Ecology, 88(11), 2007, pp. 2766–2772
? 2007 by the Ecological Society of America
QUASI-POISSON VS. NEGATIVE BINOMIAL REGRESSION:
HOW SHOULD WE MODEL OVERDISPERSED COUNT DATA?
JAY M. VER HOEF1AND PETER L. BOVENG
National Marine Mammal Laboratory, Alaska Fisheries Science Center, National Marine Fisheries Service,
7600 Sand Point Way NE, Building 4, Seattle, Washington 98115-6349 USA
parameters, and either could be used for overdispersed count data. While they often give
similar results, there can be striking differences in estimating the effects of covariates. We
explain when and why such differences occur. The variance of a quasi-Poisson model is a
linear function of the mean while the variance of a negative binomial model is a quadratic
function of the mean. These variance relationships affect the weights in the iteratively weighted
least-squares algorithm of fitting models to data. Because the variance is a function of the
mean, large and small counts get weighted differently in quasi-Poisson and negative binomial
regression. We provide an example using harbor seal counts from aerial surveys. These counts
are affected by date, time of day, and time relative to low tide. We present results on a data set
that showed a dramatic difference on estimating abundance of harbor seals when using quasi-
Poisson vs. negative binomial regression. This difference is described and explained in light of
the different weighting used in each regression method. A general understanding of weighting
can help ecologists choose between these two methods.
Quasi-Poisson and negative binomial regression models have equal numbers of
overdispersion; quasi models.
covariates; generalized linear models; harbor seals; iteratively weighted least squares;
Ecology is the science of relating organisms to their
environment. Often, data on organisms come in the form
of counts, and we would like to relate these counts to
environmental conditions. Linear regression is common-
ly used, but may not be the most appropriate for count
data, which are nonnegative integers, and hence there is
increasing interest in regression models that use Poisson
or negative binomial distributions. Count data in ecology
are often ‘‘overdispersed.’’ For a Poisson distribution,
the variance is equal to the mean. This may be quite
restrictive for biological data, which often exhibit more
variation than given by the mean. We use the term
‘‘overdispersed’’ for any data set or model where the
variance exceeds the mean. A common way to deal with
overdispersion for counts is to use a generalized linear
model framework (McCullagh and Nelder 1989), where
the most common approach is a ‘‘quasi-likelihood,’’ with
Poisson-like assumptions (that we call the quasi-Poisson
from now on) or a negative binomial model. The
objective of this statistical report is to introduce some
concepts that will help an ecologist choose between a
quasi-Poisson regression model and a negative binomial
regression model for overdispersed count data.
There are many examples of overdispersed count
models in ecology, with important applications ranging
from species richness to spatial distributions to parasit-
ism. O’Hara (2005) noted the differences between
Poisson and negative binomial distributions for species
richness, with each being appropriate only when data
were simulated from the correct model. Alexander et al.
(2000) used a negative binomial distribution with a
spatial model of parasitism. White and Bennetts (1996)
modeled bird counts with a negative binomial distribu-
tion. For trend and abundance estimation for harbor
seals, Frost et al. (1999), Small et al. (2003) and
Mathews and Pendleton (2006) used Poisson regression,
Ver Hoef and Frost (2003) used an overdispersed
Poisson regression, and Boveng et al. (2003) used
negative binomial regression.
Because overdispersion is so common, several models
have been developed for these data, including the
negative binomial, quasi-Poisson (Wedderburn 1974),
generalized Poisson (Consul 1989), and zero-inflated
(Lambert 1992) models. Relationships among some of
Manuscript received 10 January 2007; revised 28 March
2007; accepted 1 May 2007; final version received 22 May 2007.
Corresponding Editor: N. G. Yoccoz.
the distributions can be found in Joe and Zhu (2005) and
Lord et al. (2005). Despite these developments, the
quasi-Poisson and negative binomial models are used
most often, largely because they are widely available in
software and they generalize easily to the regression
case, which we outline in the next section.
The quasi-Poisson model and negative binomial
model can account for overdispersion, and both have
two parameters. Both are commonly available in
software packages such as SAS, S, S-plus, or R. A
natural question for the ecologist is: Which should I use?
In the example below, we show striking differences
between quasi-Poisson regressions and negative binomi-
al regressions for a particular harbor seal data set. There
is surprisingly little guidance in the statistical literature,
especially for the regression case. Gardner et al. (1995)
found little practical difference, but preferred a negative
binomial model when a distributional form is required.
