疾病群聚通常是指疾病的發生在某時間、空間或時空上異常地高，而疾病稀少則通常是指疾病的發生在某時間、空間或時空上異常地低，辨別時間或空間上的疾病群聚在流行病學中有其重要性，尤其可以(1)檢驗是否存在任何顯著高/低的疾病發生率群聚，(2)找到真實的疾病群聚發生的潛在危險因子，而疾病稀少代表可能找到真實的疾病稀少的保護因子或/和缺少疾病的危險因子，以及(3)加強對疾病病因學的了解。
由於人類大部分的疾病都是複雜性的疾病，通常是由多種潛在的危險因子造成，因此在進行空間疾病群聚分析時，將已知的危險因子考慮進去相當重要。我們使用the spatial scan statistic與the map-based pattern recognition procedure方法分析北卡羅來納州1974年至1978年嬰兒猝死症的數據，現今許多研究都在檢測空間疾病群聚，而本研究之主要目的為檢測是否存在顯著的疾病群聚後，辨別出疾病的危險因子並運用複線性迴歸分析估計已知的或假定的危險因子對嬰兒猝死症發生率的影響程度。
The map-based pattern recognition procedure方法使用county鄰近關係的模擬分佈，先找到高風險地區再做群聚分析。The spatial scan statistic使用似然率檢驗，運用其求出最大可能性的主要群聚及次要群聚。這兩種統計方法有些不同但都可以用來檢測空間疾病群聚，我們將其結果應用至複線性迴歸模型中作分析。
我們考慮了許多種自變數與應變數的組合，從僅分析發生率跟種族之間的線性關係，接著考慮疾病在地理上的分佈，再分別執行高/低發生率地區，接著同時考慮高/低發生率地區，最後加上其交互作用。我們總結同時加入高/低發生率地區並考慮其交互作用的迴歸模型，最適合解釋本研究中疾病發生率與種族加上疾病地理空間分佈之間的關係。為了避免迴歸係數估計結果受到單一極端值的影響，我們將Anson當作離群值排除，因為其疾病發生率為每千人9.55，而總發生率為每千人2.03，結果顯示整體的判定係數都有上升的趨勢。根據口試委員們的建議，為了符合線性迴歸的假設條件，使用Freeman-Tukey轉換針對嬰兒猝死症數據進行轉換，結果顯示其轉換過後的數值更接近常態分佈，表示進行轉換是更適合的。在最具代表性的迴歸模型中，對於the map-based pattern recognition procedure且排除離群值的轉換前結果來說，種族的作用造成中發生率地區每千人0.142病例，高發生率地區比中發生率地區增加233%，低發生率地區比中發生率地區減少307%，判定係數為0.61，對於the spatial scan statistic且Freeman-Tukey轉換後，種族的作用造成高發生率地區比中發生率地區增加65%，低發生率地區比中發生率地區減少81%，判定係數為0.47。另外我們分析性別與嬰兒猝死症的迴歸分析，結果皆未顯著，相關係數顯示兩者之間的關聯性極小。
總之我們找出疾病群聚後，重點在運用複線性迴歸分析探討了危險因子種族與性別對嬰兒猝死症發生率的效應。從結果得知種族與性別皆是嬰兒猝死症重要的危險因子，但只有種族是空間相關的危險因子，性別非空間相關的危險因子。

We use the map-based pattern recognition procedure and the spatial scan statistic to detect spatial clusters of sudden infant death syndrome in the 100 counties in North Carolina in 1974-1978. The aim of this study is to use multiple linear regression methods to estimate the spatial effects of race and gender in spatial disease clusters detected by these 2 methods. The multiple linear regression model which incorporates high/low risk regions and considers their interaction is used to explain the association between incidence of sudden infant death syndrome and race and gender in this study. The most representative multiple linear regression method which used the the map-based pattern recognition procedure and removed the outlier without the Freeman-Tukey transformation indicated that with a 10% increase of non-white live birth proportion, the number of sudden infant death syndrome will increase by 0.142 cases per 1,000 live births in medium-risk counties, significantly higher effect by 233% in high-risk counties than medium-risk counties, and significantly lower effect by 307% in low-risk counties than medium-risk counties. The same method analyzed the spatial clusters detected by the spatial scan statistic with the Freeman-Tukey transformation indicated that with an increase of 10 for the Freeman-Tukey transformed non-white live birth proportion, the number of cases will increase by 0.300 cases per 1,000 live births in medium-risk counties, significantly higher effect by 65% in high-risk counties than medium-risk counties, and significantly lower effect by 81% in low-risk counties than medium-risk counties.

