%%% Title of object:	Modeling Count Data
%%% Canonical Name:	ModelingCountData
%%% Type:	Topic
%%% Created on:	2010-08-20 20:01:09
%%% Modified on:	2010-09-04 17:34:43
%%% Creator:	jhilbe
%%% Modifier:	jkimmel
%%% Author:	jhilbe
%%% 
%%% Classification:	msc:62J12
%%% Keywords:	count, discrete, categorical, Poisson, negative binomial, NB2, NB-C, canonical, Poisson-gamma, overdispersion
%%% Synonyms:	Modeling Count Data=Discrete response models
%%% Modeling Count Data=Poisson-gamma mixture model
%%% Preamble:

\documentclass[10pt]{article}
% define commands here
%\documentclass{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}

\oddsidemargin 16.5mm
\evensidemargin 16.5mm
\textwidth 28cc
\textheight 42cc

\parskip .5mm
\parindent 2cc

\newcommand{\txtbinom}[2]{\Big(\!\!\begin{array}{c} \mbox{\rm #1} \\ \mbox {\rm #2} \end{array}\!\!\Big)}
\newcommand{\bi}[2]{\Big( \!\! \begin{array}{c} #1 \\ #2 \end{array} \!\!\Big)}
\newcommand{\fracc}[2]{\mbox{\small$\displaystyle\frac{#1}{#2}$}}
%%% Content:

\begin{document}
\begin{center}

{\large \bf MODELING COUNT DATA}\\[4mm]

{\large \it Joseph M. Hilbe}\\[2mm]

Arizona State University

\end{center}

\vspace{2cc}

Count models are a subset of discrete response regression models. Count data are distributed as
non-negative integers, are intrinsically heteroskedastic, right skewed, and have a variance
that increases with the mean. Example count data include such situations as length of hospital
stay, the number of a certain species of fish per defined area in the ocean, the number of lights
displayed by fireflies over specified time periods, or the classic case of the number of deaths
among Prussian soldiers resulting from being kicked by a horse during the Crimean War.

\textbf{Poisson regression} is the basic model from which a variety of count models are based. It is
derived from the Poisson probability mass function, which can be expressed as
\begin{equation}\label{eq1}
f(y_i;\lambda_i)=\displaystyle\frac{e^{-t_i\lambda_i}(t_i\lambda_i)^{y_i}}{y_i!},
\qquad y=0,1,2,\ldots;\;\mu>0
\end{equation}
with $y_i$ as the count response, $\lambda_i$ as the predicted count or rate parameter, and $t_i$ the area or time in
which counts enter the model. When $\lambda_i$ is understood as applying to individual counts without
consideration of size or time, $t_i=1.$ When $t_i>1,$ it is commonly referred to as an exposure, and
is modeled as an offset.

Estimation of the Poisson model is based on the log-likelihood parameterization of the
Poisson probability distribution, which is aimed at determining parameter
values making the data most likely. In exponential family form it is given as:

\begin{equation}\label{eq2}
L(\mu_i;y_i)=\sum_{i=1}^n\big\{y_i\ln(\mu_i)-\mu_i-\ln(y_i!)\big\},
\end{equation}
where $\mu_i$ is typically used to symbolize the predicted count in place of $\lambda_i.$ Equation 2, or the
deviance function based on it, is used when the Poisson model is estimated as a generalized
linear model $(GLM)$ (\textit{see} \textbf{Generalized
linear models}). When estimation employs a full maximum likelihood algorithm, $\mu_i$ is expressed
in terms of the linear predictor, $x'\beta.$ As such it appears as
\begin{equation}\label{eq3}
\mu_i=\exp(x_i\beta).
\end{equation}
In this form, the Poisson log-likelihood function is expressed as
\begin{equation}\label{eq4}
L(\beta;y_i)=\sum_{i=1}^n\big\{y_i(x_i\beta)-\exp(x_i\beta)-\ln(y_i!)\big\}.
\end{equation}
A key feature of the Poisson model is the equality of the mean and variance functions. When the
variance of a Poisson model exceeds its mean, the model is termed overdispersed. Simulation
studies have demonstrated that overdispersion is indicated when the Pearson $\chi^2$ dispersion is
greater than 1.0 (Hilbe, 2007). The dispersion statistic is defined as the Pearson $\chi^2$ divided by the
model residual degrees of freedom. Overdispersion, common to most Poisson models, biases the
parameter estimates and fitted values. When Poisson overdispersion is real, and not merely
apparent (Hilbe, 2007), a count model other than Poisson is required.

