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題名 變數遺漏值的多重插補應用於條件評估法
Multiple imputation for missing covariates in contingent valua-tion survey
作者 費詩元
Fei, Shih Yuan
貢獻者 江振東
Chiang, Jeng Tung
費詩元
Fei, Shih Yuan
關鍵詞 遺漏
多重插補法
願付價格
Missing Value
multiple imputation
WTP
日期 2008
上傳時間 2009-09-14
摘要 多數關於願付價格(WTP)之研究中,遺漏資料通常被視為完全隨機遺漏(MCAR)並刪除之。然而,研究中的某些重要變數若具有過高的遺漏比例時,則可能造成分析上的偏誤。
     收入在許多條件評估(Contingent Valuation)調查中經常扮演著一個重要的角色,同時其也是受訪者最傾向於遺漏的變項之一。在這份研究中,我們將透過模擬的方式來評估多重插補法(Multiple Imputa- tion) 於插補願付價格調查中之遺漏收入之表現。我們考慮三種資料情況:刪除遺漏資料後所剩餘之完整資料、一次插補資料、以及多重插補資料,針對這三種情況,藉由三要素混合模型(Three-Component Mixture Model)所進行之分析來評估其優劣。模擬結果顯示,多重插補法之分析結果優於僅利用刪除遺漏資料所剩餘之完整資料進行分析之結果,並且隨著遺漏比例上升,其優劣更是明顯。我們也發現多重插補法之結果也比起一次插補來的更加可靠、穩定。因此如果資料遺漏機制非完全隨機遺漏之機制時,我們認為多重插補法是一個值得信任且表現不錯的處理方法。
     此外,文中也透過「竹東及朴子地區心臟血管疾病長期追蹤研究」(Cardio Vascular Disease risk FACtor Two-township Study,簡稱CVDFACTS) 之資料來進行實證分析。文中示範一些評估遺漏機制的技巧,包括比較存活曲線以及邏輯斯迴歸。透過實證分析,我們發現插補前後的確造成模型分析及估計上的差異。
Most often, studies focus on willingness to pay (WTP) simply ignore the missing values and treat them as if they were missing completely at random. It is well-known that such a practice might cause serious bias and lead to incorrect results.
     Income is one of the most influential variables in CV (contingent valuation) study and is also the variable that respondents most likely fail to respond. In the present study, we evaluate the performance of multiple imputation (MI) on missing income in the analysis of WTP through a series of simulation experiments. Several approaches such as complete-case analysis, single imputation, and MI are considered and com-pared. We show that performance with MI is always better than complete-case analy-sis, especially when the missing rate gets high. We also show that MI is more stable and reliable than single imputation.
     As an illustration, we use data from Cardio Vascular Disease risk FACtor Two-township Study (CVDFACTS). We demonstrate how to determine the missing mechanism through comparing the survival curves and a logistic regression model fitting. Based on the empirical study, we find that discarding cases with missing in-come can lead to something different from that with multiple imputation. If the dis-carded cases are not missing complete at random, the remaining samples will be biased. That can be a serious problem in CV research. To conclude, MI is a useful method to deal with missing value problems and it should be worthwhile to give it a try in CV studies.
參考文獻 Alberini, A. (1995b) “Optimal Designs for Discrete Choice Contingent Valuation Surveys: Single-bound, Double-bound and Bivariate Models,” Journal of Environmental Economics and Management, 28:287-306.
Barbera, A.J., and McConnell, V.D. (1990) “The Impact of Environmental Regulations on Industry Productivity: Direct and Indirect Effects,” Journal of Environmental Economics and Management, 18:50-65.
Bishop, R.C., and Heberlein, T.A. (1979) “Measuring Values of Extramarket Goods: Are Indirect Measure Biased?” American Journal of Agricultural Economics, 61:926-930.
Buuren, S.V., Boshuizen, H.C., and Knook, D.L. (1999) “Multiple Imputation of Missing Blood Pressure Covariates In Survival Analysis,” Statistics In Medicine 18:681-694.
Cameron, T.A., (1988) “A New Paradigm for Valuing Non-market Goods Using Re-ferendum Data: Maximum Likelihood Estimation by Censored Logistic Regression,” Journal of Environmental Economics and Management 15, 355-379.
