poisson function

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

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.

Poisson(link = "log", trunc = -1L, LLgeneric=TRUE)
Tpoisson(link="log")
# <Poisson object>$linkfun(mu, mu_truncated = FALSE)
# <Poisson object>$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 distribution.
• 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).
• LLgeneric: For development purposes, not documented.

Returns

A family object suitable for use with glm, as stats:: family objects.

Details

Molas & Lesaffre (2010) developed expressions for deviance residuals for the truncated Poisson distribution, which were the ones implemented in spaMM until version 3.12.0. Later versions implement the (non-equivalent) definition as 2*(saturated_logLik - logLik) .

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. The mu.eta member function is that of the base poisson family, hence its mu argument represents $\mu_U$.

simulate, when applied on an object with a truncated-response family, simulates the truncated family. There is currently no clean way to override this (trying to passtype="link" to predict will not have the intended effect).

References

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

Molas M. and Lesaffre E. (2010). Hurdle models for multilevel zero-inflated data via h-likelihood. Statistics in Medicine 29: 3294-3310.

Examples

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))