可交换的两值数据的统计分析
|
摘要
在很多应用中，我们可以合理的认为来自于同一个团(cluster)的数据是可交换的。实际上，在一些团型抽样调查(cluster sample surveys)中、畸形学试验(teratological experiments)中、眼科及耳鼻喉科研究(ophthalmological and oto-laryngologic studies)中和其它的临床试验中，经常出现可交换的两值数据。本文的主要目的就是给出可交换的两值数据的统计分析的新方法。
为了解决可交换的两值数据中的统计问题，我们先给出了一类最优无偏估计方程和关于一般估计方程的解的相合性和渐近正态性的一些理论结果，并且进一步给出了当估计方程中出现缺失数据时，利用一种迭代算法来解估计方程。我们把这种算法称作AU算法。由于AU算法在以下两步中迭代：在A步中，利用完全数据函数的条件期望的近似值来代替完全数据的函数，在U步中，用一个显示表达式来更新待估参数。所以此算法使得计算带有缺失数据的估计方程的求解具有了可行性和简单性。关于此算法的收敛性和由此算法得到的估计方程解的大样本性质，我们也给了一些理论结果。而且我们还给出了AU算法中关于A步中条件期望的一种的近似形式。
基于我们得到的有关估计方程的这些理论结果，我们利用成对似然的方法(pairwise likelihood procedure)给出了一组近似最优无偏估计方程来计算带有随机簇(random cluster sizes)的可交换的两值数据的均值参数和相关参数。然后把这一方法应用到发育毒性数据分析中。通过模拟，我们得到了对于可交换的两值数据来说，我们的这一方法比GEE方法产生的结果好。
当可交换的两值数据中出现缺失数据时，我们把成对似然的方法和AU算法结合起来。我们把这一方法应用到来自于繁殖毒性研究中的一组数据。通过模拟结果，也可看出我们的这一方法是有效的，而且此方法产生的结果也比忽略了带有缺失数据的个体的完全情形分析的结果好。
In many applications, it is reasonable to expect exchangeability to model the datafrom the same cluster. In fact, exchangeable binary data are commonly encounteredin cluster sample surveys, teratological experiments, ophthalmological and otolaryngologicstudies, and other clinical trials. This dissertation aims to propose some newstatistical methods for exchangeable binary data analysis.
In order to solve the problem, we first present a general form of optimal unbiasedestimating equations and some results for the consistency and the asymptotic normalityof the solutions to general estimating equations. Then we present an iterative algorithmfor solving estimating equations in the presence of missing data. The algorithm iscalled the AU algorithm. It makes the computation very feasible and simple because ititerates between two steps: an A-step, in which the functions of the complete data arereplaced by the approximate values of their conditional expectations, and a U-step inwhich the parameters are updated using a closed-form expression. Theoretical resultsare obtained establishing convergence properties of the algorithm and the large sampleproperties of the estimators produced by the algorithm. Moreover, we give a generalform of the approximations of the conditional expectations in the A-step of the AUalgorithm.
Based on the theories of estimating equations we have obtained, we use a pairwiselikelihood procedure to give a set of approximately optimal unbiased estimatingequations for estimating the mean and variance parameters for the exchangeable binarydata with random cluster sizes. An application to developmental toxicity data analysisis given. Simulation results show that our procedure performs better than the GEEprocedure for the exchangeable binary data.
When the exchangeable binary data involve missing data, we combine the pairwiselikelihood procedure with the AU algorithm. An application is made to a data setfrom a reproductive toxicity study. Simulation results show that our method is valid and performs better than the complete-case analysis which ignores the subjects withmissing data.
为了解决可交换的两值数据中的统计问题，我们先给出了一类最优无偏估计方程和关于一般估计方程的解的相合性和渐近正态性的一些理论结果，并且进一步给出了当估计方程中出现缺失数据时，利用一种迭代算法来解估计方程。我们把这种算法称作AU算法。由于AU算法在以下两步中迭代：在A步中，利用完全数据函数的条件期望的近似值来代替完全数据的函数，在U步中，用一个显示表达式来更新待估参数。所以此算法使得计算带有缺失数据的估计方程的求解具有了可行性和简单性。关于此算法的收敛性和由此算法得到的估计方程解的大样本性质，我们也给了一些理论结果。而且我们还给出了AU算法中关于A步中条件期望的一种的近似形式。
基于我们得到的有关估计方程的这些理论结果，我们利用成对似然的方法(pairwise likelihood procedure)给出了一组近似最优无偏估计方程来计算带有随机簇(random cluster sizes)的可交换的两值数据的均值参数和相关参数。然后把这一方法应用到发育毒性数据分析中。通过模拟，我们得到了对于可交换的两值数据来说，我们的这一方法比GEE方法产生的结果好。
当可交换的两值数据中出现缺失数据时，我们把成对似然的方法和AU算法结合起来。我们把这一方法应用到来自于繁殖毒性研究中的一组数据。通过模拟结果，也可看出我们的这一方法是有效的，而且此方法产生的结果也比忽略了带有缺失数据的个体的完全情形分析的结果好。
In many applications, it is reasonable to expect exchangeability to model the datafrom the same cluster. In fact, exchangeable binary data are commonly encounteredin cluster sample surveys, teratological experiments, ophthalmological and otolaryngologicstudies, and other clinical trials. This dissertation aims to propose some newstatistical methods for exchangeable binary data analysis.
In order to solve the problem, we first present a general form of optimal unbiasedestimating equations and some results for the consistency and the asymptotic normalityof the solutions to general estimating equations. Then we present an iterative algorithmfor solving estimating equations in the presence of missing data. The algorithm iscalled the AU algorithm. It makes the computation very feasible and simple because ititerates between two steps: an A-step, in which the functions of the complete data arereplaced by the approximate values of their conditional expectations, and a U-step inwhich the parameters are updated using a closed-form expression. Theoretical resultsare obtained establishing convergence properties of the algorithm and the large sampleproperties of the estimators produced by the algorithm. Moreover, we give a generalform of the approximations of the conditional expectations in the A-step of the AUalgorithm.
Based on the theories of estimating equations we have obtained, we use a pairwiselikelihood procedure to give a set of approximately optimal unbiased estimatingequations for estimating the mean and variance parameters for the exchangeable binarydata with random cluster sizes. An application to developmental toxicity data analysisis given. Simulation results show that our procedure performs better than the GEEprocedure for the exchangeable binary data.
