1. Ecevit Eyduran Adile Tatlıyer Abdul Waheed
DETERMINATION OF THE MOST APPROPRIATE COVARIANCE STRUCTURE FOR DATA WITH MISSING OBSERVATIONS IN REPEATED MEASURES DESIGN
Ecevit Eyduran Adile Tatlıyer Abdul Waheed
Mohammad Masood Tariq
1Biometry Genetics Unit, Department of Animal Science, Faculty of Agriculture, Iğdır University, 76000, Iğdır-Turkiye
2Biometry Genetics Unit, Department of Animal Science, Faculty of Agriculture, Süleyman Demirel University, 32000, Isparta-Turkiye
3Faculty of Veterinary Sciences, Bahauddin Zakariya University, Multan, Pakistan
4Center for Advanced Studies in Vaccinology and Biotechnology (CASVAB), University of Balochistan, Quetta, Balochistan, Pakistan.

2. Introduction
Repeated measures are sequential data taken from same experimental unit over time. The most appropriate example to the sequential data in animal science is growth curve data characterizing the relationship between the quantitative trait-time for each animal.

3. Repeated measures design with two factors, treatment (between subject) and time (within subject) has been preferred considerably in literature (Littell et al., 1998).

4. Repeated measures design which is similar to split-plot design is called “within subject design” or “correlated groups design”(Gürbüz et.al.,2003 ).

5. Repeated measures designs are analyzed on the basis of classical (univariate) and advanced (multivariate) approaches.
Univariate approaches for repeated measures designs are Repeated ANOVA, Greenhouse-Geisser (G-G) and Huynh-Feldt (H-F) epsilon adjusted F test.

6. Multivariate approaches are Profile Analysis (Repeated MANOVA) and “Using Mixed model methodology” for repeated measures designs (Eyduran and Akbaş, 2010)
The most suitable approach to be used in repeated measures design is based on Spherity assumption (which relates to equality of variances between levels of repeated measures).

7. When Spherity assumption is provided, Repeated ANOVA is used reliably.
In the violation of the Spherity assumption, Greenhouse-Geisser (G-G) and Huynh-Feldt (H-F) epsilon adjusted F test can be used easily instead of Repeated ANOVA. However, multivariate approaches were superior to univariate approaches addressed above.

8. The violation of Spherity assumption is not problem for Profile Analysis (Eyduran et al., 2008).
Repeated ANOVA and Repeated MANOVA could not be specified for repeated measures designs with missing data. Therefore, the best solution to provide the specification is “Using Mixed Model in repeated measures design”.

9. In the violation of the Spherity assumption, applying “Mixed model methodology“ in repeated measures designs permit researchers to specify different covariance structures for repeated measures design with/without missing data.

10. The aims of the present investigation are:
to determine the best covariance structure in repeated measures design with/without missing data
to determine the influence of including a covariate on fitting criteria with/without missing data.

11. Material and method:
In the study, 228 Mengali lambs, single-born from dams at the age of months, were randomly selected from sheep flocks in Pakistan.

12. The first 10 observations of the example data set of Mengali Lambs

13. Statistical model:
The statistical model used for univariate analysis of variance:

14. The matrix notation of the model is:
Where β and µ are fixed vector (sex, time and sex by time interaction) and random (individual within sex) effects, respectively; X and Z are design matrices for the corresponding fixed and random effects, and e is error vector.

15. The statistical model for the mixed model analysis in data set can be defined:

16. -2 Res Log Likelihood, Bayesian Information Criteria (BIC), Akaike Information Criterion (AIC) and Burnham-Handerson Criterion (AICC) fitting criteria were used to determine the most suitable covariance structure for mixed model methodology. The smaller criterion result is better in SAS procedure.

