literature) we locate all births and cases at the centroid of the census tract of residence, due to confidentiality restrictions. 2.2 Hypothesis formulation We consider a null hypothesis where each birth is equally likely to experience the adverse outcome, so the expected incidence count of each area varies only due to heterogeneous population sizes (here, number of births). To formalize, suppose we have a geographic study region partitioned into I tracts, where ni denotes the number of persons at risk (live births) in tract i, i = 1, …,I, and is the total number of persons at risk in the study area. Consider the number of incident cases of disease in tract i as a random variable Ci with observed value ci, and denote the total number of observed cases by . We define a null hypothesis of no clustering by H0: {Ci} are independent Poisson random variables with E(Ci) = ni, i = 1,…,I (1) where  is the baseline incidence rate. That is, under Ho, the expected number of cases in each tract is the baseline rate of disease multiplied by the number at risk in tract i. When the baseline incidence rate is unknown, one obtains the null distribution of the {Ci} by conditioning on the sufficient statistic c+, defining a conditional null hypothesis as H0: C1,…,CI  c+ ~ multinomial(c+, n1/n+,…,nI/n+). (2) The alternative hypothesis depends on the model of clustering the investigator wishes to detect (Waller and Jacquez13). For this study we consider an alternative hypothesis defined by H1: E(Ci) = ni(1 + i) (3)
where i = (4)  = RR – 1 for i = 1,…,I, and RR is the relative risk of disease within the cluster6. For unknown λ we consider the conditional alternative hypothesis H1: C1,…,CI  c+ ~ multinomial(c+, 1,…,I) (5) where i = , i = 1,…, (6) 2.3 Kulldorff’s spatial scan statistic Kulldorff11 defines a spatial scan statistic to detect the most likely cluster based on locally observed likelihood ratio test statistics. The scan statistic is constructed based on circles of increasing radii centered at each tract centroid until a maximum radius, selected a priori, is attained. Kulldorff and Nagarwalla7 suggest a largest circle containing 50 per cent of the total at-risk population, and we use this definition here. Kulldorff11 refers to the collection of tracts having their centroid contained within each circle as a zone, z. The scan statistic considers each zone as a potential cluster, and identifies the zones least consistent with the null hypothesis. We note that for data consisting of regional counts, while the centroids are contained within a circle, zones actually reflect an irregular area defined by the union of polygonal regions associated with the tracts. Let nz and cz be the population size and case count, respectively, in zone z. Define and as the probability of being a case inside and outside zone z, respectively. Based on the null hypothesis H0:=versus the alternative HA:>, Kulldorff11 defines a likelihood ratio statistic proportional to ,
Figure 2. Map of the observed numbers of severe cardiac defects among reported live births in 1981 (upper left), an example “cluster” comprised of a single tract and its six nearest neighbors (upper right), and three simulated data sets based on a tripling of disease risk in the indicated cluster (bottom row). Figure 3. Choropleth maps of the statistical power of Tango’s8 test of clustering for various values of τ. Shading within each map is according to the quantiles of power values. Circles represent distance radii associated with neighborhood weights of 0.5 (inner circle) and 0.05 (outer circle).