Terceiro (2003) compared models using a Kolmogorov-
Smirnov goodness-of-fit measure, and found cases where
each model fit better. Potts and Elith (2006) found that a
zero-inflated model was better than either quasi-Poisson
or negative binomial for modeling abundance of a rare
plant species, but they point out that zero-inflation is a
special type of overdispersion that may be most
appropriate when occurrence is rare (a specific mecha-
nism creating excessive zeros).
For any given data set, information theoretic ap-
proaches such as Akaike information criteria (AIC;
Akaike 1973) or Bayesian information criteria (BIC;
Schwarz 1978) might be considered to choose between a
quasi-Poisson model and a negative binomial. These
approaches depend on a distributional form and a
likelihood; however, quasi models are only characterized
by their mean and variance, and do not necessarily have
a distributional form. For this reason, Burnham and
Anderson (2002:67) developed quasi-AIC (QAIC), but
they only used it to compare within the quasi class of
models (e.g., for subset selection of covariates), and not
between quasi models and models with distributional
forms. Nevertheless, Sileshi (2006) compared QAIC for
quasi-Poisson to AIC for negative binomial, though the
validity of this approach has not been demonstrated. In
theory, any model selection method that depends on full
distributional likelihoods, such as Bayes factors (Raftery
1995) or minimum description length (Rissanen 1978),
including the information theoretic approaches, would
not help choose between a quasi-Poisson and negative
With the lack of a demonstrated information theoretic
approach, one could adopt predictive or goodness-of-fit
criteria as used by Gardner et al. (1995), Terceiro (2003),
and Potts and Elith (2006), to choose between a quasi-
Poisson and negative binomial model. However, a good
understanding of the theoretical differences between
them can form the basis for an a priori decision based on
scientific purposes, which we explore in this article.
QUASI-POISSON AND NEGATIVE BINOMIAL REGRESSION
Both quasi and negative binomial models can be
framed as generalized linear models. Let Y be a random
variable such that
EðYÞ ¼ l
varðYÞ ¼ mPoiðlÞ ¼ hl
where E(Y) is the expectation of Y, var(Y) is the
variance of Y, l . 0 and h . 1. E(Y) is also known as
the ‘‘mean’’ of the distribution. Although l . 0, the data
themselves can be any nonnegative integer. In Eq. (1), h
is an overdispersion parameter. The close relationship
between Eq. 1 and the expectation and variance of a
Poisson distribution, along with the use of a log link
function, justify calling this a ‘‘quasi-Poisson’’ model,
denoted as Y ; Poi(l, h). The quasi-Poisson model is
characterized by the first two moments (mean and
variance [Wedderburn 1974]), but Efron (1986) and
Gelfand and Dalal (1990) showed how to create a
distribution for this model; however, it requires repar-
ameterization. Estimation often proceeds from the first
two moments and estimating equations (Lee and Nelder
2000). The quasi model formulation has the advantage
of leaving parameters in a natural, interpretable state
and allows standard model diagnostics without a loss of
efficient fitting algorithms.
We will denote the random variable Y having a
negative binomial distribution as Y ; NB(l, j), with a
parameterization such that
EðYÞ ¼ l
varðYÞ ¼ mNBðlÞ ¼ l þ jl2
where l . 0 and j . 0. Here, the overdispersion (the
amount in excess of l) is the multiplicative factor 1þjl,
which depends on l (in contrast to the quasi-Poisson).
From Eqs. 1 and 2, an important difference is that for
Poi(l, h) the variance mPoi(l) is linearly related to the
binomial) the variance mNB(l) is quadratic in the mean.
Hence, an importantdiagnosticistoplot (Yi?li)2against
li. Often, this plot is messy, so we recommend binning li
into categories and averaging (Yi? li)2within categories,
as we do in Fig. 1A. To understand the implications
further, we turn to the regression formulation.
One of the reasons these are two popular models is
that the mean for both models is a single parameter that
can vary as a function of covariates. For quasi-Poisson
regression, we assume Yi; Poi(li, h) where we let the
mean lifor the ith observation vary as a function of the
covariates for that observation. Because the mean li.
