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Institute of Mathematical Statistics

LECTURE NOTES — MONOGRAPH SERIES

Perfect sampling for posterior landmarkdistributions

with an application to the detection of diseaseclusters

Marc A. Loizeaux and Ian W. McKeagueDepartment of StatisticsFlorida State University

Tallahassee, FL 32306-4330

AbstractWe study perfect sampling for the posterior distribution in a class of

spatial point process models introduced by Baddeley and van Lieshout(1993). For Neyman-Scott cluster models, perfect sampling from theposterior is shown to be computationally feasible via a coupling-from-the-past type algorithm of Kendall and M0ller. An application to dataon leukemia incidence in upstate New York is presented.

1 IntroductionBayesian cluster models based on spatial point processes were originally in-troduced by Baddeley and van Lieshout (1993), primarily for applications incomputer vision. Disease clustering applications have also played a promi-nent role in the development of these models, as surveyed by Lawson andClarke (1999). An important special case is the Neyman-Scott process inwhich the observations arise from a superposition of inhomogeneous Poissonprocesses associated with underlying landmarks (Neyman and Scott, 1972);van Lieshout (1995) focused on this case.

Markov chain Monte Carlo (MCMC) techniques are indispensable forthe application of point process models in statistics, see, e.g., the survey ofM0ller (1999). Following the seminal work of Propp and Wilson (1996),Kendall and M0ller (2000) developed a version of perfect simulation forlocally stable point processes; see the article of M0ller (2001) in this volume.This raises the possibility of constructing perfect samplers for the posteriordistribution in Bayesian cluster models. Perfect samplers deliver an exactdraw from the target distribution; this is a distinct advantage over traditional

322 LOIZEAUX AND MCKEAGUE

MCMC schemes which are often plagued by convergence problems. For somerecent applications of perfect simulation in statistics, see Green and Murdoch(1999), M0ller and Nicholls (1999) and Casella et al. (1999).

In this article we show that the posterior in the Baddeley van Lieshoutclass of Bayesian cluster models is locally stable, provided the prior is locallystable and the likelihood satisfies some mild conditions. This has two im portant consequences: the posterior density is proper (has unit total mass),and the Kendall M0ller algorithm is potentially applicable. However, theKendall M0ller algorithm is known to be computationally feasible only un der a monotonicity condition: the Papangelou conditional intensity needs tobe attractive (favoring clustered patterns), repulsive (discouraging clusteredpatterns), or a product of such terms. We show that perfect sampling is fea sible for the Neyman Scott process when the prior satisfies this monotonicitycondition.

We present an application to data on leukemia incidence in an eightcounty area of uptstate New York during the years 1978 82. The study areaincludes 11 inactive hazardous waste sites. We assess the possibility of anincreased leukemia incidence rate in the proximity of these sites. There isan extensive literature on the analysis of these data, recent contributionsbeing Ghosh et al. (1999), who applied a hierarchical Bayes generalized lin ear model approach, and Ahrens et al. (1999), who adjusted for covariateeffects using a log linear model. Our results suggest that there is an elevatedleukemia incidence rate in the neighborhood of one of the sites.

The paper is organized as follows. In Section 2 we develop the mainresult of the paper showing that the posterior is locally stable, and examinethe Neyman Scott model in detail. Section 3 contains the application todisease clustering. Some concluding remarks are given in Section 4.

2 Bayesian cluster models2.1 Preliminaries

The basic framework comes from Carter and Prenter (1972), see also M0ller(1999). Let W be a compact subset of the plane representing the studyregion. A realization of a point process in W is a finite set of points x =# i, # 2, . . . , xn(x) C W, where n(x) is the number of points in x. If n(x) = 0,write x = 0 for the empty configuration. Let Ω denote the exponential spaceof all such finite point configurations in W, and furnish it with the σ field Tgenerated by sets of the form x : n(x ΠB) = fc, where B E B, the Borelσ field on W, and k = 0,1,2,

A standard way of constructing an Ω valued point process X is by speci fying an unnormalized density / with respect to the distribution π of the unit

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rate Poisson process on W. The unnormalized density / (or correspondingprocess X) is said to be locally stable if there is a constant K > 0 such that/ (xU £) < Kf(x) for all x G Ω, ξ e W\x. Local stability implies that thePapangelou conditional intensity

(with 0/0 = 0) is bounded.Most point processes that have been suggested for modeling spatial point

patterns are locally stable, including the Strauss (1975) process and the area interaction process of Baddeley and van Lieshout (1995). The Strauss pro cess, used later in this article, has unnormalized density / (x) = / ^ M yM ,where /? > 0, 0 < 7 < 1 and ί(x) is the number of unordered pairs of pointsin x which are within a specified distance r of each other. The Straussprocess only models repulsive pairwise interaction.

