A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use.

Abstract

In applications involving count data, it is common to encounter an excess number of zeros. In the study of outpatient service utilization, for example, the number of utilization days will take on integer values, with many subjects having no utilization (zero values). Mixed-distribution models, such as the zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB), are often used to fit such data. A more general class of mixture models, called hurdle models, can be used to model zero-deflation as well as zero-inflation. Several authors have proposed frequentist approaches to fitting zero-inflated models for repeated measures. We describe a practical Bayesian approach which incorporates prior information, has optimal small-sample properties, and allows for tractable inference. The approach can be easily implemented using standard Bayesian software. A study of psychiatric outpatient service use illustrates the methods.

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Statistical Modelling
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DOI: 10.1177/1471082X0901000404
2010 10: 421Statistical Modelling
Brian H Neelon, A James O'Malley and Sharon-Lise T Normand
outpatient psychiatric service use
A Bayesian model for repeated measures zero-inflated count data with application to


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Statistical Modelling 2010; 10(4): 421–439
A Bayesian model for repeated measures zero-inflated
count data with application to outpatient psychiatric
service use
Brian H Neelon1, A James O’Malley2and Sharon-Lise T Normand2,3
1Nicholas School of the Environment, Duke University, USA
2Department of Health Care Policy, Harvard Medical School, USA
3Department of Biostatistics, Harvard School of Public Health, USA
Abstract:
For example, in the study of outpatient service utilization, the number of utilization days will take on
integer values, with many subjects having no utilization (zero values). Mixed distribution models, such
as the zero-inflated Poisson and zero-inflated negative binomial, are often used to fit such data. A more
general class of mixture models, called hurdle models, can be used to model zero deflation as well as
zero inflation. Several authors have proposed frequentist approaches to fitting zero-inflated models for
repeated measures. We describe a practical Bayesian approach which incorporates prior information,
has optimal small-sample properties and allows for tractable inference. The approach can be easily
implemented using standard Bayesian software. A study of psychiatric outpatient service use illustrates
the methods.
Key words: Bayesian inference; hurdle model; repeated measures; zero-altered model; zero-inflated
model
In applications involving count data, it is common to encounter an excess number of zeros.
Received August 2008; revised December 2008; accepted February 2009
1 Introduction
In modelling count data, it is common to encounter a large number of zeros. For
example, in studies examining the utilization of psychiatric outpatient services, many
patients will report no utilization (zero counts). When the number of zeros is greater
(less) than expected under a standard count distribution with a fixed mean μ, the
data are said to be zero inflated (deflated).
Several classes of discrete mixture models have been proposed for zero-modified
data. These include hurdle models (Mullahy,1986; Heilbron, 1989) and zero-inflated
count models (Lambert, 1992; Greene, 1994). Hurdle models comprise a mixture of
a point mass at zero and a truncated count model for the non-zero values. Heilbron
Address for correspondence: Brian H Neelon, Children’s Environmental Health Initiative,
Box 90328 Nicholas School of the Environment, Duke University. Durham, NC 27708, USA.
E-mail: brian.neelon@duke.edu
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Brian H Neelon et al.
(1994) proposed a special type of hurdle model, called the zero-altered model, which
is convenient for testing for zero inflation or deflation.
Zero-inflated count models contain a mixture of a point mass at zero and an
untruncated count distribution. These include the zero-inflated Poisson (ZIP) and
zero-inflated negative binomial (ZINB) models. Because each part of the mixture
accommodates zeros, zero-inflated models explicitly partition the zeros into two
types: ‘structural’ zeros (e.g., those that occur, because patients are ineligible for
outpatient services) and ‘chance’ zeros (those that occur by chance among eligible
patients). Because the probability of structural zeros is non-negative, these models
do not allow for zero deflation. For a comprehensive review of zero-inflated models,
see Ridout et al. (1998).
