Multilevel modeling and panel data analysis in educational research (Case study: National examination data senior high school in West Java)

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1 Multilevel modeling and panel data analysis in educational research (Case study: National examination data senior high school in West Java) Pepi Zulvia, Anang Kurnia, and Agus M. Soleh Citation: AIP Conference Proceedings 1827, (2017); doi: / View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Cluster information of non-sampled area in small area estimation of poverty indicators using Empirical Bayes AIP Conference Proceedings 1827, (2017); / A comparative study of approximation methods for maximum likelihood estimation in generalized linear mixed models (GLMM) AIP Conference Proceedings 1827, (2017); / Preface: 2nd International Conference on Applied Statistics AIP Conference Proceedings 1827, (2017); / Robust small area estimation of poverty indicators using M-quantile approach (Case study: Sub-district level in Bogor district) AIP Conference Proceedings 1827, (2017); / Committees: 2nd International Conference on Applied Statistics AIP Conference Proceedings 1827, (2017); / Winsorization on linear mixed model (Case study: National exam of senior high school in West Java) AIP Conference Proceedings 1827, (2017); /

2 Multilevel Modeling and Panel Data Analysis in Educational Research (Case Study : National Examination Data Senior High School in West Java) Pepi Zulvia 1,a) Anang Kurnia 1,b) and Agus M Soleh 1,c) 1) Department of Statistics, Faculty of Mathematics and Natural Sciences, Bogor Agricultural University, Indonesia. a) pepizulvia17@gmail.com b) Corresponding author:anangk@apps.ipb.ac.id c) agusms@apps.ipb.ac.id Abstract. Individual and environment are a hierarchical structure consist of units grouped at different levels. Hierarchical data structures are analyzed based on several levels, with the lowest level nested in the highest level. This modeling is commonly call multilevel modeling. Multilevel modeling is widely used in education research, for example, the average score of National Examination (UN). While in Indonesia UN for high school student is divided into natural science and social science. The purpose of this research is to develop multilevel and panel data modeling using linear mixed model on educational data. The first step is data exploration and identification relationships between independent and dependent variable by checking correlation coefficient and variance inflation factor (VIF). Furthermore, we use a simple model approach with highest level of the hierarchy (level-2) is regency/city while school is the lowest of hierarchy (level-1). The best model was determined by comparing goodness-of-fit and checking assumption from residual plots and predictions for each model. Our finding that for natural science and social science, the regression with random effects of regency/city and fixed effects of the time i.e multilevel model has better performance than the linear mixed model in explaining the variability of the dependent variable, which is the average scores of UN. Keywords: hierarchical data structures, multilevel modeling, linear mixed model INTRODUCTION Background Most of social, health, and education research focused on the issue of individual interaction with its environment. Generally, individual and environment are a hierarchical structure. Stratified and multi-stage samples indicate the existence of level in data. The existence of data hierarchies is not ignorable because each hierarchy has consisted of units grouped at different levels. Therefore, a hierarchical data structure is analyzed based on several levels, with the lowest level nested in the highest level. This kind of modeling is commonly known as multilevel modeling. Multilevel modeling is a part of the general model which is often called linear mixed models. It has hierarchical data structures with dependent variable measured on the individual level and some independent variables measured on the groups level. The model could be written as two-stage system equations in which individual variation within each group is explained by an individual level equation, and variation across groups in group-specific regression Statistics and its Applications AIP Conf. Proc. 1827, ; doi: / Published by AIP Publishing /$