Cameron, T.A., and James, M.D. (1987) “Estimating Willingness to Pay from Survey Data: An Alternative Pre-Test-Market Evaluation Procedure,” Journal of Marketing Research XXIV: 389-395.
Cameron, T.A., and James, M.D. (1987) “Efficient Estimation Methods for Closed-Ended Contingent Valuation Surveys,” The Review of Economics and Statistics, 69:269-276.
Carson, R.T. (1985) “Three Essays on Contingent Valuation,” Ph.D thesis, University of Califomia, Berkeley.
Carson, R.T., Hanemann, M.W. and Steinberg, D. (1990) “A Discrete Choice Contingent Valuation Estimate of the Value of Kenai King Salmon,” Journal of Behavioral Economics, Elsevier, 19:53-68.
Carson, R.T., and Navarro, P. (1988) “A Seller`s (and Buyer`s) Guide to the Job Market for Beginning Academic Economists,” Journal of Economic Perspectives, American Economic Association, 2:137-148.
Diggle, P.J., Liang, K.Y. and Zeger, S.L. (1994), “Analysis of Longitudinal Data,” Oxford University Press.
Farewell, V. T., and Prentice, R. L (1977) “A Study of Distributional Shape in Life Testing,” Technometrics, 19:69-75.
Giorgi, R., Belot, A., Gaudart, J., and Launoy, G. (2008) “The Performance of Multiple Imputation for Missing Covariate Data within the Context of Regression Relative Survival Analysis,” Statistics In Medicine, 27: 6310–6331.
Hanemann, M.W. (1984) “Welfare Evaluation in Contingent Valuation Experiments with Discrete Responses,” American Journal of Agricultural Economics, 66:332-341.
Hanemann, M.W. (1985) “Some Issues in Continuous- and Discrete-Response Contingent Valuation Studies,” Northeast Journal of Agricultural Economics, 5-13.
Hanemann, M.W., Loomis, J., and Kanninen, B. (1991) “Statistical Efficiency of Double- Bounded Dichotomous Choice Contingent Valuation,” American Journal of Agricultural Economics, 73:1255-1263.
Hanemann, M.W., and Kanninen, B. (1998) “The Statistical Analysis of Dis-crete-Response CV Data,” in Bateman, I.J. and Willis, K.G. Valuing Environmental Preferences, Oxford University Press.
Herriges, J.A., and Shogren, J.F. (1996) “Starting Point Bias in Dichotomous Choice Valuation with Follow-Up Questioning,” Journal of Environmental Economics and Management, Elsevier, 30:112-131.
Hsu, C-H, Taylor, J., and Murray, S. (2007) “Multiple Imputation for Interval Censored Data with Auxiliary Variables,” Statistics In Medicine, 26:769-781.
Klein, J. P., and Moeschberger, M. L. (1997) “Survival Analysis: Techniques for Censored and Truncated Data,” Springer Press.
Lawless, J. F. (2003) “Statistical Models and Methods for Lifetime Data,” New Jersey John Wiley and Sons.
Little, R.A. (1988) “A Test of Missing Completely at Random for Generalised Estimating Equations with Missing Data,” Biometrika, 86:1-13.
Little, R.A., and Rubin, D.B. (1989) “The Analysis of Social Science Data with Missing Values,” Sociological Methods and Research, 18:292-326.
McConnell, K. E. (1990) “Models for Referendum Data: The Structure of Discrete Choice Models for Contingent Valuation,” Journal of Environmental Economics and Management, 18:19-35.
McFadden, D. (1974) “Conditional Logit Analysis of Qualitative Choice Behavior,” in P. Zarembka, ed., Frontiers in Econometrics, New York: Academic Press, 105-142.
Odejar, M., Ryan, M., and Mavromaras, K. (2004) “Messy Data Modelling in Health Care Contingent Valuation Studies,” Econometric Society 2004 North American Summer Meetings, 406.
Onozaka, Y. (2002) “Evaluating Alternative Methods of Dealing with Missing Obser-vations - An Economic Application,” Paper for the annual meeting of the American Agricultural Economics Association.
Paik, M.C. (1997) “Multiple Imputation for the Cox Proportional Hazards Model with Missing Covariates,” Lifetime Data Analysis, 3:289–298.
