This thesis consists of two parts. In Chapter 2, we focus on the spatial scan statistics with overdispersion and Chapter 3 is devoted to the randomized permutation test for identifying local patterns of spatial association. The spatial scan statistic has been widely used in spatial disease surveillance and spatial cluster detection. To apply it, a spatial scan test is derived under Poisson assumption. However, Poisson assumption is often violated in many real world applications due to the presence of overdispersion. This thesis extends the Poisson-based spatial scan test to a quasi-Poisson based test to account for overdispersion. The quasi-Poisson based test was evaluated numerically against the Poisson-based in both simulation and case studies. Simulation studies showed that the proposed method can substantially reduce type I error probabilities in the presence of overdispersion. In the absence of overdispersion, type I errors and power functions were almost identical between the two. The case study of infant mortality in Jiangxi, China showed that there was a cluster consistently identified by both tests, while the secondary cluster was only identified by the Poisson based test. It is recommended that a cluster detected by the Poisson based scan test should be interpreted with caution when it is not confirmed by the quasi-Poisson based test. Getis and Ord's G and Moran's I statistics, as well as their local versions G[subscript i] and I[subscript i], have been widely used in spatial data analysis. Prior research has shown that the G statistic is asymptotically normal under weak regularity conditions. In addition, sufficient conditions for the asymptotic normality of Moran's I statistic has been provided. In practice, the normal approximation and randomized permutation test are commonly used to derive the p-values for identification of local patterns of spatial association. In this thesis, we show that the normal approximation and randomized permutation test for the local G[subscript i] and I[subscript i] statistics do not always lead to reliable inference. Simulation studies are used to present various cases when normal approximation and randomized permutation test are appropriate and inappropriate to use. Particularly, when the normal approximation is not appropriate, inference based on the randomized permutation test leads to misleading results and may produce spurious spatial associations. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]