几类常用非线性回归分析中最优模型的构建与SAS智能化实现
|
摘要
【目的】针对几类常用的一重和多重非线性回归分析在具体使用中存在的难以克服的障碍，探索相应的解决方法，以获得拟合实际资料效果最好的回归模型，从而促进几类非线性回归分析在理论和方法上的进一步完善，并提升使用时的可操作性和智能化水平，以利于其推广和应用。
【内容】本研究涉及的非线性回归分析可分为固定模式和非固定模式的非线性回归分析两大类。固定模式是指原因变量和模型结构均已确定，非固定模式是指仅模型结构确定，而原因变量尚未确定。根据资料是否含有层级结构，非固定模式的非线性回归分析又可细分为两型：非固定模式的单水平与多水平非线性回归分析。
本研究中涉及的固定模式的非线性回归分析，包括多项型指数曲线类模型、S型生长曲线类模型和产量-密度曲线类模型。这些模型在实际应用中使用频率较高，但模型结构较为复杂，待估计参数较多，现有的诸多分析方法虽然也能拟合出曲线模型，但模型对资料的拟合精度远不及非线性最小二乘法拟合的效果。然而，运用非线性最小二乘法实际求解时，多基于迭代算法，它要求使用者在迭代开始前提供较为接近真实参数的初估值，否则可能无法获得收敛的结果，或者所得模型为局部最优模型而非全局最优模型。所以，如何快速得到精确的、全局最优的曲线模型，是非常值得研究的一项内容。另外，每一类曲线模型，往往不止一种，以产量-密度曲线类模型为例，它包括Bleasdale-Nelder曲线、Halliday曲线、Farazdaghi-Harris曲线等。在处理实际问题时，如何从多种曲线模型中选出最适合实际资料的那一模型，也需要作深入研究。
本研究中涉及的非固定模式的单水平非线性回归分析，包括二值结果变量、多值有序结果变量和多值名义结果变量定性资料的单水平非线性回归分析，以及计数资料的单水平非线性回归分析。这些分析方法都是在原因变量筛选的基础上，得到既精简度高又拟合度好的回归模型。然而，当前所能使用的主要变量筛选方法，包括前进法、后退法和逐步筛选法等，都在理论上存在一定程度的缺陷，无法保证所得回归模型即为最优模型。因此，提出或实现理论上最为完善的变量筛选方法是当前迫切需要解决的技术难题。另外，对于同一种资料，往往有多种回归分析方法可供使用，以二值结果变量的定性资料为例，可使用的非线性回归分析方法包括logistic回归分析、probit回归分析、互补双对数回归分析等。面对具体资料时，如何对这些回归分析方法进行比较研究，并为实际资料自动提供最好的、最适合的回归模型，也是需要重点研究的又一个技术难题。
本研究中涉及的非固定模式的多水平非线性回归分析，包括二值结果变量、多值有序结果变量和多值名义结果变量定性资料的多水平非线性回归分析，以及计数资料的多水平非线性回归分析。这些分析模型不仅包含固定效应，还包含一些随机效应，此处暂将两类效应变量统称为效应项。在实际使用时，如何对这些效应项进行组合，构造一个最优的回归模型，当前并无较好的方法，多是基于人为确定，操作起来极为不便。所以，如何灵活构建效应项的最优组合，是此类回归分析亟待解决的一个技术难点。另外，与非固定模式的单水平非线性回归分析相似，同一资料有多种可用的回归分析方法时，如何确定哪种分析方法最适合给定的实际资料，也是一项重要的研究内容。再者，多水平模型分析的研究虽已如火如荼，但其模型求解的算法仍不够完善。实际使用时，应如何选择参数估计方法，并对其进行正确的假设检验，也需要作深入研究。
本研究针对这些非线性回归分析在实际使用中存在的技术难点进行研究，并借助SAS软件的编程语言、高级编程技巧和相关过程，实现资料的智能化、自动化分析，为使用者直接提供最优的回归分析方法和分析结果。
【方法】如何实现固定模式的非线性回归分析？本研究采用曲线直线化法加非线性最小二乘法的组合策略，即先以曲线直线化法为基础，计算得到模型中参数的初始值，再采用非线性最小二乘法得到拟合效果更好的曲线模型。在曲线直线化的过程中，有些较为简单的模型经过变量变换和数学推导，即可直接进行直线回归分析；有些复杂的模型不能直接进行曲线直线化，可以先选择一个或两个变化范围较小的参数，通过设置循环变量的方式使其按一定步长在较小的可能值域内变动，这样在每次循环中这些参数都将有具体的值，此时再对曲线模型进行数学推导和变量变换，即可进行直线回归分析。直线回归分析得到的参数估计值，经某些必要的计算后，就可以作为曲线模型中相应参数的初始值。当然，在曲线直线化的过程中，如果对某些参数设置了多个可能的取值，必然会得到较多的初始值组合，进而会得到多个局部最优模型，此时可从中选出拟合效果最好的那个模型作为全局最优模型，这样可以有效地破解局部最优解的难题。
如何实现非固定模式的单水平非线性回归分析？本研究采用了完全意义上的“最优子集法”来解决常规变量筛选方法存在的理论缺陷。当前某些大型统计软件也提供了最优子集法，但仅仅能给出各种原因变量组合情形下模型对资料的拟合效果统计量的值，并未涉及模型中参数估计的结果和假设检验的结果，更未直接给出最优模型，只是提供了含各种数量原因变量的“最优子集”，而最优模型中原因变量的数目仍需由使用者确定。具体来说，本研究考虑在各种原因变量组合情形下，均进行相应模型的构建和统计分析，然后从模型精简程度和拟合优度两方面，选出一个最优模型。
如何实现非固定模式的多水平非线性回归分析？一般统计软件都未提供效应项的筛选方法，只能依靠使用者采用手工法人为调整模型，操作起来非常不便。本研究同样采用完全意义上的“最优子集法”来选择最优模型，即先以固定效应项和随机效应项进行全面组合，然后在每种组合情形下，均进行多水平非线性回归分析，最后从众多模型中，选择拟合效果最好的那个模型作为全局最优模型。当然，对于某些类型的资料，分析时还需考虑分析方法的应用前提条件不满足时如何处理。
对于同类中的多种非线性模型选择问题，可依据模型的类型和参数估计方法选择相应的拟合优度评价统计量，对各模型的拟合效果进行比较，选择拟合效果最好的那个回归模型作为最优模型。
以上方法，具体实现时，可借助SAS软件的编程语言及相应过程，如进行最优子集法分析时，可借助REG、LOGISTIC等过程产生固定效应的全面组合情形，多水平模型分析时，还需借助FACTEX等过程产生随机效应的全面组合。
【结果】本研究对各类非线性回归分析的现有分析策略、计算方法和实现途径在实际应用中存在的问题和不足进行改进，有针对性地提出了相应最优模型的选择策略，并通过编写大量的SAS程序，将这些策略付诸实施，最后以最直观、最简洁和最适用的形式呈现出来。详细来说，论文的研究结果和主要创新点包括以下四个方面。
⑴对三类固定模式的非线性回归分析进行了研究，摸索出了一套分析策略：即以变量变换和数学推导为基础，通过曲线直线化或分段曲线直线化的方式，得到曲线模型的参数初始值，然后借助相应的迭代算法完成非线性最小二乘法的精确拟合。这一策略可实现此类资料的快速、高效、精准拟合，且可行性、可操作性非常好。此外，本研究还解决了同类多种曲线模型拟合实际资料时的智能化选择问题，所编写的宏程序可以方便地调用以解决同类实际问题。
⑵对四类非固定模式（含计数资料）的单水平非线性回归分析进行研究，以当前变量筛选方法中存在的理论困境和应用难题为突破口，实现了完全意义上的“最优子集法”，确保了所得的最终模型即为全局最优模型；同时解决了多种相近功能分析方法处理同一资料时，分析方法的拟合效果比较与智能化选择问题，有效地避免了使用者处理实际资料时在分析方法选择上的盲目性和不确定性。
⑶对四类非固定模式（含计数资料）的多水平非线性回归分析进行研究，以完全意义上的“最优子集法”对效应项进行了全面组合，在拟合效果比较的基础上，自动化地给出了最优模型，破解了当前多水平模型分析时无法进行效应项筛选的难题。当然，也解决了多种相近功能分析方法处理同一资料时，分析方法的拟合效果比较与智能化选择问题，所编写的宏程序可以方便地被调用以解决同类实际问题。
⑷探索出了一条基于SAS9.2版（之前版本无效）软件进行多水平非线性回归分析时参数估计方法的选择与调整策略，即需要估计的效应项参数较少时，可直接调用操作较为方便的GLIMMIX过程，使用数值积分近似法进行参数估计，但须在假设检验时对自由度进行调整；若需要估计的效应项较多，尤其是随机效应项较多时，直接使用GLIMMIX过程中的数值积分近似法往往无法得到收敛的结果，可先使用GLIMMIX过程中的线性化法得到参数的近似精确估计值，然后以此为初始值，再使用NLMIXED过程中的数值积分近似法作精确估计。
关于GLIMMIX过程，它是SAS软件近年来力推的新过程，与使用极其繁杂的NLMIXED过程相比，具有操作简便的特点，不过该过程虽经多年丰富和发展，但至今仍未能尽善尽美，其不足之处在于：一是未提供随机效应的假设检验结果，二是固定效应的假设检验结果也不准确。使用数值积分近似法时，两过程仅因计算精度不同导致GLIMMIX过程输出的参数值及标准误的估计结果与NLMIXED过程存在非常微小的差异，但GLIMMIX过程的重大缺陷在于其对随机效应未输出自由度，对固定效应虽输出了自由度但却是错误的，故GLIMMIX过程输出的假设检验结果不能直接被采用，须在假设检验时对效应项的自由度进行调整，从而得到准确的概率值。
以上策略，以调用GLIMMIX过程并对假设检验结果稍作调整为主，特殊情形下辅用NLMIXED过程，外加相应的编程，可直接或近似直接地获得最优模型的精确估计，这极大地节省了人工调整模型时的工作量和因人而异的不确定性，相对于王济川等人提出的解决类似问题的策略不仅简化了很大的工作量，而且能够较为方便地实现以最优子集法筛选效应项并通过编程直接获得最终结果的目的。
【结论】本论文以非线性回归分析为主要研究内容，对实际使用中存在的诸多瓶颈问题、不足和不便之处，进行了探索性的实践，并获得了令人满意的结果。
在进行固定模式的非线性回归分析时，摸索出了一套高效的分析策略。这一分析策略在解决三类固定模式的非线性回归分析时，被证明是行之有效的。该策略既可快速便捷地获得参数的估计值，还可得到较传统分析方法拟合效果更优的曲线回归模型，且对其他固定模式的非线性回归分析具有一定参考和借鉴意义。
在进行非固定模式的非线性回归分析时，以完全意义上的“最优子集法”进行原因变量或效应项的筛选，得到对实际资料拟合效果最好的回归模型。这一分析策略，不但避免了常用变量筛选方法在理论上的先天缺陷，也弥补了统计软件中所谓“最优子集法”在应用上的后天不足，可以方便快捷且准确可靠地为使用者提供最优的回归模型。
此外，多种相近功能非线性回归分析方法联用，在理论和技术层面上给出获得最优分析结果的策略、借助SAS系统并加以深层次开发和巧妙利用，使灵活而又复杂的计算策略得以准确完整地实现，在当前众多分析方法孰优孰劣或适用场合不明以及统计软件智能化水平低下的情况下，不失为一个大胆而成功的尝试。当前的统计研究发展迅速，往往是多种分析方法均可实现同一分析目的，较之随意选用其中的某种分析方法，该策略所得结果能更好地诠释资料的内在规律性。
【Objective】To overcome the almost insurmountable obstacles in practicalapplications of some kinds of regularly used simple or multiple nonlinear regressionanalysis(NRA), and obtain the regression model best befitting practical data, therebyperfecting those nonlinear regression analyses both in terms of theory andmethodology and promoting the operability and intelligent implementation to anincreased popularity and more successful application of these models.
【Content】Nonlinear regression analysis in this study is divided into twocategories: fixed mode and unfixed mode. Fixed mode is a mode in which bothexplanatory variables and model structure have been determined. Unfixed mode is amode in which only model structure has been decided. According to the hierarchicalstructure of the data, nonlinear regression analysis of unfixed mode can be subdividedinto two types: single level mode and multilevel mode.
Nonlinear regression analysis of fixed mode with one independent variable inthis study includes sum of exponentials curves, sigmoidal growth curves, andyield-density curves. These models are used frequently in practical application.However, in these complicated structure models, many parameters need to beestimated. Although there are many methods which can fit these models, the precisionof fitness is vastly inferior to the effect of nonlinear least squares (NLS) method. Themethod of NLS is based upon iterative algorithm, and users must specify veryapproximate initial values of parameters, otherwise the iteration may not be able toachieve convergence or the model is just a local optimization. Therefore, it is asubject worthy of research how to obtain a precise and general optimal curve modelswiftly. Additionally, there are generally multiform models in each kind of curvemodel. Take for example yield-density curves model, it includes Bleasdale-Neldercurve model, Halliday curve model, Farazdaghi-Harris curve model. In-depth researchis needed to identify the best model from multiform models while dealing withpractical problems.
Single level nonlinear regression analysis of unfixed mode with multipleindependent variables in this study includes single level models for binary, ordinal,multinomial and count outcomes. These methods yield very simplified and nicely fitregression model by using variable selection. However, major methods of variableselection, including forward method, backward method, stepwise method, all havetheir drawbacks in terms of theory and cannot ensure that the model is the best one.Consequently, it is a technical challenge to put forward or actualize the perfectvariable selection method. Additionally, there are always several available regressionanalysis methods for each data. For example, nonlinear regression methods used todeal with qualitative data with binary dependent variable, include logistic regressionanalysis, probit regression analysis and complementary log-log regression analysis.Another technical problem that merits great attention is how to compare the effect ofseveral regression methods and provide the best models automatically.
