glm                  package:stats                  R Documentation

_F_i_t_t_i_n_g _G_e_n_e_r_a_l_i_z_e_d _L_i_n_e_a_r _M_o_d_e_l_s

_D_e_s_c_r_i_p_t_i_o_n:

     '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.

_U_s_a_g_e:

     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"), ...)

_A_r_g_u_m_e_n_t_s:

 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 'prior 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 equal to the
          number of cases.  One or more 'offset' terms can be included
          in the formula instead or as well, and if more than one is
          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. 

_D_e_t_a_i_l_s:

     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 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 and
     design matrix 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).

_V_a_l_u_e:

     '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: A version of Akaike's _An Information Criterion_, minus twice
          the maximized log-likelihood plus twice the number of
          parameters, computed by 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'.

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 '1's 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 'y' vector 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 '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.

na.action: (where relevant) information returned by 'model.frame' on
          the special handling of 'NA's.


     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.

_A_u_t_h_o_r(_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).

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     '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.

_E_x_a_m_p_l_e_s:

     ## 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)

