spaMM (version 3.2.0)

Poisson: Family function for GLMs and mixed models with Poisson and zero-truncated Poisson response.

Description

Poisson (with a capital P) is a family that specifies the information required to fit a Poisson generalized linear model. Differs from the base version stats::poisson only in that it handles the zero-truncated variant, which can be specified either as Tpoisson(<link>) or as Poisson(<link>, trunc = 0L). The truncated poisson with mean \(\mu_T\) is defined from the un-truncated poisson with mean \(\mu_U\), by restricting its response strictly positive value. \(\mu_T=\mu_U/(1-p0)\), where \(p0:=\exp(-\mu_U)\) is the probability that the response is 0.

Usage

Poisson(link = "log", trunc = -1L)
Tpoisson(link="log")
# $linkfun(mu, mu_truncated = FALSE)
# $linkinv(eta, mu_truncated = FALSE)

Arguments

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.

eta,mu

Numeric (scalar or array). The linear predictor; and the expectation of response, truncated or not depending on mu_truncated argument.

mu_truncated

Boolean. For linkinv, whether to return the expectation of truncated (\(\mu_T\)) or un-truncated (\(\mu_U\)) response. For linkfun, whether the mu argument is \(\mu_T\), or is \(\mu_U\) but has \(\mu_T\) as attribute (\(\mu_U\) without the attribute is not sufficient).

Value

A family object.

Details

The mu.eta member function is that of the base poisson family, hence ignores truncation.

predict, when applied on an object with a truncated-response family, by default returns \(\mu_T\). The simplest way to predict \(\mu_U\) is to get the linear predictor value by predict(.,type="link"), and deduce \(\mu_U\) using linkinv(.) (with default argument mu_truncated=FALSE), since getting \(\mu_U\) from \(\mu_T\) is comparatively less straightforward.

References

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

Examples

# NOT RUN {
data("scotlip")
logLik(glm(I(1+cases)~1,family=Tpoisson(),data=scotlip))
logLik(fitme(I(1+cases)~1+(1|id),family=Tpoisson(),fixed=list(lambda=1e-8),data=scotlip))
# }