr)
## End(Not run)