family objects specify characteristics of the
models used by functions such as glm. The
families implemented in the stats package include
binomial, gaussian,
Gamma, inverse.gaussian,
and poisson, which are all special cases of
the exponential family of distributions that have probability mass
or density function of the form $$f(y; \theta, \phi) =
\exp\left\{\frac{y\theta - b(\theta) - c_1(y)}{\phi/m} -
\frac{1}{2}a\left(-\frac{m}{\phi}\right) + c_2(y)\right\} \quad y
\in Y \subset \Re\,, \theta \in \Theta \subset \Re\, , \phi >
0$$ where \(m > 0\) is an observation
weight, and \(a(.)\), \(b(.)\),
\(c_1(.)\) and \(c_2(.)\) are sufficiently
smooth, real-valued functions.
The current implementation of family objects
includes the variance function (variance), the deviance
residuals (dev.resids), and the Akaike information criterion
(aic). See, also family.
The enrich method can further enrich exponential
family distributions with \(\theta\) in
terms of \(\mu\) (theta), the functions
\(b(\theta)\) (bfun), \(c_1(y)\)
(c1fun), \(c_2(y)\) (c2fun),
\(a(\zeta)\) (fun), the first two derivatives of
\(V(\mu)\) (d1variance and d2variance,
respectively), and the first four derivatives of
\(a(\zeta)\) (d1afun, d2afun,
d3afun, d4afun, respectively).
Corresponding enrichment options are also avaialble for
quasibinomial,
quasipoisson and wedderburn
families.
The quasi families are enriched with
d1variance and d2variance.
See enrich.link-glm for the enrichment of
link-glm objects.