selection of covariance structure in longitudinal...

1

Abstract

Linear mixed models with various covariance structures have becomeincreasingly popular in longitudinal data analysis because of its wideapplications. Literature is available on estimation of parameters with distinctcovariance structures. In this paper, estimation of parameters with distinctcovariance structures using ML and REML methods is considered forevaluation of models using six different information criteria with bootstrapdata. The bootstrap study is carried out to identify the most plausiblecovariance structure for longitudinal models using information criteria. Thebootstrap approach is adapted in this paper for the first time in the analysis oflongitudinal data with fourteen covariance structures. The study presentsanalysis of TLC data by considering both the nested and non-nested covariancestructure for comparison based on model selection criteria.

Keywords: Longitudinal study - Bootstrap method - Linear Mixed Model -Covariance Structures -Information Criteria - LR test.

SELECTION OF COVARIANCE STRUCTURE INLONGITUDINAL DATA BASED ON BOOTSTRAP

J. Mohanraj and M.R. SrinivasanDept. of Statistics, University of Madras, Chennai-600005

ASR Vol. 29. No. 1 March 2015

2

1. Introduction

Longitudinal study refers to an investigation where participant outcomes andpossibly treatments or exposures are collected at multiple follow-up times. Longitudinaldata analysis represents a convenient marriage of regression and time-series analysis.As with many regression data sets, longitudinal data are composed of a cross section ofsubjects. But unlike regression, longitudinal studies are observations of subjects overtime. Similarly, time-series data differ from longitudinal data as many subjects areinvolved in the study. Observing a broad cross section of subjects over time allows usto study dynamic movement, as well as cross-sectional, aspects of a problem.Longitudinal studies play a key role in epidemiology, clinical research and therapeuticevaluation. For example, to characterize normal growth and aging to assess the effectof risk factors on human health and to evaluate the effectiveness of treatments. Thereare several advantages of longitudinal data as compared to purely cross-sectional orpurely time-series data. The two important advantages are ability to study dynamicrelationships and to model the differences or heterogeneity, among the subjects. Ofcourse, longitudinal data are more complex than purely cross-sectional or times-seriesdata and so there is a price to pay in working with them. The most important drawbackis the difficulty in designing the sampling scheme to reduce the problem of subjectsleaving the study prior to its completion, known as attrition. The present study considersthe Rogan et al., (2001) Treatment of Lead-Exposed Children (TLC) data. Thelongitudinal data were analyzed with several heterogeneous covariance structures byusing a linear mixed model with Maximum Likelihood (ML) and Restricted MaximumLikelihood (REML) estimation. The main goal of this paper is selecting an appropriatecovariance structure and model selection criteria under linear mixed model and estimatesof parameters are obtained based on ML and REML approaches. This is the first attemptto study the bootstrap approach. The performances are evaluated for a set of unstructuredand a large number of structured models with longitudinal data of TLC (Rogan et. Al.,2001).

In this paper, we have studied fourteen heterogeneous covariance structures.These covariance structures are divided into two types namely, banded and non-banded.

J. Mohanraj and M.R. Srinivasan

3

The banded and non-banded covariance structure has been studied by various authorsincluding, Radhakrishnan and Monoharan (2012), Gazel Ser (2012),Chuck Kincaid(2012), Pankaj and Shukla (2011), Eyduran and Akbas (2010), Hussein and Marshadi(2007), Fitzmaurice et al. (2004), Littell et al. (2000), Keselman et al., (1999) andWolfinger (1996, 1993). The banded covariance structure was studied by, Littell et al.(2000), Ohlson and Rosen (2010) and Chaudhuri et al., (2007). The present papermainly involves using the bootstrap technique under linear mixed model setup to estimatethe parameters using ML and REML approach and identify the most appropriatecovariance structure model adapting six information criteria. The idea of using thebootstrap is to improve the performance of a model selection rule which was introducedby Efron (1983), and bootstrap sample size is taken to be same as the size of the givensample. The properties of the bootstrap have been extensively discussed by Efron andTibshirani (1986, 1993).

Our study involves fitting the linear mixed model with different covariancestructures and selecting the most appropriate covariance structure to the bootstrap sampledata by obtaining the criteria values of AIC, BIC, CAIC, HQIC, AICC and AVIC.Hussein and Marshadi (2007) have proposed the REML approach of selecting the bestamong eight different homogenous and heterogeneous covariance structures based onthe six information criteria.

2. Bootstrapping Method

The present study considers bootstrap method by Efron(1979). Thebootstrapping is a powerful statistical technique. Bootstrapping has become morepopular as computing resources have become more readily available. This is becausein order for bootstrapping to be practical a computer must be used. The bootstrapmethod has been studied by various authors including Yan and Fine (2004), Good(2001), Jin et al. (2001), Shermann and Cessie (1997), Hu and Zidek (1995), Efronand Tibshirani (1986, 1993), Paik (1988), Moulton and Zeger (1989) and Wu (1986).The purpose of this bootstrap study is to find out suitable covariance models. Mohanrajand Srinivasan (2015) have studied the linear mixed model without adapting bootstrapmethod. They have used ML and REML approach by considering fourteenheterogeneous covariance structures and applied six information criteria namely AIC,AICC, HQIC, BIC, CAIC and AVIC.

Selection of Covariance..............Data Based on Bootstrap

4

3. TLC Data

The present analysis is based on a sample of 80 subjects from the TLC datacollected by Rogan et al.,(2001). Exposure to lead can produce cognitive impairment,especially among young children and infants. A young child exposed to high levels oflead may experience various adverse health effects, including hyperactivity, hearing ormemory loss, learning disabilities, and damage to the brain and nervous system. Leadpoisoning in children is treatable in the sense that there are medical interventions,known as chelation treatments, that can help a child to excrete the lead that has beeningested. Until recently, chelation treatment of children with high levels of blood leadwas administered by injection and required hospitalization. A new chelating agent,succimer, enhances urinary excretion of lead and has the distinct advantage that it canbe given orally, rather than by injection.

