  The paper presents a new statistical method that enables the use of
systematic errors in the maximum-likelihood regression of integer-count Poisson
data to a parametric model. The method is primarily aimed at the
characterization of the goodness-of-fit statistic in the presence of the
over-dispersion that is induced by sources of systematic error, and is based on
a quasi-maximum-likelihood method that retains the Poisson distribution of the
data. We show that the Poisson deviance, which is the usual goodness-of-fit
statistic and that is commonly referred to in astronomy as the Cash statistics,
can be easily generalized in the presence of systematic errors, under rather
general conditions. The method and the associated statistics are first
developed theoretically, and then they are tested with the aid of numerical
simulations and further illustrated with real-life data from astronomical
observations. The statistical methods presented in this paper are intended as a
simple general-purpose framework to include additional sources of uncertainty
for the analysis of integer-count data in a variety of practical data analysis
situations.