There is a well-developed literature on frequentist approaches to fitting repeated
measures,zero-inflatedmodels.Hall(2000),YauandLee(2001)andMinandAgresti
(2005) have proposed various random intercept zero-inflated models. Liu and Power
(2007) proposed a zero-inflated growth model, which allowed for correlated ran-
dom intercepts and slopes for both components. Roeder et al. (1999), Dalrymple
et al. (2003) and Min and Agresti (2005) have developed finite mixture zero-inflated
models that allow for clustering of subjects based on their latent response trajectories.
Several authors have recently proposed Bayesian alternatives to fitting zero-
inflated models. The advantages of Bayesian inference are well known and include
elicitationofpriorbeliefs,avoidanceofasymptoticapproximationsandpracticalesti-
mation of functions of parameters (Gelman et al., 2004; Congdon, 2005). Rodrigues
(2003) and Ghosh et al. (2006) developed Bayesian ZIP models for cross-sectional
data, using Markov chain Monte Carlo (MCMC) with data augmentation to obtain
posterior samples. Ghosh et al. (2006) found that for small datasets with many zeros,
the Bayesian estimation outperformed maximum likelihood in terms of both bias and
precision. More recently, Fahrmeir and Osuna Echavarr´ ıa (2006) and Gschl¨ oßl and
Gzado (2008) have introduced zero-inflated models for spatially correlated data by
specifying region-specific random effects to account for geographical dependencies.
In addition, Fahrmeir and Osuna Echavarr´ ıa (2006) develop semiparametric ZIP
models using a generalized additive structure for the Poisson component.
We present a flexible Bayesian modelling strategy for longitudinal zero-inflated
count data. This approach allows for correlated random effects in both components
ofthemodel.Foreaseofexposition,wefocusonPoissonmodels,butthemethodscan
easily be extended to other count distributions, such as the negative binomial. Fol-
lowing Min and Agresti (2005), we describe three zero-inflated models: the Poisson
hurdle model, the ZIP model and the zero-altered Poisson (ZAP) model. The models
can be fit in standard Bayesian software packages such as WinBUGS (Spiegelhalter
et al., 2003).
The remainder of the paper is organized as follows: Section 2 describes the
Bayesian zero-inflated count models and discusses commonly used priors; Section 3
outlines posterior computation and reviews strategies for model assessment and com-
parison; Section 4 applies the models to an analysis of outpatient psychiatric service
use and Section 5 provides a discussion.
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Bayesian model for repeated measures zero-inflated count data
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2 Bayesian regression models for zero-inflated count data
2.1 Hurdle model
The hurdle Poisson model is a two-part mixture model consisting of a point mass at
zero followed by a truncated Poisson for the non-zero observations. For independent
and identically distributed (i.i.d.) responses, the hurdle model can be written as
P(Yi= 0) = 1 − p,
P(Yi= k) = p
k!(1 − e−μ),
where Yi denotes the response for subject i = 1,...,n and μ is the mean for an
untruncated Poisson distribution. When (1 − p) > e−μthe data are zero inflated,
when (1 − p) < e−μthere is zero deflation and when p = 1 there are no zero counts
and the model reduces to a truncated Poisson.
Model (2.2) can be easily extended to allow for predictors and repeated responses
via mixed effects regression:
logit[Pr(Yij> 0|b1i)] = logit(pij|b1i) = x?
log(μij|b2i) = x?
where xkij and zkij(k = 1,2) denote covariate vectors, β1and β2are vectors of
fixed effect coefficients and b1iand b2idenote random effect parameters. One might
allow different sets of predictors in the two components if the purpose is to obtain
the most parsimonious model, or if there is a priori scientific reason to believe that
the two components are affected by unique predictors. For illustrative purposes,
we will assume throughout that the fixed effect covariates are identical for the two
components (i.e., x1ij= x2ij= xij). We will also assume that random intercepts are
sufficient to capture the heterogeneity between subjects; i.e., b1i= b1i, b2i= b2iand
z1ij= z2ij= 1.Generalizationstomultivariaterandomeffectsforthetwocomponents
are straightforward.
