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DETERMINATION OF APPROPRIATE COVARIANCE STRUCTURES IN RANDOM SLOPE AND INTERCEPT MODEL APPLIED IN REPEATED MEASURES.

Byline: Gazel Ser

ABSTRACT

This study aims to determine variance-covariance structures of dependent variable in data set containing repeated measures and to compare covariance parameter estimation methods. To this end, random intercept and slope model which is among the special cases of linear mixed model was formed and the time variable was involved into the model in a continuous and categorical manner. Also, compound symmetry (CS), toeplitz (TOEP), first-order autoregressive (AR(1)), homogeneous variance-covariance models and unstructured (UN), heterogeneous compound symmetry (CSH), heterogeneous toeplitz (TOEPH), heterogeneous first-order autoregressive (ARH(1)), first-order ante-dependence (ANTE(1)) and unstructured correlation (UNR) heterogeneous variance-covariance models were performed in order to determine the variance-covariance structure between the repeated measures. In addition, comparison of ML and REML was carried out as covariance parameter estimation method.

Consequently, random intercept and slope model (RISM) was found to be the most appropriate one in modeling the repeated measure data when ML was used as the parameter estimation method and UN, CSH, ARH(1), TOEPH, ANTE(1), UNR as the covariance models.

Key words: Covariance structures, repeated data, linear mixed model.

INTRODUCTION

In the test designs including repeated measures, it is possible to get different features (live-weight, height at withers, body length etc in the field of stockbreeding) from variable structure test units with repeated measures made in different times for same features (Tabachnick and Fidel 2001). It is essential to properly identify the variance-covariance structure among the data in the analysis of the repeated measures (Akbas et al., 2001, Orhan et al., 2010; Eyduran and Akbas 2010). At this point, structure of general linear model (GLM) which is based on that the repeated measures in the data structure are different from each other and have a homogeneous variance is quite impractical for modeling the repeated measures.

Therefore, the linear mixed model which enables flexible modeling of variance-covariance matrix structure is recommended for the statistical analysis of the repeated measure data, without imposing any restrictive assumption on the correlated data structure of the repeated measures taken on the same experimental unit (Iyit, 2008).

In present study, random intercept and slope model was formed on the data having a repeated measure structure by involving continuous and categorical effect of the time factor into the model variable. A variable such as "time 2" was formed in order to involve the time into the model categorically (Doganay, 2007). The chest girth feature treated as the repeated measure was involved into the model as the dependent variable. To that end, compound symmetry (CS), toeplitz (TOEP), first-order autoregressive (AR(1)), homogeneous variance- covariance models and unstructured (UN), heterogeneous compound symmetry (CSH), heterogeneous toeplitz (TOEPH), heterogeneous first-order autoregressive (ARH(1)), first-order ante-dependence (ANTE(1)) and unstructured correlation (UNR) heterogeneous variance- covariance models were performed in order to determine the variance-covariance matrix structure of the dependent variable. Comparison of ML and REML was also carried out as covariance parameter estimation method.

Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) cohesion criteria were used to determine the appropriate variance-covariance structure using covariance parameter estimation methods.

MATERIALS AND METHODS

Chest girth measured every 2 weeks (14-day) from 57 Il de france lambs up to 6-months of age, as the animal material, was treated as dependent variable whereas birth type, dam age, sex and live weight of animals were involved into the model as independent variables.

RESULTS AND DISCUSSION

Cohesion criteria results of variance-covariance structures which were obtained from RISM model are given in Table 1 and parameter estimates regarding fixed effects in Table 2. ML and REML estimate values of covariance parameters regarding the random effects in the structure of RISM are given in Table 3 for UN, CSH, ARH(1), TOEPH, ANTE(1) and UNR.

UN, CSH, ARH(1), TOEPH, ANTE(1) and UNR structures were determined as the best covariance models respectively from Table 1, in modeling the change between the repeated measure data regarding the chest girth for the ML and REML covariance parameter estimation methods. CS, AR(1) and TOEP were found to be the worst covariance structures in modeling.Moreover, goodness of fit results indicated ML as the best covariance parameter estimation methods.

