# Literature Review: modeling responses multivariately in a multilevel frame work

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Chapter 2

Literature Review

This chapter binding up the assorted similar surveies related to patterning responses multivariately in a multilevel frame work. As a start, this chapter begins by puting out the recent history of univariate techniques for analysing categorical informations in a multilevel context. Then it bit by bit presents the literature available on suiting multivariate multilevel theoretical accounts for categorical and uninterrupted informations. More over this chapter reviews the grounds for ascribing losing values for partly ascertained multivariate multilevel informations sets.

2.1 The Nature of Multivariate Multilevel theoretical accounts

A multivariate multilevel theoretical account can be considered as a aggregation of multiple dependent variables in a hierarchal nature. Though the multivariate analysis increases the complexness in a multilevel context, it is an indispensable tool which facilitates to transport out a individual trial of the joint effects of some explanatory variables on several dependent variables ( Snijders & A ; Bosker ( 2000 ) . These theoretical accounts have the power of increasing the concept cogency of the analysis for complex constructs in the existent universe. See a survey on school effectivity which can be measured on three different end product variables math accomplishment, reading proficiency and wellbeing at school. These informations are collected on pupils those who are clustered within schools by connoting a hierarchal nature. Although it is surely possible to manage three results individually, it is unable to demo the overall image about school effectivity. Therefore multivariate analysis would be more preferred in these types of scenarios since it has the capableness of diminishing the type 1 mistake and increasing the statistical power ( Maeyer, Rymenans, Petegem and Bergh ) ( Draft ) .

Hierarchical natures of multivariate theoretical accounts are non like as the univariate response theoretical accounts. Let us concentrate on above illustration ; it implies a two degree multivariate theoretical account. But in world it has three degrees. In this instance, the measurings are the flat 1 units, the pupils the degree 2 units and the schools the degree three units.

2.2 Importance of Multivariate Multilevel Modeling

Multivariate multilevel informations constructions may itself show a greater complexness as it leads to concentrate the multilevel effects together with the multivariate context. Therefore the traditional statistical techniques would neglect to confront these sorts of countries since it can diminish the statistical efficiency by bring forthing overestimated standard mistakes. On the other manus misdemeanor of independency premise may do to under gauge the standard mistakes of arrested development coefficients. Therefore multivariate multilevel attacks play an of import function to acquire rid of these sorts of state of affairss by leting fluctuation at different degrees to be estimated. Furthermore Goldstein ( 1999 ) has shown that constellating provides accurate standard mistakes, assurance intervals and significance trials.

Some sum of articles have been published on multilevel patterning based on a individual response context. Multivariate multilevel construct comes into the field of statistics during the past few old ages. When people want to place the consequence of set of explanatory variables on a set of dependent variables and by sing these effects individually on response variables, so if it shows a considerable difference among those effects so it can be handled merely by agencies of a multivariate analysis ( Snijders & A ; Bosker, 2000 ) .

2.3 Software for Multivariate Multilevel Modeling

In the past decennaries, due to the inaccessibility of the package for suiting multivariate multilevel informations some research workers tend to utilize manual methods such as EM Algorithm ( Kang et al. , 1991 ) . As a consequence of developing the proficient environment, the package such as STATA, SAS and S plus are emerged in to the Statistical field by supplying facilitates to manage the multilevel informations. But none of those bundles have a capableness of suiting multivariate multilevel informations. However there is grounds in the literature that nonlinear multivariate multilevel theoretical account can be fitted utilizing bundles such as GLLAMM ( Rabe-Hesketh, Pickles and Skrondal, 2001 ) and aML ( Lillard and Panis, 2000 ) . But it was non flexible to manage this package.

Therefore MlwiN package which has become the under development since late 1980â€™s was modified at the University of Bristol in UK in order to carry through that demand. However, the usage of MlwiN for suiting multivariate multilevel theoretical accounts has been challenged by Goldstein, Carpenter and Browne ( 2014 ) who concluded that MlwiN was utile if merely when suiting the theoretical account without ascribing for the losing values. However REALCOM package was so came into the field of Statistics and provided the flexibleness to ascribe the losing values in the MLwiN environment.

MLwiN is a modified version of DOS MLn plan which uses a bid driven interface. MLwiN provides flexibleness to suiting really big and complex theoretical accounts utilizing both frequentist and Bayesian appraisal along with the losing value imputation in a user friendly interface. Some peculiar advanced characteristics which are non available in the other bundles are included in this package.

