R: Fitting Generalized Linear Models

glm {stats} | R Documentation

## 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

    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"), ...)

### 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. 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` | 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 in `data`, the variables are taken from `environment(formula)`, typically the environment from which `glm` is called.
`weights` | an optional vector of 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 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` | 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 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` | a list of parameters for controlling the fitting process. See the documentation for `glm.control` for details.
`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 only current alternative is `"model.frame"` which returns the model frame and does no fitting.
`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 dimension `n * p`, and `y` is a vector of observations of length `n`.
`contrasts` | an optional list. See the `contrasts.arg` of `model.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.
`...` | For `glm`: arguments to be passed by default to `glm.control`: see argument `control`. 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 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`.

### Value

`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` | 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 log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.
`aic` | Akaike's An Information Criterion, minus twice the maximized log-likelihood plus twice the number of coefficients (so assuming that the dispersion is known).
`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 case weights initially supplied.
`df.residual` | the residual degrees of freedom.
`df.null` | the residual degrees of freedom for the null model.
`y` | the `y` vector used. (It is a vector even for a binomial model.)
`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.

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.

### 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 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).

### 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` 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

    ## 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))

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

[Package stats version 2.6.2 Index]