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.
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")
a specification for the model link function. This can be
a name/expression, a literal character string, a length-one character
vector, or an object of class
"link-glm" (such as generated by
make.link) provided it is not specified
via one of the standard names given next.
The gaussian family accepts the links (as names)
identity, log and inverse;
the binomial family the links logit,
probit, cauchit, (corresponding to logistic,
normal and Cauchy CDFs respectively) log and
cloglog (complementary log-log);
the Gamma family the links inverse, identity
and log;
the poisson family the links log, identity,
and sqrt; and the inverse.gaussian family the links
1/mu^2, inverse, identity
and log.
The quasi family accepts the links logit, probit,
cloglog, identity, inverse,
log, 1/mu^2 and sqrt, and
the function power can be used to create a
power link function.
for all families other than quasi, the variance
function is determined by the family. The quasi family will
accept the literal character string (or unquoted as a name/expression)
specifications "constant", "mu(1-mu)", "mu",
"mu^2" and "mu^3", a length-one character vector
taking one of those values, or a list containing components
varfun, validmu, dev.resids, initialize
and name.
the function family accesses the family
objects which are stored within objects created by modelling
functions (e.g., glm).
further arguments passed to methods.
An object of class "family" (which has a concise print method).
This is a list with elements
character: the family name.
character: the link name.
function: the link.
function: the inverse of the link function.
function: the variance as a function of the mean.
function giving the deviance for each observation
as a function of (y, mu, wt), used by the
residuals method when computing
deviance residuals.
function giving the AIC value if appropriate (but NA
for the quasi- families). More precisely, this function
returns \(-2\ell + 2 s\), where \(\ell\) is the
log-likelihood and \(s\) is the number of estimated scale
parameters. Note that the penalty term for the location parameters
(typically the “regression coefficients”) is added elsewhere,
e.g., in glm.fit(), or AIC(), see the
AIC example in glm.
See logLik for the assumptions made about the
dispersion parameter.
function: derivative of the inverse-link function with respect to the linear predictor. If the inverse-link function is \(\mu = g^{-1}(\eta)\) where \(\eta\) is the value of the linear predictor, then this function returns \(d(g^{-1})/d\eta = d\mu/d\eta\).
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) and mustart (see glm).
logical function. Returns TRUE if a mean
vector mu is within the domain of variance.
logical function. Returns TRUE if a linear
predictor eta is within the domain of linkinv.
(optional) function simulate(object, nsim) to be
called by the "lm" method of simulate. It will
normally return a matrix with nsim columns and one row for
each fitted value, but it can also return a list of length
nsim. Clearly this will be missing for ‘quasi-’ families.
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 and
1, interpreted as the proportion of successful cases (with the
total number of cases given by the weights).
As a two-column 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
over-dispersion. For the binomial case see McCullagh and Nelder
(1989, pp.124--8). Although they show that there is (under some
restrictions) a model with
variance proportional to mean as in the quasi-binomial model, note
that glm does not compute maximum-likelihood estimates in that
model. The behaviour of S is closer to the quasi- variants.
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.
For binomial coefficients, choose;
the binomial and negative binomial distributions,
Binomial, and NegBinomial.
# NOT RUN {
require(utils) # for str
nf <- gaussian() # Normal family
nf
str(nf)
gf <- Gamma()
gf
str(gf)
gf$linkinv
gf$variance(-3:4) #- == (.)^2
## Binomial with default 'logit' link: Check some properties visually:
bi <- binomial()
et <- seq(-10,10, by=1/8)
plot(et, bi$mu.eta(et), type="l")
## show that mu.eta() is derivative of linkinv() :
lines((et[-1]+et[-length(et)])/2, col=adjustcolor("red", 1/4),
diff(bi$linkinv(et))/diff(et), type="l", lwd=4)
## which here is the logistic density:
lines(et, dlogis(et), lwd=3, col=adjustcolor("blue", 1/4))
stopifnot(exprs = {
all.equal(bi$ mu.eta(et), dlogis(et))
all.equal(bi$linkinv(et), plogis(et) -> m)
all.equal(bi$linkfun(m ), qlogis(m)) # logit(.) == qlogis(.) !
})
## Data from example(glm) :
d.AD <- data.frame(treatment = gl(3,3),
outcome = gl(3,1,9),
counts = c(18,17,15, 20,10,20, 25,13,12))
glm.D93 <- glm(counts ~ outcome + treatment, d.AD, family = poisson())
## Quasipoisson: compare with above / example(glm) :
glm.qD93 <- glm(counts ~ outcome + treatment, d.AD, family = quasipoisson())
# }
# NOT RUN {
glm.qD93
anova (glm.qD93, test = "F")
summary(glm.qD93)
## for Poisson results (same as from 'glm.D93' !) use
anova (glm.qD93, dispersion = 1, test = "Chisq")
summary(glm.qD93, dispersion = 1)
# }
# NOT RUN {
## Example of user-specified link, a logit model for p^days
## See Shaffer, T. 2004. Auk 121(2): 526-540.
logexp <- function(days = 1)
{
linkfun <- function(mu) qlogis(mu^(1/days))
linkinv <- function(eta) plogis(eta)^days
mu.eta <- function(eta) days * plogis(eta)^(days-1) *
binomial()$mu.eta(eta)
valideta <- function(eta) TRUE
link <- paste0("logexp(", days, ")")
structure(list(linkfun = linkfun, linkinv = linkinv,
mu.eta = mu.eta, valideta = valideta, name = link),
class = "link-glm")
}
(bil3 <- binomial(logexp(3)))
# }
# NOT RUN {
## 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, as both
## computationally and conceptually difficult:
binomial(link = "identity") ## is exactly the same as
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
# }
# NOT RUN {
y <- rbinom(100, 1, plogis(x))
# need to set a starting value for the next fit
glm(y ~ x, family = quasi(variance = "mu(1-mu)", link = "logit"), start = c(0,1))
# }