When the exchangeable binary data involve missing data, we combine the pairwiselikelihood procedure with the AU algorithm. An application is made to a data setfrom a reproductive toxicity study. Simulation results show that our method is valid and performs better than the complete-case analysis which ignores the subjects withmissing data.
引文
[1] Salsburg D. Estimating dose-response for toxic endpoints[C]. In: Morgan B J T, ed. Statistics in Toxicology. Oxford: Oxford University Press, 1996.
[2] Ryan L M. Quantitative risk assessment for developmental toxicity[J]. Biometrics, 1992, 48: 163-174.
[3] Ryan L M. Using historical controls in the analysis of developmental toxicity data[J]. Biometrics, 1992,49: 1126-1135.
[4] Lefkopoulou M, Ryan L M. Global tests for multiple binary outcomes[J]. Biometrics, 1993, 49: 975-988
[5] Liang K-Y, McCullagh P. Case studies in binary dispersion[J]. Biometrics, 1993, 49: 62-630.
[6] Bahadur R R. A representation of the joint distribution of responses to n dichotomous items[C]. In: Solomon H, ed. Studies on Item Analysis and Predication. Stanford, CA: Stanford University Press, 1961. 158-168.
[7] Cox D R. The analysis of multivariate binary data[J]. Applied Statistics, 1972, 21: 113-120.
[8] Dale J R. Global cross-ratio models for bivariate, discrete, ordered response[J]. Biomet- rics, 1986,42: 909-917.
[9] Zhu Y, Fung K Y. Statistical methods in developmental toxicity risk assessment[C]. In: Fan A, Chang L W, eds. Toxicology and Risk Assessment, Principles, Methods, and Applications. New York: Marcel Dekker, 1996. 413-446.
[10] Molenberghs G, Ryan L M. Likelihood inference for clustered multivariate binary data[J]. Environmetrics, 1999, 10: 279-230.
[11] Molenberghs G., Geys H., Declerck L., et al. Analysis of clustered multivariate data from developmental toxicity studies[C]. In: Payne R, Green P, eds. Proceedings in Computa- tional Statistics. Heidelberg: Physica Verlag, 1998. 3-14
[12] Le Cessie, Van Houwelingen J C. Logistic regression for correlated binary data[J]. Ap- plied Statistics, 1994,43: 95-108.
[13] Geys H, Molenbergh G, Lipsitz S R. A note on the comparison of pseudo-likelihood and generalized estimating equations for marginal odds ratio models[J]. Journal of Statistical Computation and Simulation, 1998, 62: 45-72.
[14] Glonek G F V, McCullagh P. Multivariate logistic models[J]. Journal of the Royal Statistical Society, Series B, 1995, 57: 533-546.
[15] Molenberghs G, Lesaffre E. Marginal modelling of correlated ordinal data using a multivariate Plackett distribution[J]. Journal of the American Statistical Association, 1994, 89: 633-644.
[16] Lang J B, Agresti A. Simultaneously modeling joint and marginal distributions of multivariate categorical responses[J]. Journal of the American Statistical Association, 1994,89: 625-632.
[17] Williams D A. The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity[J]. Biometrics, 1975, 31: 949-952.
[18] Breslow N E, Day N E. Statistical Methods in Cancer Research[M]. Oxford: Oxford University Press, 1987. Volume 2.
[19] Carr G J, Portier C J. An evalution of some methods for fitting dose-response models toquantal-response developmental toxicology data[J]. Biometrics, 1993,49: 779-791.
[20] Gaylor D W, Chen J J. Dose-response models for developmental malformations[J]. Teratology, 1993, 47: 291-297.
[21] Rai K, Van Ryzin J. A dose-response model for teratological experiments involving quan-tal experiments[J]. Biometrics, 1985,41: 1-9.
[22] Fitzmaurice G M, Laird N M. A likelihood-based method for analysing longitudinal binary response[J]. Biometrika, 1993, 80: 141-151.
[23] George E O, Bowman D. A full likelihood procedure for analysing exchangeable binary data[J]. Biometrics, 1995, 51(2): 512-523.
[24] Stefanescu C, Turnbull B W. Likelihood inference for exchangeable binary data with varying cluster sizes[J]. Biometrics, 2003, 59: 18-24.
[25] Liang K-Y, Zeger S L. Longitudinal data analysis using generalized linear models[J]. Biometrika, 1986, 73(1): 13-22.
[26] Arnold B C, Strauss, D. Pseudolikelihood estimation: some examples[J]. Sankhya B, 1991, 53: 233-243.
[27] de Finetti B. Funzione caratteristica di un fenomeno aleatorio[J]. Mathematiche e Natu-rali, 1931,4: 251-299.
[28] Draper D, Hodges J S, Mallows C L, et al. Exchangeability and data analysis[J]. Journal of the Royal Statistical Society, Series A, 1993, 156: 9-37.
[29] Mandel E, Bluestone C D, Rockette H E, et al. Duration of effusion after antibiotic treatment for acute otitis media: comparison of cefaclor and amoxicillin[J]. Pediatric Infections Diseases, 1982, 1: 310-316.
[30] Williams D A. Reader reaction: Dose-response models for teratological experiments[J]. Biometrics, 1987,43: 1013-1016.
[31] Kupper L L, Haseman J K. The use of correlated binomial model for the analysis of data from certain toxicological experiments[J]. Biometrics, 1978, 34: 69-76.
[32] Kupper L L, Portier C, Hogan M D, et al. The impact of litter effects on dose-response modeling in teratology [J]. Biometrics, 1986,42: 85-98.
[33] Prentice R L. Binary regression using an extended beta-binomial distribution with discussion of correlation induced by measurements[J]. Journal of the American Statistical Association, 1986,81(394): 321-327.
[34] Chen J J, Kodell R L, Howe R B, et al. Analysis of trinomial responses from reproductive and developmental toxicity experiments[J]. Biometrics, 1991, 47: 1049-1058.
[35] Rosner B. Multivariate methods for clustered binary data with more than one level of nesting[J]. Journal of the American Statistical Association, 1989, 84: 373-380.
[36] Chen J J, Kodell R L. Quantitative risk assessment for teratological effects[J]. Journal of the American Statistical Association, 1989, 84: 966-971.
[37] Paul S R. A clumped beta-binomial model for the analysis of clustered attribute data. Biometrics, 1979, 35: 821-824.
[38] Pack S E. Hypothesis testing for proportion with overdispersion[J]. Biometrics, 1986,42:967-972.
[39] Altham P M E. Two generalizations of the binomial distribution[J]. Applied Statistics, 1978, 27: 162-167.