Multilevel nonlinear regression analysis of unfixed mode with multipleindependent variables in this study includes multilevel models for binary, ordinal,multinomial and count outcomes. These analysis models consist of fixed effects andrandom effects, and the two kinds of effect variables are termed effect items. Inpractical use, at present a desired method is not available to assemble effect itemsrationally and construct an optimal regression model. The largely artificially selectedmethod frequently used is very cumbersome to manipulate. Accordingly, a technicalproblem calling for urgent solution is how to flexibly achieve an optimal combinationof effect items in multilevel nonlinear regression analysis. Furthermore, there areseveral regression analysis methods for each data, as in the case of single levelnonlinear regression analysis. For this reason, it is also an important research topic tofind the most appropriate analysis method for the specific data. Moreover, studies ofmultilevel modeling analysis have been booming, but there is room for improvementin algorithms for parameters estimation. In actual use, an in-depth study isimperatively needed to select the estimation method and test the hypothesis forparameters correctly.
The paper researches and overcomes the technical difficulties in nonlinearregression analysis described above by making use of programming language,advanced programming skills and corresponding procedures of SAS softwareemployed in intelligentized and roboticized analysis of data, providing users withanalysis outcome by optimal methods.
【Methods】How to do nonlinear regression analysis of fixed mode? The paperadopts a strategy of combining linearisation method with NLS method. To be specific,the estimates obtained by linearisation method are treated as starting value, andsubsequently NLS method is employed to achieve a better curve model. In the processof linearisation, some simple models are subject to linear regression analysis (LRA)directly after variable transformation and mathematical calculation. In othercomplicated models that cannot be linearized directly, one or more incomprehensiveparameters are selected as loop variables, changeable in a small range by definitestep-size. In this way, the parameters will have certain values at each loop, and thecomplicated models can be linearized after variables transformation and mathematicalcalculation, generating models that can be evaluated via linear regression analysis.After some essential calculation, parameter values estimated by LRA can be regardedas the starting values of corresponding parameter in curve models. Nevertheless, ifparameters have several set values, there will be many combinations of initial values,and consequently a plurality of local optimal models will be obtained. The best modelamong such local optimal models can be regarded as globe optimal model, where thedifficulty of local optimal solution can be resolved effectively.
How to perform single level nonlinear regression analysis of unfixed mode? Thepaper adopts normal meaning method of best regression subset to solve theoreticlimitation of ordinary variable selection methods. Although many large statisticalanalysis software packages provide best regression subset method, they only presentthe values of model fit statistic in every case of explanatory variables’ combination,where the outcome of parameter estimate and testing hypotheses is not provided, notto say an optimal model. Which means that such software packages merely presentbest subsets with different number of explanatory variables, where the appropriatenumber has to be decided by users. In this paper models are constructed and analysisperformed in every case of explanatory variables’ combination to provide an optimalmodel based on simplification and goodness of fit (GOF).
How to realize multilevel nonlinear regression analysis of unfixed mode?Generally speaking, the common statistical software packages do not provide methodof effect items selection. Hence, adjustment of model has to depend on artificialmethod by users, which is very inconvenient to manipulate. The paper adopts normalmeaning method of best regression subset to select optimal model. First, fixed effectsand random effects are combined completely. Second, multilevel nonlinear regression analysis is done in every combination situation. And finally, the best fit model isselected as globe optimal model. In addition, for some types of data considerationshould be given to the situation where the precondition for the analysis methods to beused is not met.
As for selection among several congener nonlinear models, the paper choosesrelevant GOF evaluation statistic according to the type of model and the method ofparameter estimate, compares fitting effect among models, and then selects the best fitmodel as optimal model.
By virtue of programming language and corresponding procedures of SASsoftware, such methods described above can be actualized. For example, the completecombination of fixed effects can be produced using procedure REG and procedureLOGISTIC, and the complete combination of random effects can be produced usingprocedure FACTEX in multilevel modelling analysis.
【Results】The paper attempts to address the problems and remedy thedrawbacks with the current analysis tactics, calculation methods and actualizationapproaches in the practical application of NRA, and propose tactics of optimal modelselection, following through the schemes by programming in SAS to achieve an easilyperceptible outcome. In detail, the results and major innovations of this paper aresummarized in four aspects.
Part one is devoted to research on three kinds of nonlinear regression analysis offixed mode and exploration of an analysis tactic. Specifically, the estimates obtainedby linearisation method is regarded as starting value, to be followed by use of NLSmethod based on iterative algorithm to achieve a precise model. The tactic is markedby accurate, quick and efficient fitness, and has good feasibility and maneuverability.In addition, this part aims to solve the problem of intelligent selection among severalcongener nonlinear models while dealing with the same practical data, and programsome macros that can be used expediently.
Part two explores four kinds of single level nonlinear regression analysis ofunfixed mode to address the theoretic limitation and application difficulty withcurrent variable selection methods, and to ensure that the established model is theoptimal one by adopting normal meaning method of best regression subset.Furthermore, this part actualizes the automatic comparison of fitting effect andintelligentized selection among several congener analysis methods while dealing withthe same practical data, which helps avoid user’s blindness and uncertainty in method selection effectively.
Part three researches four kinds of multilevel nonlinear regression analysis ofunfixed mode, forms entire combinations of effect items by adopting normal meaningmethod of best regression subset, and solves the problem of a lack of method of effectitems selection in multilevel modelling analysis at the present time, and provides theoptimal model automatically after comparison of fitting effect. Additionally, this partsolves the problem of automatic comparison of fitting effect and intelligent selectionamong several congener nonlinear models while dealing with the same practical data,and programs some macros that can be used expediently.
Part four explores a selection and rectification tactic of estimation method inmultilevel nonlinear regression analysis based on SAS version9.2. If there is a smallnumber of random effects, procedure GLIMMIX can be used directly to obtainaccurate parameter values based on numerical integration methods, but the degree offreedom (DF) should be revised while performing hypothesis tests. If there is a largenumber of random effects, iteration of the foregoing means often fails to beconvergented, so procedure GLIMMIX can be used firstly to obtain approximateparameter values based on linearization-based methods, followed by the adoption ofprocedure NLMIXED to obtain accurate parameter values based on numericalintegration methods by taking the estimates obtained by linearization-based method asstarting values.
Procedure GLIMMIX is promoted zealously by SAS, which is very handycompared with the complicated procedure NLMIXED. In spite of the efforts madetowards its development and perfection in recent years, it is not perfect. Thedeficiencies of procedure GLIMMIX are as follows: first, it doesn’t performhypothesis tests to random effects; second, outcomes of hypothesis tests to fixedeffects are not correct. Tiny differences can be observed in estimate of parameters andtheir standard errors between procedure GLIMMIX and procedure NLMIXED usingnumerical integration methods, which is attributable to calculation precision. But theformer procedure has some serious defects in that it doesn’t provide DF of randomeffects and provide wrong DF of fixed effects. Therefore, results of hypothesis testsobtained by procedure GLIMMIX cannot be adopted directly, and DF of randomeffects in hypothesis tests should be adjusted so as to obtain exact probability.
The primary method of the tactic described above involves using procedureGLIMMIX and adjusting its hypothesis tests result. In some special cases, procedure NLMIXED is used as an adjunct. With the addition of programming, optimal modelcan be obtained directly or almost directly. The tactic can ease the workload and theuncertainty of artificial model selection. Compared with Wang’s tactic, it substantiallyreduces workload and can fulfill the aim of attaining optimal model expediently byadopting normal meaning method of best subset.
【Conclusions】The paper takes NRA as major research content, proposes somesolutions to many pivotal problems and attempts to address the drawbacks in practicalapplication, achieving some desirable results.
An efficient analysis tactic has been proposed for NRA of fixed mode, and it hasproved effective in dealing with three kinds of NRA of fixed mode. Not only canvalue of parameter be obtained swiftly, but also fitting effect of the model obtained bythis tactic is better than the one obtained by traditional method. It can be used as aframe of reference for other NRA of fixed mode.
In performing NRA of unfixed mode, optimal regression model can be obtainedby adopting normal meaning method of best subset to select variables. The tactic isfree of the inherent limitations of ordinary variable selection methods and overcomesthe drawbacks of the so-called method of best subset in statistic software, supplyingan optimal regression model for users expediently and reliably.
At present, there are many methods that can be applied to data analysis for thesame purpose, yet it is often unclear which is the best method for practical data. Thetactic introduced here that involving several congener NRA methods simultaneouslyto gain optimal model has been a bold and successful attempt. Taking intoconsideration the fact that the intelligent level of statistic software is very low, thisflexible and complicated calculation tactic can be fully actualized with the help ofadvanced development in SAS software. In brief, results obtained by this tactic aresuperior in interpreting the inherent law of data to those obtained by an arbitrarymethod.
【内容】本研究涉及的非线性回归分析可分为固定模式和非固定模式的非线性回归分析两大类。固定模式是指原因变量和模型结构均已确定，非固定模式是指仅模型结构确定，而原因变量尚未确定。根据资料是否含有层级结构，非固定模式的非线性回归分析又可细分为两型：非固定模式的单水平与多水平非线性回归分析。
本研究中涉及的固定模式的非线性回归分析，包括多项型指数曲线类模型、S型生长曲线类模型和产量-密度曲线类模型。这些模型在实际应用中使用频率较高，但模型结构较为复杂，待估计参数较多，现有的诸多分析方法虽然也能拟合出曲线模型，但模型对资料的拟合精度远不及非线性最小二乘法拟合的效果。然而，运用非线性最小二乘法实际求解时，多基于迭代算法，它要求使用者在迭代开始前提供较为接近真实参数的初估值，否则可能无法获得收敛的结果，或者所得模型为局部最优模型而非全局最优模型。所以，如何快速得到精确的、全局最优的曲线模型，是非常值得研究的一项内容。另外，每一类曲线模型，往往不止一种，以产量-密度曲线类模型为例，它包括Bleasdale-Nelder曲线、Halliday曲线、Farazdaghi-Harris曲线等。在处理实际问题时，如何从多种曲线模型中选出最适合实际资料的那一模型，也需要作深入研究。
本研究中涉及的非固定模式的单水平非线性回归分析，包括二值结果变量、多值有序结果变量和多值名义结果变量定性资料的单水平非线性回归分析，以及计数资料的单水平非线性回归分析。这些分析方法都是在原因变量筛选的基础上，得到既精简度高又拟合度好的回归模型。然而，当前所能使用的主要变量筛选方法，包括前进法、后退法和逐步筛选法等，都在理论上存在一定程度的缺陷，无法保证所得回归模型即为最优模型。因此，提出或实现理论上最为完善的变量筛选方法是当前迫切需要解决的技术难题。另外，对于同一种资料，往往有多种回归分析方法可供使用，以二值结果变量的定性资料为例，可使用的非线性回归分析方法包括logistic回归分析、probit回归分析、互补双对数回归分析等。面对具体资料时，如何对这些回归分析方法进行比较研究，并为实际资料自动提供最好的、最适合的回归模型，也是需要重点研究的又一个技术难题。
本研究中涉及的非固定模式的多水平非线性回归分析，包括二值结果变量、多值有序结果变量和多值名义结果变量定性资料的多水平非线性回归分析，以及计数资料的多水平非线性回归分析。这些分析模型不仅包含固定效应，还包含一些随机效应，此处暂将两类效应变量统称为效应项。在实际使用时，如何对这些效应项进行组合，构造一个最优的回归模型，当前并无较好的方法，多是基于人为确定，操作起来极为不便。所以，如何灵活构建效应项的最优组合，是此类回归分析亟待解决的一个技术难点。另外，与非固定模式的单水平非线性回归分析相似，同一资料有多种可用的回归分析方法时，如何确定哪种分析方法最适合给定的实际资料，也是一项重要的研究内容。再者，多水平模型分析的研究虽已如火如荼，但其模型求解的算法仍不够完善。实际使用时，应如何选择参数估计方法，并对其进行正确的假设检验，也需要作深入研究。
本研究针对这些非线性回归分析在实际使用中存在的技术难点进行研究，并借助SAS软件的编程语言、高级编程技巧和相关过程，实现资料的智能化、自动化分析，为使用者直接提供最优的回归分析方法和分析结果。
【方法】如何实现固定模式的非线性回归分析？本研究采用曲线直线化法加非线性最小二乘法的组合策略，即先以曲线直线化法为基础，计算得到模型中参数的初始值，再采用非线性最小二乘法得到拟合效果更好的曲线模型。在曲线直线化的过程中，有些较为简单的模型经过变量变换和数学推导，即可直接进行直线回归分析；有些复杂的模型不能直接进行曲线直线化，可以先选择一个或两个变化范围较小的参数，通过设置循环变量的方式使其按一定步长在较小的可能值域内变动，这样在每次循环中这些参数都将有具体的值，此时再对曲线模型进行数学推导和变量变换，即可进行直线回归分析。直线回归分析得到的参数估计值，经某些必要的计算后，就可以作为曲线模型中相应参数的初始值。当然，在曲线直线化的过程中，如果对某些参数设置了多个可能的取值，必然会得到较多的初始值组合，进而会得到多个局部最优模型，此时可从中选出拟合效果最好的那个模型作为全局最优模型，这样可以有效地破解局部最优解的难题。
如何实现非固定模式的单水平非线性回归分析？本研究采用了完全意义上的“最优子集法”来解决常规变量筛选方法存在的理论缺陷。当前某些大型统计软件也提供了最优子集法，但仅仅能给出各种原因变量组合情形下模型对资料的拟合效果统计量的值，并未涉及模型中参数估计的结果和假设检验的结果，更未直接给出最优模型，只是提供了含各种数量原因变量的“最优子集”，而最优模型中原因变量的数目仍需由使用者确定。具体来说，本研究考虑在各种原因变量组合情形下，均进行相应模型的构建和统计分析，然后从模型精简程度和拟合优度两方面，选出一个最优模型。
如何实现非固定模式的多水平非线性回归分析？一般统计软件都未提供效应项的筛选方法，只能依靠使用者采用手工法人为调整模型，操作起来非常不便。本研究同样采用完全意义上的“最优子集法”来选择最优模型，即先以固定效应项和随机效应项进行全面组合，然后在每种组合情形下，均进行多水平非线性回归分析，最后从众多模型中，选择拟合效果最好的那个模型作为全局最优模型。当然，对于某些类型的资料，分析时还需考虑分析方法的应用前提条件不满足时如何处理。