0, it is natural to model
li¼ expðb0þ b1x1;iþ ??? þ bpxp;iÞ:
Generalizing, we can write this as the vector of mean
parameters l ¼ g?1(Xb), where g?1is the exponential
function, X is a design matrix of both continuous and
November 20072767 QUASI-POISSON VS. NEGATIVE BINOMIAL
categorical covariates, and b is a vector of parameters
(regression coefficients). The ith row x0
covariates for the ith observation. Alternatively, we
could write g(l)¼ Xb where g is the log function, and it
is called the link function. This is a fairly general
specification, and g can take on various forms, but here
we only consider the log link. For negative binomial
regression, we assume Yi; NB(li, j), where we let the
mean livary as a function of covariates. Because li. 0,
we again let g(l) ¼ Xb where g is the log link function.
One of our main questions in the choice of these
models is: How much does the use of negative binomial
vs. quasi-Poisson affect the fitting of the regression
coefficients b? The different mean/variance relations
suggest that regression coefficients might be fit differ-
ently between negative binomial and quasi-Poisson
because fitting these models uses weighted least squares,
and these weights are inversely proportion to the
variance. Thus, negative binomial and quasi-Poisson
will weight observations differently.
To see how, we use the fact that generalized linear
models and quasi models can be estimated using
iteratively weighted least squares (IWLS; see, e.g.,
iof X contains the
McCullagh and Nelder 1989:40). One iteration j þ 1 in
IWLS is given by
½jþ1?¼ ðX0W½j?XÞ?1X0W½j?˜ y½j?
where the ith element of y ˜[j]is
and W[ j]is a diagonal matrix with elements
Suppressing the iteration superscripts, for both negative
binomial and quasi-Poisson, ]g?1(gi)/]gi¼ exp(gi) ¼ li.
Thus, for quasi-Poisson we obtain the following:
categories, where the solid circles are for quasi-Poisson and the open circles are for the negative binomial. The diameters of the circles
are proportional to the number of values that were averaged. (B) Estimated weights as a function of the mean for the example data set.
For both panels, the dashed line is the quasi-Poisson regression model, and the solid line is the negative binomial regression model.
JAY M. VER HOEF AND PETER L. BOVENG 2768Ecology, Vol. 88, No. 11
W ¼ diag
where diag indicates diagonal elements only, all other
elements being zero, and for negative binomial we obtain
Eqs. 4 and 5 provide a very useful comparison
between negative binomial and quasi-Poisson. For
quasi-Poisson, weights are directly proportional to the
mean, and for negative binomial, weights have a
concave relationship to the mean; that is, very small
mean values get very little weight, but as the mean
increases, weights level off to 1/j. We find this to be the
most useful property in comparing the negative binomial
to quasi-Poisson for regression applications. We now
give an example to illustrate these differences and their
effect on a real application.
W ¼ diag
1 þ jl1
1 þ jln
Since the early 1990s, the National Marine Mammal
Laboratory has conducted annual surveys of harbor
seals in Alaska, rotating annually among five regions: (1)
southern southeast Alaska; (2) northern southeast
Alaska; (3) Gulf of Alaska; (4) Bristol Bay; and (5)
Aleutian Islands. Boveng et al. (2003) described the
methodology and reported results from a survey in the
Gulf of Alaska region. In examining data from each
region over the years, we modeled counts as a function
of date, time of day, and tide, and compared the
differences between a quasi-Poisson regression and a
negative binomial regression. In general, there were not
great differences. However, a data set from southern
Southeast Alaska in 1998 showed striking differences
that depended on the regression method.
For the southern Southeast Alaska data set, harbor
seals were counted from aircraft from 18 August to 27
August 1998 (see Plate 1). Harbor seals use traditional
haul-out sites, and 423 such sites were identified in the
survey region. Let us denote as a random variable the
count, Yij, from the ith site and the jth count. For both
quasi-Poisson and negative binomial regression, we
EðYijÞ ¼ lij
as in Eqs. 1 and 2. We allow the mean to be a positive
function of covariates,
where xijis a vector of measured covariates for the jth
count of the ith site, and b is a vector of parameters. We
fit the following model for each regression method:
ijb ¼ b0;iþ b1;ix1ijþ b2;ix2
where b0,jis an effect for each site, x1ijis day from 15
August for the jth count at the ith site, x2ijis the time of
day, in fractional hours since midnight, for the ijth
count, and x3ij is the relative tide height for the ijth
count, defined as the height of the tide (in meters, always
positive) relative to the low tide from the nearest tide
station at the time of the count. To keep the model
flexible (similar to the use of generalized additive models
by Boveng et al. 2003), we considered a cubic
polynomial for each environmental factor: date, time
of day, and relative tide height. Obviously, time itself is
not part of the ‘‘environment.’’ The effect of date is
related to seal molt, with the peak in early August (e.g.,
Ver Hoef and Frost 2003). Time of day is related to solar
gain during haul out, with peaks usually around midday
(e.g., Boveng et al. 2003). Low tide often exposes
isolated, rocky reefs that keep seals safe from terrestrial
predators, so peak haul out is often near low tide (e.g.,
Boveng et al. 2003).
