Estimating Equations for Estimation of Mcdonald Generalized Beta— Binomial Parameters

There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.

NONLINEAR STABILITY OF PLANAR SHOCK PROFILES FOR THE GENERALIZED KdV-BURGERS EQUATION IN SEVERAL DIMENSIONS

This paper is concerned with the nonlinear stability of planar shock profiles to the Cauchy problem of the generalized KdV-Burgers equation in two dimensions. Our analysis is based on the energy method developed by Goodman [5] for the nonlinear stability of scalar viscous shock profiles to scalar viscous conservation laws and some new decay estimates on the planar shock profiles of the generalized KdV-Burgers equation.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

A PRIORI ESTIMATE FOR MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions

A PRIORI ESTIMATES TO THE MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH ANISOTROPIC GROWTH CONDITIONS

In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.

ANALYSIS OF CHEBYSHEV PSEUDOSPECTRAL METHOD FOR MULTI-DIMENSIONAL GENERALIZED SRLW EQUATIONS

The Chebyshev pseudospectral approximation of the homogeneous initial boundary value problem for a class of multi-dimensional generalized symmetric regularized long wave (SRLW) equations is considered. The fully discrete Chebyshev pseudospectral scheme is constructed. The convergence of the approximation solution and the optimum error of approximation solution are obtained.

Analysis of KCSE Performance in Nakuru County: A Generalized Estimating Equations Approach

In Kenya, the Ministry of Education has set the pass mark for university entry examination as C plus and above. Using publicly available data for 2006-2010, differences in KCSE performance amongst the three types of schools in Kenya—boys only, girls only, and mixed schools—was assessed. A generalized estimating equations marginal model was applied in order to account for association between scores within a school in the five year period. To account for the missing data, multiple imputation was performed followed by estimation and inference. Results indicated that there was a significant difference between the three type of schools in their candidates’ odds of attaining the stipulated minimum university entry grade. However, the odds of success in KCSE did not depend on the year under review as was evident in the slope parameters which was not statistically significant. Although it was clear that same-gender schools perform better than mixed gender schools, there is need to weigh the social benefits of mixed-schools against respective performance in KCSE. This should guide the policy makers on the way forward with regards to the education policy in Nakuru County.

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.

A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving waves.By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are shown.To derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are used.This,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+1.Numerical experiments are given to demonstrate the theoretical results.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

Construction of an Early Warning Model for Ischemic Stroke Recurrence Based on Generalized Estimating Equation

Objective:To explore the appropriate modeling method of the early warning model of ischemic stroke recurrence in TCM.Methods:This was a prospective,multi-center and registered study conducted in 7 clinical subcenters from 8 provinces and 10 cities in China between 3rd November 2016 and 27th April,2019.1,741 patients with first-ever ischemic stroke were recruited.Univariate analysis was carried out using distance correlation coefficient,mutual information entropy,and statistical correlation test.Multivariate analysis adopted multi-factor Cox regression model and combined with expert opinions in the field of stroke to determine modeling variables.The generalized estimating equation of longitudinal data and the Cox proportional hazard regression model of cross-sectional data were used to construct and compare in the early warning model of ischemic stroke recalls.The area under the ROC curve(AUC value)was used to evaluate the early warning capability of the model.Results:The follow-up time was 1-3 years,and the median follow-up time was 1.42 years(95%CI:1.37-1.47).Recurrence events occurred in 175 cases,and the cumulative recurrence rate was 10.05%(95%CI:8.64%-11.47%).The AUC values of the TCM syndrome and TCM constitution model were 0.71809 and 0.72668 based on the generalized estimating equation and the AUC values.Conclusion:The generalized estimating equation may be more suitable for the construction of early warning models of stroke recurrence with TCM characteristics,which provides a certain reference for the evaluation of secondary prevention of ischemic stroke.

广义估计方程GEE1在婴儿生长监测回归分析中的应用

广义估计方程ＧＥＥｓ代表了纵向资料统计方法的一个划时代突破。本文把广义估计方程ＧＥＥ１应用于生长监测资料的分析领域，在进行母乳喂养与婴儿体重发育回归分析的同时，较好地解决了困扰生长监测资料分析的数据相关性问题。

应用SPSS软件实现二分类重复测量资料的GEE及GLMMs分析

目的在SPSS软件上实现分类重复测量资料的广义估计方程(GEE)和广义线性混合效应模型(GLMMs)分析。方法结合二分类重复测量的实例资料,按照SPSS 19.0软件的菜单操作过程,实现GEE与GLMMs模型的统计分析。结果在SPSS19.0软件上实现二分类重复测量资料GEE和GLMMs模型分析的菜单操作不需编程、结果直观清晰。结论 SPSS 19.0软件上可实现二分类重复测量资料的GEE与GLMMs模型分析,无需复杂的编程过程,操作简便。

