A Hierarchical Mixture Dynamic Model of School Performance in the Brazilian Mathematical Olympiads for Public Schools (OBMEP)
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1 A of School Performance in the ian Mathematical Olympiads for Public Schools (OBMEP) Alexandra M. Schmidt IM - UFRJ, Homepage: alex joint work with Caroline P. de Moraes and Helio S. Migon UFRJ, Workshop on Bayesian Multivariate Time Series Núcleo de Ciências de Dados e Decisão, INSPER,
2 OBMEP has been promoted since 2005 by the Ministries of Science and Technology, and of Education, and organized by Instituto Nacional de Matemática Pura e Aplicada (IMPA) In 2005 there were over 31,000 schools registered, comprising over 10.5 million students. In 2013 there were over 47,000 schools registered, involving nearly 19.2 million students (99,5% of the municipalities in ) The public ian educational system comprises three different types of administrative school levels: municipal, state, and federal. Any of these schools are allowed to subscribe to take part in the OBMEP. The registration is done by the schools, and each school indicates how many students will take part in the first phase of the OBMEP
3 OBMEP has been promoted since 2005 by the Ministries of Science and Technology, and of Education, and organized by Instituto Nacional de Matemática Pura e Aplicada (IMPA) In 2005 there were over 31,000 schools registered, comprising over 10.5 million students. In 2013 there were over 47,000 schools registered, involving nearly 19.2 million students (99,5% of the municipalities in ) The public ian educational system comprises three different types of administrative school levels: municipal, state, and federal. Any of these schools are allowed to subscribe to take part in the OBMEP. The registration is done by the schools, and each school indicates how many students will take part in the first phase of the OBMEP
4 OBMEP has been promoted since 2005 by the Ministries of Science and Technology, and of Education, and organized by Instituto Nacional de Matemática Pura e Aplicada (IMPA) In 2005 there were over 31,000 schools registered, comprising over 10.5 million students. In 2013 there were over 47,000 schools registered, involving nearly 19.2 million students (99,5% of the municipalities in ) The public ian educational system comprises three different types of administrative school levels: municipal, state, and federal. Any of these schools are allowed to subscribe to take part in the OBMEP. The registration is done by the schools, and each school indicates how many students will take part in the first phase of the OBMEP
5 The students are divided into three different levels: Level 1: students in the 6 th and 7 th grades of the primary school; Level 2: students in the 8 th and 9 th grades of the primary school; Level 3: students in high school. The OBMEP is performed in two phases: first, students take a multiple choice exam with 20 questions for each educational level Approximately 5% of students with the highest scores in each level of each school, are approved for the second phase of OBMEP In the second phase, students write a discursive examination comprising six questions. These tests are also separated by level of education
6 The students are divided into three different levels: Level 1: students in the 6 th and 7 th grades of the primary school; Level 2: students in the 8 th and 9 th grades of the primary school; Level 3: students in high school. The OBMEP is performed in two phases: first, students take a multiple choice exam with 20 questions for each educational level Approximately 5% of students with the highest scores in each level of each school, are approved for the second phase of OBMEP In the second phase, students write a discursive examination comprising six questions. These tests are also separated by level of education
7 Our mains aims are to Investigate the performance of schools across that have taken part of the OBMEP from 2006 until 2013 Understand what covariates influence the performance of schools in the OBMEP
8 Year No. of Schools No. of Schools Percentage of schools in Phase 1 in Phases 1 and 2 in both phases ,603 29, % ,886 35, % ,396 35, % ,851 39, % ,718 39, % ,684 39, % ,722 40, % ,145 42, % Table: Distribution of the number of schools registered for each edition and phase of the OBMEP from 2006 until 2013.