3 coefficients is explained by a groups level equation. Multilevel modeling base assumptions are similar to linear regression, such as error on individual and groups level have normal distribution [1]. Hierarchical data structures are widely used in education research, for example, the average score of National Examination (UN). UN is a standard evaluation system education in Indonesia. According to government regulation No , the results of UN can be used as consideration for development and assistance for the education sector to improve education quality. UN s results are expected to provide an education quality mapping overview at national, province, regency/city, and school level. At school level, UN s results for each test subject are expected as indicators of improvement in learning the process. The quality of a school can be reflected by UN scores and school examination (US) scores. The average scores of UN can be influenced by internal and external factors in each school. Some of the internal factors are quality of students, teacher s competence, school facilities and infrastructure. External factors include regional economics condition which is measured by influences human development index (IPM), gross regional income per capita (commonly mentioned as PDRB in Indonesia), and income per capita of each regency/city. The Government has actively made a variety of innovations for several years to improve the quality of education. One way to analyze the development of the implementation of the UN every year can be done with the data panel. Panel data is a combination of time-series and cross-section data. Time-series data is data that follows a given sample of individuals or object over time. Cross-section data is data with many individuals or objects observed at one time [2]. According to Baltagi [3], the application of panel data has several advantages. First, controlling the occurrence of heterogeneity on the individual. Second, providing more informative data, reducing collinearity between variables, more variability, more degree of freedom and greater efficiency. Third, panel data has better performance to study the dynamics of adjustment. Fourth, panel data able to detect effects better which can t be observed in time-series and the usual sectional cross. Fifth, reducing bias due to a lot of observations. UN research with panel data has been done by Safitri [4], which was conducted with linear mixture models and panel data that ignore the level of the school in regency/city. The results obtained were still relatively large, i.e for natural science and for social science. There was indications heteroskedasticity in errors and violations of freedom assumptions hierarchical data structures. Theoretically, ignoring the information for hierarchical data structures will cause problems in conceptual and statistical tests [5]. According to Ker [6], applications of multilevel analysis for educational research has some advantages. First, simple and efficient for predicting coefficient regression. Second, using the grouping information can obtain correct statistical tests. Third, a variance-covariance matrix can describe the complex cities of variation at the high level. Fourth, allowing researchers to explore the cross-level by using independent variables of each level. Fifth, minimizing the occurrence of lower variable nested at the highest level or higher variable nested at the lowest levels. This research developed multilevel and data panel modeling using the linear mixed model on educational data. A case that evaluated in this study is the average score of National Examination (UN) at Senior High Schools (SMA) in West Java from 2011 until REVIEWS OF LINEAR MIXED MODEL AND MULTILEVEL MODELING Linear mixed models is a model that combines the effect of fixed and random effects into an equation. According to Jiang [7] a mixture of linear models can be written as follows: (1) where y is an observation vector, is a regression coefficient vector (fixed effect), X is known covariate matrix, is a matrix that contains only 1 and 0, is normally distributed random effect vector, and is an error vector. Variance of and are assumed to be independent. Multilevel modeling is a part of linear mixed models with variables measured at the different level. Goldstein [8] states there are three purposes of multilevel modeling in data analysis. First, to increase the estimates for low-level effect. Second, to model the influence of cross-level hierarchy data structure. Third, to partition the variance components between a level of hierarchy data structure. The simple structure of multilevel modeling consists of 2- levels, with level 1 being the lowest level nested at the highest level, namely level 2. Multilevel modeling with 2-levels is as follow [8] : 1. Model level 1 Multilevel modeling for each group can be written as follow :

