Summary of

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Summary of A Spatial Scan Statistic by M.
Kulldorff

• Presented by Gauri S. Datta
• gauri_at_stat.uga.edu
• Mid-Year Meeting
• February 3, 2006

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Background

• Scan Statistic
• A tool to detect cluster in a Point Process
• Naus (1965 JASA) studied in one dimension
• tests if a 1-dim point process is purely random
• Point Process
• Consider a time interval a,b and a window
  At,tw of fixed width w
• ?(A) of e-mails arrived in the time window A
• n(A) nA of junk e-mails number of
  points
• Arrival times of junk e-mails define a Point
  Process

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Main Idea in Scan Statistic

• Move a window t,tw of size w lt b-a over a time
  interval a,b
• Over all possible values of t, record the maximum
  number of points in the window
• Compare this number with cut off points under the
  the hypothesis of a purely Poisson Process

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Building block of Scan Test

• Repeated use of tests for equality of two
  Binomial or Poisson populations
• Two populations are defined by the scanning
  window A and its complement Ac
• As in multiple comparison, these tests are
  dependent as one moves the scanning window

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Spatial Scan Statistic (SSS)

• Kulldorff (1997) used SSS to detect clusters in
  spatial process
• SSS can be used
• In multi-dim point process
• With variable window size
• With baseline process an inhomogeneous Poisson
  process or Bernoulli Process

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SSS (continued)

• Scanning window can be any predefined shape
• SSS is on a geographical space G with a measure ?
• In traditional point process, G is a line, ? is a
  uniform measure
• In 2-dim, G is a plane, ? a Lebesgue measure

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Examples

• Forestry
• Spatial clustering of trees.
• Want to see for clusters of a specific kind of
  trees after adjusting for uneven spatial
  distribution of all trees
• ?(A)Total of trees in region A
• nA of trees in A of specific kind

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Examples (continued)

• Epidemiology
• Interest in detecting geographical clusters of
  disease
• Need to adjust for uneven population density
• Rural vs. urban population
• For data aggregated into census districts,
  measure is concentrated at the central
  coordinates of districts

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Examples (continued)

• If interest is in space-time clusters of a
  disease, the measure will still be concentrated
  in the geographical region as in the prior
  example
• Adjusting for uneven population distribution is
  not always enough. Should take confounding
  factors into account. E.g., in epidemiology
  measure can reflect standardized expected
  incidence rate

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SS LR statistic

• For a fixed size window, scan statistic is the
  maximum of points in the window at any given
  time/geographical region
• Test Stat is equivalent to LR test statistic for
  testing H0?1?2 vs. Ha?1gt?2
• Generalization to LR test is important for
  variable window

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Generalized SS Notation/Models

• G Geographical area / study space
• A Window ½ G
• N(A) Random of points in A
• A spatial point process
• Goal to find the prominent cluster
• Two useful models for point process
• (a) Bernoulli model
• (b) Poisson model

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Standard Models for SS

• For Bernoulli model, measure ? is such that ?(A)
  is an integer for all subsets A of G
• Two states (disease point or no disease) for
  each unit
• Location of the points define a point process

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LR Test Bernoulli Model
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LR Test Bernoulli Model
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Poisson Model

• Under Poisson model, points generated by inhom.
  Poiss. Proc. There is exactly one zone Z ? G s.t.
  N(A) ? Po(pµ(A??Z) qµ(A?Zc)) for all A.
• Null hypothesis H0pq
• Alternative hypo H1 pgtq, Z ??.
• Under H0, N(A) ? Po(pµ(A)) for all A.
• - the parameter Z disappears under H0

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Poisson Model (continued)
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Poisson Model (continued)
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Poisson Model (continued)
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Choice of Zones

• How is ? selected? Possibilities
• All circular subsets
• All circles centered at any of several foci on a
  fixed grid, with a possible upper limit on size
• Same as (2) but with a fixed size
• All rectangles of fixed size and shape
• If looking for space-time clusters, use
  cylinders scanning circular geographical areas
  over variable time intervals

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Bernoulli vs. Posson Model

• Choice between a Bernoulli or Poisson model does
  not matter much if
• n(G) ltlt ?(G)
• In other cases, use the model most appropriate
  for application

