family {stats} | R Documentation

## Family Objects for Models

### Description

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

    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. The `gaussian` family accepts the links `"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 allows the links `"logit"`, `"probit"`, `"cloglog"`, `"identity"`, `"inverse"`, `"log"`, `"1/mu^2"` and `"sqrt"`. The function `power` can also be used to create a power link function for the `quasi` family.

`variance` | for all families, other than `quasi`, the variance function is determined by the family. The `quasi` family will accept the specifications `"constant"`, `"mu(1-mu)"`, `"mu"`, `"mu^2"` and `"mu^3"` for the variance function.
`object` | the function `family` accesses the `family` 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()`).

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.

### 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 residuals as a function of `(y, mu, wt)`.
`aic` | function giving the AIC value if appropriate (but `NA` for the quasi- families). See `logLik` for the assumptions made about the dispersion parameter.
`mu.eta` | function: derivative `function(eta)` dmu/deta.
`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`.
`valid.mu` | logical function. Returns `TRUE` if a mean vector `mu` is within the domain of `variance`.
`valid.eta` | logical function. Returns `TRUE` if a linear predictor `eta` is within the domain of `linkinv`.

### Author(s)

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

### 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

`glm`, `power`, `make.link`.

### Examples

    nf <- gaussian()# Normal family
    nf
    str(nf)# internal STRucture

    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)

    ## tests of quasi
    x <- rnorm(100)
    y <- rpois(100, exp(1+x))
    glm(y ~x, family=quasi(var="mu", link="log"))
    # which is the same as
    glm(y ~x, family=poisson)
    glm(y ~x, family=quasi(var="mu^2", link="log"))
    ## Not run: glm(y ~x, family=quasi(var="mu^3", link="log")) # should fail
    y <- rbinom(100, 1, plogis(x))
    # needs to set a starting value for the next fit
    glm(y ~x, family=quasi(var="mu(1-mu)", link="logit"), start=c(0,1))

[Package stats version 2.3.1 Index]