The Treatment of Lead-Exposed Children (TLC) considers a placebo-controlled,randomized trial of Succimer conducted in 647 children. Children received up to three26-day courses of Succimer or Placebo and were followed for 3 years. The blood leadlevels at week0 (baseline), week 1, week 4, and week 6 are measured for each child.The longitudinal data are analyzed using the linear mixed model approach. Theparameters of the models are estimated by ML and REML method corresponding tofourteen covariance structures.

There are four time points (week0-week6) and two treatment groups in theTLC data. Mohanraj and Srinivasan (2015) have investigated the selection of mostappropriate covariance structure for the TLC data with 80 children. Now we considerthe same data set and their analysis based on bootstrap method. Yue Li (2005) hasstated "resampling" approaches which are expected to gain more efficiency comparedwith the classical resampling methods and bootstrap.

From the dta set of 80 children a group of 20% is selected for the presentbootstrap sample study. Now 100 bootstrap samples are generated with this sample of20% of children. The analysis involves estimation of parameters and obtaining thevalues


5

of the information criteria under fourteen covariance structures have been studied foreach bootstrap samples. The linear mixed model (1) has been fitted and the values ofthe six information criteria are determined and presented Table (2). In a similar manner,sample of sizes 30%, 40% and 50% of the set of 80 childeren are chosen and for eachof these selected samples of bootstrap method has been applied. The results showingthe values of six information criteria for each of the samples of the sizes 30%, 40% and50% are presented in Tables (3 to 5).

4. Linear Mixed Model Approach

Linear Mixed effects model are mixed effects model in which both fixed andrandom effects contribute linearly to the response function. The general form of such amodel is,

ZXY . . . (1)

Y is an vector of observations, is an p × 1 vector of fixed effects, X is an n × p designmatrix for fixed effects, Z is a given n × q matrix, and is an unobservable randomvector of dimensions q × 1, is an n × 1 vector of residual and both and aredistributed as N(0,G) and N(0,R) respectively, i.e.,

0

0E

R

GVar 00

Here V),N(X~ Y and variance of Y is RZGZV ' . The model v is settingup the random-effects design matrix Z and by specifying covariance structures for Gand R. Simple random effects are a special case of the general mixed model specificationwith Z containing dummy variables, G containing variance components in a diagonalstructure, and nIR 2 , where In denotes the identity matrix. Further, general linearfixed effect model is a special case with Z = 0 and nIR 2 .

Estimation is more difficult in the mixed model than in the general linear model.Besides, as in the general linear model, we have unknown parameters in , G and R


6

as well. Least squares approach is no longer appropriate and Generalized least squares(GLS) is more appropriate by minimizing,

XYVXY 1'

However, it requires knowledge of V and thereby knowledge of G and R. Lackingsuch information, a reasonable estimation for V is obtained and the same is substitutedin the GLS for the minimization problem. The goal thus becomes finding a reasonableestimate for G and R. In many situations, the best approach is to use likelihood-basedmethods, exploiting the assumption that and are normally distributed (Hartley andRao (1967), Harville (1977), Laird and Ware (1982), Jennrich and Schluchter(1986)).Basically the Variance components analysis usually applies to a linear mixedeffect models, that is, one in which there are random and fixed effect and is consideredoften as an adjunct to these procedures. Unlike them, the variance components procedureestimates only variance components, not model regression coefficients. Variancecomponents analysis may be seen as a more computationally efficient procedure usefulfor models in special designs, such as split plot, univariate repeated measures, randomblock, and other linear mixed effect designs. Variance components procedure supportsfour methods of estimation, each of which gives somewhat different estimates: Analysisof Variance (ANOVA), Maximum Likelihood (ML), Restricted Maximum Likelihood(REML), and the Minimum Norm Quadratic Unbiased Estimator (MINQUE) method.But, the present study is restricted to comparison of two likelihood estimation methodsare namely, ML and REML.

A favorable theoretical property of ML and REML are given in (Rubin 1976;Little 1995). PROC MIXED constructs an objective function associated with ML orREML and maximizes it over all unknown parameters. Using calculus, it is possible toreduce this maximization problem over the only parameters in G and R. Thecorresponding log-likelihood functions are as follows:

2log22

1log21:, 1 nrVrVRGl T

ML


7

2log2

)(log21

21log

21:, 11 pnXVXrVrVRGl TT

REML

Where, YVXXVXXYr TT 111 )( and p is rank of X. Variance components models(Searle et al. 1992), mixed effects ANOVA models (Miller, 1977), and linear modelsfor longitudinal data (Laird and Ware, 1982) are all special cases of model(1). Whenobservation are missing in planned experiments, like incomplete block design standardlinear mixed model approach can be used to estimate the parameters and requiredhypothesis could be tested in the presence of random effects (Giesbrecht, 1989). TheMaximum likelihood (ML) and restricted maximum likelihood (REML) are the mostcommon estimation methods used for linear mixed effect models. The ML and REMLapproach to repeated measurements and the traditional treatment effects are based onthe covariance models using the linear mixed model, paying special attention to completedata. An important advantage of the linear mixed model approach is that completenessof the data on the dependent variable does not complicate the analysis.

The ML approach will first obtain the estimates of , 2 and 2

by maximizingthe likelihood function over the parameter space. The advantages of ML approach areavoidance of negative estimates for the variance components. Its disadvantages are;(a) Numerically intensive, (b) Resulting estimators are not unbiased, (c) Solving thelikelihood equations requires an iterative process which may or may not converge.Even when it converges, it may converge at a local maxima rather than a globalmaximum, (d) Tends to underestimate the variance components and (e) Distributionalproperties are not known except asymptotically.

