Posted: March 8th, 2020

# Multivariate Multilevel Modeling

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**Literature Review**

This chapter tying up the various similar studies related to modeling responses multivariately in a multilevel frame work. As a start, this chapter begins by laying out the recent history of univariate techniques for analyzing categorical data in a multilevel context. Then it gradually presents the literature available on fitting multivariate multilevel models for categorical and continuous data. More over this chapter reviews the evidence for imputing missing values for partially observed multivariate multilevel data sets.

**The Nature of Multivariate Multilevel models**

A multivariate multilevel model can be considered as a collection of multiple dependent variables in a hierarchical nature. Though the multivariate analysis increases the complexity in a multilevel context, it is an essential tool which facilitates to carry out a single test of the joint effects of some explanatory variables on several dependent variables (Snijders & Bosker (2000). These models have the power of increasing the construct validity of the analysis for complex concepts in the real world. Consider a study on school effectiveness which can be measured on three different output variables math achievement, reading proficiency and well-being at school. These data are collected on students those who are clustered within schools by implying a hierarchical nature. Although it is certainly possible to handle three outcomes separately, it is unable to show the overall picture about school effectiveness. Therefore multivariate analysis would be more preferable in these types of scenarios since it has the capability of decreasing the type 1 error and increasing the statistical power (Maeyer, Rymenans, Petegem and Bergh) (Draft).

Hierarchical natures of multivariate models are not like as the univariate response models. Let us focus on above example; it implies a two level multivariate model. But in reality it has three levels. In this case, the measurements are the level 1 units, the students the level 2 units and the schools the level three units.

**Importance of Multivariate Multilevel Modeling**

Multivariate multilevel data structures may itself present a greater complexity as it leads to focus the multilevel effects together with the multivariate context. Therefore the traditional statistical techniques would fail to face these kinds of areas since it can decrease the statistical efficiency by producing overestimated standard errors. On the other hand violation of independence assumption may cause to under estimate the standard errors of regression coefficients. Therefore multivariate multilevel approaches play an important role to get rid of these kinds of situations by allowing variation at different levels to be estimated. Furthermore Goldstein (1999) has shown that clustering provides accurate standard errors, confidence intervals and significance tests.

Some amount of articles have been published on multilevel modeling based on a single response context. Multivariate multilevel concept comes into the field of statistics during the past few years. When people want to identify the effect of set of explanatory variables on a set of dependent variables and by considering these effects separately on response variables, then if it shows a considerable difference among those effects then it can be handled only by means of a multivariate analysis (Snijders & Bosker, 2000).

**Software for Multivariate Multilevel Modeling**

In the past decades, due to the unavailability of the software for fitting multivariate multilevel data some researchers tend to use manual methods such as EM Algorithm (Kang et al., 1991). As a result of developing the technical environment, the software such as STATA, SAS and S plus are emerged in to the Statistical field by providing facilitates to handle the multilevel data. But none of those packages have a capability of fitting multivariate multilevel data. However there is evidence in the literature that nonlinear multivariate multilevel model can be fitted using packages such as GLLAMM (Rabe-Hesketh, Pickles and Skrondal, 2001) and aML (Lillard and Panis, 2000). But it was not flexible to handle this software.

Therefore MlwiN software which has become the under development since late 1980’s was modified at the University of Bristol in UK in order to fulfill that requirement. However, the use of MlwiN for fitting multivariate multilevel models has been challenged by Goldstein, Carpenter and Browne (2014) who concluded that MlwiN was useful if only when fitting the model without imputing for the missing values. However REALCOM software was then came into the field of Statistics and provided the flexibility to impute the missing values in the MLwiN environment.

MLwiN is a modified version of DOS MLn program which uses a command driven interface. MLwiN provides flexibility to fitting very large and complex models using both frequentist and Bayesian estimation along with the missing value imputation in a user friendly interface. Some particular advanced features which are not available in the other packages are included in this software.

**Univariate Multilevel Modeling vs. Multivariate Multilevel Modeling**

In general, data are often collected on multiple correlated outcomes. One major theoretical issue that has dominated the field for many years is modeling the association between risk factors and each outcome in a separate model. It may cause to statistically inefficient since it ignores outcome correlations and common predictor effects (Oman, Kamal and Ambler) (unpublished)

Therefore most of the researches tend to include all related outcomes in a single regression model within a multivariate outcome framework rather than univariate. Recently investigators have examined the comparison between Univariate and Multivariate outcomes and they have proven that Multivariate models would be preferable than several univariate models.

According to the Griffiths, Brown and Smith (2004), they conducted a study to compare univariate and multivariate multilevel models for repeated measures of use of antenatal care in Uttar Pradesh, India. In here, they examined many factors which may have a relationship to the mother’s decision to use ante-natal care services for a particular pregnancy. For that they compared Univariate multilevel logistic regression model vs. Multivariate multilevel logistic regression model. However as a result of fitting univariate models, model assumptions became violated and couldn’t get stable parameter estimates. Therefore they preferred the multivariate context rather than the univariate context after performing the analysis.