All models were fit using SAS PROC GLIMMIX
(SAS Institute 2003). The estimated overdispersion
parameter for quasi-Poisson regression was^h ¼ 25.91,
and the estimated variance parameter for negative
binomial regression was ^ j ¼ 0.7717. For negative
binomial regression, from Eq. 2, the amount of over-
dispersion changes with l; if l¼10 then 1þ ^ jl¼8.717.
For these data, overdispersion for quasi-Poisson and
negative binomial regression was equal at l ’ 32. Notice
that the amount of overdispersion is quite high. Such
overdispersion can be caused by several factors (see
Eberhardt  for a discussion), including animals
acting as a group, seals having individual responses to
covariates (e.g., an individual response to date, tide, and
so on), or ‘‘lurking covariates’’ (factors that affect all
animals but were not measured). All of these are likely to
be operating here, but the extent of each is unknown. It is
also unknown whether these factors act as a constant
overdispersion (e.g., h for the quasi-Poisson) or act as an
increasing overdispersion in the mean (e.g., 1þjl for the
negative binomial); however, see Fig. 1A and Discussion.
Using^h ¼ 25.91 and ^ j ¼ 0.7717, we plotted mPoi(l)
(Eq. 1) and mNB(l) (Eq. 2) against l (Fig. 1A). We also
plotted averaged squared residuals (Yi? ^ li)2for 10 mean
categories 0 , ^ li? 15, 15 , ^ li? 30 and so on, which
should help diagnose a linear or quadratic relationship
between the mean and variance. The diameters of the
circles are proportional to the number averaged for the
ith category, and all categories have at least 30 samples.
It appears that for small values of ^ li, the negative
binomial error structure fits slightly better, but for larger
values of ^ li, the quasi-Poisson fits much better. We can
also see from Fig. 1A that for means of less than 32
seals, the quasi-Poisson will have a higher variance, and
for means above 32, the negative binomial will have a
1ijþ b4;ix2ijþ b5;ix2
2ijþ b7;ix3ijþ b8;ix2
November 20072769QUASI-POISSON VS. NEGATIVE BINOMIAL
higher variance. What does this mean for fitting the
covariates? The same parameter estimates define the
weights, as a function of the mean, in Eqs. 4 and 5.
These weights, as a function of the mean, are given in
Fig. 1B. Clearly, a quasi-Poisson regression gives greater
overall weight to larger sites (i.e., sites with more seals)
than does a negative binomial regression. From Fig. 1B,
the negative binomial will give sites with means less than
32 more weight relative to the quasi-Poisson, while the
quasi-Poisson will give sites with means greater than 32
more weight than the negative binomial. This is due to
their different assumptions about how the variance is
related to the mean, shown in Fig. 1A. For the sequel,
we focus on the estimated effect of date, which is given
for the quasi-Poisson regression in Fig. 2A, and for the
negative binomial regression in Fig. 2B.
We created Fig. 2 over the range of the observed
dates: 18–27 August. Let x1,be the earliest recorded
observational date, and let x1,be the latest recorded
observational date, with 99 evenly spaced (fractional)
dates in between. We created the following matrix:
where ¯ x1¼ 22.8 is the mean August date, and the non-
zero columns correspond to the date parameter esti-
mates inˆb. The fitted effect of date for 101 values evenly
spaced over the range of observed dates, holding all
other effects constant, and centered around the mean, is
The variance for f1in Eq. 6 is given by the diagonal
covðf1Þ ¼ M1ˆCM0
where Cˆ¼ (X0WX)?1is the estimated covariance matrix
among the parameter estimatesˆb; that is, the final
(converged) values from Eq. (3). The 95% confidence
interval for each fitted value was formed by taking 1.96
times the standard error (square root of the estimated
variance). Then, the fitted values and confidence intervals
were put back on the original scale by exponentiation.
From Fig. 2, the multiplicative effect of date on 18
August was nearly 2.45 using negative binomial
regression, while it was only around 1.17 using quasi-
Poisson regression. This has a dramatic effect on
estimates of harbor seal abundance. As described in
Boveng et al. (2003), observed harbor seal counts are
date. (A) Fitted effect for quasi-Poisson model. (B) Fitted effect for negative binomial model for all sites. (C) Fitted effect for
negative binomial model for 226 large sites. (D) Fitted effect for negative binomial model for 197 small sites.