2.4 Univariate Multilevel Modeling vs. Multivariate Multilevel Modeling

In general, informations are frequently collected on multiple correlated outcomes. One major theoretical issue that has dominated the field for many old ages is patterning the association between hazard factors and each result in a separate theoretical account. It may do to statistically inefficient since it ignores outcome correlativities and common forecaster effects ( Oman, Kamal and Ambler ) ( unpublished )

Therefore most of the researches tend to include all related results in a individual arrested development theoretical account within a multivariate result model instead than univariate. Recently research workers have examined the comparing between Univariate and Multivariate results and they have proven that Multivariate theoretical accounts would be preferred than several univariate theoretical accounts.

Harmonizing to the Griffiths, Brown and Smith ( 2004 ) , they conducted a survey to compare univariate and multivariate multilevel theoretical accounts for perennial steps of usage of prenatal attention in Uttar Pradesh, India. In here, they examined many factors which may hold a relationship to the motherâ€™s determination to utilize ante-natal attention services for a peculiar gestation. For that they compared Univariate multilevel logistic arrested development theoretical account vs. Multivariate multilevel logistic arrested development theoretical account. However as a consequence of suiting univariate theoretical accounts, model premises became profaned and couldnâ€™t get stable parametric quantity estimations. Therefore they preferred the multivariate context instead than the univariate context after executing the analysis.

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

The history of originating the constructs related to Generalized Cochran Mantel Haenzel was streaming to the late 1950â€™s. Cochran ( 1958 ) , one of a great Statistician has foremost introduced a trial to place the independency of multiple 2 ? 2 tabular arraies by widening the general chi-square trial for independency of a individual 2-way tabular array. In here, the each tabular array consists of one or two extra variables for higher degrees to observe the multilevel nature. The trial statistic is based on the row sums of each tabular array. The premise buttocks is that the cell counts have binomial distribution.

As an extension to Cochranâ€™s work, Mantel and Haenzel ( 1959 ) extended the Cochranâ€™s trial statistic for both row and column sums by presuming the cell counts of each tabular array follows a hypergeommetric distribution. Since Cochran Mantel Hanzel ( CMH ) statistic has a major restriction on binary informations, Landis et Al ( 1978 ) generalized this trial into grip more than two degrees. However there is a major drawback of the Generalized Cochran Mantel Haenzel ( GCMH ) trial. This trial was unable to manage clustered correlative categorical informations. Liang ( 1985 ) was proposed a trial statistic for get rid of this job. However that trial statistic itself had major jobs and it was fail to utilize.

As development of the statistics field, a demand for a trial statistic capable of managing correlated informations and variables with higher degrees arouse. Zhang and Boos ( 1995 ) coming in to the field and introduced three trial statistics TElevationThyminePhosphorusand TUracilas a solution to the above jobs. However among these three trial statistics TPhosphorusand TUracilare preferred to TElevationsince these two use the single topics as the primary trying units while TElevationuse the strata as the primary sampling unit ( De Silva and Sooriyarachchi, 2012 ) .

Furthermore, by a simulation survey TPhosphorusshows better public presentation than TTocopherolby keeping its mistake values even when the strata are little and it uses the pooled calculators for discrepancy. Therefore it provides a guideline to choose TPhosphorusas the most suited statistic to execute this survey. De Silva and Sooriyarachchi ( 2012 ) developed a R plan to transport out this trial.

2.6 Missing Value Imputation in Multivariate Multilevel Framework

The job of holding losing values is frequently originating in existent universe datasets. However it contains small or no information about the losing informations mechanism ( MDM ) . Therefore patterning uncomplete informations is a really hard undertaking and may supply prejudice consequences. Therefore this major job reference to a demand of a proper mechanism to look into the missingness. As a solution to that, Rubin ( 1976 ) presented three possible ways of originating misingness. These are classified as Missing At Random ( MAR ) , Missing Wholly At Random ( MCAR ) and Missing Not At Random ( MNAR ) . Harmonizing to the Sterne et. Al ( 2009 ) , losing value imputation is necessary under the premise of losing at random. However, it can besides be done under the instance losing complete at random. On presents most statistical bundles have the capableness of placing the type of missingness.