REML estimators of the variance components are found by maximizing thatpart of the likelihood function which is invariant to fixed effects in the model. Thelikelihood function formed from Y depends only on the variance components and theREML estimates of the variance components are those values of 2

and 2 that

maximize the restricted likelihood function formed from Y. The advantages in the caseof REML are: (a) Less numerically intensive than the Maximum Likelihood Method,


8

(b) REML estimates and the ANOVA estimates agree when the data are balanced andall Method of Moments (MM) estimates of the variance components are non-negativeand (c) REML estimates tend to be less biased than the Maximum Likelihood Estimates.However the REML too has disadvantages like ML such as distributional properties ofthese estimators are not known, except asymptotically.

5. Selection of Covariance Structure

The process of analyzing data usually begins with the choice of G and R, oftenreferred as the covariance structure specification. Commenting on the importance ofthis decision, Littell et al., (2000) noted that by using incorrect covariance structurescould risk obtaining invalid estimators and inferences. Since the generalized least squaresestimator is the best linear unbiased estimator, an incorrect covariance structure willaffect the quality of the estimator. Littell et al., (2000) suggested that the first thing todo when choosing a covariance structure in repeated measures studies is to computethe Heterogeneous Compound symmetry covariance matrix and compare it to theestimates obtained using other fourteen heterogeneous covariance structures namely,Unstructured (UN), First Order Banded Unstructured (UN(1)), Second Order BandedUnstructured (UN(2)), Third order Banded Unstructured (UN(3)), First Order BandedUnstructured correlations (UNR(1)), Second Order Banded Unstructuredcorrelations(UNR(2)), Heterogeneous compound symmetry (CSH), First orderHeterogeneous Auto Regressive (ARH(1)), Heterogeneous Toeplitz (TOEPH),Heterogeneous Toeplitz with one Banded (TOEPH(1)), Heterogeneous Toeplitz withtwo Banded (TOEPH(2)), Heterogeneous Toeplitz with three Banded (TOEPH(3)),First Order-Factor Analytic (FA(1)), Hyunh-Feldt (HF) and notations of the abovestructures are given in the Table (1). The heterogeneous models are divided into twotypes, namely,

Heterogeneous for non-zero correlation models structures;

UN, CSH, ARH(1), TOEPH, FA(1) and HF

Heterogeneous for zero correlation models structures;

UN (1), UNR (1) and TOEPH (1)


9

Generally heterogeneous models have unequal variances on main diagonaland separate covariances as off diagonal, Variances are estimated for each treatmenttime covariances estimated for each pair of treatment times and UN model is the mostcomplex of the heterogeneous model. In this paper, all the covariance structures onmain diagonals are unequal.

PROC MIXED in SAS provides a very flexible environment in which modelcan be many of repeated measures data. Correlation among measurements made onsame subject or experiment unit can be modeled using random effect and through thespecification of a covariance structure. PROC MIXED provides a useful covariancestructures for modeling both time and space, including that of discrete & continuousincrements.

The selection of the most appropriate covariance structure is important in theanalysis of longitudinal data. The estimates of the parameters of the longitudinal modellargely depend upon the covariance structure. The covariance estimation is also ofinterest by itself (Diggle and Verbyla, 1998). Several authors have contributed to thestudy of covariance structures. In this paper, we have studied the selection of appropriatecovariance structure of the data of TLC trail experiment. The results of some of theresearchers are as follows by, Radhakrishnan and Monoharan (2012) have studied thegeneral linear regression model using REML approach of selecting differenthomogenous and heterogeneous with banded, non banded covariance structures andconsidered three information criteria namely AIC, AICC and BIC. The analysis basedon simulation confirmed that Unstructured covariance structure is most appropriateamong the different covariance structures. Gasel Zer (2012) have studied the problemof covariance structure relating to the data weights of animal, in which they have adaptedthe Random Intercept and Slop Model (RISM), the covariance structures consider bythem are given in two groups of covariance structures namely, heterogeneous andhomogeneous. The


10

heterogeneous covariance structure group members are UN, CSH, ARH(1), TOEPH,ANTE(1) and UNR. The homogeneous covariance structure group members are CS,TOEP and AR(1). The covariance structures are based on the analysis of weights ofanimal data and it is concluded that the heterogeneous models are better performers.Chuck Kincaid (2012) have practically described linear mixed models with variouscovariance structure, and concluded that random effect models is better fit in theseevaluation of covariance structure and these analyses were worked in PROC MIXEDdocumentation in the SAS software. Hussein and Marshadi (2007) studied simulationdata set using twelve covariance structure and model selection using six informationcriteria, and concluded that CAIC and BIC are better performer in these covariancemodel selection study. Keselman et. al., (1999) has investigated using the AIC and BICfor various conditions (e.g., covariance heterogeneity, nonnormality, unequal groupsizes) and found their performances, which concluded saying AIC is a better fit in therepeated measures data. Eyduran and Akbas (2010) have studied compare performanceof univariate and multivariate approaches by using nine covariance structures, modelselection using five information criteria and concluded multivariate approach is a betterfit in this covariance structure study. Wolfinger (1993, 1996) has discussed performanceof heterogeneous covariance model selection for the longitudinal data in linear mixedmodel. Ohlson and Rosen (2010) studied banded matrices with unequal elements exceptthat certain covariances are zero. This banded structure is a special case of the structureconsidered by Chaudhuri et. al., (2007), the basic idea is that widely separatedobservations in time or space often appear to be uncorrelated. Therefore, it is reasonableto work with a banded covariance structure, where all covariances more than 'n' stepsapart equal zero, a so called n-dependent structure. However, in general it was observedthat more the number of covariance parameters in the model the lower the efficiencyvalues in the information criteria. However for the first order banded structure UN(1),UNR(1) and TOEPH(1) have the same covariance structure.