9 To ease the computational burden of estimating our models we choose to analyze a sample from this population Distribution of the admin. level: 0.6% are federal, 42.8% are state and 26.6% are municipal Stratified random sampling scheme: strata defined by the educational level (1, 2, and 3), the administrative level of the school (federal, state or municipal), and different levels of the HDI The behavior of the HDI across is strongly related to the geographical regions. HDI was divided into 5 categories 45 strata result
10 5% of the schools that took part in the 2005 edition of the OBMEP were sampled Final sample size: n = 1, 501 schools
11 Standardization of the schools average score Let W ijt be the average score of school j within level i in year t, i = 1, 2, 3, j = 1, 2,, n i, t = 1, 2,, 8. W ijt (0, 120) Define R ijt = W ijt 120 Compute S ijt = R ijt R it S ijt ( 1, 1) (centering) Compute Y ijt = (S ijt+1) 2 (0, 1)
12 Box plots of transformations of the average schools scores (columns) in the sample, for each year, and educational level (rows) R Y R Y R Y
13 Let Z tij be an indicator variable being equal to 1 if school j, within educational level i took part in the second phase of OBMEP in year t, and 0 otherwise conditioned on Z tij = 1, let Y tij be the score of school j = 1, 2,, n i, within educational level i = 1, 2,, I, in year t = 1, 2,, T p(y tij, z tij θ tij, µ tij, φ tij ) = [ θ tij p(y tij µ tij, φ tij ) ] z tij [1 θ tij ] (1 z tij) (1)
14 Let Z tij be an indicator variable being equal to 1 if school j, within educational level i took part in the second phase of OBMEP in year t, and 0 otherwise conditioned on Z tij = 1, let Y tij be the score of school j = 1, 2,, n i, within educational level i = 1, 2,, I, in year t = 1, 2,, T p(y tij, z tij θ tij, µ tij, φ tij ) = [ θ tij p(y tij µ tij, φ tij ) ] z tij [1 θ tij ] (1 z tij) (1) θ tij represents the probability of presence in the 2nd phase of OBMEP, of school j in educational level i and year t
15 Let Z tij be an indicator variable being equal to 1 if school j, within educational level i took part in the second phase of OBMEP in year t, and 0 otherwise conditioned on Z tij = 1, let Y tij be the score of school j = 1, 2,, n i, within educational level i = 1, 2,, I, in year t = 1, 2,, T p(y tij, z tij θ tij, µ tij, φ tij ) = [ θ tij p(y tij µ tij, φ tij ) ] z tij [1 θ tij ] (1 z tij) (1) Marginalizing with respect to Z tij, we obtain p(y tij θ tij, µ tij, φ tij ) = [ θ tij p(y tij µ tij, φ tij ) ] + [1 θ tij ] (2)
16 Let Z tij be an indicator variable being equal to 1 if school j, within educational level i took part in the second phase of OBMEP in year t, and 0 otherwise conditioned on Z tij = 1, let Y tij be the score of school j = 1, 2,, n i, within educational level i = 1, 2,, I, in year t = 1, 2,, T p(y tij, z tij θ tij, µ tij, φ tij ) = [ θ tij p(y tij µ tij, φ tij ) ] z tij [1 θ tij ] (1 z tij) (1) p(y tij µ tij, φ tij ) is any distribution in the exponential family with two parameters we explore the beta, and the normal distributions, parametrized respectively, by their mean (µ tij ) and precision (φ tij ) parameters.
17 Hierarchical prior specification Let η tij = (η 1, η 2, η 3 ) tij = (g 1(θ tij ), g 2 (µ tij ), g 3 (φ tij )) be a three-dimensional vector, such that g k ( ) represents a transformation of the parameter of interest to the real line.