After placing the type of missingness, the losing value imputation comes into the field and it requires a statistical bundle to execute this. Since the losing value imputation in a hierarchal nature is small bit more advanced and it can non be done utilizing usual statistical bundles such as SPSS, SAS and R etc. Therefore Carpenter et. Al ( 2009 ) , developed the REALCOM package to execute this undertaking. However latter version of REALCOM was non deal with multilevel informations in a multivariate context. Therefore the macros related to execute this undertaking was late developed by the Bristol University squad in order to ease under this instance.

2.7 Estimation Procedure

The appraisal processs for multilevel mold are get downing late 1980â€™s. However For parametric quantity appraisal utilizing Maximum Likelihood Method, an iterative process called EM algorithm was used by early statisticians ( Raudenbush, Rowan and Kang, 1991 ) . Later on the plan HLM was developed to execute this algorithm.

The most operational processs for gauging multivariate multilevel theoretical accounts in the presence of Normal responses are Iterative Generalized Least Squares ( IGLS ) , Reweighted IGLS ( RIGLS ) and Marginal Quasi Likelihood ( MQL ) while for distinct responses are MQL and Penalized Quasi Likelihood ( PQL ) . Harmonizing to Rasbash, Steele, Browne and Goldstein ( 2004 ) all of these methods are implemented in MLwiN along with including first order or 2nd order Taylor Series enlargements. However since these methods are likelihood based frequentist methods they tend to overrate the preciseness.

Therefore more late the methods which are implemented in a Bayesian model utilizing Marcov Chain Monte Carlo methods ( Brooks, 1998 ) besides used for parametric quantity appraisal which allows capableness to utilize enlightening anterior distributions. These MCMC estimations executed in MLwiN provides consistent estimations though they require a big figure of simulations to command of holding extremely correlated ironss.

2.8 Previous researches conducted utilizing Univariate and Multivariate Multilevel Models

2.8.1 Univariate multilevel logit theoretical accounts

Before take a expression at to the literature on multivariate multilevel analysis, the literature of univariate multilevel analysis is besides be necessary to concerned since this thesis is based on some univariate multilevel theoretical accounts prior to suit multivariate multilevel theoretical accounts.

In the past decennaries, many societal Scientists used to use multilevel theoretical accounts for binay informations. Therefore it is really of import to reexamine how they have implemented their work with less engineering. As a purpose of that, Guo and Zhao ( 2000 ) was able to make a reappraisal of the methodological analysiss, hypothesis testing and hierarchal nature of the informations involve of past literature. Besides they conducted two illustrations for justify their consequences. First of all they made a comparing between estimations obtained from MQL and PQL methods which was implemented by MLn and the GLIMMIX method implemented by SAS by utilizing illustrations. They have shown that the differences in PQL 1 and PQL 2 are little when suiting binary logistic theoretical accounts. Furthermore, they have shown that PQL- 1 and PQL-2 and GLIMMIX are likely to be satisfactory for most of the past surveies undertaken in societal scientific disciplines.

Noortgate, Boeck and Meulders ( 2003 ) uses multilevel binary logit theoretical accounts for the intent of analysing Item Response Theory ( IRT ) theoretical accounts. For that they carried out an appraisal of the nine achievement marks for reading comprehension of pupils in primary schools in Belgium. They performed a multilevel analyses utilizing the cross-classified logistic multilevel theoretical accounts and used the GLIMMIX macro from SAS, every bit good as the MLwiN package. However they found that there were some convergence jobs arisen by utilizing PQL methods in MLwiN. Therefore they used SAS to carryout analysis. Furthermore they have shown that the cross-division multilevel logistic theoretical account is a really flexible to manage IRT informations and the parametric quantities can still be estimated even with the presence of imbalanced informations.

2.8.2 Multivariate Multilevel Models

In the past two decennaries a really few of researches have sought to suit the multivariate multilevel theoretical accounts to the existent universe scenarios. Among those besides all most all the researches seeking to concentrate fundamentally in educational sectors every bit good as socio economic sectors. None of them were able to concentrate these into the medical scenarios. However deficiency of multivariate multilevel analysis which presents in the field of wellness and medical scientific disciplines this chapter consists of the literatures of multivariate multilevel theoretical accounts in other Fieldss.

