| Repeated measurements datas(RM) refers to datas in which the response of each experimental unit or subject is observed on multiple occasions or under multiple conditions . The data is very common in medical field, and the responses may be all variety of variables, including quantitative variables, qualitative variables and ranked variables; for example ,in clinical medical study about II phrase hypertension patients,the blood pressure(systolic pressure and diastolic pressure) are measured regularly,the response is quantitative variable; in clinical curative study of galactophore hyperpasia patients,recording the curative preformance of the patients during treatment , the outcome variable is dichotomous response (improvement ,no improvement); in the trace study for interposition theraty of coronary heart disease , recording the effects at the time of leaving the hospital and three month,six month and nine month after leaving the hospital,the outcomes are recovery,mend,amend,little change and no change, the response is ordinal categorical variable; in other cases ,the response may be recorded as counts data,such as the number of falling sickness at unit time(year or month).According to the relationship of coefficeents of responde and covariates, RM data can be divided into linear and nonlinear ;for example ,regularly examine the blood pressure of hypertension patients ,the relationship of the outcome variable and time variable or other covariates can be fitted by linear model, then this is a linear RM data .But ,in pharmacokinetics analysis, serial blood samples are collected from each of several subjects following doses of a drug and assayed for drug concentration, in order to gaining insight into within-subject parmacokinetic processes of absorption, distribution and elimination. In most cases, the pharmacokinetic processes are nonlinear model. In HIV dynamics study, with the advent of assays capable of quantifying the concetration of viral particles in the blood, monitoring of such "viral load" measurements is now a routine feature of HIV-infected individuals, using a system of differential equations whose parameters characterize rates of production, infection and death of immune system cells and viral production and clearance. The above two example belong to nonlinear RM data ,the relationship of parameters between response and covariate is nonlinear. Recurrent event data means the subject experiences the same type of event more than once, such as heart attacks of coronary heart disease patients, recur of cancer patient, these data have the character of RM data and survival data. Above cases all don't satisfy normality and linear condition of traditional linear models. We must find another statistical model to analysis.Methods of linear modeling of RM data are well developed, their applications are very popular. Linear mixed effect model among those mehods is the most ideal method. It assume the variance-covariance to have some kind of structure, which be used to recognize heterogeneous variance and within-subject collrelation. It isn't not only more strict than univariate variance analysis, but also less strict to covariance structure than multivariante variance, can analysis balance and unbalance data ,can exploit miss data ,can model flexibly the linear model. However the theories and applications of unnormality and nonlinear model for repeated measurement data are still at the initial stage, need to be completed and popularized. An natural extention of linear mixed effect mode allow response come from an exponential family of distributions , including dispersed distribution (binomial, poisson, et all) and continuous distribution (normal ,beta and chi-square).The mean response and linear predictor are connected by link function, these is generalized linear mixed model.The types of repeated measurement datas are widespread, they are common in medical applications. Hence it is necessary to study deeply and completely all kinds of models of RM datas. The paper describe and be compared with all kinds of statistical models (linear and nonlinear models)of RM datas from two aspects ,one is different types of response (quantitative , qualitative and ordered variable),the other is the relationship of the parameters between response and covariates. The content include seven parts:First part introduce the character and variance-covariance pattern of RM data;Second part introduce the theory of linear mixed effect model;Third part introduce the theory of generalized estimate equation and applications in binary,ordered and count data:The generalized estimating equations(GEE) methodology for the analysis of RM is a marginal model approach, The GEE approach is an extension of the generalized linear model and of quasilikelihood to longitudinal data analysis, it ia a regression model for correlated repeated measurement data. The method is semiparametric in that the estimating equation are derived without full specification of the joint distribution of a subject's observations . instead, we specify only the likelihood for the (univariate) marginal distributions and a "working" covariance matrix for the vector of repeated measurements from each subject .The GEE method yields consistent and asymptotically normal solutions, even with misspecification of the time dependence, when the response is categorical variable (binary , ordered or count data), GEE is one of the most appropriate models.Forth part introduce the theory of generalized linear mixed effect model and applications in binary,ordered and count data:Generalized linear mixed effect modes are natural extention of linear mixed effect modes which can be used for both continuous and discrete longitudinal data. GLMMs are a unified approach to exponetial family regression methods with random effects. The mean response and linear predictor are connected by link function,when there isn't random effect, the model be changed a generalized linear model. All types of correlation structure in the repeated measurement data is induced by introducing a random effect.Fifth part introduce the theory of nonlinear mixed effect model and applications in nonlinear RM data ,binary,ordered and count data:Nonlinear mixed effect model not only recognize and estimate variability both between and within individuals,but also allow both fixed and random effects enter the model nonlinearly. The depentent variable is normal, binomial,poisson distribution,the applications of the most common are Pharmacokinetic, nonlinear growth curve and fit directely nolinear model for categorical response repeated measurement datas; These year ,the model enjoyed widespread attention within biological,agriculture and medical reseach community.Sixth part introduce the theory of frailty model and applications in recurrent event datas: Frailty modes are extensions of Cox proportional hazards model,which aims to account for heterogeneity caused by unmeasured covariates. The frailty has a multiplicative effect on the baseline hazard function ,namely the individual harzard rates are influenced by frailty in the multiplication way. The subgroup which frailty value is bigger than other will be susceptible to experience the event in a earlier time. Individuals in a cluster are assumed to share the same frailty which is why this model is called shared frailty model .The survival times are assumed to be conditional independent with reaponse to the shared frailty. The frailty is a random efect which is assumed to obey some kinds of distribution. The frailty distribution most often applied are the gamma distribution. The conditional frialty model combines a random effect to incorporate unobserved heterogeneity with event-based stratification(varying baseline hazards) to invorporate event dependence .They will be the ideal model for the repeated event data.Seven parts By introducing the analysis methods for nonnormal,nonindependent and nonlinear datas, deeply expound the application of generalized estimate equation, generalized linear mixed effect model and nonlinear mixed effect model in categorical data (dichotomous,ordinal,count )and of frailty model for recurrent event data, discuss the analysis methods and realization for SAS software and R software, put forward some suggestion and appraise for model construction, parameter estimate ,soft realization et all in medical applications, provide new ideal of analysis for nonnormal, nonindependent and nonlinear datas.The paper analysis and contrast GENMODE,GLIMMIX and NLMIXED procedures of SAS9.1.3 at the application in medical categorical and nonlinear repeated measurement data , conduct recurrent event data using free software R2.4.0;Using the idea of combining the model theories with example analysis , linking method researches and software realization , systematicly introduce the application of model analysis and software realization for nonnornal ,nonlinear repeated measurement datas , summarize skills and experiences about practical application by analyzing examples, systmeticly explain models of analysis and theories for nonnormal,nonindependent and nonlinear datas, proving the methodology basis for analyzing the medical datas , providing the conditions for theory models and software applications, especially light abstract statistical theory , using the idea of being base on theory but be higher than theory ,open some new situation for the practice applications of all kinds of methods ;proving some new , reliable , exact ,informative,feasible viewpoint for multivariable statistical methods to resolve practical problem by using correctly generalized estimate equation,generalized linear mixed effect model,nonlinear mixed effect model and frailty model.
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