Several methods have been used to accommodate Poisson overdispersion. Two
common methods are quasi-Poisson and negative binomial regression. Quasi-Poisson
models have generally been understood in two distinct manners. The traditional manner
has the Poisson variance being multiplied by a constant term. The second, employed in
the glm() function that is downloaded by default when installing R software, is to multiply
the standard errors by the square root of the Pearson dispersion statistic. This method
of adjustment to the variance has traditionally been referred to as scaling. Using R's
quasipoisson() function is the same as what is known in standard GLM terminology
as the scaling of standard errors.

The traditional negative binomial model is a Poisson-gamma mixture model with a
second ancillary or heterogeneity parameter, $\alpha.$ The mixture nature of the variance is
reflected in its form, $\mu_i+\alpha\mu_i^{\;2},$ or $\mu_i(1+\alpha\mu_i).$ The Poisson variance
is $\mu_i,$ and the two parameter gamma variance is $\mu_i^{\;2}/\nu.$ $\nu$ is inverted so
that $\alpha=1/\nu,$ which allows for a direct relationship between $\mu_i,$ and $\nu.$ As
a Poisson-gamma mixture model, counts are gamma distributed as they enter into the
model. $\alpha$ is the shape of the manner counts enter into the model as well as a measure of
the amount of Poisson overdispersion in the data.

The negative binomial probability mass function (\textit{see} \textbf{Geometric and negative
binomial distributions}) may be formulated as \begin{equation}\label{eq5}
f(y_i;\mu_i,\alpha)=\binom{y_i+1/\alpha-1}{1/\alpha-1}(1/(1+\alpha\mu_i))^{1/\alpha}(\alpha\mu_i/(1+\alpha\mu_i))^y_i,
\end{equation}
with a log-likelihood function specified as:
{\arraycolsep 0.4mm\begin{eqnarray}\label{eq6}
L(\mu_i;y_i,\alpha)&=&\sum_{i=1}^n\bigg\{y_i\ln\Big(\frac{\alpha\mu_i}{1+\alpha\mu_i}\Big)-\Big(\frac1\alpha\Big)\ln(1+\alpha\mu_i)\nonumber \\
&&+\ln\Gamma\Big(y_i+\frac1\alpha\Big)-\ln\Gamma(y_i+1)-\ln\Gamma\Big(\frac1\alpha\Big)\bigg\}.
\end{eqnarray}}
\vspace{-0.7cc}

\noindent In terms of $\mu=\exp(x'\beta),$ required for maximum likelihood estimation, the negative binomial
log-likelihood appears as
{\arraycolsep 0.4mm\begin{eqnarray}\label{eq7}
L(\beta;y_i,\alpha)&=&\sum_{i=1}^n\bigg\{y_i\ln\Big(\frac{\alpha\exp(x'_i\beta)}{1+\alpha\exp(x'_i\beta)}\Big)-
\Big(\frac1\alpha\Big)\ln(1+\alpha\exp(x'_i\beta)) \nonumber \\
&&+\ln\Gamma\Big(y_i+\frac1\alpha\Big)-\ln\Gamma(y_i+1)-\ln\Gamma\Big(\frac1\alpha\Big)\bigg\}.
\end{eqnarray}}
\vspace{-0.7cc}

\noindent This form of negative binomial has been termed $NB2,$ due to the quadratic nature of its variance
function. It should be noted that the $NB2$ model reduces to the Poisson when $\alpha=0.$ When $\alpha=1,$ the model is geometric, taking the shape of the discrete correlate of the continuous negative
exponential distribution. Several fit tests exist that evaluate whether data should be modeled as
Poisson or $NB2$ based on the degree to which $\alpha$ differs from 0.

When exponentiated, Poisson and $NB2$ parameter estimates may be interpreted as incidence
rate ratios. For example, given a random sample of 1000 patient observations from the German
Health Survey for the year 1984, the following Poisson model output explains the years
expected number of doctor visits on the basis of gender and marital status, both recorded as
binary (1/0) variables, and the continuous predictor, age.

\bigskip{\small
\begin{tabular}{r|rcrrrr}
\hline
docvis & IRR & OIM Std. Err. & $z$ & $P>|z|$ & \multicolumn{2}{c}{[95\% Conf.
Interval]}
\\\hline
female & 1.516855 & .054906 & 11.51 & 0.000 & 1.41297 & 1.628378
\\married & .8418408 & .0341971 & -4.24 & 0.000 & .7774145 & .9116063
\\age & 1.018807 & .0016104 & 11.79 & 0.000 & 1.015656 & 1.021968
\\ \hline
\end{tabular}}

\bigskip

\noindent The estimates may be interpreted as:

{\it Females are expected to visit the doctor some $50\%$ more times
during the year than males, holding marital status and age constant.