Fitted effect of date on harbor seal counts. The fitted effect is centered so that the multiplicative effect is 1 for the mean
JAY M. VER HOEF AND PETER L. BOVENG2770 Ecology, Vol. 88, No. 11
adjusted to conditions of the covariates that yield the
maximum counts (we call these the optimum condi-
tions). Observed counts are adjusted to optimum
conditions because it is not possible to be at every site
at low tide at noon every day of every annual survey
period due to tidal fluctuations, weather, and logistics
(Ver Hoef and Frost 2003). The optimum date for both
quasi-Poisson and negative binomial regression was the
earliest date, 18 August. However, the adjusted estimate
of harbor seal abundance using a quasi-Poisson
regression was 38884 on 18 August, while using a
negative binomial regression it was 80609. Clearly, the
choice of the regression method has a large impact on
our abundance estimate.
The difference in shapes between Fig. 2A and Fig. 2B
can be explained. The adjustment for the large sites is
strongly influenced by the smaller sites for the negative
binomial regression; from Fig. 1B, sites with means of 10
to 32 get essentially the same weight as sites with means
32 to 100. In fact, we can check this by looking at the fits
after dividing the data into the 197 sites that have raw
means of less than or equal to 32 and the 226 sites that
have raw means of greater than or equal to 32. If we
make separate adjustments for large sites using negative
binomial regression, we obtain a date effect as shown in
Fig. 2C, which is almost identical to Fig. 2A. For the 226
large sites, we get an adjusted estimate of 34239 for 18
August. If we make a separate adjustment for the small
sites using negative binomial regression, we obtain a
date effect as shown in Fig. 2D; note the change in the
scaling of the y-axis. For the 197 small sites, we get an
adjusted estimate of 84343 for 18 August, which is not
reasonable in comparison to the large sites or on
biological grounds. If we make the adjustment to 21
August for the small sites, the estimate is 5091, more
than a 15-fold drop in three days, which does not fit our
understanding of harbor seal movements or dynamics.
The preceding analysis, of splitting sites into large and
small, might suggest that combining all sites into one
analysis was wrong in the first place, and hence it was
inappropriate to use a negative binomial regression on
all of the data. However, we point out that we have
rarely witnessed such a dichotomy between large and
small sites in many other analyses of similar data. An
alternative explanation is that the difference between
large and small sites was purely random, which seems
more likely to us.
So, which is better: quasi-Poisson regression or
negative binomial regression? There is no general answer.
However, for the example that we gave, we think that the
quasi-Poisson regression is better. A diagnostic plot of
the empirical fit of the variance (using averaged squared
residuals) to mean relationship, shown in Fig. 1A,
suggests the quasi-Poisson is a better fit to the overall
Hauled-out harbor seals photographed during aerial surveys. Photo credit: NOAA National Marine Mammal
November 20072771 QUASI-POISSON VS. NEGATIVE BINOMIAL
variance-mean relationship. More importantly to us,
though, is the fact that the negative binomial gives
smaller sites more weight relative to quasi-Poisson and
allows smaller sites to have a greater effect on adjust-
ments for negative binomial regression. Our goal is to
estimate overall abundance, which is dominated by the
largersites, andweprefertohaveadjustments dominated
by the effects at those larger sites, which is what happens
for quasi-Poisson regression. Fig. 1B is crucial to our
understanding of how each regression works, and it
frames our decision. We believe this will be useful to
other ecologists when it comes time to decide between
quasi-Poissonandnegative binomialregression methods.
Ultimately, choosing among quasi-Poisson regression,
negative binomial, and other models is a model selection
problem. In the introduction, we pointed out problems
with methods based on likelihoods. However, other
approaches that do not depend on distributions, such as
cross-validation (Vehtari and Lampinen 2003) or the
methods of Gardner et al. (1995), Terceiro (2003), and
Potts and Elith (2006) could be used, in addition to
data-driven method. We believe that understanding the
difference in weighting between quasi-Poisson and
negative binomial regression provides such an example.
We thank Ward Testa and Jeff Breiwick for helpful reviews
of this manuscript. This project received financial support from
the National Marine Fisheries Service, NOAA.
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JAY M. VER HOEF AND PETER L. BOVENG2772Ecology, Vol. 88, No. 11