Rubin, D.B. (1976) “Inference and Missing Data,” Biometrika, 63:581-592.
Rubin, D.B., and Schenker, N. (1986) “Multiple Imputation for Interval Estimation from Simple Random Samples with Ignorable Nonresponse,” Journal of the American Statistical Association, 81:366-374.
Rubin, D.B. (1987) “Multiple Imputation for Nonresponse in Surveys,” New York Wiley.
Schafer, J.L. (1997) “Analysis of Incomplete Multivariate Data,” Chapman and Hall, London.
Schenker, N., Raghinathan, T.E., Chiu, P-L., Makuc, D.M., Zhang, G., and Cohen, A.J. (2006) “Multiple Imputation of Missing Income Data,” Journal of the American Statistical Association, 101:924-933.
Sellar, C., Stoll, J.R., and Chavas, J. P. (1985) “Validation of Empirical Measures of Welfare Change: A Comparison of Nonmarket Techniques”, Land Economy, 61:156-175.
Sinharay, S., Stern, S., and Russell, D. (2001) “The Use of Multiple Imputation for the Analysis of Missing Data,” Psychological Methods, 6:317-329.
Sperling, D., Setiawan, W., and Hungerford, D. (1995) “The Target Market for Methanol Fuel,” Transportation Research, 29A: 33-45.
Tsai, I-L. ,(2005), “A Three-component Mixture Model in Willingness-to-pay Analysis for General Interval Censored Data,” unpublished master’s thesis, National Chengchi University, Taiwan.
Turnbull, B.W. (1976), “The Empirical Distribution Function with Arbitrarily Grouped Censored and Truncated Data,” Journal of the Royal Statistical Society, Series B, 38:290-295.
Whitehead, J.C. (1994) “Item Nonresponse in Contingent Valuation: Should CV Re-searchers Impute Values for Missing Independent Variables?” Journal of Leisure Research, 22.
Yamaguchi, K. (1992) “Accelerated Failure-Time Regression Models with a Regression Model of Surviving Fraction: An Application to the Analysis of ‘Permanent Em-ployment’ in Japan,” Journal of the American Statistical Association, 87:284-292.
描述 碩士
國立政治大學
統計研究所
96354010
97
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0096354010
資料類型 thesis
dc.contributor.advisor 江振東zh_TW
dc.contributor.advisor Chiang, Jeng Tungen_US
dc.contributor.author (Authors) 費詩元zh_TW
dc.contributor.author (Authors) Fei, Shih Yuanen_US
dc.creator (作者) 費詩元zh_TW
dc.creator (作者) Fei, Shih Yuanen_US
dc.date (日期) 2008en_US
dc.date.accessioned 2009-09-14-
dc.date.available 2009-09-14-
dc.date.issued (上傳時間) 2009-09-14-
dc.identifier (Other Identifiers) G0096354010en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/30927-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 96354010zh_TW
dc.description (描述) 97zh_TW
dc.description.abstract (摘要) 多數關於願付價格(WTP)之研究中,遺漏資料通常被視為完全隨機遺漏(MCAR)並刪除之。然而,研究中的某些重要變數若具有過高的遺漏比例時,則可能造成分析上的偏誤。
     收入在許多條件評估(Contingent Valuation)調查中經常扮演著一個重要的角色,同時其也是受訪者最傾向於遺漏的變項之一。在這份研究中,我們將透過模擬的方式來評估多重插補法(Multiple Imputa- tion) 於插補願付價格調查中之遺漏收入之表現。我們考慮三種資料情況:刪除遺漏資料後所剩餘之完整資料、一次插補資料、以及多重插補資料,針對這三種情況,藉由三要素混合模型(Three-Component Mixture Model)所進行之分析來評估其優劣。模擬結果顯示,多重插補法之分析結果優於僅利用刪除遺漏資料所剩餘之完整資料進行分析之結果,並且隨著遺漏比例上升,其優劣更是明顯。我們也發現多重插補法之結果也比起一次插補來的更加可靠、穩定。因此如果資料遺漏機制非完全隨機遺漏之機制時,我們認為多重插補法是一個值得信任且表現不錯的處理方法。
     此外,文中也透過「竹東及朴子地區心臟血管疾病長期追蹤研究」(Cardio Vascular Disease risk FACtor Two-township Study,簡稱CVDFACTS) 之資料來進行實證分析。文中示範一些評估遺漏機制的技巧,包括比較存活曲線以及邏輯斯迴歸。透過實證分析,我們發現插補前後的確造成模型分析及估計上的差異。
zh_TW
dc.description.abstract (摘要) Most often, studies focus on willingness to pay (WTP) simply ignore the missing values and treat them as if they were missing completely at random. It is well-known that such a practice might cause serious bias and lead to incorrect results.