11

Table (1) Matrix notation of different covariance models

24434241

34233231

24232221

14131221

UN

24

23

22

21

)1(

UN

2443

342332

232221

1221

)2(

UN

24

23

22

21

)1(

UNR

244342

34233231

24232221

131221

)3(

UN

244334

3443233223

2332222112

122121

)2(

UNR

24342414

43232313

42322212

41312121

2

CSH

24

23

22

21

)1(

TOEPH

24134224

14323123213

24213222112

23112121

)3(

TOEPH

424342414

433232313

213222212

413121121

)1(

FA

24

23

24

22

24

21

24

24

232

3

22

23

21

23

24

22

23

222

2

21

22

24

21

23

21

22

212

1

222

222

222

222

HF

Model selection involves the choice of an appropriate model among a set of

candidate models and is used when there is no particular clear choice among many

different models. Model selection tools are a useful set of techniques for screening

through many different covariance models. The choice among models can be made by

comparing the maximum likelihoods for each of the covariance pattern models. Fixed

effects models and covariance models are rather different; they are usually treated

differently in modelling and selection for the covariance parameters and fixed effect

parameters.The more complex the model is (more parameters), a better fit and a higher

likelihood function value is obtained and information criteria are widely used in model

selection.


12

6. Information Criteria (IC):

There are two statistics based on the likelihood that make allowance for thenumber of covariance parameters fitted and can be used to compare two models whichfit the same fixed effects. The criteria value based on REML AND ML methods, totalnumber of cases 'n' , number of model parameters 'k' and likelihood value 'l' provide thebasis for various information criteria such as:

(1) Akaike Information Criteria (AIC):

)log(22 lKAIC

(2) Corrected Akaike Information Criteria (AICC):

)1(2)log(2

kn

nklAICC

(3) Bayesian Information Criteria (BIC):

)log()log(2 nklBIC

(4) Consistent Akaike Information Criteria(CAIC):

)1(log)log(2 nklCAIC

(5) Hannan-Quinn Information Criteria (HQIC):

knlHQIC ))log(log(2)log(2

(6) Average Information Criteria (AVIC)

AVIC = AIC + AICC + BIC + CAIC + HQIC

For REML, the value of 'n' is chosen to be total number of cases minus numberof fixed effects parameters and 'k' is number of covariance parameters and for ML, thevalue of 'n' is total number of cases and 'k' is number of fixed effects parameters plusnumber of covariance parameters.


13

Akaike Information Criterion (AIC) was developed by Akaike and Hirotugu(1974) under the name of "an information criterion" and was first published by Akaike(1973). AICC was originally proposed for linear regression (only) by Sugiura (1978)and later developed by Hurvich and Tsai (1989). The BIC was developed by Gideonand Schwarz (1978), Akaike was so impressed with Schwarz's Bayesian formalismdeveloped his own Bayesian formalism referred to as the ABIC for "a BayesianInformation Criterion" or more casually "Akaike's Bayesian Information Criterion"(Akaike, 1973). The HQIC was proposed by Hannan and Quinn (1979) proved to beuseful in the case of autoregressive models.

As k, the dimension (number of parameters) of the candidate model, increasesin comparison to n, the sample size, AIC becomes a strongly negatively biased estimateof the information and the bias is corrected by Akaike Information Criterion (AICC).The correction is of particular interest and use when the sample size is small, or whenthe number of fitted parameters is a moderate to large fraction of the sample size. Forlinear mixed model, the corrected method, AICC, is exactly unbiased. The BayesianInformation Criterion (BIC) is a criterion of model selection among a finite set ofmodels. It is based, in part, on the likelihood function likelihood function and it isclosely related to the Akaike Information Criterion (AIC). When fitting models, it ispossible to increase the likelihood by adding parameters, but doing so may result inover fitting. Both BIC and AIC resolve this problem by introducing a penalty term forthe number of parameters in the model; the penalty term is larger in BIC than in AIC.The HQIC proposed contain both of AIC and BIC and proved to be useful in thedetermination of the order of an Auto regression. Burnham and Anderson (2002) notedthat HQIC, like BIC, but unlike AIC, is not an estimator of Kullback-Leibler divergencebut Claeskens and Hjort (2008) observed that HQIC is not asymptotically efficient.The comparative studies of information criteria is a powerful tool that can be used tofind out the better ?t between various covariance models described above.

Pankaj and Shukla (2011) discussed the longitudinal data for linear mixedmodel setting using REML approach for covariance model selection based on sixinformation


14

criteria. Gazel Ser (2012) has proposed the linear mixed model for repeated measuredata to draw comparison between ML and REML approaches of covariance modelselection criteria. Radhakrishnan and Monoharan (2012) have discussed longitudinaldata for linear regression model settings using REML approach for covariance modelselection. Eyduran and Akbas (2010) have proposed univariate and multivariateapproaches to analyze the repeated measures data and determine the best covariancestructure. In this study, heterogeneous covariance structures and model selectioninformation criteria are analyzed in SAS software.

7. Analysis of Nested and Non-Nested Models Using TLC Data

The likelihood ratio test can also be used to compare models which fit thesame fixed effects and whose covariance patterns are nested. Nesting is when thecovariance pattern in the simpler model can be obtained by restricting some of theparameters in the more complex model. An example of this would be as follows:

24342414

43232313

42322212

41312121

2

CSH

24134224314

14323123213

24213222112

33123112121

TOEPH

Heterogeneous Compound Symmetry model (CSH) is nested withinHeterogeneous Toeplitz (TOEPH) model, since the Heterogeneous CompoundSymmetry model holds, then the heterogeneous Toeplitz model must necessarily hold,with 1 = 2 = 3.


15

In many ways, the likelihood ratio test is conceptually the easiest of the testconsidered in this section using for covariance model (CM). The estimate obtained byREML (or) ML denoted by

fullICCMl )( is the value of the likelihood function for themaximum of the unconstrained (full) model while

min)( ICCMl is the value of the likelihood

function of the constrained (minimum = min) model. The difference of the likelihoodratio test is computed as:

2df~))(-)((2

minICICIC CMlCMl

fullCM

Where, IC is the six information criteria and CM is the structure of the covariancemodel. This difference follows the chi square distribution with appropriate degrees offreedom (df) and this has been used comparison of the model selection criteria. Thistest is always positive (or zero) since the likelihood of the unconstrained model is atleast as high as that of the constrained model.