**Generalized Cochran Mantel Haenzel Tests for Checking Association of Multilevel Categorical Data.**

The history of arising the concepts related to Generalized Cochran Mantel Haenzel was streaming to the late 1950’s. Cochran (1958), one of a great Statistician has firstly introduced a test to identify the independence of multiple 2 × 2 tables by extending the general chi-square test for independence of a single 2-way table. In here, the each table consists of one or two additional variables for higher levels to detect the multilevel nature. The test statistic is based on the row totals of each table. The assumption behind is that the cell counts have binomial distribution.

As an extension to Cochran’s work, Mantel and Haenzel (1959) extended the Cochran’s test statistic for both row and column totals by assuming the cell counts of each table follows a hypergeommetric distribution. Since Cochran Mantel Hanzel (CMH) statistic has a major limitation on binary data, Landis et al (1978) generalized this test into handle more than two levels. However there is a major drawback of the Generalized Cochran Mantel Haenzel (GCMH) test. This test was unable to handle clustered correlated categorical data. Liang (1985) was proposed a test statistic for get rid of this problem. However that test statistic itself had major problems and it was fail to use.

As development of the statistics field, a need for a test statistic capable of handling correlated data and variables with higher levels arouse. Zhang and Boos (1995) coming in to the field and introduced three test statistics T_{EL} T_{P} and T_{U} as a solution to the above problems. However among these three test statistics T_{P} and T_{U } are preferred to T_{EL} since these two use the individual subjects as the primary sampling units while T_{EL} use the strata as the primary sampling unit (De Silva and Sooriyarachchi, 2012).

Furthermore, by a simulation study T_{P} shows better performance than T_{E} by maintaining its error values even when the strata are small and it uses the pooled estimators for variance. Therefore it provides a guideline to select T_{P} as the most suitable statistic to perform this study. De Silva and Sooriyarachchi (2012) developed a R program to carry out this test.

**Missing Value Imputation in Multivariate Multilevel Framework**

The problem of having missing values is often arising in real world datasets. However it contains little or no information about the missing data mechanism (MDM). Therefore modeling incomplete data is a very difficult task and may provide bias results. Therefore this major problem address to a need of a proper mechanism to check the missingness. As a solution to that, Rubin (1976) presented three possible ways of arising misingness. These are classified as Missing At Random (MAR), Missing Completely At Random (MCAR) and Missing Not At Random (MNAR). According to the Sterne et. Al (2009), missing value imputation is necessary under the assumption of missing at random. However, it can also be done under the case missing complete at random. On nowadays most statistical packages have the capability of identifying the type of missingness.

After identifying the type of missingness, the missing value imputation comes into the field and it requires a statistical package to perform this. Since the missing value imputation in a hierarchical nature is little bit more advanced and it cannot be done using usual statistical packages such as SPSS, SAS and R etc. Therefore Carpenter et. al (2009), developed the REALCOM software to perform this task. However latter version of REALCOM was not deal with multilevel data in a multivariate context. Therefore the macros related to perform this task was recently developed by the Bristol University team in order to facilitate under this case.

**Estimation Procedure**

The estimation procedures for multilevel modeling are starting late 1980’s. However For parameter estimation using Maximum Likelihood Method, an iterative procedure called EM algorithm was used by early statisticians (Raudenbush, Rowan and Kang, 1991). Later on the program HLM was developed to perform this algorithm.

The most operational procedures for estimating multivariate multilevel models in the presence of Normal responses are Iterative Generalized Least Squares (IGLS), Reweighted IGLS (RIGLS) and Marginal Quasi Likelihood (MQL) while for discrete responses are MQL and Penalized Quasi Likelihood (PQL). According to Rasbash, Steele, Browne and Goldstein (2004) all of these methods are implemented in MLwiN along with including first order or second order Taylor Series expansions. However since these methods are likelihood based frequentist methods they tend to overestimate the precision.

Therefore more recently the methods which are implemented in a Bayesian framework using Marcov Chain Monte Carlo methods (Brooks, 1998) also used for parameter estimation which allows capability to use informative prior distributions. These MCMC estimates executed in MLwiN provides consistent estimates though they require a large number of simulations to control of having highly correlated chains.

**Previous researches conducted using Univariate and Multivariate Multilevel Models**

- Univariate multilevel logit models

Before take a look at to the literature on multivariate multilevel analysis, the literature of univariate multilevel analysis is also be necessary to concerned since this thesis is based on some univariate multilevel models prior to fit multivariate multilevel models.

In the past decades, many social Scientists used to apply multilevel models for binay data. Therefore it is very important to review how they have implemented their work with less technology. As a aim of that, Guo and Zhao (2000) was able to do a review of the methodologies, hypothesis testing and hierarchical nature of the data involve of past literature. Also they conducted two examples for justify their results. First of all they made a comparison between estimates obtained from MQL and PQL methods which was implemented by MLn and the GLIMMIX method implemented by SAS by using examples. They have shown that the differences in PQL 1 and PQL 2 are small when fitting binary logistic models. Furthermore, they have shown that PQL- 1 and PQL-2 and GLIMMIX are probable to be satisfactory for most of the past studies undertaken in social sciences.