17. In mixed model methodology , First- Order Autoregressive (AR(1)), Heterogenous First-Order Autoregressive (ARH(1)), First-Order Ante-Dependence(ANTE(1)), Compound Symmetry (CS), Unstructured (UN), Huynth-Feldth (HF), Heterogenous Compound Symmetry (CSH), Toeplitz (TOEP), and Heterogenous Toeplitz (TOEPH) covariance structures are used.

18. In the work, applying mixed model methodology in repeated measures data was performed on the basis of three ways:
The complete data (referring to all measurements taken completely for each lamb) were exposed to all the covariance structures defined in the above. A covariate for the complete data was not defined.

19. The complete data with a covariate (birth weight) were exposed to various covariance structures mentioned above.
Deletion operation was done at different proportions (20% and 40%) of missing observations.
All the statistical evaluations were performed using MIXED procedure of SAS program.

20. Result and discussion
Estimates of fitting criteria (-2 Res Log Likelihood, AIC, AICC and BIC) for some covariance structures specified with/without covariate in general linear mixed model are given in Table 1.
Table 1. Fitting criteria results for comparing covariance structure with/without a covariate.

21. Fitting criteria
Compound Symmetry (CS) was found as the worst covariate structure with/without covariate. The second worst covariate structure is Huynth-Feldth (HF) covariate structure with/without covariate.
The best covariate structure was First-Order Ante-Dependence (ANTE(1)) with/ without covariate due to having the smallest fitting criteria.

22. Significance results
Significance results for fixed effects in general linear mixed model with/without covariate are summarized in Table 2.
Table 2. Significance results of fixed effects in mixed model approach (F and P values)

23. Significance results
According to results, all the fixed effects, and the covariate in the general linear mixed model with covariate were found significant (P<0.01) for the covariance structures, which occur their convergence.
All estimates are significant at the level of (p<0.001)

24. Fitting criteria for missing observation
Table 3. Fitting criteria results for comparing covariance structures at different missing proportions in the model with adding the covariate
As seen in Table 3, Unstructured (UN) covariate structure was not converged both with 20 (%) and 40 (%) missing observation. ANTE (1) was found to be the best covariance structure at 20 (%) and 40 (%) proportions of missing observations.

25. Significance results for missing observations
According to results, all the fixed effects, and the covariate in the general linear mixed model with covariate were found significant (P<0.001) for the covariance structures, which occur their convergence
Significance results for missing observations
Table 4. Significance results of fixed effects in mixed model approach (F and P values)

26. Table 4. Significance results of fixed effects in mixed model approach (F and P values)
In the examination of the Table 4, sex, time, sex by time interaction and covariate (birth weight-BW) effects were very significant in using mixed model for repeated measures data of missing observations (P<0.001), with the exception of UN (at %40 and %20 missing proportion) and HF at %20 missing proportion.

27. Conclusion
The present search gave some significant results in the mixed model methodology in repeated measures:
In general, adding covariate in the specified models provided much better contributions to the improvement of fitting criteria results.

28. ANTE(1) covariance structure was the best selection procedure for the repeated measures data with/without missing observations.
Covariate effect was found very significant (P<0.001) with/without missing observations.

29. Sex, time, and sex by time interaction fixed effects were very significant (P<0.001).
In conclusion, applying mixed model methodology is suggested in repeated measures data of missing observations in opposed to classical approaches.

30. References
Eyduran, E. and Y. Akbaş (2010). Comparison of diıfferent covariance structure used for experimental design with repeated measurement. J. Anim. and Plant Sci., 20(1):
Littell, R. C., Henry, R. C. and Ammerman, C. B. (1998). Statistical analysis of repeated measures data using SAS procedures. J. Anim. Sci. 76: 1216–1231.
Wang, Z. and Goonewardene, L. A. , The use of MIXED models in the analysis of animal experiments with repeated measures data. Can. J. Anim. Sci.
Gürbüz, F., E. Başpınar, H. Çamdeviren and S. Keskin,(2003). Tekrarlanan Ölçümlü Deneme Düzenlerinin Analizi. YYÜ. Matbaası, Van.130.

31. THANK YOU…