2.2 Posterior distribution

The observed point configuration which arises from the landmarks x will bedenoted y = yi,y25 ?yn(y) C W, and assumed to be non empty. Theprior and observation models are specified by point processes on W. Theprior distribution of landmarks corresponds to a point process X havingdensity pχ(x) with respect to π.

The likelihood is defined in terms of an unnormalized density / ( |x).Thus, for a given set of landmarks x, the density of the observed pointprocess Y with respect to π is

where

is the normalizing constant. We assume that / (y|x) is jointly measurable inx and y.

Prom Bayes formula, the posterior density of X with respect to π is

Px\γ=y*) oc α y(x) / (y|x)pχ(x) . (2.1)

The following theorem provides sufficient conditions for the posterior to belocally stable. We assume local stability of the prior pχ( ) and of the likeli hood / (y| ) (for each fixed y). In addition, / (y| ) is assumed to satisfy thefollowing local growth condition: there exists a constant L > 0 such that

/ ( y|χu ξ) > L / ( y|χ) (2.2)

324 LOIZEAUX AND MCKEAGUE

for all x, y E Ω, ξ G W\ x. The term 'local growth condition' is used herebecause we view it as being dual to local stability.

Theorem 2.1. Suppose pχ( ) and / (y| ) (for each y) are locally stable, and/ (y| ) satisfies the local growth condition (2.2). Then the posterior (2.1) islocally stable.

Proof. It suffices to show that aγ( ) is locally stable, because pχ ) and/ (y| ) are assumed to be locally stable, and local stability is preserved underproducts. Given ξ G W\ x,

i ίJn

> L / / (v|x)π(dv)Jn

= Lαy(x)"1,

completing the proof.

The Kendall M0ller algorithm uses the method of dominated coupling from the past to obtain perfect samples from a locally stable point processas the equilibrium distribution of a spatial birth and death process. Thealgorithm is computationally feasible if the Papangelou conditional intensityg(x, ξ) is either attractive or repulsive, or a product of such terms. In theattractive case, </(x, ξ) < g(x', ξ) whenever ξ £ x' and x C x'; in the repulsivecase g(x,£) > (x7,^) whenever ζ £ xr and x C x;.

2.3 N eyman Scott modelIn this section we focus on the Neyman Scott model in which the observationprocess Y is the superposition of n(x) independent inhomogeneous Poissonprocesses ZXi and a background Poisson noise process of intensity e > 0.The intensity h( \xi) of ZXi is specified parametrically, and the prior pχx)is assumed to be locally stable. Here and in the application in the nextsection we assume the Thomas intensity model

h(t\x) = e l l * l l W , (2.3)

where κ,σ > 0. For convenience, denote K* = κ/ (2πσ2). In this case,ί ^ x ) , where

n(x)

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is the conditional intensity at t of Y given x.We now check the relevant conditions of Theorem 2.1. To show that

/ (y| ) is locally stable, note that for ξ G W\x

n(y) / n(x

/ (y|χu£) = Π3=1

where

To check the local growth condition, note that for ξ G W\x

n(y)

/(y| xu

uniformly in y, and we can use L = 1. Thus the conditions of Theorem 2.1are satisfied, so the posterior is locally stable.

The Papangelou conditional intensity corresponding to / (y| ) is

/ (y|xU £) _ ΠA Λ ^ HVj\ξ)

for ξ G W\ x, which is clearly decreasing in x, thus repulsive.Noting that

we find that the Papangelou conditional intensity corresponding to aγ( ) is

for ξ G W\ x, which does not depend on x.We conclude that the posterior is of a feasible form for implementing

the Kendall M0ller algorithm if the Papangelou conditional intensity corre sponding to the prior is a product of repulsive or attractive components.

326 LOIZEAUX AND MCKEAGUE

3 ApplicationIn this section we present an application to data on leukemia incidence inan eight county area of uptstate New York. There is an extensive literatureon the analysis of these data, see, e.g., Waller et al. (1992, 1994), Ghosh etal. (1999) and Ahrens et al. (1999).

