此篇論文主要在探討看得到的資料有 sample selection bias 的問題時應該如何利用這些資料做正確的估計及推論 。所謂 sample selection bias 是指所能看得到的資料是具有某些特性而被收集到的 ，這些資料無法表現整個有興趣母體的真正分布 。如果我們用這些資料不考慮他的抽樣機制直接做分析則會得到一個有偏差的估計值 。文中我們利用 Heckman (1979) 的想法建立兩個模型，這兩個模型分別描述母體的回歸方程式和抽樣機制的方程式。母體模型我們討論 Logistic 、Exponential 、Normal 和 Poisson distribution 四種情形。抽樣機制我們考慮一個普遍的 Cox proportational hazard model (Cox, 1972) 。利用二階段估計法的概念 ，先估出有關抽樣機制的參數再套入 estimating equation 中估計出有關母體的參數。二階段估計法中的第二步，四種分布中，只有 Logistic distribution 有 close form 我們可以直接用 MLE 估計 。其他三種分配我們必須經由繁複的數值運算才能再藉由 MLE 估計 。因此 ，我們建議用 weighted estimating equation 和 composite likelihood method ( pairwise pseudolikelihood approach )的方法 。其中 weighted estimator 在 truncation rate 大時容易出現估計不穩定的情形 ; composite likelihood method 相對 weighted estimator 來說 bias 小很多且估計很穩定 ，只是無法估計截距項 。不過通常我們感興趣的是 covariate 對 response 的影響大小而不是截距項 ，因此 composite likelihood method 會是個很有效率的方法 。

參考文獻
Cox and Hinkley (1974). Theoretical Statistics. London:Chapman and Hall.
Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, series B, series B, 34(2):187-220.
Efron, B. (1977). The eciency of cox's likelihood function for censored data. Journal of the American Statistical Association, 72:557-565.
Feigl, P. and Zelen, M. (1965). Estimation of exponential survival probabilities with concomitant information. Biometrics, 21:826-838.
Gail, M. and Benichou, J. (2000). Encyclopedia of Epidemiologic Methods. Chichester:Wiley.
Green, W. (2003). Econometric Analysis,6th ed. Prentice Hal,Upper Saddle River.
Heckman (1979). Sampling selection bias as a specication error.Econometric, 47:153-161.
Kalbeisch, J. (1978). Likelihood methods and nonparametric tests.Journal of the American Statistical Association, 73:167-170.
Keiding, N. (1991). Age-specic incidence and prevalence:a statistical perspective. Journal of the Royal Statistical Society, Ser.A:154,371-412.
Kvam, P. (2008). Length bias in the measurements of carbon nanotubes. Technometrics, 50:462-467.
Lancaster, T. (1990). The econometric analysis of transition data.Cambridge University Press, Cambridge.
Lawless, J., Wild, C., and Kalbeisch, J. (1999). Semiparametric Methods for Response-Selective and Missing Data Problems in Regression. Journal of the Royal Statistical Society.Series B (Statistical Methodology) Vol. 61, No. 2 . 413-438.
Liang, K.-Y. and Qin, J. (1999). Generalized odds ratio model and pairwise conditional likelihood. Technical Report.
Liang, K.-Y. and Qin, J. (2000). Regression analysis under nonstandard situations: pairwise pseudolikelihood approach. Journal of the Royal Statistical Society, B:773-786.
Oakes, D. (1977). The asymptotic information in censored survival data. Biometrika, 64:441-448.
Robins, J., Rotnitzky, A., and Zhao, L. (1994). Estimation of regression coecients when some regressors are not always observed.
Journal of the American Statistical Association, 89:846-866.
Scheike, T. and Keiding, N. (2006). Design and analysis of time-to-pregnancy. Statistical Methods in Medical Research, 15:127-140.
Simon.R (1980). Length-biased sampling in etiologic studies. American Journal of Epidemiology, 111(4):444-452.
Varin, C., Reid, N., and Firth, D. (2011). An overview of composite likelihood methods. Staatistica Sinica, 21:5-42.
Wang, M.-C. (1996). Hazards regression analysis for length-biased data. Biometrika, 83:343-354.
Zelen, M. (2004). Forward and backward recurrence times and
length biased sampling:age specic models. Lifetime Data Analysis, 10:325-334.
Zelen, M. and Feinleib, M. (1969). On the theory of screening for chronic diseases. Biometrika, 56:601-614.