户主职业对农户响应林业科技服务的影响——基于江西纵向数据GEE分析

基于江西农户8年跟踪调查纵向样本数据,运用广义估计方程(GEE)Logistic回归模型,检验户主职业与农户林业科技服务响应之间关系。结果表明:户主职业对农户林业科技服务响应行为产生显著性影响,户主全职务农农户逐步成为技术采纳的弱质性农户,户主长期外出务工的农户林业科技服务响应具有异质性,户主职业的变动对农户林业科技服务响应有正向影响。提出林业科技服务供给以及协同推进发展现代林业与实施乡村振兴战略等方面政策建议。

稳健Poisson模型:两水平模型与GEE模型在相对危险度或患病率比估计中的应用比较

目的在处理具有层次结构特征的非罕见结局事件资料时,比较基于稳健Poisson模型的两水平模型和GEE模型在估计RR/PR时的应用。方法将两水平稳健Poisson模型及稳健Poisson-GEE应用到2010年欧洲社会调查资料,估计影响居民生活满意度的各因素相关的PR及95%CI,以说明两模型在理论和应用上的区别和联系。结果稳健Poisson-GEE模型的PR估计值与稳健Poisson回归模型相同,但置信区间较宽;两水平稳健Poisson模型的PR值较GEE模型为低,显示了随机效应对解释变量的混杂作用。结论两种方法均可处理具有层次结构特征的非罕见结局事件的RR/PR估计,但两水平模型比GEE可提供更多随机效应的信息,且易于扩展至更高水平或随机系数模型。

GEE与LMM模型在通脉养心丸治疗冠心病临床效果研究中的作用

[目的]本研究运用线性混合模型和广义估计方程探讨了通脉养心丸在治疗冠心病稳定性心绞痛气阴两虚证患者临床效果研究中的作用,对比研究不同模型假设下效应估计值及其变异程度的差异,探讨以上方法在临床领域重复测量数据中的适用性。[方法]通过多中心、精细化临床研究,共纳入7家临床单位83例受试者,因未完成访视脱失4例,最终纳入受试者79例,其中冠心病气阴两虚证患者39例,在西医常规治疗的基础上加服中药通脉养心丸治疗,健康人40例不予干预。比较不同时点不同组别及组内前后症状自评量表(SCL-90量表)、健康调查简表(SF-36量表)的评分情况,冠心病患者检测第0、4、8、12周水平,健康人检测第0周水平。[结果]本研究中气阴两虚组在线性混合模型中SCL-90量表的总分分别与SF-36量表的总分、时间和性别相关(P<0.05),在广义估计方程中SCL-90量表的总分分别和SF-36量表的总分、性别和舒张压相关(P<0.05)。[结论]通脉养心丸显示出其疗效,SCL-90、SF-36量表为防治冠心病稳定性心绞痛气阴两虚证提供依据。但是在该研究数据中,采用广义估计方程分析资料能取得较为满意的结果。

PA条件不成立时GEE方法中的检验问题

边际回归模型和与此有关的广义估计方程(GEE)在纵向数据分析中得到了广泛的应用.Pepe和Anderson在1994指出在边际模型和GEE方法的应用中必须满足一个重要条件,即PA条件.如果该假定不能满足,可能得不到相合估计,由此进行的统计推断的效率可能不高.本文通过简单的AR(1)模型,在理论上和数值模拟上讨论了PA条件对基于广义估计方程方法所作的关于回归系数的检验的影响.由于PA条件不成立,回归系数的GLS估计不再是渐近无偏的,构造的Wald和Score统计量的分布不再是中心卡方分布,从而对于检验的效率也产生了严重影响.

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.

个体观测次数与协变量个数都趋于无穷的二值数据GEE估计的渐近性质

广义估计方程是专用于处理纵向数据的统计模型,有效地解决了纵向数据中协变量相关的问题,得到稳健的参数估计值。但由于观测次数的不唯一,大多数文献对观测次数设置了上限,前人在假设个体观测次数为有限的条件下,证明了经典Logit广义估计方程估计的渐近性质。而随着大数据时代的到来,试验中需要对个体进行更多次观测,以便得出更为精准的结论。鉴于此,对于观测个体的数量n、协变量个数pn以及观测次数m三者都趋于无穷的情况,通过限定m的发散速度,即在mp_n^3/n和m^4p_~4n/n都趋于0和其他正则条件下,基于微分中值定理及根的存在定理,证明了经典Logit广义估计方程估计的渐近存在性和相合性;并在该估计的相合性的基础上,通过Slutsky定理进一步证明出了该估计的渐近正态性。

ROBUST ESTIMATION IN PARTIAL LINEAR MIXED MODEL FOR LONGITUDINAL DATA

In this article, robust generalized estimating equation for the analysis of partial linear mixed model for longitudinal data is used. The authors approximate the nonparametric function by a regression spline. Under some regular conditions, the asymptotic properties of the estimators are obtained. To avoid the computation of high-dimensional integral, a robust Monte Carlo Newton-Raphson algorithm is used. Some simulations are carried out to study the performance of the proposed robust estimators. In addition, the authors also study the robustness and the efficiency of the proposed estimators by simulation. Finally, two real longitudinal data sets are analyzed.