对于同类中的多种非线性模型选择问题，可依据模型的类型和参数估计方法选择相应的拟合优度评价统计量，对各模型的拟合效果进行比较，选择拟合效果最好的那个回归模型作为最优模型。
以上方法，具体实现时，可借助SAS软件的编程语言及相应过程，如进行最优子集法分析时，可借助REG、LOGISTIC等过程产生固定效应的全面组合情形，多水平模型分析时，还需借助FACTEX等过程产生随机效应的全面组合。
【结果】本研究对各类非线性回归分析的现有分析策略、计算方法和实现途径在实际应用中存在的问题和不足进行改进，有针对性地提出了相应最优模型的选择策略，并通过编写大量的SAS程序，将这些策略付诸实施，最后以最直观、最简洁和最适用的形式呈现出来。详细来说，论文的研究结果和主要创新点包括以下四个方面。
⑴对三类固定模式的非线性回归分析进行了研究，摸索出了一套分析策略：即以变量变换和数学推导为基础，通过曲线直线化或分段曲线直线化的方式，得到曲线模型的参数初始值，然后借助相应的迭代算法完成非线性最小二乘法的精确拟合。这一策略可实现此类资料的快速、高效、精准拟合，且可行性、可操作性非常好。此外，本研究还解决了同类多种曲线模型拟合实际资料时的智能化选择问题，所编写的宏程序可以方便地调用以解决同类实际问题。
⑵对四类非固定模式（含计数资料）的单水平非线性回归分析进行研究，以当前变量筛选方法中存在的理论困境和应用难题为突破口，实现了完全意义上的“最优子集法”，确保了所得的最终模型即为全局最优模型；同时解决了多种相近功能分析方法处理同一资料时，分析方法的拟合效果比较与智能化选择问题，有效地避免了使用者处理实际资料时在分析方法选择上的盲目性和不确定性。
⑶对四类非固定模式（含计数资料）的多水平非线性回归分析进行研究，以完全意义上的“最优子集法”对效应项进行了全面组合，在拟合效果比较的基础上，自动化地给出了最优模型，破解了当前多水平模型分析时无法进行效应项筛选的难题。当然，也解决了多种相近功能分析方法处理同一资料时，分析方法的拟合效果比较与智能化选择问题，所编写的宏程序可以方便地被调用以解决同类实际问题。
⑷探索出了一条基于SAS9.2版（之前版本无效）软件进行多水平非线性回归分析时参数估计方法的选择与调整策略，即需要估计的效应项参数较少时，可直接调用操作较为方便的GLIMMIX过程，使用数值积分近似法进行参数估计，但须在假设检验时对自由度进行调整；若需要估计的效应项较多，尤其是随机效应项较多时，直接使用GLIMMIX过程中的数值积分近似法往往无法得到收敛的结果，可先使用GLIMMIX过程中的线性化法得到参数的近似精确估计值，然后以此为初始值，再使用NLMIXED过程中的数值积分近似法作精确估计。
关于GLIMMIX过程，它是SAS软件近年来力推的新过程，与使用极其繁杂的NLMIXED过程相比，具有操作简便的特点，不过该过程虽经多年丰富和发展，但至今仍未能尽善尽美，其不足之处在于：一是未提供随机效应的假设检验结果，二是固定效应的假设检验结果也不准确。使用数值积分近似法时，两过程仅因计算精度不同导致GLIMMIX过程输出的参数值及标准误的估计结果与NLMIXED过程存在非常微小的差异，但GLIMMIX过程的重大缺陷在于其对随机效应未输出自由度，对固定效应虽输出了自由度但却是错误的，故GLIMMIX过程输出的假设检验结果不能直接被采用，须在假设检验时对效应项的自由度进行调整，从而得到准确的概率值。
以上策略，以调用GLIMMIX过程并对假设检验结果稍作调整为主，特殊情形下辅用NLMIXED过程，外加相应的编程，可直接或近似直接地获得最优模型的精确估计，这极大地节省了人工调整模型时的工作量和因人而异的不确定性，相对于王济川等人提出的解决类似问题的策略不仅简化了很大的工作量，而且能够较为方便地实现以最优子集法筛选效应项并通过编程直接获得最终结果的目的。
【结论】本论文以非线性回归分析为主要研究内容，对实际使用中存在的诸多瓶颈问题、不足和不便之处，进行了探索性的实践，并获得了令人满意的结果。
在进行固定模式的非线性回归分析时，摸索出了一套高效的分析策略。这一分析策略在解决三类固定模式的非线性回归分析时，被证明是行之有效的。该策略既可快速便捷地获得参数的估计值，还可得到较传统分析方法拟合效果更优的曲线回归模型，且对其他固定模式的非线性回归分析具有一定参考和借鉴意义。
在进行非固定模式的非线性回归分析时，以完全意义上的“最优子集法”进行原因变量或效应项的筛选，得到对实际资料拟合效果最好的回归模型。这一分析策略，不但避免了常用变量筛选方法在理论上的先天缺陷，也弥补了统计软件中所谓“最优子集法”在应用上的后天不足，可以方便快捷且准确可靠地为使用者提供最优的回归模型。
此外，多种相近功能非线性回归分析方法联用，在理论和技术层面上给出获得最优分析结果的策略、借助SAS系统并加以深层次开发和巧妙利用，使灵活而又复杂的计算策略得以准确完整地实现，在当前众多分析方法孰优孰劣或适用场合不明以及统计软件智能化水平低下的情况下，不失为一个大胆而成功的尝试。当前的统计研究发展迅速，往往是多种分析方法均可实现同一分析目的，较之随意选用其中的某种分析方法，该策略所得结果能更好地诠释资料的内在规律性。
【Objective】To overcome the almost insurmountable obstacles in practicalapplications of some kinds of regularly used simple or multiple nonlinear regressionanalysis(NRA), and obtain the regression model best befitting practical data, therebyperfecting those nonlinear regression analyses both in terms of theory andmethodology and promoting the operability and intelligent implementation to anincreased popularity and more successful application of these models.
【Content】Nonlinear regression analysis in this study is divided into twocategories: fixed mode and unfixed mode. Fixed mode is a mode in which bothexplanatory variables and model structure have been determined. Unfixed mode is amode in which only model structure has been decided. According to the hierarchicalstructure of the data, nonlinear regression analysis of unfixed mode can be subdividedinto two types: single level mode and multilevel mode.
Nonlinear regression analysis of fixed mode with one independent variable inthis study includes sum of exponentials curves, sigmoidal growth curves, andyield-density curves. These models are used frequently in practical application.However, in these complicated structure models, many parameters need to beestimated. Although there are many methods which can fit these models, the precisionof fitness is vastly inferior to the effect of nonlinear least squares (NLS) method. Themethod of NLS is based upon iterative algorithm, and users must specify veryapproximate initial values of parameters, otherwise the iteration may not be able toachieve convergence or the model is just a local optimization. Therefore, it is asubject worthy of research how to obtain a precise and general optimal curve modelswiftly. Additionally, there are generally multiform models in each kind of curvemodel. Take for example yield-density curves model, it includes Bleasdale-Neldercurve model, Halliday curve model, Farazdaghi-Harris curve model. In-depth researchis needed to identify the best model from multiform models while dealing withpractical problems.
Single level nonlinear regression analysis of unfixed mode with multipleindependent variables in this study includes single level models for binary, ordinal,multinomial and count outcomes. These methods yield very simplified and nicely fitregression model by using variable selection. However, major methods of variableselection, including forward method, backward method, stepwise method, all havetheir drawbacks in terms of theory and cannot ensure that the model is the best one.Consequently, it is a technical challenge to put forward or actualize the perfectvariable selection method. Additionally, there are always several available regressionanalysis methods for each data. For example, nonlinear regression methods used todeal with qualitative data with binary dependent variable, include logistic regressionanalysis, probit regression analysis and complementary log-log regression analysis.Another technical problem that merits great attention is how to compare the effect ofseveral regression methods and provide the best models automatically.
Multilevel nonlinear regression analysis of unfixed mode with multipleindependent variables in this study includes multilevel models for binary, ordinal,multinomial and count outcomes. These analysis models consist of fixed effects andrandom effects, and the two kinds of effect variables are termed effect items. Inpractical use, at present a desired method is not available to assemble effect itemsrationally and construct an optimal regression model. The largely artificially selectedmethod frequently used is very cumbersome to manipulate. Accordingly, a technicalproblem calling for urgent solution is how to flexibly achieve an optimal combinationof effect items in multilevel nonlinear regression analysis. Furthermore, there areseveral regression analysis methods for each data, as in the case of single levelnonlinear regression analysis. For this reason, it is also an important research topic tofind the most appropriate analysis method for the specific data. Moreover, studies ofmultilevel modeling analysis have been booming, but there is room for improvementin algorithms for parameters estimation. In actual use, an in-depth study isimperatively needed to select the estimation method and test the hypothesis forparameters correctly.
The paper researches and overcomes the technical difficulties in nonlinearregression analysis described above by making use of programming language,advanced programming skills and corresponding procedures of SAS softwareemployed in intelligentized and roboticized analysis of data, providing users withanalysis outcome by optimal methods.
【Methods】How to do nonlinear regression analysis of fixed mode? The paperadopts a strategy of combining linearisation method with NLS method. To be specific,the estimates obtained by linearisation method are treated as starting value, andsubsequently NLS method is employed to achieve a better curve model. In the processof linearisation, some simple models are subject to linear regression analysis (LRA)directly after variable transformation and mathematical calculation. In othercomplicated models that cannot be linearized directly, one or more incomprehensiveparameters are selected as loop variables, changeable in a small range by definitestep-size. In this way, the parameters will have certain values at each loop, and thecomplicated models can be linearized after variables transformation and mathematicalcalculation, generating models that can be evaluated via linear regression analysis.After some essential calculation, parameter values estimated by LRA can be regardedas the starting values of corresponding parameter in curve models. Nevertheless, ifparameters have several set values, there will be many combinations of initial values,and consequently a plurality of local optimal models will be obtained. The best modelamong such local optimal models can be regarded as globe optimal model, where thedifficulty of local optimal solution can be resolved effectively.
How to perform single level nonlinear regression analysis of unfixed mode? Thepaper adopts normal meaning method of best regression subset to solve theoreticlimitation of ordinary variable selection methods. Although many large statisticalanalysis software packages provide best regression subset method, they only presentthe values of model fit statistic in every case of explanatory variables’ combination,where the outcome of parameter estimate and testing hypotheses is not provided, notto say an optimal model. Which means that such software packages merely presentbest subsets with different number of explanatory variables, where the appropriatenumber has to be decided by users. In this paper models are constructed and analysisperformed in every case of explanatory variables’ combination to provide an optimalmodel based on simplification and goodness of fit (GOF).
How to realize multilevel nonlinear regression analysis of unfixed mode?Generally speaking, the common statistical software packages do not provide methodof effect items selection. Hence, adjustment of model has to depend on artificialmethod by users, which is very inconvenient to manipulate. The paper adopts normalmeaning method of best regression subset to select optimal model. First, fixed effectsand random effects are combined completely. Second, multilevel nonlinear regression analysis is done in every combination situation. And finally, the best fit model isselected as globe optimal model. In addition, for some types of data considerationshould be given to the situation where the precondition for the analysis methods to beused is not met.
As for selection among several congener nonlinear models, the paper choosesrelevant GOF evaluation statistic according to the type of model and the method ofparameter estimate, compares fitting effect among models, and then selects the best fitmodel as optimal model.
By virtue of programming language and corresponding procedures of SASsoftware, such methods described above can be actualized. For example, the completecombination of fixed effects can be produced using procedure REG and procedureLOGISTIC, and the complete combination of random effects can be produced usingprocedure FACTEX in multilevel modelling analysis.
【Results】The paper attempts to address the problems and remedy thedrawbacks with the current analysis tactics, calculation methods and actualizationapproaches in the practical application of NRA, and propose tactics of optimal modelselection, following through the schemes by programming in SAS to achieve an easilyperceptible outcome. In detail, the results and major innovations of this paper aresummarized in four aspects.
Part one is devoted to research on three kinds of nonlinear regression analysis offixed mode and exploration of an analysis tactic. Specifically, the estimates obtainedby linearisation method is regarded as starting value, to be followed by use of NLSmethod based on iterative algorithm to achieve a precise model. The tactic is markedby accurate, quick and efficient fitness, and has good feasibility and maneuverability.In addition, this part aims to solve the problem of intelligent selection among severalcongener nonlinear models while dealing with the same practical data, and programsome macros that can be used expediently.
Part two explores four kinds of single level nonlinear regression analysis ofunfixed mode to address the theoretic limitation and application difficulty withcurrent variable selection methods, and to ensure that the established model is theoptimal one by adopting normal meaning method of best regression subset.Furthermore, this part actualizes the automatic comparison of fitting effect andintelligentized selection among several congener analysis methods while dealing withthe same practical data, which helps avoid user’s blindness and uncertainty in method selection effectively.
Part three researches four kinds of multilevel nonlinear regression analysis ofunfixed mode, forms entire combinations of effect items by adopting normal meaningmethod of best regression subset, and solves the problem of a lack of method of effectitems selection in multilevel modelling analysis at the present time, and provides theoptimal model automatically after comparison of fitting effect. Additionally, this partsolves the problem of automatic comparison of fitting effect and intelligent selectionamong several congener nonlinear models while dealing with the same practical data,and programs some macros that can be used expediently.
Part four explores a selection and rectification tactic of estimation method inmultilevel nonlinear regression analysis based on SAS version9.2. If there is a smallnumber of random effects, procedure GLIMMIX can be used directly to obtainaccurate parameter values based on numerical integration methods, but the degree offreedom (DF) should be revised while performing hypothesis tests. If there is a largenumber of random effects, iteration of the foregoing means often fails to beconvergented, so procedure GLIMMIX can be used firstly to obtain approximateparameter values based on linearization-based methods, followed by the adoption ofprocedure NLMIXED to obtain accurate parameter values based on numericalintegration methods by taking the estimates obtained by linearization-based method asstarting values.
