在流行病學研究中，羅吉斯迴歸模型常用來推論風險因子和二元反應變數之間的關係。在稀有疾病常以病例對照研究改善抽樣效率。當主要的干擾變數太多時，配對病例對照設計比病例對照設計更能消除偏差。Breslow & Day(1980)使用條件概似方法消除過多的截距項。羅吉斯迴歸模型能依靠方便的統計軟體簡易且快速執行，然而常有羅吉斯迴歸模型錯置的案例，因此以模型的適合度檢定確認最後模型。我們推廣Cheng & Chen(2004)的想法，在配對病例對照資料下提出分數型態的檢定以檢定羅吉斯迴歸模型。由Arbogast & Lin (2004)的結果可知，我們提出的分數檢定會弱收斂至高斯過程。最後，進行模擬研究以評估檢定統計量的有限樣本性質，並以低出生重量研究做實例分析。

In epidemiology studies, the logistic regression model is used popularly for inferring the relation of risk factors and a binary response variable. For rare diseases, we usually take case-control studies to improve sampling efficiency. When some major confounding variables are difficult to quantity, a matched case-control design can be adopted to eliminate biased comparisons between cases and controls. Breslow & Day (1980) use the conditional likelihood to eliminate the too many intercept terms. Relying on convenient statistical software, the logistic regression models are easily and rapidly implemented. It is, however, often misplace to examine whether a logistic model fit the data well. It should be recognized that before constructing the final model, a goodness-of-fit test of the model is still an important issue. We generalized the idea of Cheng & Chen (2004), and propose a Score type test for the logistic regression model under matched case-control data. The Score test converges weakly to a centered Gaussian process by the result of Arbogast & Lin (2004).We assess the performance of the proposed test through simulation studies in finite sample and illustrate the score test by a low birth weight study.

第一章	緒論 ---------------------------------------------------------------------------01
第二章	羅吉斯迴歸模型的適合度檢定 ------------------------------------------07
第一節	羅吉斯迴歸模型的介紹 -------------------------------------------------07
第二節	迴歸參數的估計 ----------------------------------------------------------09
第三節	迴歸模型的適合度檢定 -------------------------------------------------14
第四節	1:1配對病例對照資料 ---------------------------------------------------23
第三章	模擬研究 ---------------------------------------------------------------------24
第一節	適合度檢定的數值模擬 ---------------------------------------------24
第二節	適合度檢定的實例分析 ---------------------------------------------27
第四章 結論 --------------------------------------------------------------------------31
參考文獻 -------------------------------------------------------------------------------33
附錄：原始資料 ----------------------------------------------------------------------37

表目錄
1.   分數檢定和上界檢定的模擬結果 ----------------------------------------26
2.   低出生重量研究資料的變數名稱與變數值描述 ----------------------30

[1]  Agresti, A. (2002). Categorical Data Analyasis. 2nd ed. New York: John 
    Wiley.
[2]  Allison, P.D. (1999). Logistic regression using the SAS system: theory 
    and application. Cary, N.C.:SAS Institute.
[3]  Arbogast, P.G. & Lin, D.Y. (2004). Goodness-of-fit methods for matched    
    case-control studies. The Canadian Journal of Statistics 32, 373-386.
[4]  Arbogast, P.G. & Lin, D.Y. (2005). Model-checking techniques for 
    stratified case-control studies. Statistics in Medicine 24, 299-247.
[5]  Bedrick, E. J. & Hill, J.R. (1996). Assessing the fit of the logistic 
    regression model to individual matched sets of case-control data. 
    Biometrics 52,1-9.
[6]  Breslow, N.E. & Day, N.E. (1980). Statistical Methods in Cancer 
    Research,1,The Analysis of Case-Control Studies. International 
    Agency for Research on Cancer, Lyon, France.
[7]  Breslow, N.E., Day, N.E., Halvorsen, K. T., Prentioe, R. L. & Sabai, C. 
    (1978). Estimation of multiple relative risk functions in matched 
    case-control studies. American Journal of Epidemiology 108, 299-307. 

[8]  Cheng, K.F. & 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.
[9]  Cox, D.R. (1970). The Analysis of Binary Data. London: Methuen.
[10]  Faraway, J.J. (2006). Extending the linear model with R : generalized 
     linear, mixed effects and nonparametric regression models. Boca 
     Raton, Fla.:Chapman & Hall/CRC
[11] Gail, M.H., Lubin, J.H., & Rubinstein, L.V.(1981). Likelihood 
     calculations for matched case-control studies and survival studies 
     with tied death times. Biometrika 68,703-707. 
[12]  Hirji, K.F., Mehta, C.R. & Patel, N.R. (1988). Exact inference for 
      matched case-control studies. Biometrics 44,803-814.
[13]  Holford, T. (1978). The analysis of pair-matched case-control studies, a   
      multivariate approach. Biometrics 34,665-672.
[14]  Holford, T.,White, C. & Kelsey, J.L. (1978). Multivariate analysis for 
     matched case-control studies. American Journal of Epidemiology 
     107, 245-256.
[15]  Hosmer, D.W. & Lemeshow, S. (2000). Applied logistic regression. 
      2nd ed. New York: Wiley.
[16]  Lecessie, S. & Vanhouwelingen, H.C.(1995). Testing the fit of a 
      regression model via score tests in random effects models. Biometrics  
      51, 600-614.
[17]  Liu, I., Mukherjee, B., Suesse, T., Sparrow, D. & Park, S.K.(2009).
     Graphical diagnostics to check model misspecification for the  
     proportional odds regression model. Statistics in medicine 28, 412–429.
[18]  Moolgavkar, S.H., Lustbader, E.D. & Venzon, D.J. (1985). Assessing 
      the adequacy of the logistic regression model for matched case-
      control studies. Statistics in Medicine 4, 425-435. 
[19]  Pike, M.C., Hill, A.P. & Smith, P.G.(1980). Bias and efficiency in 
     logistic analyses of stratified case-control studies. International Journal 
of Epidemiology 9, 89-95.
[20]  Pregibon, D. (1984). Data analytic methods for matched case-control 
     studies, Biometrics 40,639-651. 
[21]  Prentice, R. L. & Pyke, R. (1979). Logistic disease incidence models 
     and case-control studies. Biometrika 66, 403-411.  
[22]  Qin, J. & Zhang, B. (1997). A goodness-of-fit test for logistic 
      regression models based on case-control data. Biometrika 84,  
      609-618.        
[23]  Qin, J. (1998). Inferences for case-control and semiparametric      
      two-sample density ratio models. Biometrika 85, 619-630.  
[24]  SAS Institute. (1995). Logistic regression examples using the SAS 
     system. 1st ed. Cary, N.C. : SAS Institute.
[25]  Smith, P.G., Pike, M.C., Hill, A.P., Breslow, N.E. & Day, N.E.(1981). 
      Multivariate condictional logistic analysis of stratum matched 
      case-control studies. Applied Statistics 30, Algorithm AS162,190-197.
[26]  Woodward, M. (2005) Epidemiology: study design and data analysis. 
      2nd ed. Chapman & Hall /CRC. 
[27]  Zhang, B. (1999). A chi-squared goodness-of-fit test for logistic 
      regression models based on case-control data. Biometrika 86, 
      531-539.
[28]  Zhang, B.(2006). A score test under logistic regression models based 
     on case-control data. Statistica Neerlandica 60, 477-496.