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STATISTICAL REPORTS

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

Abstract.

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

Key words:

overdispersion; quasi models.

covariates; generalized linear models; harbor seals; iteratively weighted least squares;

INTRODUCTION

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.

1E-mail: jay.verhoef@noaa.gov

2766

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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

binomial model.

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

ð1Þ

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

ð2Þ

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

mean,whereasforNB(l,j)(whereNBstandsfornegative

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 20072767QUASI-POISSON VS. NEGATIVE BINOMIAL

STATISTICAL REPORTS

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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

ˆb

½jþ1?¼ ðX0W½j?XÞ?1X0W½j?˜ y½j?

where the ith element of y ˜[j]is

ð3Þ

˜ y½j?

i

¼ g½j?

i

þðyi? l½j?

]g?1ðg½j?

]g½j?

iÞ

iÞ

i

g½j?

i

¼ x0

iˆb

½j?

l½j?

i

¼ g?1ðx0

iˆb

½j?Þ

and W[ j]is a diagonal matrix with elements

]g?1ðg½j?

]g½j?

mðl½j?

1Þ

1

"#2

1Þ

???

]g?1ðg½j?

]g½j?

mðl½j?

nÞ

n

"#2

n Þ

:

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:

FIG. 1.

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.

(A)Estimatedvariance-to-meanrelationshipfortheexampledataset.Thecirclesareaveragedsquaredresidualsin10mean

JAY M. VER HOEF AND PETER L. BOVENG 2768Ecology, Vol. 88, No. 11

STATISTICAL REPORTS

Page 4

W ¼ diag

l2

hl1

1

???l2

n

hln

??

¼ diag

l1

h???ln

h

??

ð4Þ

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

l2

1

l1þ jl2

1

???

l2

n

lnþ jl2

n

?

¼ diag

l1

1 þ jl1

???

ln

1 þ jln

?

:

ð5Þ

EXAMPLE

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

assumed

EðYijÞ ¼ lij

as in Eqs. 1 and 2. We allow the mean to be a positive

function of covariates,

lij¼ expðx0

ijbÞ

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:

x0

ijb ¼ b0;iþ b1;ix1ijþ b2;ix2

þ b6;ix3

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 [1978] 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 b3;ix3

1ijþ b4;ix2ijþ b5;ix2

3ijþ b9;ix3

2ij

2ijþ b7;ix3ijþ b8;ix2

3ij

November 20072769 QUASI-POISSON VS. NEGATIVE BINOMIAL

STATISTICAL REPORTS

Page 5

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,[1]be the earliest recorded

observational date, and let x1,[101]be the latest recorded

observational date, with 99 evenly spaced (fractional)

dates in between. We created the following matrix:

M1[

0

...

0

???

..

???

0

...

0

x1;½1?? x1

...

x1;½101?? x1

x2

1;½1?? x2

...

x2

1

x3

1;½1?? x3

...

x3

1

0

...

0

???

..

???

0

...

0

..

1;½101?? x2

1

1;½101?? x3

1

0

B

B

@

1

C

C

A

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

given by

f1¼ M1ˆb:

ð6Þ

The variance for f1in Eq. 6 is given by the diagonal

elements of

covðf1Þ ¼ M1ˆCM0

ð7Þ

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

FIG. 2.

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

STATISTICAL REPORTS

Page 6

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.

DISCUSSION

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

PLATE 1.

Laboratory.

Hauled-out harbor seals photographed during aerial surveys. Photo credit: NOAA National Marine Mammal

November 20072771 QUASI-POISSON VS. NEGATIVE BINOMIAL

STATISTICAL REPORTS

Page 7

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

diagnosticplotsasgiveninFig.1A.Inthisarticle,wealso

pointoutthatanimportantwaytochooseanappropriate

modelisbasedonsoundscientificreasoningratherthana

data-driven method. We believe that understanding the

difference in weighting between quasi-Poisson and

negative binomial regression provides such an example.

ACKNOWLEDGMENTS

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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SUPPLEMENT

SAS code and data set for obtaining the results described in this paper (Ecological Archives E088-171-S1).

JAY M. VER HOEF AND PETER L. BOVENG2772 Ecology, Vol. 88, No. 11

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