     Income is one of the most influential variables in CV (contingent valuation) study and is also the variable that respondents most likely fail to respond. In the present study, we evaluate the performance of multiple imputation (MI) on missing income in the analysis of WTP through a series of simulation experiments. Several approaches such as complete-case analysis, single imputation, and MI are considered and com-pared. We show that performance with MI is always better than complete-case analy-sis, especially when the missing rate gets high. We also show that MI is more stable and reliable than single imputation.
     As an illustration, we use data from Cardio Vascular Disease risk FACtor Two-township Study (CVDFACTS). We demonstrate how to determine the missing mechanism through comparing the survival curves and a logistic regression model fitting. Based on the empirical study, we find that discarding cases with missing in-come can lead to something different from that with multiple imputation. If the dis-carded cases are not missing complete at random, the remaining samples will be biased. That can be a serious problem in CV research. To conclude, MI is a useful method to deal with missing value problems and it should be worthwhile to give it a try in CV studies.
en_US
dc.description.tableofcontents 1 Introduction 1
     2 Literature Review 3
     2.1 Double Bounded Dichotomous Choice Elicitation Method 3
     2.2 Three-Component Mixture Model 6
     2.3 Dealing Missing Covariate with WTP 6
     2.4 Missing Data Mechanism 7
     2.5 Multiple Imputation Method 8
     3 Preliminary Data Analysis 12
     4 Model Specifications 21
     4.1 Three-Component Mixture Model 21
     4.2 Accelerated Failure Time Model 24
     5 Simulation Studies 26
     5.1 Data Generation 26
     5.2 Missing Value Generation 30
     5.3 Evaluation 31
     5.4 Results 31
     6 Empirical Results 36
     6.1 Variable Selection 36
     6.2 Imputed-Data Analysis 38
     6.3 Model Selection and Estimation 40
     6.4 Complete-Case Analysis 44
     6.5 Evaluation on Performance of Multiple Imputation 45
     6.6 An Alternative Method to Evaluate Multiple Imputation 48
     6.7 Estimation of Mean and Median after MI 51
     6.8 Stability of MI 53
     6.9 Simulation for Verification 56
     7 Conclusion 61
     Reference 63
     Appendix 67
zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0096354010en_US
dc.subject (關鍵詞) 遺漏zh_TW
dc.subject (關鍵詞) 多重插補法zh_TW
dc.subject (關鍵詞) 願付價格zh_TW
dc.subject (關鍵詞) Missing Valueen_US
dc.subject (關鍵詞) multiple imputationen_US
dc.subject (關鍵詞) WTPen_US
dc.title (題名) 變數遺漏值的多重插補應用於條件評估法zh_TW
dc.title (題名) Multiple imputation for missing covariates in contingent valua-tion surveyen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) Alberini, A. (1995b) “Optimal Designs for Discrete Choice Contingent Valuation Surveys: Single-bound, Double-bound and Bivariate Models,” Journal of Environmental Economics and Management, 28:287-306.zh_TW
dc.relation.reference (參考文獻) Barbera, A.J., and McConnell, V.D. (1990) “The Impact of Environmental Regulations on Industry Productivity: Direct and Indirect Effects,” Journal of Environmental Economics and Management, 18:50-65.zh_TW
dc.relation.reference (參考文獻) Bishop, R.C., and Heberlein, T.A. (1979) “Measuring Values of Extramarket Goods: Are Indirect Measure Biased?” American Journal of Agricultural Economics, 61:926-930.zh_TW
dc.relation.reference (參考文獻) Buuren, S.V., Boshuizen, H.C., and Knook, D.L. (1999) “Multiple Imputation of Missing Blood Pressure Covariates In Survival Analysis,” Statistics In Medicine 18:681-694.zh_TW