The study area is comprised of 790 census tracts and leukemia incidencewas recorded by the New York Department of Health for each census tractduring the years 1978 82, see Waller et al. (1992). The study area includes11 inactive hazardous waste sites. The goal is to assess the possibility of anincreased leukemia incidence rate in the proximity of these sites.

Figure 1: Left: locations of 552 leukemia cases in upstate New York, alongwith an approximate outline of the eight county study region. The rectan gular region is 1 x 1.2 square units. Right: contour plot of the populationdensity λ(t), and the locations of the 11 hazardous waste sites.

The locations of the centroids of the census tracts are available, but pre cise locations of the leukemia cases are not. Our methods require the preciselocations, so we randomly dispersed the cases throughout their correspond ing census tracts; if there was exactly one case in a tract, we placed it atthe centroid, see the left panel of Figure 1. (Sensitivity analysis indicatesthat this approximation makes no difference to our conclusions.) In someinstances a case could not be associated with a unique census tract, resultingin fractional counts. Our approach does not accomodate this type of data,

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so we follow Ghosh et al. (1999) and group the 790 census tracts into 281blocks in order to identify most of the cases with a specific block. Less than10% of all cases could not be identified with a specific block, and such casesare excluded from our analysis.

Figure 2: Posterior intensity map for the leukemia data based on theNeyman Scott model with e = 5.2 x 10~4, σ = 0.01, K* = 0.23e, and aStrauss prior with interaction radius r = 0.1, βx = 0.5 andLocations of the 11 hazardous waste sites are included.

= 0.1.

Our analysis is based on the Neyman Scott model with the leukemiaintensity rate specified by

n(x)

2 = 1

where X(t) adjusts for population density, e > 0 and h(t\x) is the Thomasintensity (2.3). Our earlier treatment of the Neyman Scott model extendswithout change to this form of the model because λ(ί) does not depend onx. We use a Strauss prior for the landmarks x. For X(t) we used a smoothedversion of the population density based on the 1980 U.S. census, see theright panel of Figure 1; this plot also gives the locations of the 11 inactivehazardous waste sites suspected of causing elevated leukemia incidence rates.

Figure 2 gives the posterior intensity for the landmarks based on thedata shown in the left panel of Figure 1; we used 1000 samples drawn using

328 LOIZEAUX AND MCKEAGUE

Site 6

Site 9

/i l l . . 1

Ί

Site 10 5. Site 11

I• hi

I|

J r. \ H\ . I . . .

Figure 3: Posterior observed (solid lines) and expected (dotted lines) prob abilities of at least one landmark within a given distance (in kms) of eachwaste site.

the Kendall M0ller algorithm. Note that one of the waste sites (site 1) islocated close to an area of high posterior intensity.

To assess the significance of an elevated leukemia rate in the neighbor hood of a given site, we compare the 'observed' with the 'expected' posteriorlandmark distribution. The relevant null hypothesis here is that the leukemiacases form an inhomogeneous Poisson process with intensity ρ\ (t), where pis the average leukemia rate throughout the study region. To sample fromthe null distribution, we generated an artificial data set using independent

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Poisson counts for each census tract, then analyzed the artificial data thesame way as the original data.

In Figure 3 we compare the observed and expected posterior probabilitiesof at least one landmark within a given distance (0-7.2 kms) of each wastesite. The 20 dotted lines correspond to samples from the null distribution,and the 5 solid lines correspond to the data (with the leukemia cases ran-domly dispersed throughout their corresponding census tracts). The plotsprovide evidence of elevated leukemia rates in the neighborhood of site 1.

4 ConclusionIn this article we have developed perfect sampling for the posterior distribu-tion in Bayesian cluster models for spatial point processes. We have isolatedconditions under which perfect sampling using the Kendall-M0ller algorithmis applicable. The algorithm is shown to be feasible under mild conditionson the prior and the likelihood, and, in particular, for the useful special caseof the Neyman-Scott model when the prior is repulsive.

We are currently working on a more detailed study of this topic in whichwe examine an extended formulation of the Baddeley-van Lieshout clus-ter model and provide other examples in which perfect sampling from theposterior is feasible.

Acknowledgements. We thank George Casella, John Staudenmayer andLance Waller for help with the leukemia incidence data. We also thank Jes-per M0ller for several useful comments. The project was partially supportedby NSA Grant MDA904-99-1-0070 and NSF Grant 9971784. Equipmentsupport was provided under ARO Grant DAAG55-98-1-0102 and NSF Grant9871196.

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