Harmonizing to the old surveies of instruction, Xin Ma ( 2001 ) examined the association between the academic accomplishments and the background of pupils in Canada by sing three degrees of involvement. For that the three degree Hierarchical Linear Model ( HLM ) was developed in order to accomplish his ends. This work allows him to pull the decisions that both pupils and schools were differentially successful in different capable countries and it was more obvious among pupils than among schools. However the success of this survey is based on some strong premises about the priors of studentâ€™s cognitive accomplishments.

Exclusive of the field of instruction Raudenbush, Johnson and Sampson ( 2003 ) carried out a survey in Chicago to find the condemnable behaviour at individual degree every bit good as at vicinity degree with regard to some personal features. For this intent they use a Rasch theoretical account with random effects by presuming conditional independency along with the additives.

Furthermore, Yang, Goldstein, Browne and Woodhouse ( 2002 ) developed a multivariate multilevel analysis of analysing scrutiny consequences via a series of theoretical accounts of increasing complexness. They used scrutiny consequences of two mathematics scrutinies in England in 1997 and analyzed them at single and institutional degree with regard to some pupils characteristics. By get downing from a simpler theoretical account of multivariate normalcy without sing the institutional random effects, they bit by bit increased the complexness of the theoretical account by adding institutional degrees together with the multivariate responses. When closely looked at, there work shows that the pick of topic is strongly associated with the public presentation.

Along with this growing of applications of multivariate multilevel theoretical accounts, researches may be given to use those in to the other Fieldss such as Forestry etc. Hall and Clutter ( 2004 ) presented a survey sing patterning the growing and output in forestry based on the cut pine in U.S.A. In their work, they developed a methodological analysis to suit nonlinear assorted consequence theoretical account in a multivariate multilevel frame work in order to place the effects of the several plot-level lumber measure features for the output of lumber volume.

In add-on to that they besides developed a methodological analysis to bring forth anticipations and anticipation intervals from those theoretical accounts. Then by utilizing their developments they have predicted lumber growing and output at the secret plan single and population degree.

Grilli and Rampichini ( 2003 ) carried out a survey to pattern ordinal response variables harmonizing to the pupils evaluation informations which were obtained from a study of class quality carried out by the University of Florence in 2000-2001 academic old ages. For that they developed an alternate specification to the multivariate multilevel probit ordinal response theoretical accounts by trusting on the fact that responses may be viewed as an extra silent person underside degree variable. However they non yet measure the efficiency of that method since they were non implemented it utilizing standard package.

When sing the groundss of the recent applications of these theoretical accounts the literature shows that Goldstein and Kounali ( 2009 ) late conducted a survey on kid goon growing with regard to the aggregation of growing measurings and grownup features. For that they extended the latent normal theoretical account for multilevel informations with assorted response types to the ordinal categorical responses with holding multiple classs for covariates. Since information consists of counts they bit by bit developed the theoretical account by get downing a theoretical account with presuming a Poison distribution. However since the informations are non follow precisely a Poisson distribution they treated the counts as an ordered classs to acquire rid of that job.

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Frank, Cerda and Rendon ( 2007 ) did a survey to place whether the residential location have an impact to the wellness hazard behaviours of Latino immigrants as they are increasing well in every twelvemonth. For that they used a Multivariate Multilevel Rasch theoretical account for the informations obtained by Los Angelis household and vicinity study based on two indices of wellness hazard behaviours along with their usage of drugs and engagement for hazard based activities. They get downing this effort by patterning the behaviour of striplings as a map of the features related to both single and neighborhood.According to the survey they found that there is an association between increased wellness hazard behaviours with the above state mean degrees of Latinos and poorness peculiarly for those who born in U.S.A.

Another application of multivariate multilevel theoretical accounts was carried out Subramanian, Kim and Kawachi ( 2005 ) in U.S.A. Their chief purpose was to place the person and community degree factors for the wellness and felicity of persons. For that they performed a multivariate multilevel arrested development analysis on the informations obtained by a study which was held on 2000. Their findings reflect that those who have hapless wellness and sadness have a high relationship with the single degree covariates

By looking at the available literature, it can be seen that there are some sum of surveies conducted on instruction and societal scientific disciplines in other states but none of the surveies conducted sing wellness and medical scientific disciplines. Therefore it is indispensable to execute a survey by analysing the mortality rates of some violent death diseases which are spread in worldwide to understand hazard factors and forms associated with these diseases in order to supply better penetrations about the disease to the populace every bit good as to the responsibly policy shapers.

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