dc.relation.reference (參考文獻) Cameron, T.A., (1988) “A New Paradigm for Valuing Non-market Goods Using Re-ferendum Data: Maximum Likelihood Estimation by Censored Logistic Regression,” Journal of Environmental Economics and Management 15, 355-379.zh_TW
dc.relation.reference (參考文獻) Cameron, T.A., and James, M.D. (1987) “Estimating Willingness to Pay from Survey Data: An Alternative Pre-Test-Market Evaluation Procedure,” Journal of Marketing Research XXIV: 389-395.zh_TW
dc.relation.reference (參考文獻) Cameron, T.A., and James, M.D. (1987) “Efficient Estimation Methods for Closed-Ended Contingent Valuation Surveys,” The Review of Economics and Statistics, 69:269-276.zh_TW
dc.relation.reference (參考文獻) Carson, R.T. (1985) “Three Essays on Contingent Valuation,” Ph.D thesis, University of Califomia, Berkeley.zh_TW
dc.relation.reference (參考文獻) Carson, R.T., Hanemann, M.W. and Steinberg, D. (1990) “A Discrete Choice Contingent Valuation Estimate of the Value of Kenai King Salmon,” Journal of Behavioral Economics, Elsevier, 19:53-68.zh_TW
dc.relation.reference (參考文獻) Carson, R.T., and Navarro, P. (1988) “A Seller`s (and Buyer`s) Guide to the Job Market for Beginning Academic Economists,” Journal of Economic Perspectives, American Economic Association, 2:137-148.zh_TW
dc.relation.reference (參考文獻) Diggle, P.J., Liang, K.Y. and Zeger, S.L. (1994), “Analysis of Longitudinal Data,” Oxford University Press.zh_TW
dc.relation.reference (參考文獻) Farewell, V. T., and Prentice, R. L (1977) “A Study of Distributional Shape in Life Testing,” Technometrics, 19:69-75.zh_TW
dc.relation.reference (參考文獻) Giorgi, R., Belot, A., Gaudart, J., and Launoy, G. (2008) “The Performance of Multiple Imputation for Missing Covariate Data within the Context of Regression Relative Survival Analysis,” Statistics In Medicine, 27: 6310–6331.zh_TW
dc.relation.reference (參考文獻) Hanemann, M.W. (1984) “Welfare Evaluation in Contingent Valuation Experiments with Discrete Responses,” American Journal of Agricultural Economics, 66:332-341.zh_TW
dc.relation.reference (參考文獻) Hanemann, M.W. (1985) “Some Issues in Continuous- and Discrete-Response Contingent Valuation Studies,” Northeast Journal of Agricultural Economics, 5-13.zh_TW
dc.relation.reference (參考文獻) Hanemann, M.W., Loomis, J., and Kanninen, B. (1991) “Statistical Efficiency of Double- Bounded Dichotomous Choice Contingent Valuation,” American Journal of Agricultural Economics, 73:1255-1263.zh_TW
dc.relation.reference (參考文獻) Hanemann, M.W., and Kanninen, B. (1998) “The Statistical Analysis of Dis-crete-Response CV Data,” in Bateman, I.J. and Willis, K.G. Valuing Environmental Preferences, Oxford University Press.zh_TW
dc.relation.reference (參考文獻) Herriges, J.A., and Shogren, J.F. (1996) “Starting Point Bias in Dichotomous Choice Valuation with Follow-Up Questioning,” Journal of Environmental Economics and Management, Elsevier, 30:112-131.zh_TW
dc.relation.reference (參考文獻) Hsu, C-H, Taylor, J., and Murray, S. (2007) “Multiple Imputation for Interval Censored Data with Auxiliary Variables,” Statistics In Medicine, 26:769-781.zh_TW
dc.relation.reference (參考文獻) Klein, J. P., and Moeschberger, M. L. (1997) “Survival Analysis: Techniques for Censored and Truncated Data,” Springer Press.zh_TW
dc.relation.reference (參考文獻) Lawless, J. F. (2003) “Statistical Models and Methods for Lifetime Data,” New Jersey John Wiley and Sons.zh_TW
dc.relation.reference (參考文獻) Little, R.A. (1988) “A Test of Missing Completely at Random for Generalised Estimating Equations with Missing Data,” Biometrika, 86:1-13.zh_TW
