病例對照研究被廣泛地使用在探討稀有疾病的研究中。當干擾變數難以量化時，實務上採用配對設計控制干擾變數，以調整因干擾作用產生的偏差。在配對病例對照研究中，常使用羅吉斯迴歸模型推論風險因子和二元反應變數的關係。由於模型包含大量的層效應參數，Breslow & Day (1980)提出條件概似法來消除過多的截距項參數。Chen & Wang (2013) 針對模型的適合度檢定提出動差形式檢定統計量，該檢定統計量是由任意的可測函數的條件最大概似估計量和無母數最大概似估計量的差建構而來。Chen & Wang (2013)採用的可測函數為共變量的平方，本文考慮可測函數為多項式函數、指標函數、對數函數，進一步以模擬研究評估不同的可測函數在動差形式適合度檢定的表現，最後透過實際資料說明新方法的操作。

Case-control studies have been widely applied to investigate rare diseases. When confounding variables are difficult to be quantified, matching designs are used to adjust the confounding effects to reduce the bias in practice. Matched case-control studies often use the logistic regression model to fit the relationship between the risk factors and binary response variable. As a result of highly stratum-effect parameters, Breslow & Day (1980) adopted the conditional approach to eliminate the intercepts. Chen & Wang (2013) proposed a moment-type goodness-of-fit test. This test statistic was constructed based on the discrepancy between the conditional maximum likelihood estimator and nonparametric maximum likelihood estimator of any measurable function. Chen & Wang (2013) set the measurable function as the square of the covariates. This study considers the measurable function to be the polynomial function, indicator function and logarithmic function. Further, the performances of the different testing functions of the moment-type goodness-of-fit test are assessed through simulation studies. Finally, a real dataset is used to illustrate the implement of the proposed method.

目錄
第一章 緒論	1
第二章 配對病例對照研究下羅吉斯迴歸模型分析	5
第一節 參數估計	5
第二節 動差形式檢定統計量	7
第三節 檢定統計量的近似分配	9
第三章 實例分析	11
第四章 模擬研究	14
第五章 結論       27
參考文獻	29
附錄一	31
附錄二	33
附錄三	34
附錄四：靜脈血栓栓塞配對病例對照資料	36


圖目錄
圖 1：BMI的常態分佈的Q-Q圖	26


表目錄
表 1：靜脈血栓栓塞資料的變數名稱與紀錄方式	13
表 2：條件羅吉斯迴歸模型配適結果	13
表 3：在顯著水準 之下，對照組共變量服從Normal(4,1)，且病例組共變量服從Normal(5,1)的檢定統計量Q的經驗型一誤差比率	17
表 4：在顯著水準 之下，對照組共變量服從Normal(4,1) ，且病例組共變量服從Normal(5,1)的檢定統計量Q的經驗檢定力	18
表 5：在顯著水準 之下，對照組共變量服從Normal(12,1.2) ，且病例組共變量服從Normal(13,1.2)的檢定統計量Q的經驗型一誤差比率	19
表 6：在顯著水準 之下，對照組共變量服從Normal(12,1.2) ，且病例組共變量服從Normal(13,1.2)的檢定統計量Q的經驗檢定力	20
表 7：在顯著水準 之下，對照組共變量服從Lognormal (0,1) ，且病例組共變量服從Lognormal (1,1)的檢定統計量Q的經驗型一誤差比率	21
表 8：在顯著水準 之下，對照組共變量服從Lognormal (0,1) ，且病例組共變量服從Lognormal (1,1)的檢定統計量Q的經驗檢定力	22
表 9：在顯著水準 之下，對照組共變量服從Exp (0.5) ，且病例組共變量服從Exp (1)的檢定統計量Q的經驗型一誤差比率	23
表 10：在顯著水準 之下，對照組共變量服從Exp (0.5) ，且病例組共變量服從Exp (1)的檢定統計量Q的經驗檢定力	24
表 11：由靜脈血栓栓塞資料重抽之檢定統計量Q的經驗檢定力	25

Aitchison, J. and Silvey, S. D. (1958). Maximum-likelihood estimation of parameters subject to restraints. The Annals of Mathematical Statistics, 39, 813-828.
Arbogast, P. G. and Lin, D. Y. (2004). Goodness-of-fit methods for matched case-control studies. The Canadian Journal of Statistics, 32, 373-386.
Bedrick, E. J. and Hill, J. R. (1996). Assessing the fit of the logistic regression model to individual matched sets of case-control data. Biometrics, 52, 1-9.
Breslow, N. E. and Day, N. E. (1980). Statistical Methods in Cancer Research, 1, The Analysis of Case-Control Studies. International Agency for Research on Cancer, Lyon, France.
Chen, L. C. & Wang, J. Y. (2013). Testing the fit of the logistic model for matched case-control studies. Computational Statistics and Data Analysis, 57, 309-319.
Cheng, K. F. and Chen, L. C. (2004). Testing goodness-of-fit of a logistic regression model with case-control data. Journal of Statistical Planning and Inference, 124, 409-422.
Cox, D. R. (1970). The Analysis of Binary Data. London: Methuen.
Fokianos, K., Peng, A. and Qin, J. (1999). A generalized-moments specification test for the logistic link. The Canadian Journal of Statistics, 27, 735-750.
Horowitz, H. and Neumann, G. R. (1992). A generalized moments specification test of the proportional hazards model. Journal of the American Statistical Association, 87, 234-240.
Hosmer, D. W. and Lemeshow, S. (1980). Goodness-of-fit tests for the multiple logistic regression model. Communications in Statistics-Theory and Methods, 9, 1043-1069. 
Hosmer, D. W. and Lemeshow, S. (2000). Applied Logistic Regression. Wiley, New York.
Le Cessie. S. and Van Houwelingen, H. C. (1995) Testing the fit of a regression model via score tests in random effects models. Biometrics, 51, 600-614.
Moolgavkar, S. H., Lustbader, E. D. and Venzon, D. J. (1984). A geometric approach to nonlinear regression diagnostics with application to matched case-control studies. The Annals of Statistics, 12, 816-826.
Moolgavkar, S. H., Lustbader, E. D. and Venzon, D. J. (1985). Assessing the adequacy of the logistic regression model for matched case-control studies. Statistics in Medicine, 4, 425-435.
Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica, 5, 1047-1070.
Pregibon, D. (1984). Data analytic methods for matched case-control studies, Biometrics, 40, 639-651.
Prentice, R. L. and Pyke, R. (1979). Logistic disease incidence models and case-control studies. Biometrics, 66, 403-411.
Qin, J. and Zhang, B. (1997). A goodness-of-fit test for logistic regression models based on case-control data. Biometrika, 84, 609-618.
White, H. (1994) Estimation, Inference and Specification Analysis. Cambridge University Press.
Woodward, M. (2005) Epidemiology: Study Design and Data Analysis, 2nd ed. London: Chapman & Hall/CRC.
Zhang, B. (1999). A chi-squared goodness-of-fit test for logistic regression models based on case-control data. Biometrika, 86, 531-539.