The SAS codes and the results corresponding to various sizes of the bootstrapsamples and the analysis of performance based on information criteria using fourteencovariance structures are given in APPENDIX-I and APPENDIX-II.

8. Results of the Analysis:

The present study considers the analysis of TLC data based on the generallinear fixed effect model which is a special case of linear mixed model given in equation(1) with Z = 0 and R = 2In. The heterogeneous models with maximum number ofestimated parameters, namely UN, CSH, TOEPH and FA(1)were found to be the goodperformers compared to the other heterogeneous covariance structures. In particularUN and CSH covariance structures have outperformed in these entire heterogeneousmodels. The results are presented in the Table (2 to 5) given in APPENDIX-II. Showthe performance of the covariance structures based on the estimation method, ML andREML. Goodness of fit results indicated REML as the most appropriate covarianceparameter estimation approach. It is true for a single sample data study but cannot beseen directly in the bootstrap samples. The AIC, AICC, HQIC, BIC, CAIC, AVIC chosencriteria showed heterogeneous covariance models as the best model group as UN, CSHFA(1) and HF.


16

The values of AIC, AICC and HQIC criteria based on ML and REML estimationof parameters strongly recommend for these four covariance structures and in particularUN covariance model, because the other three covariance models of the likelihooddifferences are smaller (p-value) than UN covariance model and their informationcriteria. The computed values of BIC, CAIC and AVIC criteria based on ML andREML estimation of parameters strongly recommend CSH covariance structure, alsothese three information criteria have given higher likelihood values compared to thevalues for AIC, AICC and HQIC. In these heterogeneous covariance groups of modelselection criteria has shown the banded covariance model is weakest models in theheterogeneous covariance model group namely, UN(1), UN(2),UN(3), UNR(1),UNR(2), UNR(3), TOEPH(1), TOEPH(2) and TOEPH(3). More critically, the estimatesof all fourteen covariance structures are very similar and the differences are unlikely toaffect the results of the primary analyses.

Now for each of the fourteen Covariance structures and corresponding to eachof the six information criteria, the values of Likelihood Ratio (LR) test statistics areobtained. The Differences in the values of LR test statistics and values of the probability(p value) are given in Table (2-5). This has been done for each of the four bootstrapsample data with difference percentages. For example in the case of 40% bootstrapsample data set, the difference in the values of the LR test statistic between the twocovariance structures CSH and UN with respect to each of the six information criteriaare given below.

0.6692)(~2.3)7.1051*5.9.1054*5(.2 25 p

CSHAIC

0.9770)(~8.)5.1056*5.3.1057*5(.2 25 p

CSHAICC

0.9998)(~1.0)7.1062*5.8.1062*5(.2 25 p

CSHHQIC

0.8703)(~3.5)8.1076*5.1.1082*5(.2 25 p

UNBIC

0.4146)(~3.10)8.1089*5.1.1100*5(.2 25 p

UNCAIC

0.9935)(~3.2)62.1070*5.3.1068*5(.2 25 p

UNAVIC


17

Each one of the above difference of the LR statistics is compared to a chi-squareddistribution with 5 (or 10-5) degrees of freedom (df). The value of the first difference(3.2) and the p value (0.6692) show that

CSHAIC of Heterogeneous Compound Symmetrycovariance structure does not provide an adequate fit. The Unstructured covariancestructure (p =1.00) provides an adequate fit based on the values given in Tables (2-5).In a similar manner the values of the second and third differences (0.8 and 0.1) andtheir p values (p = 0.9770, 0.9998) showed that

CSHAICC andCSHHQIC of Heterogeneous

Compound Symmetry covariance structure does not provide an adequate fit andunstructured (p =1.000, 1.000) covariance structure provides an adequate fit to thevarious bootstrap sample sizes.

The value of the fourth difference (5.3) and the p value (0.8703) show that

UNBIC of unstructured covariance structure does not provide an adequate fit. TheHeterogeneous Compound Symmetry covariance structure (p = 1.00) provides anadequate fit based on the values given in Tables (2-5). In a similar manner the values ofthe fifth and sixth differences (10.3 and 2.3) and their p values (p = 0.4146 and 0.9935)showed that

UNCAIC andUNAVIC of unstructured covariance structure does not provide

an adequate fit and Heterogeneous Compound Symmetry (p =1.00, 1.00) covariancestructure provides an adequate fit to the various bootstrap sample sizes. In the tables(2 to 5) values of p =1.00 can be seen dirtectly for the other p values are not seenexplicitly since this values corresponds to the results of the four bootstrap samplesizes.

In the above first three difference of the LR test statistics,CSHAIC ,

CSHAICC and

CSHHQIC indicate that the UN Covariance structure is a significant improvement overthe simpler CSH covariance model. However,

UNBIC ,UNCAIC and

UNAVIC indicate thatthe CSH covariance structure is a significant improvement over the simpler UNCovariance model. In those heterogeneous model with p values (1.00) indicate mostappropriate model fit (refer to the Table (2), Table (3), Table(4) and Table(5) inAppendix-II). The comparison of the nested and non-nested cases in the heterogeneousmodels of the observed 2 value at the appropriate degrees of freedom is approximatelysignificant as indicated by the high p-values (1.00).


18

The bootstrap study as carried out has no unanimity and this could be seenselecting a covariance structure based on information criteria for various sizes ofbootstrap samples. For example, the values of AIC, corresponding to ML and REML inTable (2) show that UN is an adequate fit in 40% and 36%. The CSH is an adequate fitin 28% and 28%. The FA(1) is an adequate fit in 32% and 24%. In a similar manner foreach of the other five information criteria, the percentage of bootstrap sample providingadequate fit (1.00) to the corresponding models can be obtained. The values of theTables (3-5) provide details about the percentage giving the adequate fit with respectto the corresponding model for bootstrap sample of sizes 30%, 40% and 50%. Theresults are varying to situations when the percentages of data observations increasedfrom 30%, 40% and 50%.

