More generally, it was shown that IWLS computations yield ML estimates
for data from the regular exponential family---
see Charnes, Frome,
and Yu (JASA,1976) . This result is important
since it showed that statistical software that supported weighted
least squares could be used to obtain ML estimates for arbitrary
regression functions when the dependent variable was in the regular
exponential family (e.g., Poisson, Binomial,Gamma and Gaussian
distributions.) . For a review of Poisson and
Binomial Regression Model see Regression
Methods for Binomial and Poisson Distributed Data(2MB PDF)
Another important area where Poisson regression methods are used is in
the analysis of data organized in a life-table type of format that is
encountered in epidemiologic follow-up studies ---see
Frome(Biometrics,1983) Nelder-Frome comments
(Biometrics,1986)
and
Frome and Checkoway (AJE,1985).
Poisson regression yields ML estimates,
their asymptotic covariance matrix, hypothesis testing procedures, and
regression diagnostics that are now widely used in cohort studies and
other epidemiologic applications that involve event rates---see
Frome and Morris(1989),
Frome et al(1990)
Frome et al(1997)
A third area where Poisson regression methods are of importance is in
the analysis of chromosome aberrations. In many situations the number
of chromosome aberrations follows the Poisson distribution and the
dose-response relations of biologic interest are intrinsically
nonlinear in the unknown parameters. Frome and DuFrain
(Biometrics,1986)
show how Poisson regression methods could be used to analyze
cytogenetic dose-response data. The examples include regression functions that
are linear or nonlinear (as opposed to log-linear).
An efficient algorithm based on Fisher's exact test was developed
(Frome,1982). This test is used to verify the assumption of Poisson
variation in situations where the data may be sparse.
The effective use of Poisson regression in over 30 publications and citations
to above references are presented here, demonstrating the value of these
methods to researchers in many areas that routinely deal with discrete
data and regression models.
Most of the computaions described in the papers on this page can be
done using R. R is a language and environment for statistical computing and
graphics. It compiles and runs on a wide variety of UNIX platforms and
similar systems (including FreeBSD and Linux), Windows, and MacOS. R is
available as Free Software under the terms of the Free Software
Foundation's GNU General Public License in source code form and as
binaries. Visit the R Homepage and read What
is R?( see Menu on left) and the FAQs under Documentation.
In many situations of practical interest the dependent variable in an
experiment or observational study is a count that follows the Poisson
distribution. The relation between the count or event rate and the
explanatory variables is described by a regression function that is,
in general, nonlinear in the unknown parameters. The equivalence of
maximum likelihood (ML) and iterative weighted least squares (IWLS)
was first shown for a nonlinear survival curve model
that is widely used in radiation biology by
Frome and Beauchamp(Biometrics,1968)
The equivalence of IWLS and ML estimation for general
nonlinear regression functions for Poisson distributed data was
presented by
Frome 1972 ,and
Frome, Kutner
and Beauchamp (JASA,1973)---hence the term Poisson regression.
A new graph traversal algorithm has been developed that is used for
model selection for large multidimensional tables encountered in
epidemiology and other disciplines---Ostrouchov and Frome(1993).