\medskip

Married patients are expected to visit the doctor some $16\%$ fewer times
during the year than unmarried patients, holding gender and age constant.

\medskip

For a one year increase in age, the rate of visits to the doctor increases
by some $2\%,$ with marital status and gender held constant.}

\medskip

It is important to understand that the canonical form of the negative binomial, when considered
as a $GLM,$ is not $NB2.$ Nor is the canonical negative binomial model, $NB$-$C,$ appropriate to
evaluate the amount of Poisson overdispersion in a data situation. The $NB$-$C$ parameterization of
the negative binomial is directly derived from the negative binomial log-likelihood as expressed in
Equation 6. As such, the link function is calculated as $\ln(\alpha\mu/(1+\alpha\mu)).$ The
inverse link function, or mean, expressed in terms of $x'\beta$, is $1/(\alpha(\exp(-x'\beta)-1)).$

When estimated as a $GLM,$ $NB$-$C$ can be amended to $NB2$ form by substituting $\ln(\mu)$ and
$\exp(x'\beta)$ respectively for the two above expressions. Additional amendments need to be made to
have the $GLM$-estimated $NB2$ display the same parameter standard errors as are calculated
using full maximum likelihood estimation. The $NB$-$C$ log-likelihood, expressed in terms of $\mu,$ is
identical to that of the $NB2$ function. However, when parameterized as $x'\beta,$ the two differ, with
the $NB$-$C$ appearing as
{\arraycolsep 0.4mm\begin{eqnarray}\label{eq8}
L(\beta;y_i,\alpha)&=&\sum_{i=1}^n\{y_i(x_i\beta)+(1/\alpha)\ln(1-\exp(x_i\beta)) \nonumber\\
&&+\ln\Gamma(y_i+1/\alpha)-\ln\Gamma(y_i+1)-\ln\Gamma(1/\alpha)\}
\end{eqnarray}}

\vspace{-0.7cc}

\noindent The $NB$-$C$ model better fits certain types of count data than $NB2,$ or any other variety of count model. However, since its fitted values are not on the log scale, comparisons cannot be made to
Poisson or $NB2.$

The $NB2$ model, in a similar manner to the Poisson, can also be overdispersed if the model
variance exceeds its nominal variance. In such a case one must attempt to determine the source
of the extra correlation and model it accordingly.

The extra correlation that can exist in count data, but which cannot be accommodated by
simple adjustments to the Poisson and negative binomial algorithms, has stimulated
the creation of a number of enhancements to the two base count models. The differences
in these enhanced models relates to the attempt of identifying the various sources of overdispersion.

For instance, both the Poisson and negative binomial models assume that there exists the
possibility of having zero counts. If a given set of count data excludes that possibility, the
resultant Poisson or negative binomial model will likely be overdispersed. Modifying the log-likelihood
function of these two models in order to adjust for the non-zero distribution of counts will
eliminate the overdispersion, if there are no other sources of extra correlation. Such models are
called, respectively, zero-truncated Poisson and zero-truncated negative binomial models.

Likewise, if the data consists of far more zero counts that allowed by the distributional
assumptions of the Poisson or negative binomial models, a zero-inflated set of models may need to be
designed. Zero-inflated models are mixture models, with one part consisting of a 1/0 binary
response model, usually a logistic regression, where the probability of a zero count is estimated
in difference to a non-zero-count. A second component is generally comprised of a Poisson or negative
binomial model that estimates the full range of count data, adjusting for the overlap in estimated
zero counts. The point is to 1) determine the estimates that account for zero counts, and 2) to
estimate the adjusted count model data.

Hurdle models are another type mixture model designed for excessive zero counts. However,
unlike the zero-inflated models, the hurdle-binary model estimates the probability of being a non-zero
count in comparison to a zero count; the hurdle-count component is estimated on the basis of a
zero-truncated count model. Zero-truncated, zero-inflated, and hurdle models all address abnormal
zero-count situations, which violate essential Poisson and negative binomial assumptions.

Some of the more recently developed count modlels include finite mixture models and exact Poisson
regression. Finite mixture models allow the count response to have been created from two or more
separate generating mechanisms. For example, a portion of the counts may have a Poisson
distribution with a mean .5, with another portion having a Poisson distribution with a mean of 4.
A response may consist of two separate underlying distributions. Such a model allows estimation
of a more complex structures of counts than do standard Poisson and negative binomial models. Exact
Poisson models are not based on the asymptotic methods characteristic of maximum likelihood or
generalized linear models estimation; rather they are based on the construction of a statistical
distribution that can be thoroughly emumerated. This highly iterative technique allows appropriate
estimation of parameters and confidence intervals for small and unbalanced data which would
otherwise not be able to be modeled using conventional estimation methods.

