糖尿病為臺灣普遍的慢性代謝疾病之一，而心血管疾病為造成糖尿病患者死亡的重要原因。由於心血管疾病的發生與空氣汙染、溫度等環境因素可能有關，因此患者的病發時間也許會因為所在的地理位置產生空間相關，這樣的資料稱為空間存活資料，本研究則針對此類型的資料作探討。為了在考慮空間相關下，建立存活時間與感興趣的解釋變數之間的關係，本研究為右設限空間存活資料提出考慮空間相關下的比例風險模型。我們的模型假設存活時間服從Cox比例風險模型 (Cox proportional hazards model，簡稱Cox模型)，並對存活時間做常態轉換，再以內在條件自迴歸模型 (intrinsic conditional autoregressive model，簡稱ICAR模型) 建立常態轉換後存活時間的空間相關結構，最後利用空間半參數估計方程式 (spatial semiparametric estimating equations) 同時估計迴歸係數及空間未知參數。我們透過模擬研究來評估本研究提出的模型表現，並將此模型應用於第II型糖尿病患者的資料。

Type 2 diabetes mellitus (T2DM) is a common chronic diseases predisposing to cardio-vascular disease (CVD), which is the principal cause of death among diabetic patients. Due to the physical environment, there is likely to be a spatial component to the time of onset of CVD. This study aims to model the factors that affect the survival time while taking the spatial correlation into account. We develop a spatial Cox model with intrinsic conditional autoregressive (ICAR) dependence for spatial survival data. The proposed model is based on Cox proportional hazards model, and the joint distribution of the transformed survival outcomes is assumed to be the multivariate normal with a spatial correlation structure which is implied by an ICAR prior. The parameters of interest, including the regression coefficients and the spatial correlation parameter, are estimated by solving spatial semi-parametric estimating equations. The paper presents a simulation study to evaluate the performance of the proposed model, and then applied the method to analyze the data of type 2 diabetic patients in Taiwan.

Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Second Edition: CRC Press.
Breslow, N. E. (1972). Discussion of Professor Cox’s paper. Journal of the Royal Statistical Society. Series B (Methodological), 34, 216-217.
Chiu, W. C., Ho, W. C., Liao, D. L., Lin, M. H., Chiu, C. C., Su, Y. P., et al. (2015). Progress of Diabetic Severity and Risk of Dementia. The Journal of Clinical Endocrinology & Metabolism, 100(8), 2899-2908.
Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), 187-220.
Cressie, N. A. C. (1993). Statistics for Spatial Data: J. Wiley.
Grambsch, P. M., and Therneau, T. M. (1994). Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika, 81(3), 515-526.
Kaplan, E. L., and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457-481.
Li, L., Hanson, T., and Zhang, J. (2015). Spatial Extended Hazard Model with Application to Prostate Cancer Survival. Biometrics, 71(2), 313-322.
Li, Y., and Lin, X. (2006). Semiparametric Normal Transformation Models for Spatially Correlated Survival Data. Journal of the American Statistical Association, 101(474), 591-603.
Mantel, N. (1966). Evaluation of Survival Data and Two New Rank Order Statistics Arising in Its Consideration. Cancer Chemother Rep, 50, 163-170.
Prentice, R. L., and Cai, J. (1992). Covariance and Survivor Function Estimation Using Censored Multivariate Failure Time Data. Biometrika, 79(3), 495-512.
Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging: Springer New York.