In two level modeling, random effect and error's normality assumption is one of the basic assumptions. Violating this assumption leads to incorrect inference about coefficients of the model. In this paper, to resolve this problem, we use skew normal distribution instead of normal distribution for random and error components. Also, we show that ignoring positive (negative) skewness in the model causes overestimating (underestimating) in intercept estimation and underestimating (overestimating) in slope estimation by a simulation study. Finally, we use this model to study relationship between shift work and blood cholesterol.

In some medical studies, we may have several measurements on each patient. Sometimes these longitudinal data may be measured for several response variables, in this case, although the responses can be modeled separately, such an approach reduces the power and efficiency in estimating the effects of auxiliary variables on the response variable. In the analysis of such data, in addition to the analysis of the dependence between repeated measures related to each of the response variables, the dependence between the responses should also be considered. Among the methods used in recent years to model multivariate data is the copulafunction. One of the most important advantages of using the copula function compared to the longitudinal multivariate modeling of the data in the classic way is that, in addition to the normal distribution, any other distribution other than the normal can be considered as marginal distributions. Also, marginal distributions can even have different distributions. In situations where the data have a multivariate structure, one of the ways to form multivariate distributions is to use vine pair-copula function. In this study, we form a multivariate longitudinal structure by using the vine pair copula functions and compare these models with the model obtained from the fitting of the multivariate normal copula function. Then we will introduce the best model using the Akaike information criterion and at the end we will use the presented model on the data of the estimation of the effect of nutrition on growth.

To accurately estimate the longitudinal dispersion coefficient is important and indispensable in river modeling. Many theoretical as well as empirical formulations have been proposed to determine the longitudinal dispersion coefficient, but these have not been put into consideration because of their great error, and as well the complexity of the phenomenon. The main aim followed in the present paper is to investigate the method as well as equations developed for dispersion coefficient estimation and assessment of the accuracy of these methods in comparison with real data and developing an accurate methodology for dispersion coefficient determination making use of such soft computing techniques as, neural, genetic programming and Neuron-Fuzzy Inference System. ANFIS approach ended up with the excellent results of: R2 = 0.87, RMSE = 72.21, CRM = 0.103 and EF=0.75 as compared with the existing predictors of dispersion coefficient. In total ANFIS approach is hereby proposed as a most acceptable technique for estimating the longitudinal dispersion coefficient.

In some longitudinal studies, especially in social, economic, medical and other fields, there may be two interested responses with two different scales at a time where they may be correlated with each other. Also, considering the nature of longitudinal studies, each of the responses associated with a subject over time can also be correlated. So two correlation structure should be considered simultaneously in the data analysis. In a longitudinal study, some subjects may not be available for any reason (such as displacement, death and others), In a longitudinal study, some subjects may withdraw for any reason (such as displacement, death, etc.) and their information is not available. In this case, joint modeling of longitudinal data and drop-out event is more desirable than separate modeling of either one. In this paper, the mathematics modeling of this type of data under drop-out mechanism is presented using Bayesian approach. A Simulations study and a real data analysis is used to evaluate the performance of the proposed model. This model includes the presented models for complete data as a special case when there is no drop-out in the data set. Also, some tests for choosing the best fitted model to data are performed in the real data analysis.

Marginal model is one of the approaches that can be used in analyzing longitudinal data. In this model, to obtain valid inference, correlations between responses of the same individuals are considered as parameters to be estimated. The various marginal models for analyzing longitudinal data with binary responses such as marginal models with marginal odds ratio. conditional odds ratio, dependence ratio, multivariate probit and the method of generalized estimating equations (GEE) are reviewed and compared. Some residuals for examining the goodness of fit of these models are presented. In an empirical example, these models are fitted to some data.

The aim of this study was to teach medical students the multivariate linear mixed statistical model in multiple longitudinal data: A case study of child development data. The research method was descriptive-analytical and based on longitudinal data. The statistical population of the study consisted of 100 students who were randomly selected and participated in two training workshops based on statistical modeling of t and normal mixed multivariate linear distributions based on children's growth data (height, weight and head circumference). After the training course, they participated in practical test. Mean and standard deviation and statistical modeling of multivariate mixed linear t-distribution were used to analyze the data. The results showed that the amount of height variable parameters according to MtLMM and MnLMM models for breastfed infants was significantly higher than formula (P <0. 05). Also, the estimation of weight variable parameters for infants who used formula was significantly higher (P <0. 05) compared to infants who consumed only formula. The trainings provided to the two groups led to a significant increase in students' learning (P <0. 05). The students stated that teaching a multivariate statistical model in longitudinal data revealed the important of using this model in medical research and life sciences.