[40] Catalano P J, Ryan L M. Bivariate latent variable models for clustered discrete and con- tinuous outcomes[J]. Journal of the American Statistical Association, 1992, 87: 651-658.
[41] Zeger S L, Liang K-Y. Longitudinal data analysis for discrete continuous outcomes[J].Biometrics, 1986,42: 121-130.
[42] Lipsitz S R, Ibrahim J G, Zhao L P. A weighted estimating equation for missing covariate data with properties similar to maximum likelihood[J]. Journal of the American Statisti- cal Association, 1999,94(448): 1147-1160.
[43] Prentice R L, Zhao L P. Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses[J]. Biometrics, 1991,47: 825-839.
[44] McCullagh P, Nelder J A. Generalized linear models[M]. New York: Chapman and Hall,1989.
[45] Moore D F. Asymptotic properties of moment estimators for overdispersed counts and proportions[J]. Biometrika, 1986, 73: 583-588.
[46] Willams D A. Estimation of bias using the beta-binomial distribution in teratology[J]. Biometrics, 1988, 44: 305-309.
[47] Williams D A. Extra-binomial variation in logistic linear models[J]. Applied Statistics, 1982, 31: 144-158.
[48] Zhao L P, Prentice R L. Correlated binary regression using a quadratic exponential model[J]. Biometrika, 1990, 77: 642-648.
[49] Bowman D, Chen J J, George E O. Estimating variance functions in developmental toxi- city studies[J]. Biometrics, 1995, 51: 1523-1528.
[50] Lipsitz S R, Laird N M, Harrington D P. Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association[J]. Biometrika, 1991, 78:153-160.
[51] Liang K-Y, Hanfelt J. On the use of the quasi-likelihood method in teratological experi-ments[J]. Biometrics, 1994, 50: 872-880.
[52] Chen J J, Li L. Dose-response modeling of trinomial responses from developmental ex- periments[J]. Statistica Sinica, 1994, 4: 265-274.
[53] Nelder J A, Pregibon D. An extended quasi-likelihood function[J]. Biometrika, 1987, 74:221-232.
[54] Rotnitzky A, Jewell N P. Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data[J]. Biometrika, 1990, 77: 485-497.
[55] Ryan L M. The use of generalized estimating equations for risk assessment in developmental toxicity[J]. Risk Analysis, 1992, 12: 439-447.
[56] Carey V, Zeger S L, Diggle P. Modelling multivariate binary data with alternating logistic regressions[J]. Biometrika, 1993, 80(3): 517-526.
[57] Wedderburn R W M. Quasi-likelihood functions, generalized linear models and the Gaussian method[J]. Biometrika, 1974, 61: 439-447.
[58] Liang K-Y, Zeger S L, Qaqish B. Multivariate regression analyses for categorical data (with discussion)[J]. Journal of the Royal Statistical Society, Series B, 1992,54(1): 3-40.
[59] Cohen B H. Chronic obstructive pulmonary disease: a challenge in genetic epidemiol- ogy[J]. American Journal of Epidemiology, 1980, 112: 274-288.
[60] Boos D D. Analysis of dose-response data in the presence of extrabinomial variation[J]. Applied Statistics, 1993,42: 173-183.
[61] Lipsitz S R, Fitzmaurice G M. Estimating equations for measures of association between repeated binary responses[J]. Biometrics, 1996, 52: 903-912
[62] Zucker D, Wittes J. Testing the effect of treatment in experiments with correlated binary outcomes[J]. Biometrics, 1992,48: 695-710.
[63] Bowman D, George E O. A saturated model for analyzing exchangeable binary data:applications to clinical and developmental toxicity studies[J]. Journal of the Ameri- can Statistical Association, 1995, 90(431): 871 -879.
[64] Xu J L, Prorok P C. Modelling and analysing exchangeable binary data with random cluster sizes[J]. Statistics in Medicine, 2003, 22: 2401-2416.
[65] Stefanescu C, Turnbull B W. Likelihood inference for exchangeable binary data with varying cluster sizes[J]. Biometrics, 2003,59: 18-24.
[66] Aerts M, Geys H, Molenberghs G, Ryan L M. Topics in modelling of clustered data[M]. London: Chapman and Hall, 2002.
[67] Kuk A Y C. A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions[J]. Applied Statistics, 2004, 53: 369-386.
[68] Davidian M, Carroll R J. Variance function estimation[J]. Journal of the American Statistical Association, 1987,82: 1079-1091
[69] Gourieroux C, Monfort A, Trognon A. Pseudo-maximum likelihood methods: theory[J]. Econometrica, 1984, 52: 681-700.
[70] Geys H, Molenberghs G, Ryan L. Pseudo-likelihood modelling of multivariate outcomes in developmental toxicology[J]. Journal of the American Statistical Association, 1999, 94: 734-745.
[71] Nott D J, Rydén T. Pairwise likelihood methods for inference in image models[J]. Biomtrika, 1999, 86(3): 661-676.
[72] Kuk A Y C, Nott D J. A pairwise likelihood approach to analyzing correlated binary data[J]. Statistics & Probability Letters, 2000,47: 329-335.
[73] Elashoff M, Ryan L. An EM algorithm for estimating equations[J]. Journal of Computational and Graphical Statistics, 2004, 13(1): 48-65.
[74] Rosen O, Jiang W X, Tanner M A. Mixtures of marginal models[J]. Biometrika, 2000,87(2): 391-404.
[75] Robins J M, Rotnitzky A, Zhao L P. Estimation of regression coefficient when some regressors are not always observed[J]. Journal of the American Statistical Association, 1994, 89(427): 846-866.
[76] Robins J M, Rotnitzky A, Zhao L P. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data[J]. Journal of the American Statistical.Association, 1995, 90(429): 106-121.
[77] Godambe V, Kale B. Estimating functions: an overview[C]. In: Godambe V, ed. Estimating Functions. Oxford: Clarendon Press, 1991.
[78] Liang K-Y, Zeger S L. Inference based on estimating functions in the presence of nuisance parameters[J]. Statistical Science, 1995, 10: 158-173.
[79] Liang K-Y, Liu X H. Estimating equations in generalized linear models with measure- ment error[C]. In: Godambe V, ed. Estimating Functions. Oxford: Clarendon Press, 1991.47-63.
[80] Godambe V P. An optimum property of regular maximum likelihood estimation[J]. Annals of Mathematical Statistics, 1960, 31: 1208-1212.
[81] Kale B K. An extension of Cramer-Rao inequality for statistical estimation functions[J]. Skandinavisk Aktuarietid, 1962, 45: 60-82.