Both AIC and BIC cohesion criteria showed a tendency to rank the heterogeneous variance-covariance models as the best model group for the repeated measure data regarding the chest girth.

Table 1. Goodness of fit results

Estimation Method for Covariance###Covariance###-2 Res. Log###AIC###BIC2

###Parameters###Structure###Likelihood

###ML###CS,AR(1)###3068.6###3110.6###3153.5

###REML###3077.6###3081.6###3085.7

###ML###TOEP###3065.7###3109.7###3154.6

###REML###3074.1###3080.1###3086.3

###ML###UN, CSH, ARH(1),###3008.9###3019.3###3027.5

###REML###TOEPH,###3011.3###3054.9###3101.9

###-ANTE(1),UNR

Table 2. The results of significance of fixed effects

Covariance###Sex###Dam Age###Birth Type###Time 2###Live Weight

Structures###Fw###pW###F###P###F###P###F###P###F###P

###4.12###0.0429 2.69 0.0302 19.26 less than .0001 30.77 less than .0001 764.73 less than .0001

###4.16###0.0419 2.63 0.0333 18.99 less than .0001 29.91 less than .0001 729.24 less than .0001

###4.50###0.0342 2.71 0.0293 19.26 less than .0001 32.21 less than .0001 744.15 less than .0001

###4.56###0.0331 2.66 0.0320 18.99 less than .0001 31.38 less than .0001 710.89 less than .0001

CSH,ARH(1),TOEPH,UTN,###1.02###0.3128 1.10 0.3543 5.40###0.0204 43.83 less than .0001 571.52 less than .0001

ANTE(1), UNR###0.96###0.3272 0.95 0.4353 4.79###0.0290 43.46 less than .0001 528.20 less than .0001

CONCLUSION

In this study, the RISM model was applied on repeated measure data. Results revealed that UN, CSH, TOEPH, ARH(1), ANTE(1) and UNR heterogeneous variance- covariance models were the best model group in modeling the variance- covariance matrix structure regarding the dependent variable and CS, TOEP and AR(1) homogeneous variance- covariance models were the worst model group. When the heterogeneous variance-covariance models were used as the best model group, the percent of random-effect terms in the model to explain overall change in the dependent variable was 56.9% and 60.5%, this percent declined to 55% and 59.2% when the homogeneous variance- covariance models which were found to be the worst model group were used.

CS, TOEP and AR(1) covariance models were thereby found to be the weakest model in explaining the overall change in the structure of the dependent variable. It is quite important to accurately identify the covariance structures between the repeated measures. Because, covariance structures between the repeated measures have an important effect on the estimates to be made.

REFERENCES

Akbas Y., M. Z. Firat and c. Yakupoglu (2001).Comparison of different models used in the analysis of repeated measurements in animal science and their sas applications. Agricultural Information Technology Symposium. 20-22

September 2001, Sutcu Imam University, Kahramanmaras. Antonio, K. and Beirlant, J. (2007). Actuarial statistics with generalized linear mixed models. Mathematics and Economics 40: 58-76.

Doganay, B. (2007). Mixed Effects Models for Analyzing Longitudinal Studies. Master's Thesis. University of Ankara, Turkey

Eyduran, E. and Akbas, Y. (2010). Comparison of different covariance structure used for experimental design with repeated measurement. The J. Anim. and Plant Sci., 20(1): 44-51.

Iyit N. (2008). Constitution of Linear Mixed Models in the Analysis of Correlated Data. Ph.D. Thesis. University of Selcuk, Turkey.

Kincaid, C. (2005). Guidelines for selecting the covariance structure in mixed model analysis. Statistics and Data Analysis 30: 1-8.

Orhan. H., E. Eyduran, and Y. Akbas (2010). Defining the best covariance structure for sequential variation on live weights of anatolian Merinos male lambs. The J. Anim. and Plant Sci. 20(3):158-163.

Tabachnick B. G. and L. S. Fidel (2001). Using Multivariate Statistics. Allyn and Bacon, USA.
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Article Details
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Author:Ser, Gazel
Publication:Journal of Animal and Plant Sciences
Article Type:Report
Geographic Code:9PAKI
Date:Sep 30, 2012
Words:1331
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