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.

Usage

bell(link = "log")

Arguments

link

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.

Value

An object of class "family" (which has a concise print method). This is a list with elements

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 for each observation as a function of (y, mu, wt), used by the residuals method when computing deviance residuals.
aic: 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.
mu.eta: 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 \(d(g^{-1})/d\eta = d\mu/d\eta\).
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) and mustart (see glm).
validmu: logical function. Returns TRUE if a mean vector mu is within the domain of variance.
valideta: logical function. Returns TRUE if a linear predictor eta is within the domain of linkinv.
simulate: (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.
dispersion: (optional since R version 4.3.0) numeric: value of the dispersion parameter, if fixed, or NA_real_ if free.

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 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.

Note

The link and variance arguments have rather awkward semantics for back-compatibility. The recommended way is to supply them as quoted character strings, but they can also be supplied unquoted (as names or expressions). Additionally, they can be supplied as a length-one character vector giving the name of one of the options, or as a list (for link, of class "link-glm"). 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.

Author

The design was inspired by S functions of the same names described in Hastie & Pregibon (1992) (except quasibinomial and quasipoisson).

Examples

library(bellreg)
data(faults)
fit <- glm(nf ~ lroll, data = faults, family = bell("log"))
summary(fit)
#> 
#> Call:
#> glm(formula = nf ~ lroll, family = bell("log"), data = faults)
#> 
#> Coefficients:
#>              Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) 0.9852513  0.3336642   2.953 0.003149 ** 
#> lroll       0.0019093  0.0004923   3.878 0.000105 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for bell family taken to be 1)
#> 
#>     Null deviance: 38.820  on 31  degrees of freedom
#> Residual deviance: 23.175  on 30  degrees of freedom
#> AIC: 181.92
#> 
#> Number of Fisher Scoring iterations: 4
#>