Other violations of the distributional assumptions of Poisson and negative binomial
probability distributions exist. The table below summarizes major types of violations that
have resulted in the creation of specialized count models.

\bigskip

{\bf Table 1.} Models to adjust for violations of Poisson/NB distributional assumptions

{\small\begin{tabular}{ll}
\hline
\hline
Response & example models
\\ \hline
1: no zeros & zero-truncated models $(ZTP;\,ZTNB)$
\\2: excessive zeros & zero-inflated $(ZIP;\,ZINB);$ hurdle models
\\3: truncated & truncated count models
\\4: censored & econometric and survival censored count models
\\5: panel & $GEE;$ fixed, random, and mixed effects count models
\\6: separable & sample selection, finite mixture models
\\7: two-responses & bivariate count models
\\8: other & quantile, exact, and Bayesian count models
\\ \hline
\hline
\end{tabular}}

\bigskip

\noindent Alternative count models have also been constructed based on an adjustment to the Poisson
variance function, $\mu.$ We have previously addressed two of these. Table 2 provides a summary
of major types of adjustments.

%\newpage

\bigskip

{\bf Table 2.} Methods to directly adjust the variance (from Hilbe, 2007)

{\small\begin{tabular}{ll}
\hline
\hline
Variance function & example models
\\ \hline
0: $\mu$ & Poisson
\\1: $\mu(\Phi)$ & quasi-Poisson; scaled SE; robust SE
\\2: $\mu(1+\alpha)$ & linear $NB$ $(NB1)$
\\3: $\mu(1+\mu)$ & geometric
\\4: $\mu(1+\alpha\mu)$ & standard $NB$ $(NB2);$ quadratic $NB$
\\5: $\mu(1+(\alpha\nu)\mu)$ & heterogeneous $NB$ $(NH$-$H)$
\\6: $\mu(1+\alpha\mu^\rho)$ & generalized $NB$ $(NB$-$P)$
\\7: $V[R]V'$ & generalized estimating equations
\\ \hline
\hline
\end{tabular}}

\bigskip

\noindent The four texts listed in the {\it References} below are specifically devoted to describing the theory and variety of count models, and are currently regarded as standard resources on the subject. A number of journal articles and book chapters have been written on the subject. Other texts dealing with
discrete response models in general, as well as texts on generalized linear
models (\textit{see} \textbf{Generalized linear models}), also have descriptions of count models, although
only a few go beyond examining basic Poisson and negative binomial regression.

\vspace{1.5cc}

\noindent{\bf References}
\newcounter{ref}
\begin{list}{\small [\,\arabic{ref}\,]}{\usecounter{ref} \leftmargin 4mm \itemsep -1mm}
{\small

\item Cameron, A. C. and P. K. Trivedi (1986). Econometric models based on count data: Comparisons and
applications of some estimators, {\it Journal of Applied Econometrics}, 1: 29-53.

\item Cameron, A. C., P. K. Trivedi (1998). {\it Regression analysis of count data}. New York:
Cambridge University Press.

\item Hilbe, J. M, (1993). Log-negative binomial regression as a generalized linear model, {\it Technical
report COS 93/94-5-26}, Department of Sociology, Arizona State University.

\item Hilbe, J. M. (2007). {\it Negative binomial regression}. Cambridge, UK: Cambridge University Press.

\item Hilbe, J. M. (2011). {\it Negative binomial regression}. 2nd edition. Cambridge, UK: Cambridge University Press. In press.

\item Hilbe, J. M. and W. H. Greene (2007). Count response regression models, in (eds) C.R. Rao, J.P. Miller,
and D.C. Rao, {\it Epidemiology and Medical Statistics}, Elsevier Handbook of Statistics Series, London: Elsevier.

\item Hinde, J. and C. G. B. Demetrio (1998). Overdispersion: models and estimation, {\it Computational
Statistics and Data Analysis}, Vol 27, 2: 151-170.

\item Lawless, J. F. (1987). Negative binomial and mixed Poisson regression, {\it Canadian Journal of
Statistics}, 15, 3: 209-225.

\item Long, J. S. (1997). {\it Regression models for categorical and limited dependent variables}. Thousand
Oaks, CA: Sage.

\item Simon, L. J. (1960). The negative binomial and Poisson distributions compared. {\it Proceedings
of the casualty and actuarial society} XLVII: 20-24.


\item Winkelmann, R. (2008). {\it Econometric Analysis of Count Data}. 5th edition, Heidelberg, Ger: Springer.
}
\end{list}

Reprinted with permission from Lovric, Miodrag (2011), International Encyclopedia of Statistical Science. Heidelberg: Springer Science +Business Media, LLC
\end{document}