Noortgate, Boeck and Meulders (2003) uses multilevel binary logit models for the purpose of analyzing Item Response Theory (IRT) models. For that they carried out an assessment of the nine achievement targets for reading comprehension of students in primary schools in Belgium. They performed a multilevel analyses using the cross-classified logistic multilevel models and used the GLIMMIX macro from SAS, as well as the MLwiN software. However they found that there were some convergence problems arisen by using PQL methods in MLwiN. Therefore they used SAS to carryout analysis. Furthermore they have shown that the cross-classification multilevel logistic model is a very flexible to handle IRT data and the parameters can still be estimated even with the presence of unbalanced data.

- Multivariate Multilevel Models

In the past two decades a very few of researches have sought to fit the multivariate multilevel models to the real world scenarios. Among those also all most all the researches trying to focus basically in educational sectors as well as socio economic sectors. None of them were able to focus these into the medical scenarios. However lack of multivariate multilevel analysis which presents in the field of health and medical sciences this chapter consists of the literatures of multivariate multilevel models in other fields.

According to the previous studies of education, Xin Ma (2001) examined the association between the academic achievements and the background of students in Canada by considering three levels of interest. For that the three level Hierarchical Linear Model (HLM) was developed in order to achieve his goals. This work allows him to draw the conclusions that both students and schools were differentially successful in different subject areas and it was more obvious among students than among schools. However the success of this study is based on some strong assumptions about the priors of student’s cognitive skills.

Exclusive of the field of education Raudenbush, Johnson and Sampson (2003) carried out a study in Chicago to determine the criminal behavior at person level as well as at neighborhood level with respect to some personal characteristics. For this purpose they use a Rasch model with random effects by assuming conditional independence along with the additives.

Moreover, Yang, Goldstein, Browne and Woodhouse (2002) developed a multivariate multilevel analysis of analyzing examination results via a series of models of increasing complexity. They used examination results of two mathematics examinations in England in 1997 and analyzed them at individual and institutional level with respect to some students features. By starting from a simpler model of multivariate normality without considering the institutional random effects, they gradually increased the complexity of the model by adding institutional levels together with the multivariate responses. When closely looked at, there work shows that the choice of subject is strongly associated with the performance.

Along with this growth of applications of multivariate multilevel models, researches may tend to apply those in to the other fields such as Forestry etc. Hall and Clutter (2004) presented a study regarding modeling the growth and yield in forestry based on the slash pine in U.S.A. In their work, they developed a methodology to fit nonlinear mixed effect model in a multivariate multilevel frame work in order to identify the effects of the several plot-level timber quantity characteristics for the yield of timber volume.

In addition to that they also developed a methodology to produce predictions and prediction intervals from those models. Then by using their developments they have predicted timber growth and yield at the plot individual and population level.

Grilli and Rampichini (2003) carried out a study to model ordinal response variables according to the students rating data which were obtained from a survey of course quality carried out by the University of Florence in 2000-2001 academic years. For that they developed an alternative specification to the multivariate multilevel probit ordinal response models by relying on the fact that responses may be viewed as an additional dummy bottom level variable. However they not yet assess the efficiency of that method since they were not implemented it using standard software.

When considering the evidences of the recent applications of these models the literature shows that Goldstein and Kounali (2009) recently conducted a study on child hood growth with respect to the collection of growth measurements and adult characteristics. For that they extended the latent normal model for multilevel data with mixed response types to the ordinal categorical responses with having multiple categories for covariates. Since data consists of counts they gradually developed the model by starting a model with assuming a Poison distribution. However since the data are not follow exactly a Poisson distribution they treated the counts as an ordered categories to get rid of that problem.

Frank, Cerda and Rendon (2007) did a study to identify whether the residential location have an impact to the health risk behaviors of Latino immigrants as they are increasing substantially in every year. For that they used a Multivariate Multilevel Rasch model for the data obtained by Los Angelis family and neighborhood survey based on two indices of health risk behaviors along with their use of drugs and participation for risk based activities. They starting this attempt by modeling the behavior of adolescents as a function of the characteristics related to both individual and neighborhood .According to the study they found that there is an association between increased health risk behaviors with the above country average levels of Latinos and poverty particularly for those who born in U.S.A.

Another application of multivariate multilevel models was carried out Subramanian, Kim and Kawachi (2005) in U.S.A. Their main aim was to identify the individual and community level factors for the health and happiness of individuals. For that they performed a multivariate multilevel regression analysis on the data obtained by a survey which was held on 2000. Their findings reflect that those who have poor health and unhappiness have a high relationship with the individual level covariates

By looking at the available literature, it can be seen that there are some amount of studies conducted on education and social sciences in other countries but none of the studies conducted regarding health and medical sciences. Therefore it is essential to perform a study by analyzing the mortality rates of some killing diseases which are spread in worldwide to understand risk factors and patterns associated with these diseases in order to provide better insights about the disease to the public as well as to the responsibly policy makers.

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