對於一般的測量誤差問題中，過去的文獻都假設為無差別性測量誤差 (nondifferential measurement error)，其定義為測量誤差與應變數 (dependent variable) 獨立，也就是說測量誤差不帶有應變數的任何訊息。實驗的過程中真實自變數重覆測量的次數常會與反應變數有關。在這種情況下，若基於希望將收集到的資料做高效率的使用而單純地使用全部觀測值的平均，就會產生有差別性測量誤差 (differential measurement error) 的問題。

在本論文中，我們考慮兩個統計模型：(i) 計數資料的零截段模型與 (ii) 結合 Cox 的風險比例模型 (proportional hazards model) 與隨機效應模型的聯合建模 (joint modeling) 之模型。在這兩個模型中，有差別性測量誤差的問題會自然產生。針對這個問題，我們提出誤差增量 (error augmentation) 估計方法，此方法除了能解決有差別性測量誤差的問題外，也可以提高原先估計方法的效率。

In the context of measurement error problems, most of the literatures assumed that the measurement error is nondifferential, that is, the measurement error is independent to the response variable. In other words, measurement error contains no information for the response variable. Such assumption may be plausible for many applications in practice. Nevertheless, there are occasions that number of repeat measurements depends on the response variable and hence the accuracy of averaged surrogate depends on the response variable, too. This situation induces a differential measurement error problem and there was no satisfactory analysis for the problem so far in general.

In this thesis, we consider two regression models: (i) a zero-value truncated model for count data and (ii) a joint modeling for the Cox proportional hazards model and a random effect model. The differential measurement error problems arise naturally in these two models. In this thesis, we propose the error augmentation method. It could not only solve the problem brought by differential measurement errors, but also enhance the efficiency of the original estimating method.

1 緒論     1

2 測量誤差校正方法     7
2.1 傳統測量誤差模型與校正方法回顧     7
2.1.1 校正分數法 (Corrected Score-Nakamura, 1992)     7
2.1.2 條件分數法 (Conditional Score-Stefanski 與 Carroll, 1987)     8
2.1.3 模擬外插法 (Simulation Extrapolation-Cook 與 Stefanski, 1994)     9
2.1.4 當測量誤差變異數為未知時三明治法的修正     11
2.2 校正方法的討論     12

3 計數資料的零截斷模型     15
3.1 模型介紹與條件分數函數估計方法     15
3.2 誤差增量法     17
3.3 參數估計量漸近分布與效率改進     20

4 長期性研究的比例涉險模型     23
4.1 模型介紹     23
4.2 條件分數估計方法     26
4.3 誤差增量法與效率改進     27
4.4 參數估計量漸近分布     29
4.5 討論     30

5 統計模擬     31
5.1 零截斷模型     31
5.2 長期性研究的比例涉險模型     33

6 實例分析     37
6.1 雪霸自然保護區野生動物資源調查：巢鼠資料分析     37
6.2 ACTG 175 研究資料     41

7 結論     45

參考文獻     47

附錄 A 零截斷模型的模擬結果     51
附錄 B 聯合建模的模擬結果     59

[1] Brown, B. W., and Mariano, R. S. (1993). Stochastic simulations for inference in nonlinear errors-in-variables models. Handbook of Statistics, Vol. 11, 611-627. North Holland: New York.

[2] Carroll, R. J., Gail, M. H., and Lubin, J. H. (1993). Case-control studies with errors in predictors. Journal of the American Statistical Association 88, 177-191.

[3] Carroll, R. J., Ruppert, D., Stefanski, L. A., and Crainiceanu, C. M. (2006). Measurement Errors in Nonlinear Models: A Modern Perspective (2nd ed.). Chapman & Hall, London.

[4] Cook, J. R. and Stefanski, L. A. (1994). Simulation-extrapolation estimation in parametric measurement errors models. Journal of the American Statistical Association 89, 1314-1328.

[5] Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Society, Series B 34, 187-220.

[6] Cox, D. R. (1975). Partial likelihood. Biometrika 62, 269-276.

[7] Fuller, W. A. (1987) Measurement Error Models. New York: John Wiley & Sons.

[8] Hammer, S. M., Katezstein, D. A., Hughes, M. D., Gundaker, H., Schooley, R. T., Haubrich, R. H., Henry, W. K., Lederman, M. M., Phair, J. P., Niu, M., Hirsch, M. S., and Merigan, T. C., for the AIDS Clinical Group Study 175 Study Team. (1996). A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter. New England Journal of Medicine 335, 1081-1089.

[9] Huang, Y. H., Hwang, W. H., and Chen, F. Y. (2011). Differential measurement errors in zero-truncated regression models for count data. Biometrics 62, 1037-1043.

[10] Hwang, W. H., Huang, S. Y. H., and Wang, C. Y. (2007). Effects of measurement error and conditional score estimation in capture recapture models. Statistics Sinica 17, 301-316.

[11] Imai, K. and Yamamoto, T. (2010). Causal inference with differential measurement error: Nonparametric identification and sensitivity analysis. American Journal of Political Science 54, 543-560.

[12] Jones D. Y., Schatzkin A., Green, S. B., Block G., Brinton L. A., Ziegler R. G., Hoover R., and Taylor P. R. (1987) Dietary fat and breast cancer in the National Health and Nutrition Examination Survey I: Epidemiologic Follow-up Study. Journal of the National Cancer Institute 79, 465-471.

[13] Laird, N. M. and Ware, J. H. (1982) Random effects models for longitudinal data. Biometrics 38, 963-947.

[14] Long, S. J. (1997) Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, California: Sage Publications.

[15] Ma, Y. and Tsiatis, A. A. (2006) Closed form semiparametric estimators for measurement error models. Statistica Sinica 16, 183-193.

[16] McCullagh, P., and Nelder, J. A. (1989) Generalized Linear Models} (2nd ed.). London: Chapman & Hall.

[17] Nakamura, T. (1992). Proportional hazards model with covariates subject to measurement error. Biometrics 48, 829-838.

[18] Stefanski, L. A. (1985) The effects of measurement error in parameter estimation. Biometrika 72, 385-389.

[19] Stefanski, L. A. and Carroll, R. J. (1985) Covariate measurement error in logistic regression. Annals of Statistics 13, 1335-1351.

[20] Stefanski, L. A. and Carroll, R. J. (1987). Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika 74, 703-716.

[21] Stefanski, L. A. (1989). Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. Communications in Statistics, Series A 18, 4335-4358.

[22] Terza, J. V. (1985) A Tobit type estimator for the censored Poisson regression model. Economics Letters 18, 361-365.

[23] Tosteson T., Stefanski L. A., and Schafer D.W. (1989) A measurement error model for binary and ordinal regression. Statistics in Medicine 8, 1139-1147.

[24] Tsiatis, A. A. and Davidian, M. (2001). A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika 88, 447-458.

[25] Wang, C. Y. (2006). Corrected score estimator for joint modeling of longitudinal and failure time data. Statistica Sinica 16, 235-253.

[26] Wu M. L., Whittemore, A. S., and Jung, D. L. (1986) Errors in reported dietary intakes. American Statistical Association 97, 955-964.

[27] Xiao Song, Marie Davidian, and Anastasios A. Tsiatis (2002). A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data. Biometrics 58, 742-753.