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. 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.

`variance`

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`.

`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()`).

For the `binomial` and `quasibinomial` families the response can be specified in one of three ways:

1. As a factor: ‘success’ is interpreted as the factor not having the first level (and hence usually of having the second level).

2. 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`).

3. 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.

### 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 ll + 2 s, where ll 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 = ginv(eta) where eta is the value of the linear predictor, then this function returns d(ginv(eta))/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.

### 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]`.)

### 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`.

For binomial coefficients, `choose`; the binomial and negative binomial distributions, `Binomial`, and `NegBinomial`.

### Examples

    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())

    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)

    ## 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)))

    ## 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
    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))

[Package stats version 3.6.0 Index]