spaMM (version 3.2.0)

negbin: Family function for GLMs and mixed models with negative binomial and zero-truncated negative binomial response.

Description

family object that specifies the information required to fit a negative binomial generalized linear model, with known or unknown underlying Gamma shape parameter. The zero-truncated variant can be specified either as Tnegbin(.) or as negbin(., trunc = 0L).

Usage

negbin(shape = stop("negbin's 'shape' must be specified"), link = "log", trunc = -1L)
Tnegbin(shape = stop("negbin's 'shape' must be specified"), link = "log")
# (the shape parameter is actually not requested unless this is used in a glm() call)

Arguments

shape

Shape parameter of the underlying Gamma distribution, given that the negbin family can be represented as a Poisson-Gamma mixture, where the conditional Poisson mean is \(\mu\) times a Gamma random variable with mean 1 and shape shape (as produced by rgamma(., shape=shape,scale=1/shape)).

link

log, sqrt or identity link, specified by any of the available ways for GLM links (name, character string, one-element character vector, or object of class link-glm as returned by make.link).

trunc

Either 0L for zero-truncated distribution, or -1L for default untruncated dsitribution.

Value

A family object.

Details

shape is the \(k\) parameter of McCullagh and Nelder (1989, p.373) and the theta parameter of Venables and Ripley (2002, section 7.4). The latent Gamma variable has mean 1 and variance 1/shape, and the negbin with mean \(mu\) has variance \(mu+mu^2\)/shape. The negbin family is sometimes called the NegBin1 model (as the first, historically) in the literature on negative binomial models, and sometimes the NegBin2 model (because its variance is a quadratic function of its mean).

spaMM does not handle models with the ``other'' negative-binomial response family where the variance is a linear function of the mean, because this is not an exponential-family model. However, it can approximate it, through a Laplace approximation and a bit of additional programming, as a Poisson-Gamma mixture model with an heteroscedastic Gamma random-effect, specified e.g. as (weights-1|.) where the weights need to be updated iteratively as function of predicted response. File test-negbin1.R in the /test directory provides one example. Other mean-variance relationship can be handled in the same way.

The name NB_shape should be used to set values of shape in control arguments of the fitting functions (e.g., fitme(.,init=list(NB_shape=1))).

References

McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd edition. London: Chapman & Hall.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S-PLUS. Fourth Edition. Springer.

Examples

# NOT RUN {
## Fitting negative binomial model with estimated scale parameter:
data("scotlip")
fitme(cases~I(prop.ag/10)+offset(log(expec)),family=negbin(), data=scotlip)
negfit <- fitme(I(1+cases)~I(prop.ag/10)+offset(log(expec)),family=Tnegbin(), data=scotlip)
simulate(negfit,nsim=3)
# }