dc.relation.reference (參考文獻) Little, R.A., and Rubin, D.B. (1989) “The Analysis of Social Science Data with Missing Values,” Sociological Methods and Research, 18:292-326.zh_TW
dc.relation.reference (參考文獻) McConnell, K. E. (1990) “Models for Referendum Data: The Structure of Discrete Choice Models for Contingent Valuation,” Journal of Environmental Economics and Management, 18:19-35.zh_TW
dc.relation.reference (參考文獻) McFadden, D. (1974) “Conditional Logit Analysis of Qualitative Choice Behavior,” in P. Zarembka, ed., Frontiers in Econometrics, New York: Academic Press, 105-142.zh_TW
dc.relation.reference (參考文獻) Odejar, M., Ryan, M., and Mavromaras, K. (2004) “Messy Data Modelling in Health Care Contingent Valuation Studies,” Econometric Society 2004 North American Summer Meetings, 406.zh_TW
dc.relation.reference (參考文獻) Onozaka, Y. (2002) “Evaluating Alternative Methods of Dealing with Missing Obser-vations - An Economic Application,” Paper for the annual meeting of the American Agricultural Economics Association.zh_TW
dc.relation.reference (參考文獻) Paik, M.C. (1997) “Multiple Imputation for the Cox Proportional Hazards Model with Missing Covariates,” Lifetime Data Analysis, 3:289–298.zh_TW
dc.relation.reference (參考文獻) Rubin, D.B. (1976) “Inference and Missing Data,” Biometrika, 63:581-592.zh_TW
dc.relation.reference (參考文獻) Rubin, D.B., and Schenker, N. (1986) “Multiple Imputation for Interval Estimation from Simple Random Samples with Ignorable Nonresponse,” Journal of the American Statistical Association, 81:366-374.zh_TW
dc.relation.reference (參考文獻) Rubin, D.B. (1987) “Multiple Imputation for Nonresponse in Surveys,” New York Wiley.zh_TW
dc.relation.reference (參考文獻) Schafer, J.L. (1997) “Analysis of Incomplete Multivariate Data,” Chapman and Hall, London.zh_TW
dc.relation.reference (參考文獻) Schenker, N., Raghinathan, T.E., Chiu, P-L., Makuc, D.M., Zhang, G., and Cohen, A.J. (2006) “Multiple Imputation of Missing Income Data,” Journal of the American Statistical Association, 101:924-933.zh_TW
dc.relation.reference (參考文獻) Sellar, C., Stoll, J.R., and Chavas, J. P. (1985) “Validation of Empirical Measures of Welfare Change: A Comparison of Nonmarket Techniques”, Land Economy, 61:156-175.zh_TW
dc.relation.reference (參考文獻) Sinharay, S., Stern, S., and Russell, D. (2001) “The Use of Multiple Imputation for the Analysis of Missing Data,” Psychological Methods, 6:317-329.zh_TW
dc.relation.reference (參考文獻) Sperling, D., Setiawan, W., and Hungerford, D. (1995) “The Target Market for Methanol Fuel,” Transportation Research, 29A: 33-45.zh_TW
dc.relation.reference (參考文獻) Tsai, I-L. ,(2005), “A Three-component Mixture Model in Willingness-to-pay Analysis for General Interval Censored Data,” unpublished master’s thesis, National Chengchi University, Taiwan.zh_TW
dc.relation.reference (參考文獻) Turnbull, B.W. (1976), “The Empirical Distribution Function with Arbitrarily Grouped Censored and Truncated Data,” Journal of the Royal Statistical Society, Series B, 38:290-295.zh_TW
dc.relation.reference (參考文獻) Whitehead, J.C. (1994) “Item Nonresponse in Contingent Valuation: Should CV Re-searchers Impute Values for Missing Independent Variables?” Journal of Leisure Research, 22.zh_TW
dc.relation.reference (參考文獻) Yamaguchi, K. (1992) “Accelerated Failure-Time Regression Models with a Regression Model of Surviving Fraction: An Application to the Analysis of ‘Permanent Em-ployment’ in Japan,” Journal of the American Statistical Association, 87:284-292.zh_TW