We note that, in most of the bootstrap sample data sets whenever UN model isfind to provide adequate fit, FA(1) model also gives an adequate fit to the data. Also,whenever CSH model is find to provide adequate fit, HF model also gives an adequatefit in most of bootstrap samples. Generally UN and CSH covariance model performanceare good in information criteria, as observed in the range of 50% to 80% of results andotherwise FA(1) and HF covariance structures are contained 20 % to 50% of results.

The estimates of the parameters of all fourteen covariance structures are verysimilar and the differences are unlikely to affect the results of the primary analyses.Therefore, statistical analysis justify in using the more complex covariance patternUN, CSH, FA (1) and HF has a significant better fit in comparison to other covariancemodels.

9. Conclusion

In this bootstrap study, the linear mixed model was applied to the analysis ofTLC data of longitudinal study. The results show that performance of REML approachseems to be more efficient than ML approach. The conclusions based on differentbootstrap sample data sets of various percentages (20%, 30%, 40% and 50%) are notsimilar but much different. These are based on results presented in Tables (2-5). The


19

overall results reveal that heterogeneous covariance models are the most appropriatemodel group with the covariance structures as in UN, CSH, FA(1) and HF. These fourcovariance models performed well and UN and CSH structures outperform with respectto all the information criteria.

The model selection criteria play an important role in selecting the appropriatecovariance structures. The study has shown that though well known criteria namely,AIC, AICC and HQIC suggest UN covariance structure as the most appropriate andbetter than CSH, FA(1) and HF models. The CAIC, BIC and AVIC strongly suggestCSH covariance structure as the most appropriate and better than UN, FA (1) and HF models.The banded covariance models are weakest among the heterogeneous covariancestructures namely UN (1), UN (2), UN (3), UNR (1), UNR (2), TOEPH (1), TOEPH(2) and TOEPH (3). The results given in Tables (2-5) show that, the information criteriaAIC, AICC and HQIC indicate that the model with unstructured (UN) covariancestructure is the most appropriate. While the information criteria BIC, CAIC and AVICshow the selection of the model with Heterogeneous Compound Symmetry (CSH)covariance structure as most appropriate. In conclusion we observe that, the analysisof four various sizes of bootstrap samples sets from the longitudinal data based onlinear mixed model has shown that UN and CSH covariance structure to be moreappropriate in dealing with nested and non-nested covariance structures.


20

References:

[1] Akaike, H. (1973): Information theory and an extension of the maximumlikelihood principle, In: Petrov, B.N., Csaki, F. (Eds.), Second InternationalSymposium on Information Theory, 261-281.

[2] Akaike and Hirotugu (1974): A new look at the statistical model identi?cation,IEEE Transactions on Automatic Control, 19, 716-723.

[3] Ali Hussein and AL Marshadi (2007): The New Approach to Guide the Selectionof the Covariance Structure in Mixed Model. Research Journal of Medicineand Medical Sciences, 2:2, 88-97.

[4] Burnham, K. P. and Anderson, D. R. (2002): Model Selection and MultimodelInference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag.

[5] Chaudhuri, S., Drton, M. and Richardson, T. S. (2007): Estimation of acovariance matrix with zeros. Biometrika, 94(1):199-216.

[6] Claeskens, G. and Hjort, N. L. (2008): Model Selection and Model Averaging,Cambridge.

[7] Chuck Kincaid (2012): Guidelines for Selecting the Covariance Structure inMixed Model Analysis, COMSYS Information Technology Services, Inc.,Portage, MI.

[8] Diggle, P. and Verbyla, A. (1998): Nonparametric estimation of covariancestructure in longitudinal data, Biometrics, 54, 401-415.

[9] Efron, B. (1979): Bootstrap methods: another look at the jackknife. Annals ofStatistics, 7, 1-26.

[10] Efron and Gong (1983), A Leisurely Look at the Bootstrap, the Jackknife, andCross Validation, The American Statistician, Vol.37, No.1, 36-48.

[11] Efron, B. and Tibshirani, R. J. (1986): Bootstrap Methods for Standard Errors,Confidence Intervals, and Other Measures of Statistical Accuracy.


21

[12] Efron, B. and Tibshirani, R. J. (1993): An Introduction to the Bootstrap,Chapman & Hall, New York.

[13] Eyduran, E. and Akbas, Y. (2010): Comparison of different covariance structureused for experimental design with repeated measurement. Journal of Animal& Plant Sciences, 20:1, 44-51.

[14] Garrett Fitzmaurice, M., Laird Nan, M. and James H. Ware, (2004): AppliedLongitudinal Analysis, John Wiley & Sons, Inc., 326-328.

[15] Gazel Ser (2012): Determination of appropriate covariance structures in randomslope and intercept model applied in repeated measures, Journal of Animal &Plant Sciences, 22:3, 552-555.

[16] Gideon and Schwarz (1978): Estimating the dimension of a model. The AnnalsOF Statistics, vol.6, no.2, 461-464.

[17] Giesbrecht, F. G. (1989): A General Structure for the Class of Mixed LinearModels, Southern Cooperative Series Bulletin 343, Louisiana Station, BatonRouge. Agricultural Experiment Station, Baton Rouge.

[18] Good, P. I. (2001): Resampling Methods: a practical guide to data analysis,Birkhauser.

[19] Hannan, E. J. and Quinn, B. G. (1979): The Determination of the Order of anAutoregression, Journal of the Royal Statistical Society, B, 41, 190-195.

[20] Harville, D. A. (1977): Maximum likelihood approaches to variance componentsestimation and to related problems, Journal of the American StatisticalAssociation, 72, 320-338.

[21] Hartley, H. O. and Rao, J. N. K. (1967): Maximum likelihood estimation forthe mixed analysis of variance model, Biometrika, 54, 93-108.

[22] Hurvich, C. M. and Tsai, C. L. (1989): Regression and time series modelselection in small samples, Biometrika, 76, 297-307.


22

[23] Hu, F. and Zidek, J. V. (1995): A bootstrap based on the estimating equationsof the linear model, Biometrika, 82, 263-275.

[24] Jennrich, R. I. and Schluchter, M. D. (1986): Unbalanced repeated-measuresmodels with structured covariance matrices, Biometrics, 42, 805-820.

[25] Jin, Z., Ying, Z. and Wei, L. J. (2001): A resampling method by perturbing theminimand, Biometrika, 88, 381-390.

[26] Keselman, H. J., Algina. J., Kowalchuk. R. K. and Wolfinger, R. D. (1999):The analysis of repeated measurements: a comparison of mixed-modelSatterthwaite F-test and a nonpooled adjusted degrees of freedom multivariatetest. communication statistics theory and methods, 28, 2967-2999.

[27] Laird, Nan M. and Ware, James H. (1982): Random-Effects Models forLongitudinal Data. Biometrics, 38 (4): 963-974.

[28] Little, R. J. (1995): Modeling the Dropout Mechanism in Repeated MeasuresStudies. Journal of the American Statistical Association, 90, 1112-1121.

[29] Littlell. R C., Milliken, G. A., Stroup, W. and Wolfinger, R. D. (2000): SASsystem for mixed models.4th edn.SAS Institute, Cary, NC, 437-432.

[30] Littell, R.C., Pendergast, J. and Natarajan. R. (2000): Modeling covariances t r u c t u r e i n t h e a n a l y s i s o f r e p e a t e d m e a s u r e s d a t a . Statistics in Medicine, 19:1793-1819.

[31] Miller, J. J. (1977): Asymptotic properties of maximum likelihood estimatesin the mixed model of the analysis of variance. Ann. of Statistics, 5, 746-762.

[32] Mohanraj, J. and Srinivasan, M. R. (2015): Assessment of Covariance Structurein Longitudinal Analysis. Communicated for publication.

[33] Moulton, L. H. and Zeger, S. L. (1989): Analyzing repeated measures ongeneralized linear models via the bootstrap. Biometrics, 45, 381-394.


23

[34] Ohlson, M. and Von Rosen, D. (2010): Explicit estimators in the Growth Curvemodel with a patterned covariance matrix, Journal of Multivariate Analysis,Volume 101, Issue 5, 1284-1295.

[35] Paik, M. C. (1988): Repeated measurement analysis for nonnormal data insmall samples, Communications in Statistics, Simulation and Computation,17, 1155-1171.

[36] Pankaj Tiwari and Gaurav Shukla (2011): Approach of Linear Mixed Model inLongitudinal Data Analysis Using SAS, Journal of Reliability and StatisticalStudies, 0974-8024.

[37] Radhakrishnan Nair,K. and Manoharan, M.(2012): On selection of covariancestructure in modelling longitudinal data, Application of Reliability Theory andSurvival Analysis,

[38] Rogan, W. J., Dietrich, K. N., Ware, J. H., Dockery, D. W., Salganik, M.,Radcliffe, J., Jones, R. L., Ragan, N. B., Chisolm, J. J. and Rhoads, G. G.(2001): The effect of chelation therapy with succimer on neuro psychologicaldevelopment in children exposed to lead, New England Journal of Medicine,1421-1426.

[39] Rubin, D. B. (1976). Inference and missing data (with discussion). Biometrika63 581-592.

[40] Searle, S. R., Casella, G. and McCulloch, C. E. (1992): Variance Components,Wiley and Sons, New York.

[41] Sherman, M. and le Cessie, S. (1997): A comparison between bootstrap methodsand generalized estimating equations for correlated outcomes in generalizedlinear models, Communications in Statistics, Simulation and Computation, 26,901-925.

[42] Sugiura, N. (1978): Further analysis of the data by Akaike's information criterionand the finite corrections. Communications in Statistics - Theory and MethodsA7, 13-26.


24

[43] Wolfinger, R. D. (1993): Covariance structure selection in general mixedmodels, Communication in Statistics, Simulation, 22, 1079-1106.

[44] Wolfinger, R. D. (1996): Heterogeneous Variance-Covariance Structures forRepeated Measures. Journal of Agricultural, Biological, and EnvironmentalStatistics, 1, 205-230.

[45] Wu, C. F. J. (1986): Jackknife, bootstrap and other resampling methods inregression analysis (with discussion), Annals of Statistics, 14, 1261-95.

[46] Yan, J. and Fine, J. (2004): Estimating equations for association structures.Statistics in Medicine 23, 859-874.

[47] Yue Li (2005): Resampling Methods For Longitudinal Data Analysis, PublishedThesis.


25

% macro mix;% LET rept = 5;% LET set name = TLC20;% do i = 1 %to & rept;data TLC; set & set name (where = (samp = &i)); y = l0; time = 0; output; y = l1; time = 1; output; y = l4; time = 4; output; y = l6; time = 6; output; drop l0 l1 l4 l6; run;

proc print data = tlc(obs = 1) NOOBS; var samp; run;

proc mixed noclprint = 10 order = data data = tlc;class id grp time; model y = grp time grp*time / s chisq ;repeated time / type = UN subject = id r;run;

% end;% mend;% mixmiss

APPENDIX-I

SAS CODES IN COVARIANCE STRUCTURES ANALYSIS


26

APPENDIX-IITable (2) % of most appropriate model for 14 covariance structures

corresponding to 20% bootstrap data (p < 1.00)

S.NOCovarianceStructure

EstimationMethod

AIC(%)

)( 2df AICLR

AICC(%))( 2

dfCAICLR

HQIC(%))( 2

df HQICLR

BIC (%)

)( 2df BICLR

CAIC (%))( 2

df CAICLR

AVIC(%))( 2

df AVICLR

1UN

MLREML

40%(1.00)36%(1.00)

40%(1.00)40%(1.00)

36%(1.00)36%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

10%(1.00)10%(1.00)

2UN(1)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

3UN(2)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

4UN(3)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

5UNR(1)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

6UNR(2)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

7CSH

MLREML

28%(1.00)28%(1.00)

8%(1.00)12%(1.00)

32%(1.00)20%(1.00)

52%(1.00)48%(1.00)

56%(1.00)56%(1.00)

52%(1.00)44%(1.00)

8ARH(1)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

4%(1.00)4%(1.00)

0%(1.00)0%(1.00)

9TOEPH

MLREML

4%(1.00)4%(1.00)

4%(1.00)4%(1.00)

4%(1.00)4%(1.00)

4%(1.00)0%(1.00)

0%(1.00)0%(1.00)

4%(1.00)0%(1.00)

10TOEPH(1)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

11TOEPH(2)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

12TOEPH(3)

MLREML

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

13FA(1)

MLREML

32%(1.00)24%(1.00)

24%(1.00)24%(1.00)

24%(1.00)20%(1.00)

20%(1.00)24%(1.00)

8%(1.00)8%(1.00)

2%(01.00)24%(1.00)

14HF

MLREML

12%(1.00)16%(1.00)

28%(1.00)24%(1.00)

16%(1.00)24%(1.00)

24%(1.00)32%(1.00)

36%(1.00)36%(1.00)

24%(1.00)32%(1.00)


27

Table (3) % of most appropriate model for 14 covariance structurescorresponding to 30% bootstrap data (p<1.00)

S.No.CovarianceStructure

EstimationMethod

AIC(%)

)( 2df AICLR

AICC(%)

)( 2df

CAICLR

HQIC(%))( 2

df HQICLR

BIC (%)

)( 2df BICLR

CAIC (%)

)( 2df CAICLR

AVIC(%)

)( 2df AVICLR

1 UNML

REML48%(1.00)48%(1.00)

32%(1.00)36%(1.00)

36%(1.00)32%(1.00)

0%(1.00)4%(1.00)

0%(1.00)0%(1.00)

12%(1.00)20%(1.00)

2 UN(1)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

3 UN(2)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

4 UN(3)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

5 UNR(1)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

6 UNR(2)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

7 CSHML

REML16%(1.00)16%(1.00)

20%(1.00)16%(1.00)

16%(1.00)16%(1.00)

48%(1.00)40%(1.00)

48%(1.00)52%(1.00)

32%(1.00)24%(1.00)

8 ARH(1)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

9 TOEPHML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

10 TOEPH(1)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

11 TOEPH(2)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

12 TOEPH(3)ML

REML0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

13 FA(1)ML

REML32%(1.00)32%(1.00)

24%(1.00)32%(1.00)

32%(1.00)28%(1.00)

16%(1.00)16%(1.00)

12%(1.00)4%(1.00)

20%(1.00)20%(1.00)

14 HFML

REML12%(1.00)12%(1.00)

28%(1.00)20%(1.00)

20%(1.00)28%(1.00)

36%(1.00)40%(1.00)

40%(1.00)44%(1.00)

32%(1.00)36%(1.00)


28


S.NOCovariance

Structure

Estimation

method

AIC(%)

)( 2df AICLR

AICC(%)

)( 2df

CAICLR

HQIC(%)

)( 2df HQICLR

BIC (%)

)( 2df BICLR

CAIC (%)

)( 2df CAICLR

AVIC(%)

)( 2df AVICLR

1 UNML

REML

56%(1.00)

60%(1.00)

52%(1.00)

56%(1.00)

48%(1.00)

40%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

28%(1.00)

24%(1.00)

2 UN(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

3 UN(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

4 UN(3)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

5 UNR(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

6 UNR(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

7 CSHML

REML

28%(1.00)

24%(1.00)

36%(1.00)

24%(1.00)

44%(1.00)

36%(1.00)

60%(1.00)

64%(1.00)

68%(1.00)

64%(1.00)

52%(1.00)

56%(1.00)

8 ARH(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

9 TOEPHML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

10 TOEPH(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

11 TOEPH(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

12 TOEPH(3)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

13 FA(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

4%(1.00)

4%(1.00)

4%(1.00)

16%(1.00)

12%(1.00)

8%(1.00)

12%(1.00)

12%(1.00)

4%(1.00)

14 HFML

REML

20%(1.00)

20%(1.00)

20%(1.00)

20%(1.00)

20%(1.00)

20%(1.00)

24%(1.00)

24%(1.00)

24%(1.00)

30%(1.00)

20%(1.00)

20%(1.00)


29


S.NOCovariance

Structure

Estimation

method

AIC(%)

)( 2df AICLR

AICC(%)

)( 2df

CAICLR

HQIC(%)

)( 2df HQICLR

BIC (%)

)( 2df BICLR

CAIC (%)

)( 2df CAICLR

AVIC(%)

)( 2df AVICLR

1 UNML

REML

72%(1.00)

68%(1.00)

64%(1.00)

60%(1.00)

48%(1.00)

44%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

40%(1.00)

32%(1.00)

2 UN(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

3 UN(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

4 UN(3)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

5 UNR(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

6 UNR(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

7 CSHML

REML

16%(1.00)

20%(1.00)

20%(1.00)

24%(1.00)

28%(1.00)

28%(1.00)

64%(1.00)

72%(1.00)

68%(1.00)

80%(1.00)

48%(1.00)

48%(1.00)

8 ARH(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

9 TOEPHML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

10 TOEPH(1)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

11 TOEPH(2)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

12 TOEPH(3)ML

REML

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

0%(1.00)

13 FA(1)ML

REML

8%(1.00)

12%(1.00)

8%(1.00)

12%(1.00)

8%(1.00)

12%(1.00)

20%(1.00)

12%(1.00)

12%(1.00)

8%(1.00)

12%(1.00)

12%(1.00)

14 HFML

REML

16%(1.00)

12%(1.00)

16%(1.00)

12%(1.00)

20%(1.00)

16%(1.00)

20%(1.00)

16%(1.00)

20%(1.00)

16%(1.00)

20%(1.00)

16%(1.00)


selection of covariance structure in longitudinal...

Documents