4 (2) where is an dependent variable vector for individual i ( ) in group j (), is a independent variable matrix for individual level in group j, is a regression coefficient vector associated with, and is a vector of individual level error in group j, which is assumed to be normally distributed. 2. Model level 2 The regression coefficient in level 1,,, in model (2) has values that varies between groups. Variations of values will be explained by establishing the level 2 model. Level 2 model is made for each regression coefficient as a response to-r using explanatory variables in level 2. Modeling on level 2 can be written in the form as follow: (3) where is a vector of dependent variable for group j, R is a matrix of independent variable for groups level in groups j, is a vector of regression coefficient associated with, and is a vector of an error term representing a unique effect associated with groups j. If the Equation 3 is substituted to Equation 2, then it follows: (4) with is fixed effect and [ ] is random effect, E( ) =, dan Var( ) =. Assumption of can be written as follow: 1. E ( ) = E ( ) = 0 2. Cov (, ) = Cov (, ) = Cov (, ) = 0,, dan 3. Var ( ) =, with is residual variance model level 1 groups j 4. Var ( ) =, with and is residual variance of model level 2 for regression coefficient to-r groups-m. Hox [5] states that matrix R in Equation 4 is an intermediate variable to connect Y and X. Therefore, the variation of relationship between Y and X depends on the value of R. The interpretation of regression coefficients in model level 1 and level 2 depends on the positive and negative signs of the coefficients regression. If regression coefficient model level 1 and level 2 have similar signs, it means that both regression coefficients have influence proportional to Y. In a factor interactions in the model as a slope variance variable X. RESEARCH METHOD UN and US data for all SMA in West Java were accessed from Educational Research Center, Ministry of Education and Culture. Data of national education standard score were collected from National Accreditation Institution. Data of human development index, gross regional income per-capita, and income percapita of each regency/city were taken from Statistics Indonesia Website. Level 1, public school (all SMANs), dependent variables were content standard scores (X 1 ), process standard scores (X 2 ), graduation competency standard scores (X 3 ), educator scores (X 4 ), school facilities scores (X 5 ), management score (X 6 ), financing standard scores (X 7 ), assessment standard scores (X 8 ), and average scores of US for all SMANs in West Java (X 9 ). Level 2, regency/city (West Java), dependent variables were IPM scores (R 1 ), PDRB (R 2 ), income per-capita of each regency/city (R 3 ) and Y is average scores of UN all SMANs for both natural science and social science. Identification relationships between independent and dependent variable by checking correlation coefficient and variance factor inflation (VIF). If coefficient correlation has value more than 50% and VIF value more than 2, it is indicated that there is an occurrence of multicollinearity. To analyze multicollinearity one can use dimension reduction method such as principal component analysis. The number of principal components to retain is determined from cumulative proportion and Eigenvalue. Cumulative proportion more than 0.7 or Eigenvalue more than 1 suffices to obtain information from independent variables represented by principal components [9]. In this research,

5 there would be four models compared i.e first, modeling with school effect. School was set as fixed effect in following model: (5) where is dependent variable for school-i, (p = 1, 2, k) are parameter model, is independent variable, is fixed effect in school-i, and is an error. Equation 5 known as Model 1. Second, modeling with school effect and time effect. School and time were set as fixed effect in following model: (6) where is dependent variable for school-i in time-t, are parameter model, is independent variable, is fixed effect in school-i, is fixed effect in time-t, and is an error. Equation 6 known as Model 2. Third, multilevel modeling with regency/city effect and panel data. Base on the Equation 4 that by using multilevel model with regency/city effect and panel data, so then it follows: Fourth, if there is not regency/city effect then model (7) become : (7) (8) where is dependent variable, is intercept, (q = 1, 2,,r) is parameters model for regency/city, a independent variable from regency/city j, is parameter model for school, a independent variable from school in regency/city j in time-t, is fixed effect of time, is an error from regency/city ( and is an error (. In this model, there were 17 regencies and 9 cities in West Java. Equation 7 known as Model 3 and Equation 8 known as Model 4. Best model for average scores of UN was determined by goodness-of-fit and by checking assumption from residual plots and predictions for each model. Best model was the one with the smallest value of error estimator i.e root mean square error (RMSE), root mean square error prediction (RMSEP) and Likelihood estimator i.e Akaike's Information Criterion (AIC) and Bayesian Information Criterion (BIC). RESULT AND DISCUSSION Based on data obtained in , there were 421 accredited SMANs which consisted of 328 SMANs located in 17 regencies and 93 SMANs located in 9 cities in West Java. Generally, SMAN has two class categories, which are class of Natural Sciences (IPA) and class of Social Sciences (IPS). From a total of 421 SMANs in West Java, there were only 2 SMANs that didn t have IPS class, SMAN 1 Karang Bahagia in Bekasi regency and SMAN 28 Garut in Garut. IPA and IPS often have different characteristics, so the analysis was separated for those two class categories. The distribution of average scores UN in is presented in Figure 1. Figure 1 shows a decrease in the average score of UN every year. The extreme decline occurred in 2013 and 2014, where the average UN score in 2013 smaller than the previous year. The decline may be due to policy changes in the implementation of the UN. Technical changes are clearly visible in the test package which is divided into 20 different packages in one class, where each student gets a different packages