Procedure GLIMMIX is promoted zealously by SAS, which is very handycompared with the complicated procedure NLMIXED. In spite of the efforts madetowards its development and perfection in recent years, it is not perfect. Thedeficiencies of procedure GLIMMIX are as follows: first, it doesn’t performhypothesis tests to random effects; second, outcomes of hypothesis tests to fixedeffects are not correct. Tiny differences can be observed in estimate of parameters andtheir standard errors between procedure GLIMMIX and procedure NLMIXED usingnumerical integration methods, which is attributable to calculation precision. But theformer procedure has some serious defects in that it doesn’t provide DF of randomeffects and provide wrong DF of fixed effects. Therefore, results of hypothesis testsobtained by procedure GLIMMIX cannot be adopted directly, and DF of randomeffects in hypothesis tests should be adjusted so as to obtain exact probability.
The primary method of the tactic described above involves using procedureGLIMMIX and adjusting its hypothesis tests result. In some special cases, procedure NLMIXED is used as an adjunct. With the addition of programming, optimal modelcan be obtained directly or almost directly. The tactic can ease the workload and theuncertainty of artificial model selection. Compared with Wang’s tactic, it substantiallyreduces workload and can fulfill the aim of attaining optimal model expediently byadopting normal meaning method of best subset.
【Conclusions】The paper takes NRA as major research content, proposes somesolutions to many pivotal problems and attempts to address the drawbacks in practicalapplication, achieving some desirable results.
An efficient analysis tactic has been proposed for NRA of fixed mode, and it hasproved effective in dealing with three kinds of NRA of fixed mode. Not only canvalue of parameter be obtained swiftly, but also fitting effect of the model obtained bythis tactic is better than the one obtained by traditional method. It can be used as aframe of reference for other NRA of fixed mode.
In performing NRA of unfixed mode, optimal regression model can be obtainedby adopting normal meaning method of best subset to select variables. The tactic isfree of the inherent limitations of ordinary variable selection methods and overcomesthe drawbacks of the so-called method of best subset in statistic software, supplyingan optimal regression model for users expediently and reliably.
At present, there are many methods that can be applied to data analysis for thesame purpose, yet it is often unclear which is the best method for practical data. Thetactic introduced here that involving several congener NRA methods simultaneouslyto gain optimal model has been a bold and successful attempt. Taking intoconsideration the fact that the intelligent level of statistic software is very low, thisflexible and complicated calculation tactic can be fully actualized with the help ofadvanced development in SAS software. In brief, results obtained by this tactic aresuperior in interpreting the inherent law of data to those obtained by an arbitrarymethod.
引文
[1] Vittinghoff E, Shiboski SC, Glidden DV, et al. Regression methods inbiostatistics: linear,logistic,survival, and repeated measures models. New York:Springer-Verlag,2005:69-156
[2] Forthofer RN, Lee ES, Hernandez M. Biostatistics:a guide to design, analysis,and discovery.2ndedition. San Diego: Elsevier Inc.,2007:349-385
[3] Bates DM, Watts DG. Nonlinear regression analysis and its applications. NewYork: John Wiley&Sons, Inc.,1988:32-66
[4] Fox J, Weisberg S. Nonlinear regression and nonlinear least squares in R:appendix to an R companion to applied regression.2ndedition. http://socserv.mcmaster.ca/jfox/Books/Companion/appendix/Appendix-Nonlinear-Regression.pdf,2010:1-20
[5]方开泰,全辉,陈庆云.实用回归分析.北京:科学出版社,1988:168-183
[6]赵香兰.临床药代动力学——基础与应用.郑州:郑州大学出版社,2002:52-68
[7] Urso R, Blardi P, Giorgi G. A short introduction to pharmacokinetics. Eur RevMed Pharmacol Sci,2002,6(2-3):33-34
[8]梁文权.生物药剂学与药物动力学.第2版.北京:人民卫生出版社,2006:205-225
[9] Seber GAF, Wild CJ. Nonlinear Regression. New York: John Wiley&Sons,Inc.,2003:325-341
[10] Cobaleda C, García-Sastre A, Villar E. Fusion between Newcastle disease virusand erythrocyte ghosts using octadecyl Rhodamine B fluorescence assayproduces dequenching curves that fit the sum of two exponentials.J. Biol. Chem,1993,300(pt2):347-354
[11] Imbimbo BP, Martinelli P, Rocchetti M, et al. Efficiency of different criteria forselecting pharmacokinetic multiexponential equations. Biopharm Drug Dispos,1991,12(2):139-147
[12] Steltner H, Vogel M, Sorichter S, et al. Analysis of forced expired volumesignals using multi-exponential functions. Med Biol Eng Comput.,2001,39(2):190-194
[13] Nahashon SN, Aggrey SE, Adefope NA, et al. Growth characteristics of pearlgray guinea fowl as predicted by the richards, gompertz, and logistic models.Poultry Science,2006,85:359-363
[14] Forni S, Piles M, Blasco A, et al. Comparison of different nonlinear functions todescribe Nelore cattle growth. Journal of animal science,2009,87:496-506
[15] Vitezica ZG, Marie-Etancelin C, Bernadet MD, et al. Comparison of nonlinearand spline regression models for describing mule duck growth curves. PoultryScience,2010,89:1778-1784
[16] Harless C, DiStefano JJ III. Automated expert multiexponential biomodelinginteractively over the internet. Computer Methods and Programs inBiomedicine,2005,79:169-178
[17] Russell S, Distefano JJ Ⅲ. W3MAMCAT: a world wide web based tool formammillary and catenary compartmental modeling and expert systemdistinguishability. Comput Methods Programs Biomed.2006,83(1):34-42
[18] Graybill FA, Iyer HK. Regression analysis: concerpts and application.California:Duxbury Press,1994,501-550
[19] Li RZ, Lin DK. Variable selection for screening experiments. Qual TechnolQuant Manag.2009,6(3):271–280
[20] Walter S, Tiemeier H. Variable selection: current practice in epidemiologicalstudies. Eur J Epidemiol,2009,24:733–736
[21] Pourhoseingholi A, Pourhoseingholi MA, Vahedi M, et al. Relation betweendemographic factors and type of gastrointestinal cancer using probit and logitRegression. Asian Pacific Journal of Cancer Prevention,2008,l9:753-755
[22] Huang J, Huang B. Evaluating the proportion of treatment effect explained by acontinuous surrogate marker in logistic or probit regression models. StatBiopharm Res,2010,2:229–238
[23] Scott SC, Goldberg MS, Mayo NE. Statistical assessment of ordinal outcomesin comparative studies. J Clin Epidemiol,1997,50:45-55
[24] Javali SB, Pandit PV. A comparision of ordinal regression models in an analysisof factors associated with periodontal disease. Journal of Indian society ofperiodontology.2010,14(3):155-159
[25] Abreu MNS, Siqueira AL, Cardoso CS,et al. Ordinal logistic regression models:application in quality of life studies. Cad. Saúde Pública, Rio de Janeiro,2008,24(Sup4):S581-S591
[26] Likelihood ratio test for equal slopes assumption(ordinal regression algorithms).http://publib.boulder.ibm.com/infocenter/spssstat/v20r0m0/index.jsp?topic=%2Fcom.ibm.spss.statistics.help%2Falg_plum_statistics_chisquare_equal-slopes.htm
[27] Greenland S. Principles of multilevel modelling. International Journal ofEpidemiology,2000,29:158–167
[28] Greenland S. Multilevel modeling and model averaging. Scand J Work EnvironHealth,1999,25(suppl4):43-48
[29] Feldens CA, Kramer PF, Ferreira SH,et al.Exploring factors associated withtraumatic dental injuries in preschool children: a Poisson regression analysis.Dental Traumatology,2010,26(2):143-148.
[30] Hutchinson MK, Holtman MC. Analysis of count data using Poisson regression.Research in Nursing&Health,2005,28(5):408-418
[31] Henríquez FC, Castrillón SV. The use of a mixed Poisson model for tumourcontrol probability computation in non homogeneous irradiations. AustralasPhys Eng Sci Med,2011,34:267–272
[32] Gibbons RD, Segawa E, Karabatsos G. Mixed-effects Poisson regressionanalysis of adverse event reports:The relationship between antidepressants andsuicide. Stat Med,2008,27(11):1814–1833
[33] http://www.bristol.ac.uk/cmm/learning/mmsoftware/tables.html
[34] Albright JJ, Marinova DM. Estimating multilevel models using SPSS,Stata,SAS, and R. http://www.indiana.edu/~statmath/stat/all/hlm/hlm.pdf,2010:1-35
[35] Wang JC, Xie HY, Fisher JH. Multilevel models:application using SAS.Beijing:Higher Education Press,2009:113-174
[36] SAS Institute Inc. SAS/STAT9.22user’s guide. Cary, NC:SAS Institute Inc.2010:4892-4965
[37] Johnson ML. Nonlinear least-squares fitting methods. Methods Cell Biol.2008,84:781-805
[38] Jennrich RI, Ralston ML. Fitting nonlinear models to data. Ann. Rev. Biophys.Bioeng.,1979,8:195-238
[39] Hagquist C, Stenbeck M. Goodness of fit in regression analysis-R2and G2Reconsidered. Quality Quantity,1998,32:229-245
[40]郑青山,丁学庭.用相关指数作曲线拟合优度比较.中国卫生统计,1995,12(5):62-63
[41]杨树勤.中国医学百科全书(医学统计学).上海:上海科学技术出版社,1985:171-176
[42] Glatting G, Kletting P, Reske SN, et al. Choosing the optimal fit function:comparison of the Akaike information criterion and the F-test. Med Phys.2007,34(11):4285-4292
[43] Ludden TM, Beal SL, Sheiner LB. Comparison of the Akaike informationcriterion, the Schwarz criterion and the f test as guides to model selection. JPharmacokinet Biopharm.1994,22(5):431-445
[44]薛仲三.医学统计方法和原理（内部资料）.军事医学科学院印,1984:276-287
[45] Wu G. Fit fluctuating blood drug concentration:a beginner's first note.Pharmacol Res.1996,33(6):379-383
[46] Rygiewicz PT, Bledsoe CS, Glass ADM. A comparison of methods fordetermining compartmental analysis parameters. Plant Physiol,1984,76:913-917
[47] Trappey CV, Wu HY. An evaluation of the time-varying extended logistic,simple logistic, and Gompertz models for forecasting short product lifecycles.Advanced Engineering Informatics,2008,22:421–430
[48]胡良平. SAS统计分析教程.北京:电子工业出版社,2010:223-229
[49] Winsor CP. Gompertz curve as a growth curve. Proceedings of the NationalAcademy of Sciences,1932,18:1-8
[50]杨义群,吴良欢,张火法. Gompertz曲线与logistic增长曲线之比较.生物数学学报,1996,11(3):166-168
[51]王济川,郭志刚. Logistic回归模型——方法与应用.北京:高等教育出版社,2001:19-243
[52] SAS Institute Inc. SAS/STAT9.2user’s guide. Cary, NC:SAS Institute Inc.2008:2078-2419,3255-3470,4338-4431,5252-5350
[53] SAS Institute Inc. SAS/STAT9.3user’s guide. Cary, NC:SAS Institute Inc.2011:2607-2801
[54] Bingham NH, Fry JM. Regression: linear models in statistics. London:Springer-Verlag,2010:157-160
[55] Hosmer DW, Lemeshow S. Applied logistic regression.2ndEdition. NewYork:John Wiley&Sons, Inc.,2000:116-134
[56]郭祖超.医用数理统计分析方法.第3版.北京:人民卫生出版社,1988:446-449
[57] Posada D, Buckley TR. Model selection and model averaging in phylogenetics:advantages of Akaike information criterion and Bayesian approaches overlikelihood ratio tests. Syst. Biol.,2004,53(5):793–808
[58] Bozdogan H. Model selection and Akaike’s information criterion(AIC):thegeneral theory and its analytical extensions. Psychometrika,1987,52:345-370
[59] Bozdogan H. Akaike's information criterion and recent developments ininformation complexity. J Math Psychol,2000,44(1):62-91
[60] Wagenmakers EJ, Farrell S. AIC model selection using Akaike weights.Psychon Bull Rev,2004,11(1):192-196
[61] Acquah HG. Comparison of Akaike information criterion (AIC) and Bayesianinformation criterion (BIC) in selection of an asymmetric price relationship.Journal of Development and Agricultural Economics,2010,2(1):001-006
[62] Jones RH. Bayesian information criterion for longitudinal and clustered data.Stat Med,2011,30(25):3050-3056
[63] Cox DR, Snell EJ. The analysis of binary data.2ndedition. London:Chapmanand Hall,1989:1-146
[64] Nagelkerke NJD. A note on a general definition of the coefficient ofdetermination. Biometrika,1991,78:691-692
[65] Nick TG., Campbell KM. Logistic regression. Methods Mol Biol,2007,404:273-301
[66] Huet S, Bouvier A, Poursat M-A, et al. Statistical tools for nonlinearregression:a practical guide with S-PLUS and R examples.2ndedition. NewYork: Springer-Verlag,2004:153-198