Carlin, R. F., & Moon, R. Y. (2017). Risk Factors, Protective Factors, and Current Recommendations to Reduce Sudden Infant Death Syndrome: A Review. JAMA Pediatrics, 171(2), 175-180.
Centers for Disease Control. (1990). Guidelines for Investigating Clusters of Health Events. In C. f. D. Control (Ed.), Morbidity and Mortality Weekly Report (vol. 39, pp. 1-23).
Cliff, A. D., & Ord, J. K. (1973). Spatial Autocorrelation. Pion Press, London.
Cressie, N. (1992). Smoothing Regional Maps Using Empirical Bayes Predictors. Geographical Analysis, 24, 75-95.
Cressie, N., & Chan, N. H. (1989). Spatial Modeling of Regional Variables. Journal of the American Statistical Association, 84, 393-401.
Cressie, N. & Read, T. R. C. (1985). Do Sudden Infant Death Syndrome Come in Clusters? Statistics and Decisions, Supplement Issue NO. 2, 333-349.
Freeman, M, F., & Tukey, J, W. (1950). Transformations Related to the Angular and the Square Root. The Annals of Mathematical Statistics, 21(4), 607-611.
Grimson, R. C., Wang, K. C., & Johnson, P. W. C. (1981). Searching for hierarchical clusters of disease: Spatial patterns of sudden infant death syndrome. Social Science & Medicine. Part D: Medical Geography, 15(2), 287-293.
Hjalmars, U., Kulldorff, M., Gustafsson, G., & Nagarwalla, N. (1996). Childhood leukaemia in Sweden: using GIS and a spatial scan statistic for cluster detection. Statistics in Medicine, 15(7-9), 707-715.
Kulldorff, M. (1997). A Spatial Scan Statistic. Communications in Statistics – Theory and Methods, 26(6), 1481-1496.
Kulldorff, M. (1999). An Isotonic Spatial Scan Statistic for Geographical Disease Surveillance. Journal of the National Institute of Public Health, 48, 94-101.
Kulldorff, M. (2001). Prospective Time Periodic Geographical Disease Surveillance Using A Scan Statistic. Journal of the Royal Statistical Society Series A, 164, 61-72.
Kulldorff, M., Athas, W. F., Feurer, E. J., Miller, B. A., & Key, C. R. (1998). Evaluating cluster alarms: a space-time scan statistic and brain cancer in Los Alamos, New Mexico. American Journal of Public Health, 88(9), 1377-1380.
Kulldorff, M., Feuer, E. J., Miller, B. A., & Freedman, L. S. (1997). Breast cancer clusters in the northeast United States: a geographic analysis. American Journal of Epidemiology, 146(2), 161-170.
Kulldorff, M., Huang, L., Pickle, L., & Duczmal, L. (2006). An elliptic spatial scan statistic. Statistics in Medicine, 25(22), 3929-3943.
Kulldorff, M. & Nagarwalla, N. (1995). Spatial disease clusters: detection and inference. Statistics in Medicine, 14(8), 799-810.
Mantel N. (1967). The detection of disease clustering and a generalized regression approach. American Association for Cancer Research, 27(2):209-20.
Mostashari, F., Kulldorff, M., Hartman, J. J., Miller, J. R., & Kulasekera, V. (2003). Dead bird clusters as an early warning system for West Nile virus activity. Emerging Infectious Diseases, 9(6), 641-646.
Mosteller, F., & Youtz, C. (1961). Tables of the Freeman-Tukey Transformations for the Binomial and Poisson Distributions. Biometrika, 48(3/4), 433-440.
Naus, J. I. (1965). The Distribution of the Size of the Maximum Cluster of Points on a Line. Journal of the American Statistical Association, 60(310), 532-538.