family
Family Objects for Models
Family objects provide a convenient way to specify the details of the
models used by functions such as glm
. See the
documentation for glm
for the details on how such model
fitting takes place.
 Keywords
 models
Usage
family(object, ...)
binomial(link = "logit")
gaussian(link = "identity")
Gamma(link = "inverse")
inverse.gaussian(link = "1/mu^2")
poisson(link = "log")
quasi(link = "identity", variance = "constant")
quasibinomial(link = "logit")
quasipoisson(link = "log")
Arguments
 link
 a specification for the model link function. This can be
a name/expression, a literal character string, a lengthone character
vector or an object of class
"linkglm"
(such as generated bymake.link
) provided it is not specified via one of the standard names given next.The
gaussian
family accepts the links (as names)identity
,log
andinverse
; thebinomial
family the linkslogit
,probit
,cauchit
, (corresponding to logistic, normal and Cauchy CDFs respectively)log
andcloglog
(complementary loglog); theGamma
family the linksinverse
,identity
andlog
; thepoisson
family the linkslog
,identity
, andsqrt
and theinverse.gaussian
family the links1/mu^2
,inverse
,identity
andlog
.The
quasi
family accepts the linkslogit
,probit
,cloglog
,identity
,inverse
,log
,1/mu^2
andsqrt
, and the functionpower
can be used to create a power link function.  variance
 for all families other than
quasi
, the variance function is determined by the family. Thequasi
family will accept the literal character string (or unquoted as a name/expression) specifications"constant"
,"mu(1mu)"
,"mu"
,"mu^2"
and"mu^3"
, a lengthone character vector taking one of those values, or a list containing componentsvarfun
,validmu
,dev.resids
,initialize
andname
.  object
 the function
family
accesses thefamily
objects which are stored within objects created by modelling functions (e.g.,glm
).  ...
 further arguments passed to methods.
Details
family
is a generic function with methods for classes
"glm"
and "lm"
(the latter returning gaussian()
).
For the binomial
and quasibinomial
families the response
can be specified in one of three ways:
 As a factor: ‘success’ is interpreted as the factor not having the first level (and hence usually of having the second level).
 As a numerical vector with values between
0
and1
, interpreted as the proportion of successful cases (with the total number of cases given by theweights
).  As a twocolumn integer matrix: the first column gives the number of successes and the second the number of failures.
The quasibinomial
and quasipoisson
families differ from
the binomial
and poisson
families only in that the
dispersion parameter is not fixed at one, so they can model
overdispersion. For the binomial case see McCullagh and Nelder
(1989, pp.\ifelse{latex}{\out{~}}{ } 1248). Although they show that there is (under some
restrictions) a model with
variance proportional to mean as in the quasibinomial model, note
that glm
does not compute maximumlikelihood estimates in that
model. The behaviour of S is closer to the quasi variants.
Value

An object of class
 family
 character: the family name.
 link
 character: the link name.
 linkfun
 function: the link.
 linkinv
 function: the inverse of the link function.
 variance
 function: the variance as a function of the mean.
 dev.resids
 function giving the deviance residuals as a function
of
(y, mu, wt)
.  aic
 function giving the AIC value if appropriate (but
NA
for the quasi families). SeelogLik
for the assumptions made about the dispersion parameter.  mu.eta
 function: derivative
function(eta)
$d\mu/d\eta$.  initialize
 expression. This needs to set up whatever data
objects are needed for the family as well as
n
(needed for AIC in the binomial family) andmustart
(seeglm
.  validmu
 logical function. Returns
TRUE
if a mean vectormu
is within the domain ofvariance
.  valideta
 logical function. Returns
TRUE
if a linear predictoreta
is within the domain oflinkinv
.  simulate
 (optional) function
simulate(object, nsim)
to be called by the"lm"
method ofsimulate
. It will normally return a matrix withnsim
columns and one row for each fitted value, but it can also return a list of lengthnsim
. Clearly this will be missing for ‘quasi’ families.
"family"
(which has a concise print method).
This is a list with elements
Note
The link
and variance
arguments have rather awkward
semantics for backcompatibility. The recommended way is to supply
them is as quoted character strings, but they can also be supplied
unquoted (as names or expressions). In addition, they can also be
supplied as a lengthone character vector giving the name of one of
the options, or as a list (for link
, of class
"linkglm"
). The restrictions apply only to links given as
names: when given as a character string all the links known to
make.link
are accepted.
This is potentially ambiguous: supplying link = logit
could mean
the unquoted name of a link or the value of object logit
. It
is interpreted if possible as the name of an allowed link, then
as an object. (You can force the interpretation to always be the value of
an object via logit[1]
.)
References
McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.
Dobson, A. J. (1983) An Introduction to Statistical Modelling. London: Chapman and Hall.
Cox, D. R. and Snell, E. J. (1981). Applied Statistics; Principles and Examples. 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.
See Also
For binomial coefficients, choose
;
the binomial and negative binomial distributions,
Binomial
, and NegBinomial
.
Examples
library(stats)
require(utils) # for str
nf < gaussian() # Normal family
nf
str(nf)
gf < Gamma()
gf
str(gf)
gf$linkinv
gf$variance(3:4) # == (.)^2
## quasipoisson. compare with example(glm)
counts < c(18,17,15,20,10,20,25,13,12)
outcome < gl(3,1,9)
treatment < gl(3,3)
d.AD < data.frame(treatment, outcome, counts)
glm.qD93 < glm(counts ~ outcome + treatment, family = quasipoisson())
glm.qD93
anova(glm.qD93, test = "F")
summary(glm.qD93)
## for Poisson results use
anova(glm.qD93, dispersion = 1, test = "Chisq")
summary(glm.qD93, dispersion = 1)
## Example of userspecified link, a logit model for p^days
## See Shaffer, T. 2004. Auk 121(2): 526540.
logexp < function(days = 1)
{
linkfun < function(mu) qlogis(mu^(1/days))
linkinv < function(eta) plogis(eta)^days
mu.eta < function(eta) days * plogis(eta)^(days1) * binomial()$mu_eta
valideta < function(eta) TRUE
link < paste0("logexp(", days, ")")
structure(list(linkfun = linkfun, linkinv = linkinv,
mu.eta = mu.eta, valideta = valideta, name = link),
class = "linkglm")
}
binomial(logexp(3))
## in practice this would be used with a vector of 'days', in
## which case use an offset of 0 in the corresponding formula
## to get the null deviance right.
## Binomial with identity link: often not a good idea.
## Not run: binomial(link = make.link("identity"))
## tests of quasi
x < rnorm(100)
y < rpois(100, exp(1+x))
glm(y ~ x, family = quasi(variance = "mu", link = "log"))
# which is the same as
glm(y ~ x, family = poisson)
glm(y ~ x, family = quasi(variance = "mu^2", link = "log"))
## Not run: glm(y ~ x, family = quasi(variance = "mu^3", link = "log")) # fails
y < rbinom(100, 1, plogis(x))
# needs to set a starting value for the next fit
glm(y ~ x, family = quasi(variance = "mu(1mu)", link = "logit"), start = c(0,1))