[67]胡良平.医学统计学——运用三型理论进行现代回归分析.北京:人民军医出版社,2010:158-174
[68] Agresti A. Categorical data Analysis.2ndEdition. New York:John Wiley&Sons,Inc.,2002:245-250
[69] Hedeker D, Gibbons RD. MIXOR: a computer program for mixed-effectsordinal regression analysis. Comput Methods Programs Biomed,1996,49(2):157-176
[70] Martuzzi M, Elliott P. Estimating the incidence rate ratio in cross-sectionalstudies using a simple alternative to logistic regression. Ann Epidemiol,1998,8(1):52-55
[71] Penman AD, Johnson WD. Complementary log-log regression for theestimation of covariate-adjusted prevalence ratios in the analysis of data fromcross-sectional studies. Biom J,2009,51(3):433-442
[72] Xu HP, Craig BA. Likelihood analysis of multivariate probit models using aparameter expanded MCEM algorithm. Technometrics,2010,52(2):340-348
[73] Zhang X, Boscardin WJ, Belin TR. Bayesian analysis of multivariate nominalmeasures using multivariate multinomial probit models. Comput Stat Data Anal,2008,52(7):3697–3708
[74] Maura ES, Charles SD, Gary GK. Categorical data analysis using the SASsystem.2ndEdition. Cary, NC: SAS Institute Inc.,2000:257-270
[75] Liang KY. Generalized linear models, estimating functions and multivariateextensions.1999:43-55. http://www.stat.nthu.edu.tw/EngWeb/Teachers/kyliang930225.pdf
[76] Agresti A著.张淑梅,王睿,曾莉,译.属性数据分析引论.第2版.北京:高等教育出版社,2008:146-150,267-271
[77] David GK, Klein M. Logistic regression a self-learning text.3rdedition. NewYork: Springer-Verlag,2010:464-488
[78] Bender R, Grouven U. Ordinal logistic regression in medical research. J R CollPhysicians Lond,1997,31(5):546-551
[79] Lee J. Cumulative logit modelling for ordinal response variables: applicationsto biomedical research. Comput Appl Biosci.1992,8(6):555-562
[80] Warner P. Ordinal logistic regression. J Fam Plann Reprod Health Care,2008,34(3):169-170
[81] Kim JH. Assessing practical significance of the proportional odds assumption.Statistics&Probability Letters,2003,65(3):233-239
[82] Ananth CV, Kleinbaum DG. Regression models for ordinal responses:a reviewof methods and applications. International Journal of Epidemiology,1997,26(6):1323-1333
[83] Powers DA, Yu X. Statistics methods for categorical data analysis. AcademicPress, Inc.,1999:210-222
[84] Abreu MN, Siqueira AL, Caiaffa WT. Ordinal logistic regression inepidemiological studies. Rev Saúde Pública2009,43(1):183-194
[85] Zayeri F, Kazemnejad A, Khanafshar N, et al. Modeling repeated ordinalresponses using a family of power transformations: application to neonatalhypothermia data. BMC Medical Research Methodology,2005,5:art29
[86] McCullagh P. Regression models for ordinal data.Journal of the RoyalStatistical Society. Series B(Methodological),1980,42(2):109-142
[87] Peugh JL. A practical guide to multilevel modeling. J Sch Psychol,2010,48(1):85-112
[88] Goldstein H, Browne W, Rasbash J. Multilevel modelling of medical data. StatMed,2002,21(21):3291-3315
[89] Smith RB. Multilevel modeling of social problems: a causal perspective. NewYork: Springer,2011:1-22
[90] West SG, Ryu E, Kwok OM, et al. Multilevel modeling: current and futureapplications in personality research. J Pers,2011,79(1):2-50
[91] Kahn JH. Multilevel modeling: Overview and applications to research incounseling psychology. Journal of Counseling Psychology,2011,58(2):257-271
[92] Clarke P. When can group level clustering be ignored?Multilevel models versussingle-level models with sparse data. J Epidemiol Community Health,2008,62:752–758
[93] Raudenbush SW, Bryk AS. Hierarchical linear models: applications and dataanalysis methods. Thousand Oaks: Sage Publications Inc.,2001:1-15
[94] Goldstein H. Multilevel statistics models.4th edition. West Sussex: John Wiley&Sons ltd.,2010:1-14
[95] Gelman A, Hill J. Data analysis using regression and multilevel/hierarchicalmodels. Cambridge: Cambridge University Press,2006:246-247,325-342
[96] Nelder JA, Wedderburn RWM. Generalized linear models. Journal of Royalstatistical Society, Series A (General),1972,135(3):370-384
[97] Breslow NE, Clayton DG. Approximate inference in generalized linear mixedmodels. Journal of the American Statistical Association,1993,88(421):9-25
[98] Dean CB, Nielsen JD. Generalized linear mixed models: a review and someextensions. Lifetime Data Anal,2007,13(4):497-512
[99] Song XY, Lee SY. Model comparison of generalized linear mixed models. StatMed,2006,25(10):1685-1698
[100] Tuerlinckx F, Rijmen F, Verbeke G, et al. Statistical inference in generalizedlinear mixed models: a review. Br J Math Stat Psychol,2006,59(Pt2):225-255
[101]李丽霞,郜艳晖,张丕德,等.广义线性混合效应模型及其应用.现代预防医学,2007,34(11):2103-2104
[102] Schabenberger O. Introducing the GLIMMIX Procedure for Generalized LinearMixed Models. http://www2.sas.com/proceedings/sugi30/196-30.pdf
[103] Goldstein H. Nonlinear multilevel models, with an application to discreteresponse data. Biometrika,1991,78(1):45-51
[104] Dahlquist G, Bj rk. Numerical methods in scientific computing, volume I.Philadelphia:Socity for Industrial and Applied Mathematics,2008:521-608
[105] Hox JJ. Multilevel analysis:techniques and applications.2ndedition. New York:Routledge,2010:4-7,54-78,151-158
[106] McGraw KO, Wong SP. Forming inferences about some intraclass correlationcoefficients. Psychological Methods,1996,1(l):30-46
[107] Campbell MK, Grimshaw JM, Piaggio G. Intraclass correlation coefficient.Wiley Encyclopedia of Clinical Trials,2008:1-8
[108] Donner A. A review of inference procedures for the intraclass correlationcoefficient in the one-way random effects model. International StatisticalReview,1986,54(1):67-82
[109] Ibrahim JG, Zhu HT, Garcia RI, et al. Fixed and random effects selection inmixed effects models. Biometrics,2011,67(2):495–503
[110] Bondell HD, Krishna A, Sujit K. Ghosh. Joint variable selection for fixed andrandom effects in linear mixed-effects models. Biometrics,2010,66(4):1069-1077
[111] Jiang JM. Linear and generalized linear mixed models and their applications.New York: Springer-Verlag,2007:199-204
[112] Littell RC, Milliken GA, Stroup WW, et al. SAS for mixed models.2ndedition.SAS Institute Inc.,2006:13-552
[113] Flom PL, McMahon JM, Pouget ER. Using PROC NLMIXED and PROCGLMMIX to analyze dyadic data with a dichotomous dependent variable. SASGlobal Forum2007Proceedings.2007:179
[114] Witte JS, Greenland S, Kim LL, et al. Multilevel modeling in epidemiologywith GLIMMIX. Epidemiology,2000,11(6):684-688
[115] Capanu M, Begg CB. Hierarchical modeling for estimating relative risks of raregenetic variants: properties of the pseudo-likelihood method. Biometrics,2011,67(2):371-380
[116] Wolfinger R, O’Connell M. Generalized linear mixed models:a pseudo-likelihood approach. Journal of Statistical Computatuin and Simulation,1993,48:233-243
[117] McMahon JM, Pouget ER, Tortu S. A guide for multilevel modeling of dyadicdata with binary outcomes using SAS PROC NLMIXED. Comput Stat DataAnal.2006,50(12):3663–3680
[118] Richardson DB. A simple approach for fitting linear relative rate models in SAS.American Journal of Epidemiology.2008,168(11):1333-1338
[119] Patefield M. Fitting non-linear structural relationships using SAS procedureNLMIXED. The Statistician.2002,51:355–366
[120] Zheng M. Fitting linear and nonlinear growth curve models using PROCNLMIXED. SAS Global Forum2010. Paper260-2010
[121]茆诗松,周纪芗,陈颖.北京:中国统计出版社,2004:231-286
[122]王万中.试验的设计与分析.北京:高等教育出版社,2004:192-231
[123] Plan EL, Maloney A,Trocóniz IF, et al. Performance in population models forcount data, part I: maximum likelihood approximations. J PharmacokinetPharmacodyn,2009,36:353–366
[124] Shun ZM, Mccullagh P. Laplace approximation of high dimensional integrals. J.R. Statist. Soc. B,1995,57(4):749-760
[125] Wolfinger R. D. Laplace’s approximation for nonlinear mixed models.Biometrika,1993,80(4):791-795
[126] Golub GH, Welsch JH. Calculation of Gaussian quadrature rules. MathematicalComputing,1969,23:221-230
[127] Liu L, Huang X. The use of Gaussian quadrature for estimation in frailtyproportional hazards models. Stat Med,2008,27(14):2665-2683
[128] Liu Q, Pierce DA. A note on Gauss-Hermite quadrature. Biometrika,1994,81(3):624-629
[129] Pinheiro JC, Chao EC. Efficient Laplacian and adaptive Guassian quadraturealgorithms for multilevel generalized linear mixed models. Journal ofComputatonal and Graphical Statistics,2006,15(1):58-81
[130] Winkelmann R. Econometric analysis of count data.5th edition. New York:Springer-Verlag,2008:285-287
[131] Kreft IGG., De Leeuw J. Introducing multilevel modeling. London: SagePublications Inc.,1998:105-114
[132] Green MJ, Medley G., Browne WJ, et al. Use of posterior predictiveassessments to evaluate model fit in multilevel logistic regression. Vet.Res.,2009,40:30-39
[133] Kinney SK, Dunson DB. Fixed and random effects selection in linear andlogistic models. Biometrics,2007,63(3):690-698
[134] Longford NT. Logistic regression with random coefficient. ComputationalStatistics and Data Analysis,1994,17(1):1-15
[135] De Leeuw J, Meijer E. Handbook of multilevel analysis. New York: Springer-Verlag,2007:237-274
[136] Liang LM, Chen WM, Sham PC, et al. Variance components linkage analysiswith repeated measurements.Human Heredity,2008,67:237–247
[137] McCulloch CE. Maximum likelihood variance components estimation forbinary data. Journal of the American Statistical Association,1994,89(425):330-335
[138] SPSS Inc. IBM SPSS advanced statistics19. SPSS Inc:140-142
[139] SAS Institute Inc. SAS/QC9.2user’s guide. Cary, NC:SAS Institute Inc.2008:661-747
[140] Gibbons RD, Hedeker D. Application of random-effects probit regressionmodels. J Consult Clin Psychol.1994,62(2):285-296
[141] Turechek WW, Madden LV. A generalized linear modeling approach forcharacterizing disease incidence in a spatial hierarchy. Phytopathology.2003,93(4):458-466
[142] Raman R, Hedeker D. A mixed-effects regression model for three-level ordinalresponse data. Stat Med.2005,24(21):3331-3345
[143] Hedeker D, Gibbons RD. A random-effects ordinal regression model formultilevel analysis. Biometrics,1994,50(4):933-944
[144]王济川,谢海义,姜宝法.多层统计分析模型——方法与应用.北京:高等教育出版社,2008:44-80
[145] Gibbons R. D., Hedeker D. Random effects probit and logistic regressionmodels for three-level data. Biometrics.1997,53(4):1527-1537
[146] Hedeker D. A mixed-effects multinomial logistic regression model. Statistics inMedicine,2003,22:1433–1446
[147]张岩波,何大卫,梁执群.多分类反应变量的多水平多项式模型及其应用.中国卫生统计,2000,17(3):151-153
[148]夏结来,徐雷.计数资料的统计分析模型.疾病控制杂志,2003,7(2):81-84
[149]曾平,赵晋芳,刘桂芬. Poisson回归中过度离散的检验方法.中国卫生统计,2011,28(2):211-212
[150] Consul PC, Famoye F. Generalized poisson regression model. Communicationsin statistics:theroy and methods.1992,21:89-109
[151] Cameron AC, Trivedi PK. Regression Analysis of Count Data. New York:Cambridge University Press,1998:3-18
[152] Zelterman D. Models for discrete data. Revised edition. New York: OxfordUniversity Press Inc.,2006:12-20
[153] Waltoft BL. A SAS-macro for estimation of the cumulative incidence usingPoisson regression. Computer methods and programs in biomedicine,2009,93:140–147
[154] Ver Hoef JM, Boveng PL.Quasi-Poisson vs. negative binomial regression: howshould we model overdispersed count data? Ecology.2007,88(11):2766-2772
[155] Lindén A, M ntyniemi S. Using the negative binomial distribution to modeloverdispersion in ecological count data. Ecology.2011,92(7):1414-1421
[156] Montgomery DC, Runger GC. Applied statistics and probability for engineers.3rdedition. New York: John Wiley&Sons, Inc.,2002:72-96
[157]茆诗松.统计手册.北京:科学出版社,2006:1004-1007
[158] Vives J, Losilla JM, Rodrigo MF, et al. Overdispersion tests in count-dataanalysis.Psychol Rep.2008,103(1):145-160
[159] Cox DR. Some remarks on over-dispersion. Biometrika.1983,70:269-274
[160] Xiang L, Lee AH. Sensitivity of test for overdispersion in Poisson regression.Biometrical Journal.2005,47(2):167-176
