rqpois: Sampling of overdispersed Poisson data with constant overdispersion
Description
rqpois() samples overdispersed Poisson data with constant overdispersion from
the negative-binomial distribution such that the quasi-Poisson assumption is fulfilled.
The following description of the sampling process is based on the parametrization
used by Gsteiger et al. 2013.
Usage
rqpois(n, lambda, phi, offset = NULL)
Value
a data.frame containing the sampled observations and the offsets
Arguments
n
defines the number of clusters (\(I\))
lambda
defines the overall Poisson mean (\(\lambda\))
phi
dispersion parameter (\(\Phi\))
offset
defines the number of experimental units per cluster (\(n_i\))
Details
It is assumed that the dispersion parameter (\(\Phi\))
is constant for all \(i=1, ... I\) clusters, such that the variance becomes
$$var(y_i) = \Phi n_i \lambda$$
For the sampling \(\kappa_i\) is defined as
$$\kappa_i=(\Phi-1)/(n_i \lambda)$$
where \(a_i=1/\kappa_i\) and \(b_i=1/(\kappa_i n_i \lambda)\). Then, the Poisson means
for each cluster are sampled from the gamma distribution
$$\lambda_i \sim Gamma(a_i, b_i)$$
and the observations per cluster are sampled to be
$$y_i \sim Pois(\lambda_i).$$
Please note, that the quasi-Poisson assumption is not in contradiction with the
negative-binomial distribution, if the data structure is defined by the number
of clusters only (which is the case here) and the offsets are all the same
\(n_h = n_{h´} = n\).
References
Gsteiger, S., Neuenschwander, B., Mercier, F. and Schmidli, H. (2013):
Using historical control information for the design and analysis of clinical
trials with overdispersed count data. Statistics in Medicine, 32: 3609-3622.
tools:::Rd_expr_doi("10.1002/sim.5851")