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A Useful Result

• An important result on most likely cluster
  based on these models is given in the paper. It
  states that as long as the points within the zone
  constituting the most likely cluster are located
  where they are, H_0 will be rejected irrespective
  of the other points in G. If a cluster is located
  in Seattle, locations of the points in the east
  coast of U.S. do not matter (Theorem 1)

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Computations and MC

• To find the value of ?, we need to calculate LR
  maximized over collection of zones in H1. Seems
  like a daunting task since of zones could be
  infinite.
• of observed points finite
• For a fixed of points, likelihood decreases as
  µ(Z) increases

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Computations (contd)

• If the circle size increases for a fixed foci,
  need to recalculate likelihood whenever a new
  point enters the circle. For a finite points,
  of recalcing likelihood for each foci is finite.
• Distribution of ? is difficult. MC simulation
  used to generate histogram of ? . Under H0,
  replicate the data sets conditional on nG .

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Application of SSS to SIDS

• Bernoulli and Poisson models are illustrated
  using the SIDS data from NC
• For 100 counties in NC, total of live births
  and of SIDS cases for 1974-84.
• Live births range from 567 to 52345
• Location of county seats are the coordinates.
  Measure is the of live births in a county

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Application to SIDS (continued)

• Zones for scanning window are circles centered at
  a county coordinate point including at most half
  of the total population
• Zones are circular only wrt the aggregated data.
  As circles around a county seat are drawn, other
  counties will either be completely part of a zone
  or else not at all, depending on whether its
  county seat is within the circle or not

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Bernoulli model for SIDS

• Bernoulli model is very natural. Each birth can
  correspond to at most one SID. Table 1 summarizes
  the results of the analysis.
• From Figure 1, the most likely cluster A,
  consists of Bladen, Columbus, Hoke, Robeson, and
  Scotland.
• Using a conservative test, a secondary cluster is
  B, consists of Halifax, Hartford and Northampton
  counties.

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Poisson model for SIDS

• For a rare disease SIDS, Poisson model gives a
  close approximation to Bernoulli. Results are
  reported in Table 1
• Both models detect the same cluster
• P-values for the primary cluster are same for
  both the models p-values for the secondary
  cluster are very close

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Application to SIDS (continued)
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Two significant clusters based on SSS
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SSS adjusted for Race

• For SIDS one useful covariate is race
• Race is related to SIDS through unobserved
  covariates such as quality of housing, access to
  health care
• Overall incidence of SIDS for white children is
  1.512 per 1000 and for black children is 2.970
  per 1000.

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SSS race-adjusted (continued)

• Racial distribution differs widely among the
  counties in NC
• This analysis leads to the same primary cluster
  (see Figure 2)
• Previous secondary cluster disappeared but a
  third secondary cluster C emerges. Cluster C
  consists of a bunch of counties in the western
  part of the state

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Application to SIDS (continued)
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SSS to SIDS adjusted for race
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A Bayesian alternative to SSS

• Scott and Berger (2006) Idea of Bayesian
  multiple testing.
• Observe Xj ? N(µj, s2), j1,,M,
• To determine which µj are nonzero ? we have M
• (conditionally) independent tests, each
  testing
  
• H0jµj 0 vs. H1j µj ? 0
• p0 prior probability that µj is zero
• Crucial point here let data estimate p0 .
• SB use the hierarchical model
• 1. Xjµj , s2, ?j N(?jµj, s2),
  independently
• 2. µj t2 I.I.D. N(0, t2 ), ?j p0
  I.I.D. Bern (1-p0)
• 3. (t2 , s2) p (t2 , s2) (t2 s2)-2, p0
  p(p0)
• Several choices for p(p0) Uniform,
  Beta(a,1)
• SB computed posterior probability ?j 1.

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Modification of SB Model

• Assume Xj ? N(µj, s2), j1,,M,
• To determine which µj are positive ? we have M
• (conditionally) independent tests, each
  testing
  
• H0jµj 0 vs. H1j µj gt 0
• As before
• 1. Xjµj , s2, ?j N(?jµj, s2),
  independently
• 2. µj µ(-j), ?, t2 N(??qjkµk, t2 ),
  CAR
• ?j pj Ind. Bern (1-pj)
• 3. (t2, s2, ?) p (t2 , s2, ?) (t2 s2)-2
• 4. CAR model on logit(pj)
• Compute posterior probability of µj gt0.