[82] Kimball B F. Sufficient statistical estimation functions for the parameters of the distribution of maximum values[J]. Annals of Mathematical Statistics, 1946, 17: 299-309.
[83] Wilks S S. Mathematical statistics[M]. New York: Wiley, 1948.
[84]Kale B K.Essential uniqueness of optimal estimating functions[J].Journal of Statistical Planning and Inference,1987,17:405-407.
[85]McLeish D L,Small C G.The theory and applications of statistical inference functions[C].In:Springer Lecture Notes in Statistics NO.44.Newyork:Springer Verlag,1988.
[86]Godambe V P,Heyde C C.Quasi-likelihood and optimal estimation[J].International Statistical Review,1987,55(3):231-244.
[87]Wilks S S.Shortest average confidence intervals from large samples[J].Annals of Mathematical Statistics,1938,9:166-175.
[88]Durbin J.Estimation of parameters in time series regression models[J].Journal of the Royal Statistical Society,Series B,1960,22:139-153.
[89]Ferreira P E.Multiparametric estimating equations[J].Annals of the Institute of Statistical Mathematics,1982,34A:423-431.
[90]Chandrasekar B,Kale B K.Unbiased statistical estimation functions in presence of nuisance parameter[J].Journal of Statistical Planning and Inference,1984,9:45-54.
[91]Godambe V P,Thompson M E.Estimating equations in the presence of nuisance parameters[J],the Annals of Statistics,1974,2:568-571.
[92]Godambe V P.Conditional likelihood and unconditional optimum estimating equations[J].Biometrika,1976,63:277-284.
[93]Subramanyam A,Naik-Nimbalkar U V.Optimal unbiased statistical estimating functions for Hilbert space valued parameters[J].Journal of Statical Planning and Inference,1990,24:95-105.
[94]Heyde C C.Quasi-likelihood and Its Application:A General Approach to Optimal Parameter Estimation[M].New York:Springer,1997.
[95]Hanfelt J J,Liang K-Y.Approximate likelihood ratios for general estimating functions[J].Biometrika,1995,82:461-477.
[96]Godambe V P.The foundations of finite sample estimation in stochastic processes[J].Biometrika,1985,72:419-428.
[97]Cox D R.Partial likelihood[J].Biometrika,1975,62:269-276.
[98] Huber P J. The behavior of maximum likelihood estimates under nonstandard condi- tions[J]. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability[J], 1967, 1: 221-233.
[99] Carroll R J, Ruppert D, Stefanski L A. Measurement Error in Nonlinear ModelsfM]. New York: Chapman and Hall, 1995.
[100] Drum M, McCullagh P, Prentice R L. Comment on "Regrssion models for discrete longitudinal response,"by Fitznaurice, Laird and Rotnitzky[J]. Statistical Science, 1993, 8:300-309.
[101] Newey W K, McFadden D. Estimation in large samples[C]. In: McFadden D, Engler R,eds. Handbook of Econometrics, Vol. 4. Amsterdam: Noth-Holland, 1993.
[102] Manski C F. Analog estimation methods in econometrics[M]. New York: Chapman and Hall, 1988.
[103] Gallant A R, White H. A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models[M]. Oxford: Basil Blackwell, 1988.
[104] Little R J, Rubin D B. Statistical Analysis With Missing Data[M]. New York: John Wiley, 1987.
[105] Dempster A P, Laird N M, Rubin D B. Maximum likelihood estimation from incomplete data via the EM algorithm (with disscussion)[J]. Journal of the Royal Statistical Society, Series B, 1977, 39: 1-38.
[106] McKendrick A G. Applications of mathematics to medical prohlems[J]. Proceedings of the Edinburgh Mathematical Society, 1926,44: 98-130.
[107] Hartley H O. Maximum likelihood estimation from incomplete data[J]. Biometrics, 1958, 14: 174-194.
[108] Baum L E, Petrie T, Soules G, et al. A maximization tehnique occurring in the statistical analysis of probabilistic functions of Markov chains[J]. Annals of Mathematical Statistics, 1970,41: 164-171.
[109] Orchard T, Woodbury M A. A missing information principle: theory and applications[J]. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1972, 1: 697-715.
[110] Sundberg R. Maximum likelihood theory for incomplete data from an exponential family[J]. Scandinavian Journal of Statistics, 1974, 1: 49-58.
[111] Beale E M L, Little R J A. Missing values in multivariate analysis[J]. Journal of the Royal Statistical Society, Series B, 1975, 37: 129-145.
[112] Meilijson I. A fast improvement to the EM algorithm on its own terms[J]. Journal of the Royal Statistical Society, Series B, 1989, 51: 127-138.
[113] McLachlan G J, Krishnan T. The EM algorithm and extensions[M]. New York: Wiley,1997.
[114] Meng X L, Rubin D B. Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm[J]. Journal of the American Statistical Association, 1991, 86: 899-909.
[115] Meng X L, Rubin D B. Maximum likelihood estimation via the ECM algorithm: a general framework[J]. Biomtrika, 1993, 80(2): 267-278.
[116] Ruud P A. Extensions of estimation methods using the EM algorithm[J]. Journal of Econometrics, 1991,49: 305-341.
[117] Wu C F J. On the convergence properties of the EM algorithm[J]. The Annals of Statistics, 1983, 11(1): 95-103.
[118] Heyde C C, Morton R. Quasi-likelihood and generalizing the EM algorithm[J]. Journal of the Royal Statistical Society, Series B, 1996, 58(2): 317-327.
[119] Paik M C. The EM algorithm for the quasi-likelihood regression model[J]. Communication in Statistics, Part A-Theory and Methods, 1995, 24(6): 1403-1430.
[120] Robins J M, Rotnitzky A. Semiparametric efficiency in multivariate regression models with missing data[J]. Journal of the American Statistical Association, 1995, 90(429):122-129.
[121] Paik M C. The generalized estimating equation approach when data are not missing completely atrandom[J]. Journal of the American Statistical Association, 1997, 92(440):1320-1329.
[122] Robins J M. Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers[J]. Proceedings of the Biopharma-ceutical Section, American Statistical Association, 1993, 24-33.
[123] Robins J M, Hsieh F, Newey W. Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates[J]. Journal of the Royal Statistical Society, Series B, 1995, 57: 409-424.
[124] Sattten G A, Datta S. The S-U algorithm for missing data problems[J]. Computational Statistics, 2000, 15: 243-277.
[125] Ortega J M. Numerical Analysis: a Second Course[M]. New York: Academic Press,1972.
[126] Reilly M, Pepe, MS. A mean score method for missing and auxiliary covariate data in regression models[J]. Biometrika, 1995, 82(2): 299-314.
[127] Prentice R L. Correlated binary regression with covariates specific to each binary observation[J]. Biometrics, 1988,44: 1033-1048.
[128] Chen H Y, Little R. A testing of missing completely at random for generalised estimating equations with missing data[J]. Biometrika, 1999, 86(1): 1-13.
[129] Bishop YMM, Fienberg S E, Holland P W. Discrete Multivariate Analysis:Theory and Practice[M]. Massachusetts, Cambridge: MIT Press, 1975.
[130] Lindsay B G. Composite likelihood methods[C]. In: Prabhu N U, ed. Statistical Infer- ence from Stochastic Process, Contemporary Mathematics 80. Providence, RI: American Mathematical Society, 1988. 221-239.
[131] HjortN L, Omre H. Topics in spatial statistics (with discussion)[J]. Scandinavian Journal of Statistics, 1994, 21: 289-357.
[132] Heagerty P J, Lele S R. A composite likelihood approach to binary spatial data[J]. Journal of the American Statistical Association, 1998,93: 1099-1111.
[133] Chen J J, Gaylor D W. Correlations of developmental end points observed after 2,4,5- trichlorophenoxyacetic acid exposure in mice[J]. Teratology, 1992,45: 241-246.
[134] Holson J F, Gaines T B, Nelson C J, et al. Developmental toxicity of 2,4,5-trichlorophenoxyacetic acid I: dose-response studies in four inbred strains and one out- bred stock of mice[J]. Fundamentals of Applied Toxicology, 1991, 19: 286-297.
[135] de Finetti B. Theory of Probability[M]. New York: Wiley, 1974.
[136] Darroch J N. The mantel-haenszel test and tests of marginal symmetry: fixed-effects and mixed models for a categorical response[J]. International Statistical Review, 1981, 49:285-307.
[137] Ryan L M, Catalano P J, Kimmel C A, et al. Relationship between fetal weight and malformation in developmental toxicity studies[J]. Teratology, 1991, 44: 215-223.
[138] Perreault S, Jeffay S, Poss P, et al. Use of the fungicide carbendazim as a model com- pound to determine the impact of acute chemical exposure during oocyte maturation and fertilization on pregnancy outcome in the hamster[J]. Toxicology and Applied Pharma- cology, 1992, 114: 13-29.
[139] Diggle P, Kenward M G. Informative dropout in longitudinal data analysis (with discussion)[J]. Applied Statistics, 1994,43: 49-94.
[140] Heyting A, Tolboom JTBM, Essers J G A. Statistical handling of drop-outs in longitudinal clinical trials[J]. Statistics in Medicine, 1992, 11: 2043-2062.
[141] Heitjan D F, Rubin D B. Ignorability and coarse data[J].The Annals of Statistics, 1991,19: 2244-2253.
[142] Kim M, Degruttola V, Lagakos S. Analyzing doubly censored data with covariates, with application to AIDS[J]. Biometrics, 1993,49: 13-22.
[143] Pepe M S, Fleming T R. A nonparametric method for dealing with mismeasured covariate data[J]. Journal of the American Statistical Association, 1991, 86: 108-113.
[144] Manel L A, DeRouen T A. A covariance estimator for GEE with improved small sample properties[J]. Biometrics, 2001, 57: 126-134.
[2] Ryan L M. Quantitative risk assessment for developmental toxicity[J]. Biometrics, 1992, 48: 163-174.
[3] Ryan L M. Using historical controls in the analysis of developmental toxicity data[J]. Biometrics, 1992,49: 1126-1135.
[4] Lefkopoulou M, Ryan L M. Global tests for multiple binary outcomes[J]. Biometrics, 1993, 49: 975-988
[5] Liang K-Y, McCullagh P. Case studies in binary dispersion[J]. Biometrics, 1993, 49: 62-630.
[6] Bahadur R R. A representation of the joint distribution of responses to n dichotomous items[C]. In: Solomon H, ed. Studies on Item Analysis and Predication. Stanford, CA: Stanford University Press, 1961. 158-168.
[7] Cox D R. The analysis of multivariate binary data[J]. Applied Statistics, 1972, 21: 113-120.
[8] Dale J R. Global cross-ratio models for bivariate, discrete, ordered response[J]. Biomet- rics, 1986,42: 909-917.
[9] Zhu Y, Fung K Y. Statistical methods in developmental toxicity risk assessment[C]. In: Fan A, Chang L W, eds. Toxicology and Risk Assessment, Principles, Methods, and Applications. New York: Marcel Dekker, 1996. 413-446.
[10] Molenberghs G, Ryan L M. Likelihood inference for clustered multivariate binary data[J]. Environmetrics, 1999, 10: 279-230.
[11] Molenberghs G., Geys H., Declerck L., et al. Analysis of clustered multivariate data from developmental toxicity studies[C]. In: Payne R, Green P, eds. Proceedings in Computa- tional Statistics. Heidelberg: Physica Verlag, 1998. 3-14
[12] Le Cessie, Van Houwelingen J C. Logistic regression for correlated binary data[J]. Ap- plied Statistics, 1994,43: 95-108.
[13] Geys H, Molenbergh G, Lipsitz S R. A note on the comparison of pseudo-likelihood and generalized estimating equations for marginal odds ratio models[J]. Journal of Statistical Computation and Simulation, 1998, 62: 45-72.
[14] Glonek G F V, McCullagh P. Multivariate logistic models[J]. Journal of the Royal Statistical Society, Series B, 1995, 57: 533-546.
[15] Molenberghs G, Lesaffre E. Marginal modelling of correlated ordinal data using a multivariate Plackett distribution[J]. Journal of the American Statistical Association, 1994, 89: 633-644.
[16] Lang J B, Agresti A. Simultaneously modeling joint and marginal distributions of multivariate categorical responses[J]. Journal of the American Statistical Association, 1994,89: 625-632.
[17] Williams D A. The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity[J]. Biometrics, 1975, 31: 949-952.
[18] Breslow N E, Day N E. Statistical Methods in Cancer Research[M]. Oxford: Oxford University Press, 1987. Volume 2.
[19] Carr G J, Portier C J. An evalution of some methods for fitting dose-response models toquantal-response developmental toxicology data[J]. Biometrics, 1993,49: 779-791.
[20] Gaylor D W, Chen J J. Dose-response models for developmental malformations[J]. Teratology, 1993, 47: 291-297.
[21] Rai K, Van Ryzin J. A dose-response model for teratological experiments involving quan-tal experiments[J]. Biometrics, 1985,41: 1-9.
[22] Fitzmaurice G M, Laird N M. A likelihood-based method for analysing longitudinal binary response[J]. Biometrika, 1993, 80: 141-151.
[23] George E O, Bowman D. A full likelihood procedure for analysing exchangeable binary data[J]. Biometrics, 1995, 51(2): 512-523.
[24] Stefanescu C, Turnbull B W. Likelihood inference for exchangeable binary data with varying cluster sizes[J]. Biometrics, 2003, 59: 18-24.
[25] Liang K-Y, Zeger S L. Longitudinal data analysis using generalized linear models[J]. Biometrika, 1986, 73(1): 13-22.
[26] Arnold B C, Strauss, D. Pseudolikelihood estimation: some examples[J]. Sankhya B, 1991, 53: 233-243.
[27] de Finetti B. Funzione caratteristica di un fenomeno aleatorio[J]. Mathematiche e Natu-rali, 1931,4: 251-299.
[28] Draper D, Hodges J S, Mallows C L, et al. Exchangeability and data analysis[J]. Journal of the Royal Statistical Society, Series A, 1993, 156: 9-37.
[29] Mandel E, Bluestone C D, Rockette H E, et al. Duration of effusion after antibiotic treatment for acute otitis media: comparison of cefaclor and amoxicillin[J]. Pediatric Infections Diseases, 1982, 1: 310-316.
[30] Williams D A. Reader reaction: Dose-response models for teratological experiments[J]. Biometrics, 1987,43: 1013-1016.
[31] Kupper L L, Haseman J K. The use of correlated binomial model for the analysis of data from certain toxicological experiments[J]. Biometrics, 1978, 34: 69-76.
[32] Kupper L L, Portier C, Hogan M D, et al. The impact of litter effects on dose-response modeling in teratology [J]. Biometrics, 1986,42: 85-98.
[33] Prentice R L. Binary regression using an extended beta-binomial distribution with discussion of correlation induced by measurements[J]. Journal of the American Statistical Association, 1986,81(394): 321-327.
[34] Chen J J, Kodell R L, Howe R B, et al. Analysis of trinomial responses from reproductive and developmental toxicity experiments[J]. Biometrics, 1991, 47: 1049-1058.
[35] Rosner B. Multivariate methods for clustered binary data with more than one level of nesting[J]. Journal of the American Statistical Association, 1989, 84: 373-380.
[36] Chen J J, Kodell R L. Quantitative risk assessment for teratological effects[J]. Journal of the American Statistical Association, 1989, 84: 966-971.
[37] Paul S R. A clumped beta-binomial model for the analysis of clustered attribute data. Biometrics, 1979, 35: 821-824.
[38] Pack S E. Hypothesis testing for proportion with overdispersion[J]. Biometrics, 1986,42:967-972.
[39] Altham P M E. Two generalizations of the binomial distribution[J]. Applied Statistics, 1978, 27: 162-167.
[40] Catalano P J, Ryan L M. Bivariate latent variable models for clustered discrete and con- tinuous outcomes[J]. Journal of the American Statistical Association, 1992, 87: 651-658.
[41] Zeger S L, Liang K-Y. Longitudinal data analysis for discrete continuous outcomes[J].Biometrics, 1986,42: 121-130.
[42] Lipsitz S R, Ibrahim J G, Zhao L P. A weighted estimating equation for missing covariate data with properties similar to maximum likelihood[J]. Journal of the American Statisti- cal Association, 1999,94(448): 1147-1160.
[43] Prentice R L, Zhao L P. Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses[J]. Biometrics, 1991,47: 825-839.
[44] McCullagh P, Nelder J A. Generalized linear models[M]. New York: Chapman and Hall,1989.
[45] Moore D F. Asymptotic properties of moment estimators for overdispersed counts and proportions[J]. Biometrika, 1986, 73: 583-588.
[46] Willams D A. Estimation of bias using the beta-binomial distribution in teratology[J]. Biometrics, 1988, 44: 305-309.
[47] Williams D A. Extra-binomial variation in logistic linear models[J]. Applied Statistics, 1982, 31: 144-158.
[48] Zhao L P, Prentice R L. Correlated binary regression using a quadratic exponential model[J]. Biometrika, 1990, 77: 642-648.
[49] Bowman D, Chen J J, George E O. Estimating variance functions in developmental toxi- city studies[J]. Biometrics, 1995, 51: 1523-1528.
[50] Lipsitz S R, Laird N M, Harrington D P. Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association[J]. Biometrika, 1991, 78:153-160.
[51] Liang K-Y, Hanfelt J. On the use of the quasi-likelihood method in teratological experi-ments[J]. Biometrics, 1994, 50: 872-880.
[52] Chen J J, Li L. Dose-response modeling of trinomial responses from developmental ex- periments[J]. Statistica Sinica, 1994, 4: 265-274.
[53] Nelder J A, Pregibon D. An extended quasi-likelihood function[J]. Biometrika, 1987, 74:221-232.
[54] Rotnitzky A, Jewell N P. Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data[J]. Biometrika, 1990, 77: 485-497.
[55] Ryan L M. The use of generalized estimating equations for risk assessment in developmental toxicity[J]. Risk Analysis, 1992, 12: 439-447.
[56] Carey V, Zeger S L, Diggle P. Modelling multivariate binary data with alternating logistic regressions[J]. Biometrika, 1993, 80(3): 517-526.
[57] Wedderburn R W M. Quasi-likelihood functions, generalized linear models and the Gaussian method[J]. Biometrika, 1974, 61: 439-447.
[58] Liang K-Y, Zeger S L, Qaqish B. Multivariate regression analyses for categorical data (with discussion)[J]. Journal of the Royal Statistical Society, Series B, 1992,54(1): 3-40.
[59] Cohen B H. Chronic obstructive pulmonary disease: a challenge in genetic epidemiol- ogy[J]. American Journal of Epidemiology, 1980, 112: 274-288.
[60] Boos D D. Analysis of dose-response data in the presence of extrabinomial variation[J]. Applied Statistics, 1993,42: 173-183.
[61] Lipsitz S R, Fitzmaurice G M. Estimating equations for measures of association between repeated binary responses[J]. Biometrics, 1996, 52: 903-912
[62] Zucker D, Wittes J. Testing the effect of treatment in experiments with correlated binary outcomes[J]. Biometrics, 1992,48: 695-710.
[63] Bowman D, George E O. A saturated model for analyzing exchangeable binary data:applications to clinical and developmental toxicity studies[J]. Journal of the Ameri- can Statistical Association, 1995, 90(431): 871 -879.
[64] Xu J L, Prorok P C. Modelling and analysing exchangeable binary data with random cluster sizes[J]. Statistics in Medicine, 2003, 22: 2401-2416.
[65] Stefanescu C, Turnbull B W. Likelihood inference for exchangeable binary data with varying cluster sizes[J]. Biometrics, 2003,59: 18-24.
[66] Aerts M, Geys H, Molenberghs G, Ryan L M. Topics in modelling of clustered data[M]. London: Chapman and Hall, 2002.
[67] Kuk A Y C. A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions[J]. Applied Statistics, 2004, 53: 369-386.
[68] Davidian M, Carroll R J. Variance function estimation[J]. Journal of the American Statistical Association, 1987,82: 1079-1091
[69] Gourieroux C, Monfort A, Trognon A. Pseudo-maximum likelihood methods: theory[J]. Econometrica, 1984, 52: 681-700.
[70] Geys H, Molenberghs G, Ryan L. Pseudo-likelihood modelling of multivariate outcomes in developmental toxicology[J]. Journal of the American Statistical Association, 1999, 94: 734-745.
[71] Nott D J, Rydén T. Pairwise likelihood methods for inference in image models[J]. Biomtrika, 1999, 86(3): 661-676.
[72] Kuk A Y C, Nott D J. A pairwise likelihood approach to analyzing correlated binary data[J]. Statistics & Probability Letters, 2000,47: 329-335.
[73] Elashoff M, Ryan L. An EM algorithm for estimating equations[J]. Journal of Computational and Graphical Statistics, 2004, 13(1): 48-65.
[74] Rosen O, Jiang W X, Tanner M A. Mixtures of marginal models[J]. Biometrika, 2000,87(2): 391-404.
[75] Robins J M, Rotnitzky A, Zhao L P. Estimation of regression coefficient when some regressors are not always observed[J]. Journal of the American Statistical Association, 1994, 89(427): 846-866.
[76] Robins J M, Rotnitzky A, Zhao L P. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data[J]. Journal of the American Statistical.Association, 1995, 90(429): 106-121.
[77] Godambe V, Kale B. Estimating functions: an overview[C]. In: Godambe V, ed. Estimating Functions. Oxford: Clarendon Press, 1991.
[78] Liang K-Y, Zeger S L. Inference based on estimating functions in the presence of nuisance parameters[J]. Statistical Science, 1995, 10: 158-173.
[79] Liang K-Y, Liu X H. Estimating equations in generalized linear models with measure- ment error[C]. In: Godambe V, ed. Estimating Functions. Oxford: Clarendon Press, 1991.47-63.
[80] Godambe V P. An optimum property of regular maximum likelihood estimation[J]. Annals of Mathematical Statistics, 1960, 31: 1208-1212.
[81] Kale B K. An extension of Cramer-Rao inequality for statistical estimation functions[J]. Skandinavisk Aktuarietid, 1962, 45: 60-82.
[82] Kimball B F. Sufficient statistical estimation functions for the parameters of the distribution of maximum values[J]. Annals of Mathematical Statistics, 1946, 17: 299-309.
[83] Wilks S S. Mathematical statistics[M]. New York: Wiley, 1948.
[84]Kale B K.Essential uniqueness of optimal estimating functions[J].Journal of Statistical Planning and Inference,1987,17:405-407.
[85]McLeish D L,Small C G.The theory and applications of statistical inference functions[C].In:Springer Lecture Notes in Statistics NO.44.Newyork:Springer Verlag,1988.
[86]Godambe V P,Heyde C C.Quasi-likelihood and optimal estimation[J].International Statistical Review,1987,55(3):231-244.
[87]Wilks S S.Shortest average confidence intervals from large samples[J].Annals of Mathematical Statistics,1938,9:166-175.
[88]Durbin J.Estimation of parameters in time series regression models[J].Journal of the Royal Statistical Society,Series B,1960,22:139-153.
[89]Ferreira P E.Multiparametric estimating equations[J].Annals of the Institute of Statistical Mathematics,1982,34A:423-431.
[90]Chandrasekar B,Kale B K.Unbiased statistical estimation functions in presence of nuisance parameter[J].Journal of Statistical Planning and Inference,1984,9:45-54.
[91]Godambe V P,Thompson M E.Estimating equations in the presence of nuisance parameters[J],the Annals of Statistics,1974,2:568-571.
[92]Godambe V P.Conditional likelihood and unconditional optimum estimating equations[J].Biometrika,1976,63:277-284.
[93]Subramanyam A,Naik-Nimbalkar U V.Optimal unbiased statistical estimating functions for Hilbert space valued parameters[J].Journal of Statical Planning and Inference,1990,24:95-105.
[94]Heyde C C.Quasi-likelihood and Its Application:A General Approach to Optimal Parameter Estimation[M].New York:Springer,1997.
[95]Hanfelt J J,Liang K-Y.Approximate likelihood ratios for general estimating functions[J].Biometrika,1995,82:461-477.
[96]Godambe V P.The foundations of finite sample estimation in stochastic processes[J].Biometrika,1985,72:419-428.
[97]Cox D R.Partial likelihood[J].Biometrika,1975,62:269-276.
[98] Huber P J. The behavior of maximum likelihood estimates under nonstandard condi- tions[J]. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability[J], 1967, 1: 221-233.
[99] Carroll R J, Ruppert D, Stefanski L A. Measurement Error in Nonlinear ModelsfM]. New York: Chapman and Hall, 1995.
[100] Drum M, McCullagh P, Prentice R L. Comment on "Regrssion models for discrete longitudinal response,"by Fitznaurice, Laird and Rotnitzky[J]. Statistical Science, 1993, 8:300-309.
[101] Newey W K, McFadden D. Estimation in large samples[C]. In: McFadden D, Engler R,eds. Handbook of Econometrics, Vol. 4. Amsterdam: Noth-Holland, 1993.
[102] Manski C F. Analog estimation methods in econometrics[M]. New York: Chapman and Hall, 1988.
[103] Gallant A R, White H. A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models[M]. Oxford: Basil Blackwell, 1988.
[104] Little R J, Rubin D B. Statistical Analysis With Missing Data[M]. New York: John Wiley, 1987.