[161] Yang Z, Hardin JW, Addy CL, et al. Testing approaches for overdispersion inpoisson regression versus the generalized poisson model. Biometrical Journal.2007,49(4):565-584
[162] Trocóniz IF, Plan EL, Miller R, et al. Modelling overdispersion and Markovianfeatures in count data. J Pharmacokinet Pharmacodyn,2009,36:461–477
[163] Coxe S, West SG, Aiken LS. The analysis of count data: a gentle introduction topoisson regression and its alternatives. Journal of Personality Assessment,2009,91(2):121-136
[164] Gardner W, Mulvey EP, Shaw EC. Regression analyses of counts and rates:Poisson, overdispersed Poisson, and negative binomial models. PsycholBull,1995,118(3):392-404
[165]陈峰.医用多元统计分析方法.第二版.北京:中国统计出版社,2007:114-131
[166]杨珉,李晓松.医学和公共卫生研究常用多水平统计模型.北京:北京大学出版社,2007:106-118
[167] Li BY, Lingsma HF, Steyerberg EW, et al. Logistic random effects regressionmodels: a comparison of statistical packages for binary and ordinal outcomes.BMC Medical Research Methodology,2011,11:art77
[168] Austin PC. Estimating multilevel logistic regression models when the numberof clusters is low: a comparison of different statistical software procedures. TheInternational Journal of Biostatistics,2010,6(1):art16
[169] Box GEP. Science and statistics. J. Am. Stat. Assoc.,1976,71:791–799
[170] Burnham KP, Anderson DR. Model selection and multimodel inference: apractical information-theoretic approach.2ndedition. New York: Springer-Verlag,2003:20-23
[1] Wang JC, Xie HY, Fisher JH. Multilevel models:application using SAS.Beijing:Higher Education Press,2009:30
[2] Jennrich RI, Ralston ML. Fitting nonlinear models to data. Ann. Rev. Biophys.Bioeng.,1979,8:195-238
[3]郑青山,丁学庭.用相关指数作曲线拟合优度比较.中国卫生统计,1995,12(5):62-63
[4] Graybill FA, Iyer HK. Regression analysis:concerpts and application. California:Duxbury Press,1994:501-550
[5]胡良平.现代统计学与SAS应用.北京:军事医学科学院,2000:263
[6]杨树勤.中国医学百科全书(医学统计学).上海:上海科学技术出版社,1985:171-176
[7] Akaike H. A New Look at Statistical Model Identification. IEEE Transactions onAutomatic Control,1974,19:716–723
[8] Hurvich CM, Tsai C-L. Regression and Time Series Model Selection in SmallSamples. Biometrika,1989,76:297–307
[9] Schwarz G. Estimating the Dimension of a Model. Annals of Statistics,1978,6:461–464
[10]Hannan EJ, Quinn BG. The Determination of the Order of an Autoregression.Journal of the Royal Statistical Society, Series B,1979,41:190–195
[11]张家放.医用多元统计方法.武汉:华中科技大学出版社,2002:97-99
[12]金丕焕,陈峰.医用统计方法.第三版.上海:复旦大学出版社,2009:358-364
[13]方开泰,全辉,陈庆云.实用回归分析.北京:科学出版社,1988:168-234
[14]薛仲三.医学统计方法和原理（内部资料）.军事医学科学院印,1984:276-287
[15]梁文权.生物药剂学与药物动力学.第二版.北京:人民卫生出版社,2006:205-225
[16]Seber GAF, Wild CJ. Nonlinear Regression. New York: John Wiley&Sons,Inc.,2003:325-341
[17]Winsor CP. Gompertz curve as a growth curve. Proceedings of the NationalAcademy of Sciences,1932,18:1-8
[18]胡良平.医学统计学——运用三型理论进行现代回归分析.北京:人民军医出版,2010:57-58
[19]Johnson ML. Nonlinear least-squares fitting methods. Methods Cell Biol.2008,84:781-805
[20]郑洲顺,普乐.非线性最小二乘问题的一种迭代解法.数学理论与应用.2002,22(1):43-45
[21]SAS Institute Inc. SAS/STAT9.22User’s Guide. Cary, NC:SAS Institute Inc.2010:4892-4965
[22]王济川,郭志刚. Logistic回归模型——方法与应用.北京:高等教育出版社,2001:19-243
[23]Agresti A. Categorical Data Analysis.2ndEdition. New York:John Wiley&Sons,Inc.,2002:245-250
[24]Penman AD, Johnson WD. Complementary log-log regression for the estimationof covariate-adjusted prevalence ratios in the analysis of data from cross-sectionalstudies. Biom J,2009,51(3):433-442
[25]Warner P. Ordinal logistic regression. J Fam Plann Reprod Health Care,2008,34(3):169-170
[26]Kim JH. Assessing practical significance of the proportional odds assumption.Statistics&Probability Letters,2003,65(3):233-239
[27]Powers DA, Xie Y. Statistics methods for categorical data analysis. AcademicPress, Inc.,1999:210-222
[28]McCullagh P. Regression models for ordinal data. Journal of the Royal StatisticalSociety, Series B(Methodological),1980,42(2):109-142
[29]Maura ES, Charles SD, Gary GK. Categorical Data Analysis Using the SASSystem.2ndEdition. Cary, NC: SAS Institute Inc.,2000:257-270
[30]Zelterman D. Models for Discrete Data. Revised edition. New York: OxfordUniversity Press Inc.,2006:12-20
[31]茆诗松.统计手册.北京:科学出版社,2006:1004-1007
[32]Vives J, Losilla JM, Rodrigo MF, et al. Overdispersion tests in count-dataanalysis.Psychol Rep.2008,103(1):145-160
[33]Coxe S, West SG, Aiken LS. The analysis of count data: a gentle introduction topoisson regression and its alternatives. Journal of Personality Assessment,2009,91(2):121-136
[34]Trocóniz IF, Plan EL, Miller R, et al. Modelling overdispersion and Markovianfeatures in count data. J Pharmacokinet Pharmacodyn,2009,36:461–477
[35]Lindén A, M ntyniemi S.Using the negative binomial distribution to modeloverdispersion in ecological count data. Ecology.2011,92(7):1414-1421
[36]Nelder JA, Wedderburn RWM. Generalized linear models. Journal of Royalstatistical Society, Series A (General),1972,135(3):370-384
[37]陈峰.医用多元统计分析方法.第二版.北京:中国统计出版社,2007:114-131
[38]SAS Institute Inc.. SAS/STAT9.3User’s Guide. Cary, NC:SAS Institute Inc.,2011:2607-2801
[39]杨珉,李晓松.医学和公共卫生研究常用多水平统计模型.北京:北京大学出版社,2007:106-118
[40]Kinney SK, Dunson DB. Fixed and random effects selection in linear and logisticmodels. Biometrics,2007,63(3):690-698
[41]De Leeuw J, Meijer E. Handbook of multilevel analysis. New York: Springer-Verlag,2007:237-274
[42]Gibbons RD, Hedeker D. Application of random-effects probit regression models.J Consult Clin Psychol.1994,62(2):285-296
[43]Turechek WW, Madden LV. A generalized linear modeling approach forcharacterizing disease incidence in a spatial hierarchy. Phytopathology.2003,93(4):458-466
[44]Hedeker D, Gibbons RD. A random-effects ordinal regression model formultilevel analysis. Biometrics,1994,50(4):933-944
[45]Gibbons RD, Hedeker D. Random effects probit and logistic regression modelsfor three-level data. Biometrics,1997,53(4):1527-1537
[46]Hedeker D. A mixed-effects multinomial logistic regression model. Statistics inMedicine,2003,22:1433–1446
[47]张岩波,何大卫,梁执群.多分类反应变量的多水平多项式模型及其应用.中国卫生统计,2000,17(3):151-153
[48]Gelman A, Hill J. Data analysis using regression and multilevel/hierarchicalmodels. Cambridge: Cambridge University Press,2006:246-247,325-342
[49]Hox JJ. Multilevel analysis:techniques and applications.2ndedition. New York:Routledge,2010:4-7,54-78,151-158
[50]Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics,1982,38(4):963-974.
[51]谢远涛,杨娟.广义Gamma分布簇广义线性混合模型的构建.统计研究,2010,27(10):75-80
[52]Dean CB, Nielsen JD. Generalized linear mixed models: a review and someextensions. Lifetime Data Anal,2007,13(4):497-512
[53]Tuerlinckx F, Rijmen F, Verbeke G, et al. Statistical inference in generalizedlinear mixed models: a review. Br J Math Stat Psychol,2006,59(Pt2):225-255
[54]Goldstein H. Nonlinear multilevel models, with an application to discreteresponse data. Biometrika,1991,78(1):45-51
[55]Agresti A著.张淑梅,王睿,曾莉,译.属性数据分析引论.第二版.北京:高等教育出版社,2008:267-271
[56]王济川,谢海义,姜宝法.多层统计分析模型——方法与应用.北京:高等教育出版社,2008:128-132
[57]杨志雄,袁岱菁.非线性混合效应模型和广义线性模型拟合随机效应logistic回归的应用比较.中国卫生统计,2011,28(3):321-323
[58]Vonesh EF. A note on the use of Laplace’s approximation for nonlinear mixed-effects models. Biometrika,1996,83(2):447–452
[59]Liu Q, Pierce DA. A note on Gauss-Hermite quadrature. Biometrika,1994,81(3):624-629
[60]Winkelmann R. Econometric analysis of count data.5thedition. New York:Springer-Verlag,2008:285-287
[61]Pinheiro JC, Chao EC. Efficient Laplacian and adaptive Guassian quadraturealgorithms for multilevel generalized linear mixed models. Journal ofComputatonal and Graphical Statistics,2006,15(1):58-81
[62]Pinheiro JC, Bates DM. Approximations to the Loglikelihood Function in theNonlinear Mixed Effects Model. Journal of Computational and GraphicalStatistics,1995,4(1):12-35
[2] Forthofer RN, Lee ES, Hernandez M. Biostatistics:a guide to design, analysis,and discovery.2ndedition. San Diego: Elsevier Inc.,2007:349-385
[3] Bates DM, Watts DG. Nonlinear regression analysis and its applications. NewYork: John Wiley&Sons, Inc.,1988:32-66
[4] Fox J, Weisberg S. Nonlinear regression and nonlinear least squares in R:appendix to an R companion to applied regression.2ndedition. http://socserv.mcmaster.ca/jfox/Books/Companion/appendix/Appendix-Nonlinear-Regression.pdf,2010:1-20
[5]方开泰,全辉,陈庆云.实用回归分析.北京:科学出版社,1988:168-183
[6]赵香兰.临床药代动力学——基础与应用.郑州:郑州大学出版社,2002:52-68
[7] Urso R, Blardi P, Giorgi G. A short introduction to pharmacokinetics. Eur RevMed Pharmacol Sci,2002,6(2-3):33-34
[8]梁文权.生物药剂学与药物动力学.第2版.北京:人民卫生出版社,2006:205-225
[9] Seber GAF, Wild CJ. Nonlinear Regression. New York: John Wiley&Sons,Inc.,2003:325-341
[10] Cobaleda C, García-Sastre A, Villar E. Fusion between Newcastle disease virusand erythrocyte ghosts using octadecyl Rhodamine B fluorescence assayproduces dequenching curves that fit the sum of two exponentials.J. Biol. Chem,1993,300(pt2):347-354
[11] Imbimbo BP, Martinelli P, Rocchetti M, et al. Efficiency of different criteria forselecting pharmacokinetic multiexponential equations. Biopharm Drug Dispos,1991,12(2):139-147
[12] Steltner H, Vogel M, Sorichter S, et al. Analysis of forced expired volumesignals using multi-exponential functions. Med Biol Eng Comput.,2001,39(2):190-194
[13] Nahashon SN, Aggrey SE, Adefope NA, et al. Growth characteristics of pearlgray guinea fowl as predicted by the richards, gompertz, and logistic models.Poultry Science,2006,85:359-363
[14] Forni S, Piles M, Blasco A, et al. Comparison of different nonlinear functions todescribe Nelore cattle growth. Journal of animal science,2009,87:496-506
[15] Vitezica ZG, Marie-Etancelin C, Bernadet MD, et al. Comparison of nonlinearand spline regression models for describing mule duck growth curves. PoultryScience,2010,89:1778-1784
[16] Harless C, DiStefano JJ III. Automated expert multiexponential biomodelinginteractively over the internet. Computer Methods and Programs inBiomedicine,2005,79:169-178
[17] Russell S, Distefano JJ Ⅲ. W3MAMCAT: a world wide web based tool formammillary and catenary compartmental modeling and expert systemdistinguishability. Comput Methods Programs Biomed.2006,83(1):34-42
[18] Graybill FA, Iyer HK. Regression analysis: concerpts and application.California:Duxbury Press,1994,501-550
[19] Li RZ, Lin DK. Variable selection for screening experiments. Qual TechnolQuant Manag.2009,6(3):271–280
[20] Walter S, Tiemeier H. Variable selection: current practice in epidemiologicalstudies. Eur J Epidemiol,2009,24:733–736
[21] Pourhoseingholi A, Pourhoseingholi MA, Vahedi M, et al. Relation betweendemographic factors and type of gastrointestinal cancer using probit and logitRegression. Asian Pacific Journal of Cancer Prevention,2008,l9:753-755
[22] Huang J, Huang B. Evaluating the proportion of treatment effect explained by acontinuous surrogate marker in logistic or probit regression models. StatBiopharm Res,2010,2:229–238
[23] Scott SC, Goldberg MS, Mayo NE. Statistical assessment of ordinal outcomesin comparative studies. J Clin Epidemiol,1997,50:45-55
[24] Javali SB, Pandit PV. A comparision of ordinal regression models in an analysisof factors associated with periodontal disease. Journal of Indian society ofperiodontology.2010,14(3):155-159
[25] Abreu MNS, Siqueira AL, Cardoso CS,et al. Ordinal logistic regression models:application in quality of life studies. Cad. Saúde Pública, Rio de Janeiro,2008,24(Sup4):S581-S591
[26] Likelihood ratio test for equal slopes assumption(ordinal regression algorithms).http://publib.boulder.ibm.com/infocenter/spssstat/v20r0m0/index.jsp?topic=%2Fcom.ibm.spss.statistics.help%2Falg_plum_statistics_chisquare_equal-slopes.htm
[27] Greenland S. Principles of multilevel modelling. International Journal ofEpidemiology,2000,29:158–167
[28] Greenland S. Multilevel modeling and model averaging. Scand J Work EnvironHealth,1999,25(suppl4):43-48
[29] Feldens CA, Kramer PF, Ferreira SH,et al.Exploring factors associated withtraumatic dental injuries in preschool children: a Poisson regression analysis.Dental Traumatology,2010,26(2):143-148.