Under a fully Bayesian approach, prior distributions are assumed for all model
parameters. A common prior distribution for the fixed effects is an independent,
multivariate normal distribution: β1and β2∼ Np(β0,σ2
known hyperparameters and Ipdenotes the p × p identity matrix. A large prior
variance, such as σ2
β1and β2. For the random effects, a typical prior is the bivariate normal distribution
with unstructured covariance
?b1i
where ρ denotes corr(b1i,b2i). Prior specification is completed by assigning a prior
to ?, such as the diffuse inverse-Wishart (I-W) distribution, ? ∼ I-W(2,I2). Some
0 ? p ? 1,
μke−μ
(2.1)
k = 1,...,∞, 0 < μ < ∞,
(2.2)
1ijβ1+ z?
j = 1,...,ni, i = 1,...,n,
1ijb1i,
(2.3)
(2.4)
2ijβ2+ z?
2ijb2i,
βIp), where β0and σ2
βare
β= 103, can be used to express lack of prior information about
bi=
b2i
?
∼ N2(0,?),
? =
?
σ2
1
ρσ1σ2
σ2
ρσ1σ2
2
?
,
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Brian H Neelon et al.
authors have noted that I-W priors can yield biased and imprecise model estimates,
particularly if the number of clustering units is small (Browne and Draper, 2000;
Lambert et al., 2005). As an alternative, one can specify a product normal parame-
terization for the random effects (Spiegelhalter, 1998; Cooper et al., 2007)
b1i∼ N(0,η2
with diffuse hyperpriors such as ψ ∼ N(0,103) and η2
0.001). The components of ? can then be expressed as
1) and b2i|b1i∼ N(ψb1i,η2
2),
1,η2
2∼ Inv-Gamma(0.001,
σ2
1= η2
1,σ2
2= ψ2η2
1+ η2
2,ρ =
ψη2
1
?
η2
1(ψ2η2
1+ η2
2).
Because |ψ| ∈ [0,∞), the product normal prior makes no assumptions about the
relative magnitudes of σ2
Under prior independence, the joint posterior distribution of the model parame-
ters is
π(β1,β2,b,?|y) ∝ f(y|β1,β2,b,?)π(β1)π(β2)π(b|?)π(?)
n?
×π(β1)π(β2)π(b|?)π(?),
where f(y|·) denotes the likelihood for y given the model parameters; dij is an
indicator that yij> 0; b is a 2n-stacked vector of random effects with prior π(b|?)
N2n(0,In⊗ ?); pij = pij(xij|β1,b1i) and μij = μij(xij|β2,b2i) are functions of the
covariates and the fixed and random effects; and π(β1), π(β2) and π(?) denote prior
densities for β1, β2and ?, respectively.
1and σ2
2.
=
i=1
ni ?
j=1
(1 − pij)1−dij
?
pijμyij
yij!(1 − e−μij)
ije−μij
?dij
(2.5)
d=
2.2 ZIP model
The ZIP model (Lambert, 1992) consists of a degenerate distribution at zero mixed
with an untruncated Poisson distribution. For i.i.d. data, the ZIP model can be
written as
P(Yi= 0) = (1 − p) + pe−μ,
P(Yi= k) = pμke−μ
k!
where the quantity 1− pdenotes the probability of a structural zero. When p = 1, the
probability of zero inflation is zero, and the model reduces to the ordinary Poisson
with Pr(Yi = 0) = e−μ. Conversely, when p < 1, e−μ< (1 − p) + pe−μand the
zeros are inflated. Thus, unlike the hurdle model, the ZIP model allows only for zero
0 ? p ? 1,
(2.6)
,
k = 1,...,∞, 0 < μ < ∞,
(2.7)
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