6 (a) (b) FIGURE 1. Scatterplot of Average score of UN versus years for a) IPA b) IPS Relationship and linear direction between average score of UN as an independent variable and eight components of national educations standard (SNP) as dependent variables were explained from the correlation coefficient. Based on correlation coefficient test, some independent variables had correlation values more than 50%, indicating the relationships between variables were not independent, which could lead to multicollinearity. To overcome this, principal component analysis was used. Principal component analysis was used so that the new variables would be independent. These new variables a linear combination of the original variables called principal components. The correlation matrix was used as a basis for the establishment of the principal components. TABLE 1. Principal component analysis results for IPA and IPS ( ) IPA PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 Eigenvalue Proportion Cumulative IPS Eigenvalue Proportion Cumulative Table 1 gives the eigenvalue for each component. It could be seen that only first two principal components had eigenvalue more than 1 with cumulative proportion 62.3%. This means that only 62.3% of dependent variables information could be represented by principal components. In addition, the selection of principal component can also be based on the cumulative proportion above 0.70 [9]. In this study, there were three principal components with eigenvalue approaching 1 and cumulative proportion more than 70%. Based on Table 1, the proportion of cumulative for the two majors three main components were This means that about 72% of the information of the dependent variables could be represented by three principal components, while the rest were not analyzed by the model. Modeling Table 2 shows a summary of estimation each model was constructed based on the data analysis procedures. Where ( is principal component in school-i, is estimate for dummy variable school-i effect which consisted of 420 schools for IPA and 418 schools for IPS, is estimate dummy variable time-t consisted of 3 years. Multilevel model consist of ( is dependent variable for regency/city j. Goodness of each model could be seen from RMSE, RMSEP, AIC, and BIC values. The best estimation method was determined by the value of error estimators and smallest maximum likelihood estimators

7 TABLE 2. Summary of estimation each model for IPA and IPS IPA IPS Coefficient Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 Intercept Appendix 1 Appendix Appendix 2 Appendix Appendix 3 Appendix 5 Appendix 7 - Appendix 4 Appendix 6 Appendix 8 RMSE * * RMSEP * * AIC * * BIC * * Based on Table 2, the smallest RMSE for IPA was in model 2, whereas the smallest RMSEP and maximum likelihood estimators values were in model 3. Therefore, model 3, multilevel model with regency/city effect, would be regarded as the best one. For IPS, generally, model 3 had the smallest RMSEP and AIC values. It means for IPS, regency/city didn t affect the results of UN at every school. Checking assumptions was imperative to obtain the best model. Assumptions violation could be seen from residual plots and predictions for each model. IPA Model 1 Model 2 Model 3 Model 4 IPS Model 1 Model 2 Model 3 Model 4 FIGURE 2. Residual plots and predictions of each model for IPA and IPS Based on the residual plot and prediction in Figure 2, model 3 had a variety of heterogeneous although IPA and IPS were still seen their interconnections among multiple schools. Overall, by checking the assumptions it could be inferred that model 3 was the best model. Although for social science, the development of each model that didn t meaningful the average scores of UN because results weren t quite well. Generally, model comparison concluded that the best model for science and social studies majors were model 3. Model 3 was a multilevel modeling and data