[30] Hutchinson MK, Holtman MC. Analysis of count data using Poisson regression.Research in Nursing&Health,2005,28(5):408-418
[31] Henríquez FC, Castrillón SV. The use of a mixed Poisson model for tumourcontrol probability computation in non homogeneous irradiations. AustralasPhys Eng Sci Med,2011,34:267–272
[32] Gibbons RD, Segawa E, Karabatsos G. Mixed-effects Poisson regressionanalysis of adverse event reports:The relationship between antidepressants andsuicide. Stat Med,2008,27(11):1814–1833
[33] http://www.bristol.ac.uk/cmm/learning/mmsoftware/tables.html
[34] Albright JJ, Marinova DM. Estimating multilevel models using SPSS,Stata,SAS, and R. http://www.indiana.edu/~statmath/stat/all/hlm/hlm.pdf,2010:1-35
[35] Wang JC, Xie HY, Fisher JH. Multilevel models:application using SAS.Beijing:Higher Education Press,2009:113-174
[36] SAS Institute Inc. SAS/STAT9.22user’s guide. Cary, NC:SAS Institute Inc.2010:4892-4965
[37] Johnson ML. Nonlinear least-squares fitting methods. Methods Cell Biol.2008,84:781-805
[38] Jennrich RI, Ralston ML. Fitting nonlinear models to data. Ann. Rev. Biophys.Bioeng.,1979,8:195-238
[39] Hagquist C, Stenbeck M. Goodness of fit in regression analysis-R2and G2Reconsidered. Quality Quantity,1998,32:229-245
[40]郑青山,丁学庭.用相关指数作曲线拟合优度比较.中国卫生统计,1995,12(5):62-63
[41]杨树勤.中国医学百科全书(医学统计学).上海:上海科学技术出版社,1985:171-176
[42] Glatting G, Kletting P, Reske SN, et al. Choosing the optimal fit function:comparison of the Akaike information criterion and the F-test. Med Phys.2007,34(11):4285-4292
[43] Ludden TM, Beal SL, Sheiner LB. Comparison of the Akaike informationcriterion, the Schwarz criterion and the f test as guides to model selection. JPharmacokinet Biopharm.1994,22(5):431-445
[44]薛仲三.医学统计方法和原理（内部资料）.军事医学科学院印,1984:276-287
[45] Wu G. Fit fluctuating blood drug concentration:a beginner's first note.Pharmacol Res.1996,33(6):379-383
[46] Rygiewicz PT, Bledsoe CS, Glass ADM. A comparison of methods fordetermining compartmental analysis parameters. Plant Physiol,1984,76:913-917
[47] Trappey CV, Wu HY. An evaluation of the time-varying extended logistic,simple logistic, and Gompertz models for forecasting short product lifecycles.Advanced Engineering Informatics,2008,22:421–430
[48]胡良平. SAS统计分析教程.北京:电子工业出版社,2010:223-229
[49] Winsor CP. Gompertz curve as a growth curve. Proceedings of the NationalAcademy of Sciences,1932,18:1-8
[50]杨义群,吴良欢,张火法. Gompertz曲线与logistic增长曲线之比较.生物数学学报,1996,11(3):166-168
[51]王济川,郭志刚. Logistic回归模型——方法与应用.北京:高等教育出版社,2001:19-243
[52] SAS Institute Inc. SAS/STAT9.2user’s guide. Cary, NC:SAS Institute Inc.2008:2078-2419,3255-3470,4338-4431,5252-5350
[53] SAS Institute Inc. SAS/STAT9.3user’s guide. Cary, NC:SAS Institute Inc.2011:2607-2801
[54] Bingham NH, Fry JM. Regression: linear models in statistics. London:Springer-Verlag,2010:157-160
[55] Hosmer DW, Lemeshow S. Applied logistic regression.2ndEdition. NewYork:John Wiley&Sons, Inc.,2000:116-134
[56]郭祖超.医用数理统计分析方法.第3版.北京:人民卫生出版社,1988:446-449
[57] Posada D, Buckley TR. Model selection and model averaging in phylogenetics:advantages of Akaike information criterion and Bayesian approaches overlikelihood ratio tests. Syst. Biol.,2004,53(5):793–808
[58] Bozdogan H. Model selection and Akaike’s information criterion(AIC):thegeneral theory and its analytical extensions. Psychometrika,1987,52:345-370
[59] Bozdogan H. Akaike's information criterion and recent developments ininformation complexity. J Math Psychol,2000,44(1):62-91
[60] Wagenmakers EJ, Farrell S. AIC model selection using Akaike weights.Psychon Bull Rev,2004,11(1):192-196
[61] Acquah HG. Comparison of Akaike information criterion (AIC) and Bayesianinformation criterion (BIC) in selection of an asymmetric price relationship.Journal of Development and Agricultural Economics,2010,2(1):001-006
[62] Jones RH. Bayesian information criterion for longitudinal and clustered data.Stat Med,2011,30(25):3050-3056
[63] Cox DR, Snell EJ. The analysis of binary data.2ndedition. London:Chapmanand Hall,1989:1-146
[64] Nagelkerke NJD. A note on a general definition of the coefficient ofdetermination. Biometrika,1991,78:691-692
[65] Nick TG., Campbell KM. Logistic regression. Methods Mol Biol,2007,404:273-301
[66] Huet S, Bouvier A, Poursat M-A, et al. Statistical tools for nonlinearregression:a practical guide with S-PLUS and R examples.2ndedition. NewYork: Springer-Verlag,2004:153-198
[67]胡良平.医学统计学——运用三型理论进行现代回归分析.北京:人民军医出版社,2010:158-174
[68] Agresti A. Categorical data Analysis.2ndEdition. New York:John Wiley&Sons,Inc.,2002:245-250
[69] Hedeker D, Gibbons RD. MIXOR: a computer program for mixed-effectsordinal regression analysis. Comput Methods Programs Biomed,1996,49(2):157-176
[70] Martuzzi M, Elliott P. Estimating the incidence rate ratio in cross-sectionalstudies using a simple alternative to logistic regression. Ann Epidemiol,1998,8(1):52-55
[71] Penman AD, Johnson WD. Complementary log-log regression for theestimation of covariate-adjusted prevalence ratios in the analysis of data fromcross-sectional studies. Biom J,2009,51(3):433-442
[72] Xu HP, Craig BA. Likelihood analysis of multivariate probit models using aparameter expanded MCEM algorithm. Technometrics,2010,52(2):340-348
[73] Zhang X, Boscardin WJ, Belin TR. Bayesian analysis of multivariate nominalmeasures using multivariate multinomial probit models. Comput Stat Data Anal,2008,52(7):3697–3708
[74] Maura ES, Charles SD, Gary GK. Categorical data analysis using the SASsystem.2ndEdition. Cary, NC: SAS Institute Inc.,2000:257-270
[75] Liang KY. Generalized linear models, estimating functions and multivariateextensions.1999:43-55. http://www.stat.nthu.edu.tw/EngWeb/Teachers/kyliang930225.pdf
[76] Agresti A著.张淑梅,王睿,曾莉,译.属性数据分析引论.第2版.北京:高等教育出版社,2008:146-150,267-271
[77] David GK, Klein M. Logistic regression a self-learning text.3rdedition. NewYork: Springer-Verlag,2010:464-488
[78] Bender R, Grouven U. Ordinal logistic regression in medical research. J R CollPhysicians Lond,1997,31(5):546-551
[79] Lee J. Cumulative logit modelling for ordinal response variables: applicationsto biomedical research. Comput Appl Biosci.1992,8(6):555-562
[80] Warner P. Ordinal logistic regression. J Fam Plann Reprod Health Care,2008,34(3):169-170
[81] Kim JH. Assessing practical significance of the proportional odds assumption.Statistics&Probability Letters,2003,65(3):233-239
[82] Ananth CV, Kleinbaum DG. Regression models for ordinal responses:a reviewof methods and applications. International Journal of Epidemiology,1997,26(6):1323-1333
[83] Powers DA, Yu X. Statistics methods for categorical data analysis. AcademicPress, Inc.,1999:210-222
[84] Abreu MN, Siqueira AL, Caiaffa WT. Ordinal logistic regression inepidemiological studies. Rev Saúde Pública2009,43(1):183-194
[85] Zayeri F, Kazemnejad A, Khanafshar N, et al. Modeling repeated ordinalresponses using a family of power transformations: application to neonatalhypothermia data. BMC Medical Research Methodology,2005,5:art29
[86] McCullagh P. Regression models for ordinal data.Journal of the RoyalStatistical Society. Series B(Methodological),1980,42(2):109-142
[87] Peugh JL. A practical guide to multilevel modeling. J Sch Psychol,2010,48(1):85-112
[88] Goldstein H, Browne W, Rasbash J. Multilevel modelling of medical data. StatMed,2002,21(21):3291-3315
[89] Smith RB. Multilevel modeling of social problems: a causal perspective. NewYork: Springer,2011:1-22
[90] West SG, Ryu E, Kwok OM, et al. Multilevel modeling: current and futureapplications in personality research. J Pers,2011,79(1):2-50
[91] Kahn JH. Multilevel modeling: Overview and applications to research incounseling psychology. Journal of Counseling Psychology,2011,58(2):257-271
[92] Clarke P. When can group level clustering be ignored?Multilevel models versussingle-level models with sparse data. J Epidemiol Community Health,2008,62:752–758
[93] Raudenbush SW, Bryk AS. Hierarchical linear models: applications and dataanalysis methods. Thousand Oaks: Sage Publications Inc.,2001:1-15
[94] Goldstein H. Multilevel statistics models.4th edition. West Sussex: John Wiley&Sons ltd.,2010:1-14
[95] Gelman A, Hill J. Data analysis using regression and multilevel/hierarchicalmodels. Cambridge: Cambridge University Press,2006:246-247,325-342
[96] Nelder JA, Wedderburn RWM. Generalized linear models. Journal of Royalstatistical Society, Series A (General),1972,135(3):370-384
[97] Breslow NE, Clayton DG. Approximate inference in generalized linear mixedmodels. Journal of the American Statistical Association,1993,88(421):9-25
[98] Dean CB, Nielsen JD. Generalized linear mixed models: a review and someextensions. Lifetime Data Anal,2007,13(4):497-512
[99] Song XY, Lee SY. Model comparison of generalized linear mixed models. StatMed,2006,25(10):1685-1698
[100] Tuerlinckx F, Rijmen F, Verbeke G, et al. Statistical inference in generalizedlinear mixed models: a review. Br J Math Stat Psychol,2006,59(Pt2):225-255
[101]李丽霞,郜艳晖,张丕德,等.广义线性混合效应模型及其应用.现代预防医学,2007,34(11):2103-2104
[102] Schabenberger O. Introducing the GLIMMIX Procedure for Generalized LinearMixed Models. http://www2.sas.com/proceedings/sugi30/196-30.pdf
[103] Goldstein H. Nonlinear multilevel models, with an application to discreteresponse data. Biometrika,1991,78(1):45-51
[104] Dahlquist G, Bj rk. Numerical methods in scientific computing, volume I.Philadelphia:Socity for Industrial and Applied Mathematics,2008:521-608
[105] Hox JJ. Multilevel analysis:techniques and applications.2ndedition. New York:Routledge,2010:4-7,54-78,151-158
[106] McGraw KO, Wong SP. Forming inferences about some intraclass correlationcoefficients. Psychological Methods,1996,1(l):30-46
[107] Campbell MK, Grimshaw JM, Piaggio G. Intraclass correlation coefficient.Wiley Encyclopedia of Clinical Trials,2008:1-8
[108] Donner A. A review of inference procedures for the intraclass correlationcoefficient in the one-way random effects model. International StatisticalReview,1986,54(1):67-82
[109] Ibrahim JG, Zhu HT, Garcia RI, et al. Fixed and random effects selection inmixed effects models. Biometrics,2011,67(2):495–503
[110] Bondell HD, Krishna A, Sujit K. Ghosh. Joint variable selection for fixed andrandom effects in linear mixed-effects models. Biometrics,2010,66(4):1069-1077
[111] Jiang JM. Linear and generalized linear mixed models and their applications.New York: Springer-Verlag,2007:199-204
[112] Littell RC, Milliken GA, Stroup WW, et al. SAS for mixed models.2ndedition.SAS Institute Inc.,2006:13-552
[113] Flom PL, McMahon JM, Pouget ER. Using PROC NLMIXED and PROCGLMMIX to analyze dyadic data with a dichotomous dependent variable. SASGlobal Forum2007Proceedings.2007:179
[114] Witte JS, Greenland S, Kim LL, et al. Multilevel modeling in epidemiologywith GLIMMIX. Epidemiology,2000,11(6):684-688
[115] Capanu M, Begg CB. Hierarchical modeling for estimating relative risks of raregenetic variants: properties of the pseudo-likelihood method. Biometrics,2011,67(2):371-380
[116] Wolfinger R, O’Connell M. Generalized linear mixed models:a pseudo-likelihood approach. Journal of Statistical Computatuin and Simulation,1993,48:233-243
[117] McMahon JM, Pouget ER, Tortu S. A guide for multilevel modeling of dyadicdata with binary outcomes using SAS PROC NLMIXED. Comput Stat DataAnal.2006,50(12):3663–3680
[118] Richardson DB. A simple approach for fitting linear relative rate models in SAS.American Journal of Epidemiology.2008,168(11):1333-1338
[119] Patefield M. Fitting non-linear structural relationships using SAS procedureNLMIXED. The Statistician.2002,51:355–366
[120] Zheng M. Fitting linear and nonlinear growth curve models using PROCNLMIXED. SAS Global Forum2010. Paper260-2010
[121]茆诗松,周纪芗,陈颖.北京:中国统计出版社,2004:231-286
[122]王万中.试验的设计与分析.北京:高等教育出版社,2004:192-231
[123] Plan EL, Maloney A,Trocóniz IF, et al. Performance in population models forcount data, part I: maximum likelihood approximations. J PharmacokinetPharmacodyn,2009,36:353–366