[105] Dempster A P, Laird N M, Rubin D B. Maximum likelihood estimation from incomplete data via the EM algorithm (with disscussion)[J]. Journal of the Royal Statistical Society, Series B, 1977, 39: 1-38.
[106] McKendrick A G. Applications of mathematics to medical prohlems[J]. Proceedings of the Edinburgh Mathematical Society, 1926,44: 98-130.
[107] Hartley H O. Maximum likelihood estimation from incomplete data[J]. Biometrics, 1958, 14: 174-194.
[108] Baum L E, Petrie T, Soules G, et al. A maximization tehnique occurring in the statistical analysis of probabilistic functions of Markov chains[J]. Annals of Mathematical Statistics, 1970,41: 164-171.
[109] Orchard T, Woodbury M A. A missing information principle: theory and applications[J]. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1972, 1: 697-715.
[110] Sundberg R. Maximum likelihood theory for incomplete data from an exponential family[J]. Scandinavian Journal of Statistics, 1974, 1: 49-58.
[111] Beale E M L, Little R J A. Missing values in multivariate analysis[J]. Journal of the Royal Statistical Society, Series B, 1975, 37: 129-145.
[112] Meilijson I. A fast improvement to the EM algorithm on its own terms[J]. Journal of the Royal Statistical Society, Series B, 1989, 51: 127-138.
[113] McLachlan G J, Krishnan T. The EM algorithm and extensions[M]. New York: Wiley,1997.
[114] Meng X L, Rubin D B. Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm[J]. Journal of the American Statistical Association, 1991, 86: 899-909.
[115] Meng X L, Rubin D B. Maximum likelihood estimation via the ECM algorithm: a general framework[J]. Biomtrika, 1993, 80(2): 267-278.
[116] Ruud P A. Extensions of estimation methods using the EM algorithm[J]. Journal of Econometrics, 1991,49: 305-341.
[117] Wu C F J. On the convergence properties of the EM algorithm[J]. The Annals of Statistics, 1983, 11(1): 95-103.
[118] Heyde C C, Morton R. Quasi-likelihood and generalizing the EM algorithm[J]. Journal of the Royal Statistical Society, Series B, 1996, 58(2): 317-327.
[119] Paik M C. The EM algorithm for the quasi-likelihood regression model[J]. Communication in Statistics, Part A-Theory and Methods, 1995, 24(6): 1403-1430.
[120] Robins J M, Rotnitzky A. Semiparametric efficiency in multivariate regression models with missing data[J]. Journal of the American Statistical Association, 1995, 90(429):122-129.
[121] Paik M C. The generalized estimating equation approach when data are not missing completely atrandom[J]. Journal of the American Statistical Association, 1997, 92(440):1320-1329.
[122] Robins J M. Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers[J]. Proceedings of the Biopharma-ceutical Section, American Statistical Association, 1993, 24-33.
[123] Robins J M, Hsieh F, Newey W. Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates[J]. Journal of the Royal Statistical Society, Series B, 1995, 57: 409-424.
[124] Sattten G A, Datta S. The S-U algorithm for missing data problems[J]. Computational Statistics, 2000, 15: 243-277.
[125] Ortega J M. Numerical Analysis: a Second Course[M]. New York: Academic Press,1972.
[126] Reilly M, Pepe, MS. A mean score method for missing and auxiliary covariate data in regression models[J]. Biometrika, 1995, 82(2): 299-314.
[127] Prentice R L. Correlated binary regression with covariates specific to each binary observation[J]. Biometrics, 1988,44: 1033-1048.
[128] Chen H Y, Little R. A testing of missing completely at random for generalised estimating equations with missing data[J]. Biometrika, 1999, 86(1): 1-13.
[129] Bishop YMM, Fienberg S E, Holland P W. Discrete Multivariate Analysis:Theory and Practice[M]. Massachusetts, Cambridge: MIT Press, 1975.
[130] Lindsay B G. Composite likelihood methods[C]. In: Prabhu N U, ed. Statistical Infer- ence from Stochastic Process, Contemporary Mathematics 80. Providence, RI: American Mathematical Society, 1988. 221-239.
[131] HjortN L, Omre H. Topics in spatial statistics (with discussion)[J]. Scandinavian Journal of Statistics, 1994, 21: 289-357.
[132] Heagerty P J, Lele S R. A composite likelihood approach to binary spatial data[J]. Journal of the American Statistical Association, 1998,93: 1099-1111.
[133] Chen J J, Gaylor D W. Correlations of developmental end points observed after 2,4,5- trichlorophenoxyacetic acid exposure in mice[J]. Teratology, 1992,45: 241-246.
[134] Holson J F, Gaines T B, Nelson C J, et al. Developmental toxicity of 2,4,5-trichlorophenoxyacetic acid I: dose-response studies in four inbred strains and one out- bred stock of mice[J]. Fundamentals of Applied Toxicology, 1991, 19: 286-297.
[135] de Finetti B. Theory of Probability[M]. New York: Wiley, 1974.
[136] Darroch J N. The mantel-haenszel test and tests of marginal symmetry: fixed-effects and mixed models for a categorical response[J]. International Statistical Review, 1981, 49:285-307.
[137] Ryan L M, Catalano P J, Kimmel C A, et al. Relationship between fetal weight and malformation in developmental toxicity studies[J]. Teratology, 1991, 44: 215-223.
[138] Perreault S, Jeffay S, Poss P, et al. Use of the fungicide carbendazim as a model com- pound to determine the impact of acute chemical exposure during oocyte maturation and fertilization on pregnancy outcome in the hamster[J]. Toxicology and Applied Pharma- cology, 1992, 114: 13-29.
[139] Diggle P, Kenward M G. Informative dropout in longitudinal data analysis (with discussion)[J]. Applied Statistics, 1994,43: 49-94.
[140] Heyting A, Tolboom JTBM, Essers J G A. Statistical handling of drop-outs in longitudinal clinical trials[J]. Statistics in Medicine, 1992, 11: 2043-2062.
[141] Heitjan D F, Rubin D B. Ignorability and coarse data[J].The Annals of Statistics, 1991,19: 2244-2253.
[142] Kim M, Degruttola V, Lagakos S. Analyzing doubly censored data with covariates, with application to AIDS[J]. Biometrics, 1993,49: 13-22.
[143] Pepe M S, Fleming T R. A nonparametric method for dealing with mismeasured covariate data[J]. Journal of the American Statistical Association, 1991, 86: 108-113.
[144] Manel L A, DeRouen T A. A covariance estimator for GEE with improved small sample properties[J]. Biometrics, 2001, 57: 126-134.