8 panel to the average score of the UN that takes into account the level of regency/city. Results of average score of the UN obtained from each school would be affected by local conditions. CONCLUSION The best model to capture the diversity of each school for the average scores of UN for all SMANs in West Java is multilevel model with regency/city effect and effect of time. The obtained model is limited for analyzing four years of data observation UN in West Java. REFERENCES 1. C. J. M. Mass and J. J. Hox, (2004). Robustness issues in multilevel regression analysis. Statistica Neerlandica. Utrecht University, Netherlands Vol. 58, nr. 2, C. Hsiao, (2003). Analysis of Panel Data 2 nd edition. Cambridge University Press, New York 3. B. H. Baltagi, (2005). Econometrics Analysis of Panel Data Ed ke-3.john Wiley & Sons Ltd., England 4. K. A. Safitri, A. Kurnia and K. A. Notodiputro, (2015). On Modeling The Average Scores Of National Examination In West Java, in Proceeding of International Conference On Research, Implementation And Education Of Mathematics And Sciences 2015 (ICRIEMS 2015), (Yogyakarta State University, Yogyakarta, Indonesia, 2015), M215-M J. J. Hox, (2010). Multilevel Analysis Techniques and Applications : Quantitative Methodology Series 2 nd Edition. Routledge, New York 6. H. W. Ker, (2014). Application of Hierarchical Linear Model/Linear Mix-Effects Models in School Effectiveness Research. Universal Journal of Educational Research 2(2), Horizon Research Publishing, J. Jiang, (2007). Linear and Generalized Linear Mixed Models and Their Applications. Springer, New York 8. H. Goldstein, (2011). Multilevel Statistical Model 4 th Edition. John Wiley & Sons Ltd., England 9. I. T. Jolliffe, (2002). Principal Component Analysis, Second Edition. Springer, New York 10. D. Anggara, Indahwati, A. Kurnia, (2015). Generalized Linear Mixed Models Approaches To Modeling Panel Data : Application To Poverty In East Nusa Tenggara. Global Journal of Pure and Applied Mathematics, Research India Publication. ISSN Volume 11, Number 5,

9 APPENDIX 1. School effect for major IPA in the model 1 No. School Estimation Std. Error 1 SMA NEGERI 1 ARJAWINANGUN, KABUPATEN CIREBON SMA NEGERI 1 ASTANAJAPURA, KABUPATEN CIREBON SMA NEGERI 1 BABAKAN MADANG, KABUPATEN BOGOR SMAN CAMPAKA, KABUPATEN PURWAKARTA APPENDIX 2. School effect for major IPS in the model 1 No. School Estimation Std. Error 1 SMA NEGERI 1 ARJAWINANGUN, KABUPATEN CIREBON SMA NEGERI 1 ASTANAJAPURA, KABUPATEN CIREBON SMA NEGERI 1 BABAKAN MADANG, KABUPATEN BOGOR SMAN CAMPAKA, KABUPATEN PURWAKARTA APPENDIX 3. School effect and time effect for major IPA in the model 2 No. School Estimation Std. Error 1 SMA NEGERI 1 ARJAWINANGUN, KABUPATEN CIREBON SMA NEGERI 1 ASTANAJAPURA, KABUPATEN CIREBON SMA NEGERI 1 BABAKAN MADANG, KABUPATEN BOGOR SMAN CAMPAKA, KABUPATEN PURWAKARTA No. Years Estimation Std. Error 1 T T T APPENDIX 4. School effect and time effect for major IPS in the model 2 No. School Estimation Std. Error 1 SMA NEGERI 1 ARJAWINANGUN, KABUPATEN CIREBON SMA NEGERI 1 ASTANAJAPURA, KABUPATEN CIREBON SMA NEGERI 1 BABAKAN MADANG, KABUPATEN BOGOR SMAN CAMPAKA, KABUPATEN PURWAKARTA

10 No. Years Estimation Std. Error 1 T T T APPENDIX 5. Time effect for major IPA in the model 3 No. Years Estimation Std. Error 1 TA TA TA APPENDIX 6. Time effect for major IPS in the model 3 No. Years Estimation Std. Error 1 TA TA TA APPENDIX 7. Time effect for major IPA in the model 4 No. Years Estimation Std. Error 1 TA TA TA APPENDIX 8 Time effect for major IPS in the model 4 No. Years Estimation Std. Error 1 TA TA TA

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