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 denotes the number of persons at risk (live births) in tract ni 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 as a random variable with observed value and i Ci ci, denote the total number of observed cases by . We define a null hypothesis of no clustering by {Ci} are independent Poisson random variables with = ni, = (1) H0: E(Ci) i 1,…,I where  is the baseline incidence rate. That is, under the expected number of cases Ho, 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 defining a conditional null hypothesis c+, as ~ multinomial(c+, (2)  H0: C1,…,CI c+ n1/n+,…,nI/n+). 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 = + (3) H1: E(Ci) ni(1 i)
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)
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