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ISDS 2012 Conference Abstracts

Spatial Scan Statistics for Models with Excess Zeros
and Overdispersion
Max Sousa de Lima2, Luiz H. Duczmal*1 and Letícia P. Pinto1

1Universidade Federal de Minas Gerais, Belo Horizonte, Brazil; 2Universidade Federal do Amazonas, Manaus, Brazil

Objective
To propose a more realistic model for disease cluster detection,

through a modification of the spatial scan statistic to account simul-
taneously for inflated zeros and overdispersion.

Introduction
Spatial Scan Statistics [1] usually assume Poisson or Binomial dis-

tributed data, which is not adequate in many disease surveillance sce-
narios. For example, small areas distant from hospitals may exhibit
a smaller number of cases than expected in those simple models.
Also, underreporting may occur in underdeveloped regions, due to
inefficient data collection or the difficulty to access remote sites.
Those factors generate excess zero case counts or overdispersion, in-
ducing a violation of the statistical model and also increasing the type
I error (false alarms). Overdispersion occurs when data variance is
greater than the predicted by the used model. To accommodate it, an
extra parameter must be included; in the Poisson model, one makes
the variance equal to the mean.

Methods
Tools like the Generalized Poisson (GP) and the Double Poisson

[2] may be a better option for this kind of problem, modeling sepa-
rately the mean and variance, which could be easily adjusted by co-
variates. When excess zeros occur, the Zero Inflated Poisson (ZIP)
model is used, although ZIP’s estimated parameters may be severely
biased if nonzero counts are too dispersed, compared to the Poisson
distribution. In this case the Inflated Zero models for the Generalized
Poisson (ZIGP), Double Poisson (ZIDP) and Negative Binomial
(ZINB) could be good alternatives to the joint modeling of excess
zeros and overdispersion. By one hand, Zero Inflated Poisson (ZIP)
models were proposed using the spatial scan statistic to deal with the
excess zeros [3]. By the other hand, another spatial scan statistic was
based on a Poisson-Gamma mixture model for overdispersion [4]. In
this work we present a model which includes inflated zeros and
overdispersion simultaneously, based on the ZIDP model. Let the pa-
rameter p indicate the zero inflation. As the the remaining parameters
of the observed cases map and the parameter p are not independent,
the likelihood maximization process is not straightforward; it be-
comes even more complicated when we include covariates in the
analysis. To solve this problem we introduce a vector of latent vari-
ables in order to factorize the likelihood, and obtain a facilitator for
the maximization process using the E-M (Expectation-Maximization)
algorithm. We derive the formulas to maximize iteratively the likeli-
hood, and implement a computer program using the E-M algorithm
to estimate the parameters under null and alternative hypothesis. The
p-value is obtained via the Fast Double Bootstrap Test [5].

Results
Numerical simulations are conducted to assess the effectiveness

of the method. We present results for Hanseniasis surveillance in the
Brazilian Amazon in 2010 using this technique. We obtain the most
likely spatial clusters for the Poisson, ZIP, Poisson-Gamma mixture
and ZIDP models and compare the results.

Conclusions
The Zero Inflated Double Poisson Spatial Scan Statistic for dis-

ease cluster detection incorporates the flexibility of previous models,
accounting for inflated zeros and overdispersion simultaneously.

The Hanseniasis study case map, due to excess of zero cases counts
in many municipalities of the Brazilian Amazon and the presence of
overdispersion, was a good benchmark to test the ZIDP model. The
results obtained are easier to understand compared to each of the pre-
vious spatial scan statistic models, the Zero Inflated Poisson (ZIP)
model and the Poisson-Gamma mixture model for overdispersion,
taken separetely. The E-M algorithm and the Fast Double Bootstrap
test are computationally efficient for this type of problem.

Keywords
Scan statistics; Zero inflated; Overdispersion; Expectation-Maxi-
mization algorithm

Acknowledgments

The authors acknowledge the grants provided by FAPEAM and CNPq.

References

[1] Kulldorff, M. (1999). Spatial scan statistics: Models, calculations and
applications, in J. Glaz & N. Balakrishnan (eds), Scan Statistics and
Applications, Springer Netherlands, pp. 303–322.

[2] Efron, B. (1986) Double Exponential Families and Their Use in Gen-
eralized Linear Regression, Journal of the American Statistical Asso-
ciation, 81, pp. 709-721. 

[3] Cançado A., da Silva C. and da Silva M.(2011) A zero-inflated Pois-
son-based spatial scan statistic. Emerging Health Threats Journal,
2011;4: 

[4] Zhang T., Zhang Z.; Lin G.(2012) Spatial scan statistics with overdis-
persion. Statistics in Medicine,31(8):762-774.

[5] Davidson, R. and J. G. MacKinnon (2001). “Improving the reliability
of bootstrap tests”, Queen’s University Institute for Economic Re-
search Discussion Paper No. 995, revised.

*Luiz H. Duczmal
E-mail: duczmal@ufmg.br

Online Journal of Public Health Informatics * ISSN 1947-2579 * http://ojphi.org * 5(1):e88, 2013