[124] Shun ZM, Mccullagh P. Laplace approximation of high dimensional integrals. J.R. Statist. Soc. B,1995,57(4):749-760
[125] Wolfinger R. D. Laplace’s approximation for nonlinear mixed models.Biometrika,1993,80(4):791-795
[126] Golub GH, Welsch JH. Calculation of Gaussian quadrature rules. MathematicalComputing,1969,23:221-230
[127] Liu L, Huang X. The use of Gaussian quadrature for estimation in frailtyproportional hazards models. Stat Med,2008,27(14):2665-2683
[128] Liu Q, Pierce DA. A note on Gauss-Hermite quadrature. Biometrika,1994,81(3):624-629
[129] Pinheiro JC, Chao EC. Efficient Laplacian and adaptive Guassian quadraturealgorithms for multilevel generalized linear mixed models. Journal ofComputatonal and Graphical Statistics,2006,15(1):58-81
[130] Winkelmann R. Econometric analysis of count data.5th edition. New York:Springer-Verlag,2008:285-287
[131] Kreft IGG., De Leeuw J. Introducing multilevel modeling. London: SagePublications Inc.,1998:105-114
[132] Green MJ, Medley G., Browne WJ, et al. Use of posterior predictiveassessments to evaluate model fit in multilevel logistic regression. Vet.Res.,2009,40:30-39
[133] Kinney SK, Dunson DB. Fixed and random effects selection in linear andlogistic models. Biometrics,2007,63(3):690-698
[134] Longford NT. Logistic regression with random coefficient. ComputationalStatistics and Data Analysis,1994,17(1):1-15
[135] De Leeuw J, Meijer E. Handbook of multilevel analysis. New York: Springer-Verlag,2007:237-274
[136] Liang LM, Chen WM, Sham PC, et al. Variance components linkage analysiswith repeated measurements.Human Heredity,2008,67:237–247
[137] McCulloch CE. Maximum likelihood variance components estimation forbinary data. Journal of the American Statistical Association,1994,89(425):330-335
[138] SPSS Inc. IBM SPSS advanced statistics19. SPSS Inc:140-142
[139] SAS Institute Inc. SAS/QC9.2user’s guide. Cary, NC:SAS Institute Inc.2008:661-747
[140] Gibbons RD, Hedeker D. Application of random-effects probit regressionmodels. J Consult Clin Psychol.1994,62(2):285-296
[141] Turechek WW, Madden LV. A generalized linear modeling approach forcharacterizing disease incidence in a spatial hierarchy. Phytopathology.2003,93(4):458-466
[142] Raman R, Hedeker D. A mixed-effects regression model for three-level ordinalresponse data. Stat Med.2005,24(21):3331-3345
[143] Hedeker D, Gibbons RD. A random-effects ordinal regression model formultilevel analysis. Biometrics,1994,50(4):933-944
[144]王济川,谢海义,姜宝法.多层统计分析模型——方法与应用.北京:高等教育出版社,2008:44-80
[145] Gibbons R. D., Hedeker D. Random effects probit and logistic regressionmodels for three-level data. Biometrics.1997,53(4):1527-1537
[146] Hedeker D. A mixed-effects multinomial logistic regression model. Statistics inMedicine,2003,22:1433–1446
[147]张岩波,何大卫,梁执群.多分类反应变量的多水平多项式模型及其应用.中国卫生统计,2000,17(3):151-153
[148]夏结来,徐雷.计数资料的统计分析模型.疾病控制杂志,2003,7(2):81-84
[149]曾平,赵晋芳,刘桂芬. Poisson回归中过度离散的检验方法.中国卫生统计,2011,28(2):211-212
[150] Consul PC, Famoye F. Generalized poisson regression model. Communicationsin statistics:theroy and methods.1992,21:89-109
[151] Cameron AC, Trivedi PK. Regression Analysis of Count Data. New York:Cambridge University Press,1998:3-18
[152] Zelterman D. Models for discrete data. Revised edition. New York: OxfordUniversity Press Inc.,2006:12-20
[153] Waltoft BL. A SAS-macro for estimation of the cumulative incidence usingPoisson regression. Computer methods and programs in biomedicine,2009,93:140–147
[154] Ver Hoef JM, Boveng PL.Quasi-Poisson vs. negative binomial regression: howshould we model overdispersed count data? Ecology.2007,88(11):2766-2772
[155] Lindén A, M ntyniemi S. Using the negative binomial distribution to modeloverdispersion in ecological count data. Ecology.2011,92(7):1414-1421
[156] Montgomery DC, Runger GC. Applied statistics and probability for engineers.3rdedition. New York: John Wiley&Sons, Inc.,2002:72-96
[157]茆诗松.统计手册.北京:科学出版社,2006:1004-1007
[158] Vives J, Losilla JM, Rodrigo MF, et al. Overdispersion tests in count-dataanalysis.Psychol Rep.2008,103(1):145-160
[159] Cox DR. Some remarks on over-dispersion. Biometrika.1983,70:269-274
[160] Xiang L, Lee AH. Sensitivity of test for overdispersion in Poisson regression.Biometrical Journal.2005,47(2):167-176
[161] Yang Z, Hardin JW, Addy CL, et al. Testing approaches for overdispersion inpoisson regression versus the generalized poisson model. Biometrical Journal.2007,49(4):565-584
[162] Trocóniz IF, Plan EL, Miller R, et al. Modelling overdispersion and Markovianfeatures in count data. J Pharmacokinet Pharmacodyn,2009,36:461–477
[163] Coxe S, West SG, Aiken LS. The analysis of count data: a gentle introduction topoisson regression and its alternatives. Journal of Personality Assessment,2009,91(2):121-136
[164] Gardner W, Mulvey EP, Shaw EC. Regression analyses of counts and rates:Poisson, overdispersed Poisson, and negative binomial models. PsycholBull,1995,118(3):392-404
[165]陈峰.医用多元统计分析方法.第二版.北京:中国统计出版社,2007:114-131
[166]杨珉,李晓松.医学和公共卫生研究常用多水平统计模型.北京:北京大学出版社,2007:106-118
[167] Li BY, Lingsma HF, Steyerberg EW, et al. Logistic random effects regressionmodels: a comparison of statistical packages for binary and ordinal outcomes.BMC Medical Research Methodology,2011,11:art77
[168] Austin PC. Estimating multilevel logistic regression models when the numberof clusters is low: a comparison of different statistical software procedures. TheInternational Journal of Biostatistics,2010,6(1):art16
[169] Box GEP. Science and statistics. J. Am. Stat. Assoc.,1976,71:791–799
[170] Burnham KP, Anderson DR. Model selection and multimodel inference: apractical information-theoretic approach.2ndedition. New York: Springer-Verlag,2003:20-23
[1] Wang JC, Xie HY, Fisher JH. Multilevel models:application using SAS.Beijing:Higher Education Press,2009:30
[2] Jennrich RI, Ralston ML. Fitting nonlinear models to data. Ann. Rev. Biophys.Bioeng.,1979,8:195-238
[3]郑青山,丁学庭.用相关指数作曲线拟合优度比较.中国卫生统计,1995,12(5):62-63
[4] Graybill FA, Iyer HK. Regression analysis:concerpts and application. California:Duxbury Press,1994:501-550
[5]胡良平.现代统计学与SAS应用.北京:军事医学科学院,2000:263
[6]杨树勤.中国医学百科全书(医学统计学).上海:上海科学技术出版社,1985:171-176
[7] Akaike H. A New Look at Statistical Model Identification. IEEE Transactions onAutomatic Control,1974,19:716–723
[8] Hurvich CM, Tsai C-L. Regression and Time Series Model Selection in SmallSamples. Biometrika,1989,76:297–307
[9] Schwarz G. Estimating the Dimension of a Model. Annals of Statistics,1978,6:461–464
[10]Hannan EJ, Quinn BG. The Determination of the Order of an Autoregression.Journal of the Royal Statistical Society, Series B,1979,41:190–195
[11]张家放.医用多元统计方法.武汉:华中科技大学出版社,2002:97-99
[12]金丕焕,陈峰.医用统计方法.第三版.上海:复旦大学出版社,2009:358-364
[13]方开泰,全辉,陈庆云.实用回归分析.北京:科学出版社,1988:168-234
[14]薛仲三.医学统计方法和原理（内部资料）.军事医学科学院印,1984:276-287
[15]梁文权.生物药剂学与药物动力学.第二版.北京:人民卫生出版社,2006:205-225
[16]Seber GAF, Wild CJ. Nonlinear Regression. New York: John Wiley&Sons,Inc.,2003:325-341
[17]Winsor CP. Gompertz curve as a growth curve. Proceedings of the NationalAcademy of Sciences,1932,18:1-8
[18]胡良平.医学统计学——运用三型理论进行现代回归分析.北京:人民军医出版,2010:57-58
[19]Johnson ML. Nonlinear least-squares fitting methods. Methods Cell Biol.2008,84:781-805
[20]郑洲顺,普乐.非线性最小二乘问题的一种迭代解法.数学理论与应用.2002,22(1):43-45
[21]SAS Institute Inc. SAS/STAT9.22User’s Guide. Cary, NC:SAS Institute Inc.2010:4892-4965
[22]王济川,郭志刚. Logistic回归模型——方法与应用.北京:高等教育出版社,2001:19-243
[23]Agresti A. Categorical Data Analysis.2ndEdition. New York:John Wiley&Sons,Inc.,2002:245-250
[24]Penman AD, Johnson WD. Complementary log-log regression for the estimationof covariate-adjusted prevalence ratios in the analysis of data from cross-sectionalstudies. Biom J,2009,51(3):433-442
[25]Warner P. Ordinal logistic regression. J Fam Plann Reprod Health Care,2008,34(3):169-170
[26]Kim JH. Assessing practical significance of the proportional odds assumption.Statistics&Probability Letters,2003,65(3):233-239
[27]Powers DA, Xie Y. Statistics methods for categorical data analysis. AcademicPress, Inc.,1999:210-222
[28]McCullagh P. Regression models for ordinal data. Journal of the Royal StatisticalSociety, Series B(Methodological),1980,42(2):109-142
[29]Maura ES, Charles SD, Gary GK. Categorical Data Analysis Using the SASSystem.2ndEdition. Cary, NC: SAS Institute Inc.,2000:257-270
[30]Zelterman D. Models for Discrete Data. Revised edition. New York: OxfordUniversity Press Inc.,2006:12-20
[31]茆诗松.统计手册.北京:科学出版社,2006:1004-1007
[32]Vives J, Losilla JM, Rodrigo MF, et al. Overdispersion tests in count-dataanalysis.Psychol Rep.2008,103(1):145-160
[33]Coxe S, West SG, Aiken LS. The analysis of count data: a gentle introduction topoisson regression and its alternatives. Journal of Personality Assessment,2009,91(2):121-136
[34]Trocóniz IF, Plan EL, Miller R, et al. Modelling overdispersion and Markovianfeatures in count data. J Pharmacokinet Pharmacodyn,2009,36:461–477
[35]Lindén A, M ntyniemi S.Using the negative binomial distribution to modeloverdispersion in ecological count data. Ecology.2011,92(7):1414-1421
[36]Nelder JA, Wedderburn RWM. Generalized linear models. Journal of Royalstatistical Society, Series A (General),1972,135(3):370-384
[37]陈峰.医用多元统计分析方法.第二版.北京:中国统计出版社,2007:114-131
[38]SAS Institute Inc.. SAS/STAT9.3User’s Guide. Cary, NC:SAS Institute Inc.,2011:2607-2801
[39]杨珉,李晓松.医学和公共卫生研究常用多水平统计模型.北京:北京大学出版社,2007:106-118
[40]Kinney SK, Dunson DB. Fixed and random effects selection in linear and logisticmodels. Biometrics,2007,63(3):690-698
[41]De Leeuw J, Meijer E. Handbook of multilevel analysis. New York: Springer-Verlag,2007:237-274
[42]Gibbons RD, Hedeker D. Application of random-effects probit regression models.J Consult Clin Psychol.1994,62(2):285-296
[43]Turechek WW, Madden LV. A generalized linear modeling approach forcharacterizing disease incidence in a spatial hierarchy. Phytopathology.2003,93(4):458-466
[44]Hedeker D, Gibbons RD. A random-effects ordinal regression model formultilevel analysis. Biometrics,1994,50(4):933-944
[45]Gibbons RD, Hedeker D. Random effects probit and logistic regression modelsfor three-level data. Biometrics,1997,53(4):1527-1537
[46]Hedeker D. A mixed-effects multinomial logistic regression model. Statistics inMedicine,2003,22:1433–1446
[47]张岩波,何大卫,梁执群.多分类反应变量的多水平多项式模型及其应用.中国卫生统计,2000,17(3):151-153
[48]Gelman A, Hill J. Data analysis using regression and multilevel/hierarchicalmodels. Cambridge: Cambridge University Press,2006:246-247,325-342
[49]Hox JJ. Multilevel analysis:techniques and applications.2ndedition. New York:Routledge,2010:4-7,54-78,151-158
[50]Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics,1982,38(4):963-974.
[51]谢远涛,杨娟.广义Gamma分布簇广义线性混合模型的构建.统计研究,2010,27(10):75-80
[52]Dean CB, Nielsen JD. Generalized linear mixed models: a review and someextensions. Lifetime Data Anal,2007,13(4):497-512
[53]Tuerlinckx F, Rijmen F, Verbeke G, et al. Statistical inference in generalizedlinear mixed models: a review. Br J Math Stat Psychol,2006,59(Pt2):225-255
[54]Goldstein H. Nonlinear multilevel models, with an application to discreteresponse data. Biometrika,1991,78(1):45-51
[55]Agresti A著.张淑梅,王睿,曾莉,译.属性数据分析引论.第二版.北京:高等教育出版社,2008:267-271
[56]王济川,谢海义,姜宝法.多层统计分析模型——方法与应用.北京:高等教育出版社,2008:128-132
[57]杨志雄,袁岱菁.非线性混合效应模型和广义线性模型拟合随机效应logistic回归的应用比较.中国卫生统计,2011,28(3):321-323
[58]Vonesh EF. A note on the use of Laplace’s approximation for nonlinear mixed-effects models. Biometrika,1996,83(2):447–452
[59]Liu Q, Pierce DA. A note on Gauss-Hermite quadrature. Biometrika,1994,81(3):624-629
[60]Winkelmann R. Econometric analysis of count data.5thedition. New York:Springer-Verlag,2008:285-287
[61]Pinheiro JC, Chao EC. Efficient Laplacian and adaptive Guassian quadraturealgorithms for multilevel generalized linear mixed models. Journal ofComputatonal and Graphical Statistics,2006,15(1):58-81
[62]Pinheiro JC, Bates DM. Approximations to the Loglikelihood Function in theNonlinear Mixed Effects Model. Journal of Computational and GraphicalStatistics,1995,4(1):12-35
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |