MODEL SELECTION UNDER COVRIANCE STRUCTURE FOR LONGITUDINAL DATA ANALYSIS

BY

J. MOHANRAJ, M.SC.,M.SC.,M.PHIL.,MBA., .

DPh.

CONTENT Objective Longitudinal Data Linear mixed model Covariance structures Maximum Likelihood(ML) and Restricted Maximum

Likelihood(REML) Information criteria Likelihood ratio test Results and Conclusion Reference

OBJECTIVE Linear mixed model with various covariance structures are popular in longitudinal

data analysis. The ML and REML methods are used in estimating the parameters of the model. The evaluation of the models has been studied under six different information criteria, namely AIC, AICC, HQIC, BIC, CAIC and AVIC. We have analyzed the Rogan et al., (2001) data relating to Treatment of Lead-Exposed children (TLC). The result of this investigation showed that the most appropriate covariance structure and attempted on varies of the longitudinal data setup.

LONGITUDINAL DATADefinition:

The term Longitudinal studies refers to situations in which data are collected on the same subjects (or) experimental units, over several occasions.

Purpose of longitudinal study:The longitudinal data collected are based on repeated

measurement on the same individual over different point of time. This set of repeated measurement will be useful to study the developing pattern of change over time.

Purpose of covariance structure analysis:The covariance among repeated measure increases

efficiency of the estimates. These study of the covariance structure determines the model for the mean response of the variable over the time.

Problems considered in our study,

The above objective is attempted in this five longitudinal data setup,

1) Normal TLC data – Table(1)

2) Normal TLC data with bootstrap method– Table(2).3) Create, Missing at Random (MAR) in TLC data with bootstrap method–

Table(2).4) Create, Missing complete at Random (MCAR) in TLC data with bootstrap

method– Table(2).5) Normal TLC data – Table(1).

EXAMPLE OF LONGITUDINAL DATA

Table(1,2,5) Original TLC data Table (3,4) MAR and MCAR in TLC data (Create data set)

(S) = Succimer group, (P) = Placebo group, (-) = Missing value.

Missing At Random (MAR): MAR data of missing values contained the both of dependent and independent variable.

Missing Completely At Random (MCAR): MCAR data of missing values contained only the independent variable.

IDGroup week0 week1 week4 week6

1 P 31.9 27.9 27.3 34.2

2 S 29.6 15.8 23.7 22.6

3 S 21.5 6.5 7.1 16

4 P 26.2 27.8 25.3 24.8

5 S 21.8 12 17.1 19.2

ID Group week0 week1 week4 week6

1 P 31.9 27.9 27.3 34.2

2 - 29.6 15.8 23.7 -

3 S 21.5 6.5 7.1 16

4 - 26.2 - 25.3 24.8

5 S 21.8 12 - 19.2

Generally the missing data analysis deals with two types of techniques namely, missing data with data replacement and missing data with deletion technique. But in this study is restricted only to the deletion technique. Again, the case deletion technique has two different deletion methods namely, Listwise deletion and Pairwise deletion. We have studied both the Pairwise deletion technique, because loss of information is very lower than compared to listwise deletion technique.

LINEAR MIXED MODEL (LMM) Linear mixed model provides flexibility in fitting models with various combinations of fixed

effect and random effect and is often used to analyze data in a broad spectrum of areas in including longitudinal study.

Y is an vector of observations, is an vector of fixed effects X is an design matrix for fixed effects, Z is a given matrix, and is an unobservable random vector of dimensions , is an vector of residual and both     and   are MND distributed with

The variance of Y is, therefore, .The model is setting up the random-effects design matrix Z and by specifying covariance structures for G and R. Simple random effects are a special case of the general mixed model specification with Z containing dummy variables, G containing variance components in a diagonal structure, and where   denotes the  identity matrix. The general linear model is a further special case with Z=0 and .

In this study considered only the fixed effect model

ZXY

1n 1p pnqn 1q

R)MND(0,~ G),MND(0,~,,0

0Var and V),ˆMND(X~Y ,

00

ieR

GE

RZZGV

nIR 2

nIR 2

nI

nn

MAXIMUM LIKELIHOOD (ML) AND RESTRICTED MAXIMUM LIKELIHOOD (REML)Variance Components procedure supports two methods of

estimation, both of which gives different estimates: Maximum Likelihood (ML), Restricted Maximum Likelihood (REML).

Where, and p is rank of X. The ML and REML has been studied Searle, et. al., (1992). Millar. (1977), Laird and Ware (1982), Schabenberger. (2004), Fitzmaurice (2004), Gazel Ser (2012).

)2log(22

1log21:),( 1 nrVrVRGlML

)log()pn(XVXlogrVrVlog:)R,G(l TT

REML 222

121

21 11

YVXXVXXYr TT 111 )(

LIST OF COVARIANCE STRUCTURES In this paper performance of covariance study considered fourteen covariance

structures are namely,1. Unstructured (UN)2. First Order Banded Unstructured (UN(1))3. Second Order Banded Unstructured (UN(2))4. Third Order Banded Unstructured (UN(3))5. First Order Banded Unstructured correlations (UNR(1))6. Second Order Banded Unstructured correlations (UNR(2))7. Heterogeneous compound symmetry (CSH)8. First Order Heterogeneous Auto Regressive (ARH(1)) 9. Heterogeneous Toeplitz (TOEPH)10. Heterogeneous Toeplitz with one Banded (TOEPH(1))11. Heterogeneous Toeplitz with two Banded (TOEPH(2))12. Heterogeneous Toeplitz with three Banded (TOEPH(3))13. First Order Factor Analytic (FA(1))14. Hyunh-Feldt (HF)

LIST OF INFORMATION CRITERIA Akaike information criteria (AIC):

Corrected Akaike information criteria (AICC):

Hannan-Quinn information criteria (HQIC):

Bayesian information criteria (BIC):

Consistent Akaike information criteria(CAIC):

Average Information Criteria (AVIC):

)1(2)log(2

knnklAICC

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

)log()log(2 nklBIC

)1log()log(2 nklCAIC

CAICBICHQICAICAICAVIC C

)log(22 lKAIC

LIKELIHOOD RATIO TEST In many ways, the likelihood ratio test is conceptually the easiest of the test

considered in this section using for covariance model. The estimate obtained by maximum likelihood.

is the value of the likelihood function for the maximum of the unconstrained model and is the value when the constraints are imposed. The likelihood ratio test is obtained by taking the difference computed as:

and the likelihood ratio difference are used in comparison of the model selection criteria. with df as number of covariance parameters. This test is always positive (or zero) since the likelihood of the unconstrained model is at least as high as that of the constrained model. The LR statistic is distributed asymptotically as a Chi-Squared distribution with (n-1) Degrees of freedom equal to the number of constraints.

The information criteria and LR test are applied for first four problem only, but not applied for fifth problem. The fifth problem, the Detection of influence observation in the longitudinal data study, models and method of estimator is same, but covariance structures are evaluated by two diagnostic measures are namely, Cook’s distance and likelihood distance (ML and REML).

fullIC)CM(l

minIC)CM(l

22 dfminICIC CMlCMlfullCM

FIRST PROBLEM- ORIGINAL TLC DATA

The result in this analysis evaluated the performance of ML and REML approach and has shown that the performance of REML approach seems to be more efficient as compared to ML approach.

The overall result revealed that heterogeneous covariance models are the best model group in modeling the covariance structures are UN, CSH, TOEPH, FA(1), ARH(1) and HF performed well among the class of fourteen covariance structures.

The banded covariance models are weakest models in the heterogeneous covariance models namely, UN(1), UN(2), UN(3), UNR(1), UNR(2), TOEPH(1), TOEPH(2) and TOEPH(3).

The overall study suggest that CSH is a better performer based on the uniformity in the results for all six criterion considered for study. Hence in conclusion, the longitudinal study based on mixed model has shown that CSH covariance structure seems to be more appropriate in dealing with nested and non-nested data structure.

Result TableCovarianceStructures

Estimationmethods Parameter

(df) AIC AICC HQIC BIC CAIC AVIC

UNML

REML 1010

2038.32009.7

2040.62010.4

2055.52019.2

2081.22033.5

2099.22043.5

2062.92023.2

UN(1)ML

REML 44

2156.12124.2

2157.12124.4

2167.62128.1

2184.72133.8

2196.72137.8

2172.42129.6

UN(2)ML

REML 77

2082.42052.5

20842052.9

2096.72059.2

2118.12069.2

2133.12076.2

2102.82060.0

UN(3)ML

REML 99

2065.92036.5

2067.92037.1

2082.12045.1

2106.42057.9

2123.42066.9

2089.12048.7

UNR(1)ML

REML 44

2156.12124.2

2157.12124.4

2167.62128.1

2184.72133.8

2196.72137.8

2172.42129.6

UNR(2)ML

REML 77

2082.42052.5

2084.02052.9

2096.72059.2

2118.12069.2

2133.12076.2

2102.82062.0

CSHML

REML 55

2039.82010.8

2041.02011.0

2052.22015.6

2070.82022.8

2083.82027.8

2057.52017.6

ARH(1)ML

REML 55

2051.62022.4

2052.82022.6

2064.12027.2

2082.62034.3

2095.62039.3

2069.32029.1

TOEPHML

REML 77

2039.42010.6

2041.02011.0

2053.82017.3

2075.22027.3

2090.22034.3

2059.92020.1

TOEPH(1)ML

REML 44

2156.12124.2

2157.12124.4

2167.62128.1

2184.72133.8

2196.72137.8

2172.42129.6

TOEPH(2)ML

REML 55

2079.12049.1

2080.32049.3

2091.52053.9

2110.02061.0

2123.02066.0

2096.72055.8

TOEPH(3)ML

REML 66

2066.92037.4

2068.32037.6

2080.32043.1

2100.32051.7

2114.32057.7

2086.02045.5

FA(1)ML

REML 88

2040.12011.3

2041.92011.8

2055.42019.0

2078.22030.4

2094.22038.4

2061.92022.1

HFML

REML 55

2061.62032.1

2062.82032.3

2074.02036.9

2092.52044.0

2105.52049.0

2079.22038.8

Conclusion

SECOND PROBLEM- ORIGINAL TLC DATA WITH BOOTSTRAP

CovarianceStructure

Parameter(df)

AIC %(p- value)

AICC %(p- value)

HQIC %(p- value)

BIC %(p- value)

CAIC %(p- value)

AVIC %(p- value)

UN 1010

48%(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)

UN(1) 44

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)

UN(2) 77

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)

UN(3) 99

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)

UNR(1) 44

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)

UNR(2) 77

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)

CSH 55

16%(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)

ARH(1) 55

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)

TOEPH 77

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)

TOEPH(1) 44

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)

TOEPH(2) 55

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)

TOEPH(3) 66

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)

FA(1) 88

32%(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)

HF 55

12%(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)

Hence in conclusion, the longitudinal study based on mixed model has shown that UN and CSH covariance structure seems to be more appropriate in dealing with nested non-nested data structure.

Result Table Conclusion

THIRD PROBLEM- CREATE, MISSING AT RANDOM (MAR) IN TLC DATA WITH BOOTSTRAP

CovarianceStructure

Paramete(df)

AIC %(p- value)

AICC %(p- value)

HQIC %(p- value)

BIC %(p- value)

CAIC %(p- value)

AVIC %(p- value)

UN 1010

63.3%(1.00)63.3%(1.00)

46.6%(1.00)

50%(1.00)

26.6%(1.00)

23.3%(1.00)

6.6%(1.00)6.6%(1.00)

3.3%(1.00)3.3%(1.00)

16.6%(1.00)16.6%(1.00)

UN(1) 44

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)

UN(2) 77

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)

UN(3) 99

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)

UNR(1) 44

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)

UNR(2) 77

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)

CSH 55

13.3%(1.00)16.6%(1.00)

23.3%(1.00)

20%(1.00)

30%(1.00)33.3%(1.00

)

46.6%(1.00)

43.3%(1.00)

60%(1.00)60%(1.00)

36.6%(1.00)36.6%(1.00)

ARH(1) 55

0%(1.00)0%(1.00)

0% (1.00)0% (1.00)

0% (1.00)0% (1.00)

3.3%1.00)3.3%1.00)

3.3%1.00)3.3%1.00)

3.3%1.00)3.3%1.00)

TOEPH 77

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)

TOEPH(1) 44

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)

TOEPH(2) 55

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)

TOEPH(3) 66

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)

FA(1) 88

26.6%(1.00)26.6%(1.00)

23.3%(1.00)

23.3%(1.00)

13.3%(1.00)

16.6%(1.00)

16.6%(1.00)

13.3%(1.00)

6.6%(1.00)6.6%(1.00)

16.6%(1.00)16.6%(1.00)

HF 55

13.3%(1.00)1.00

(13.3%)

20%(1.00)16.6%(1.00

)

26.6%(1.00)

26.6%(1.00)

30%(1.00)33.3%(1.00

)33.3%(1.00)33.3%(1.00)

30%(1.00)30%(1.00)

Hence in conclusion, the longitudinal study based on mixed model has shown that CSH covariance structure seems to be more appropriate in dealing with nested non-nested data structure except that UN is a better performer under AIC and AICC. The results are observed similar to the complete longitudinal data with CSH structure performs well in general except in the case of AIC and AICC with UN structure as a good choice.

Conclusion Result Table

FOURTH PROBLEM- CREATE, MISSING COMPLETE AT RANDOM (MCAR) IN TLC DATA WITH BOOTSTRAP

CovarianceStructure

Paramete(df)

AIC %(p- value)

AICC %(p- value)

HQIC %(p- value)

BIC %(p- value)

CAIC %(p- value)

AVIC %(p- value)

UN 1010

53%(1.00)56%(1.00)

53%(1.00)56%(1.00)

33%(1.00)36%(1.00)

0%(1.00)0%(1.00)

0%(1.00)0%(1.00)

20%(1.00)16%(1.00)

UN(1) 44

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)

UN(2) 77

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)

UN(3) 99

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)

UNR(1) 44

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)

UNR(2) 77

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)

CSH 55

10%(1.00)10%(1.00)

16%(1.00)13%(1.00)

33%(1.00)36%(1.00)

50%(1.00)46%(1.00)

50%(1.00)50%(1.00)

30%(1.00)40%(1.00)

ARH(1) 55

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%(1.00)6%(1.00)

0%(1.00)0%(1.00)

TOEPH 77

10%(1.00)3%(1.00)

6%(1.00)3%(1.00)

6%(1.00)6%(1.00)

6%(1.00)6%(1.00)

0%(1.00)0%(1.00)

3%(1.00)3%(1.00)

TOEPH(1) 44

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)

TOEPH(2) 55

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)

TOEPH(3) 66

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)

FA(1) 88

10%(1.00)13%(1.00)

16%(1.00)16%(1.00)

13%(1.00)13%(1.00)

16%(1.00)20%(1.00)

10%(1.00)6%(1.00)

26%(1.00)20%(1.00)

HF 55

20%(1.00)20%(1.00)

20%(1.00)20%(1.00)

23%(1.00)23%(1.00)

26%(1.00)30%(1.00)

36%(1.00)40%(1.00)

23%(1.00)26%(1.00)

The results are observed similar to the complete longitudinal data with UN and CSH structures performs well in the all classes of criteria.

Result

FIFTH PROBLEM-“DETECTION OF INFLUENCE OBSERVATIONS USING DIFFERENT COVARIANCE STRUCTURE IN LONGITUDINAL ANALYSIS ”. ORIGINAL TLC DATA Result Table Conclusion

20 34 46 20 34 46 20 34 46 20 34 46

UN36.7 2.15 7.24 36.03 2.09 7.06 2.441 0.212 0.12 2.38 0.182 0.117

UN(1)9.84 0.69 2.04 9.6 0.7 2 0.193 0.236 0.203 0.182 0.242 0.198

UN(2)19.18 3.06 6.99 18.42 2.97 6.81 0.119 0.063 0.084 0.116 0.061 0.082

UN(3)16.86 3.31 7.41 13.84 3.22 7.22 0.138 0.061 0.09 0.135 0.059 0.088

UNR(1)9.97 2.9 2.61 9.71 2.82 2.53 0.095 0.077 0.082 0.093 0.075 0.08

UNR(2)19.18 3.06 6.99 18.44 2.97 6.81 0.119 0.063 0.084 0.116 0.061 0.082

CSH44.74 6.58 8.9 43 6.55 8.86 4.607 2.279 3.688 4.2 2.28 3.667

ARH(1)47.91 6.55 8.89 46.42 6.48 8.84 4.591 2.365 3.902 4.287 2.358 3.884

TOEPH16.09 0.97 2.07 15.76 0.94 2.04 3.727 0.54 0.261 3.721 0.533 0.253

TOEPH(1)9.84 0.74 2.04 9.6 0.7 2 0.193 0.236 0.203 0.182 0.242 0.198

TOEPH(2)16.2 3.01 3.26 15.77 2.92 3.17 0.119 0.063 0.084 0.116 0.061 0.082

TOEPH(3)16.02 3.08 3.06 15.34 2.99 2.97 0.143 0.059 0.08 0.14 0.058 0.078

FA(1)37.62 1.03 7.25 36.91 1.03 7.04 1.626 0.932 0.251 1.514 0.937 0.257

HF9.58 2.08 2.73 9.49 2.03 2.66 0.146 0.119 0.048 0.144 0.091 0.048

ANTE(1)22.82 2.04 6.6 22.4 1.99 6.46 0.334 0.469 0.62 0.327 0.436 0.615

Likelihood distance COOK DISTANC

ML REML ML REML The diagnostics measures (Likelihood based distance

(MLD and REMLD), Cook’s Distance) are play an important role in this detection of influence observations study. The observations are evaluated by these diagnostic measures and corresponding to fourteen covariance structures. The two diagnostic measure partially detected (20, 34 and 46) these three observations are more influential in this TLC data set. In more partially proposed, the 20th observation is more influential in this TLC data set.

The two diagnostic measure are proposed UN, CSH, ARH (1) and FA(1) covariance structures are performed well and particularly the CSH and ARH (1) covariance structures are outperforming in this detection of influence observations study.

CONCLUSION REML is more better estimator in the first four problem. ML is more better estimator in the first four problem. Banded models weakest model in these context. UN and CSH models are outperforming in all the five data set.

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