18 Hierarchical prior specification Let η tij = (η 1, η 2, η 3 ) tij = (g 1(θ tij ), g 2 (µ tij ), g 3 (φ tij )) be a three-dimensional vector, such that g k ( ) represents a transformation of the parameter of interest to the real line. g 1 (θ tij ) = log θ tij 1 θ tij g 2 (µ tij ) = log µ tij 1 µ tij, if y tij beta(a tij, b tij ), where a tij = µ tij φ tij b tij = (1 µ tij )φ tij (Ferrari & Cribari-Neto, 2004) µ tij, if y tij N(µ tij, φ 1 tij ) g 3 (φ tij ) = log φ tij
19 Hierarchical prior specification Let η tij = (η 1, η 2, η 3 ) tij = (g 1(θ tij ), g 2 (µ tij ), g 3 (φ tij )) be a three-dimensional vector, such that g k ( ) represents a transformation of the parameter of interest to the real line. We propose a hierarchical DLM (Gamerman and Migon, 1993) for g k ( ) (η k ) tij = g k ( ) = β k ti Xk tij + δk j, with β k ti = F k αk t + v k ti, vk ti N p k (0, V k i ), t = 1,, T, α k t = G k α k t 1 + ωk t, ω k t N pk (0, W k ) α k 0 N(m k 0, Ck 0)
20 Likelihood function Let y be the vector comprising the average scores of the schools stacked across the different educational levels and years, z a vector of 0s and 1s indicating if school j, within educational level i, took part in the second phase of OBMEP in year t. When we assume a beta distribution for y tij then n T I i f (y Θ) = θ Γ(φ tij ) tij ( ) y [µ ) tijφ tij 1] [(1 µtij )φ (1 y tij 1] tij tij t=1 i=1 j=1 Γ(µ tij φ tij )Γ (1 µ tij )φ + [1 θ tij], tij where Γ(.) is the usual Gamma function. When we assume y tij follows a normal distribution n T I i φ 1/2 { tij f (y Θ) = θ tij exp φ } ( ) tij 2 y tij µ tij + [1 θ tij ] t=1 i=1 j=1 2π 2
21 Posterior Distribution Following the Bayes theorem, the posterior distribution of Θ, p(θ y), is proportional to the likelihood function times the prior distribution. As we assume independence among the hyperparameters, it follows that 3 I T [ p(θ y) f (y Θ) p(β k it αk t, Vk i )p(αk t αk t 1 )] p k 1, Wk p(v k im ) p(wk m ) p(α k 0 mk 0, Ck 0 ) k=1 i=1 t=1 m=0 The posterior does not have a closed analytical form regardless of the assumption of the distribution of the response variable We make use of Markov chain Monte Carlo (MCMC) methods to obtain samples from the posterior distribution above
22 DAG representation of the proposed model b2 a2 b3 a3 a1 b1 X 2 tij Ytij X 1 tij X 3 tij Vi 2 Vi 3 Vi 1 β 2 ti µtij δj 2 Ztij θtij δj 1 φtij β 3 ti β 1 ti school j level i α 2 t 1 α 2 t α 2 t+1 α 3 t 1 α 3 t α 3 t+1 α 1 t 1 α 1 t α 1 t+1 year t W 2 W 3 W 1 aw bw
23 Fitted models Distribution Model of p(y tij µ tij, φ tij ) g 1 (θ tij ) g 2 (µ tij ) g 3 (φ tij ) M1 Beta logit(θ tij ) = X 1 tij β1 tij logit(µ tij ) = X 2 tij β2 tij log(φ tij ) = X 3 tij β3 tij M2 Beta logit(θ tij ) = X 1 tij β1 tij logit(µ tij ) = X 2 tij β2 tij + δ2 j log(φ tij ) = X 3 tij β3 tij M3 Beta logit(θ tij ) = X 1 tij β1 tij + δ1 j logit(µ tij ) = X 2 tij β2 tij + δ2 j log(φ tij ) = X 3 tij β3 tij M4 Normal logit(θ tij ) = X 1 tij β1 tij µ tij = X 2 tij β2 tij log(φ tij ) = X 3 tij β3 tij M5 Normal logit(θ tij ) = X 1 tij β1 tij µ tij = X 2 tij β2 tij + δ2 j log(φ tij ) = X 3 tij β3 tij M6 Normal logit(θ tij ) = X 1 tij β1 tij + δ1 j µ tij = X 2 tij β2 tij + δ2 j log(φ tij ) = X 3 tij β3 tij X 1 tij = (1, ADM, HDI, LIB, LAB, NEL) tij X 2 tij = (1, ADM, HDI, LIB, LAB, BOYS, NEL) tij X 3 tij = (1, nstudent tij)
24 Model comparison Fitted Distribution of Inclusion of School Model p(y tij µ tij, φ tij ) random effect (νj k ) p D DIC RPS LogS DSS M1 Beta M2 Beta logit(µ tij ) M3 Beta logit(θ tij ), logit(µ tij ) M4 Normal M5 Normal µ tij M6 Normal logit(θ tij ), µ tij Table: Model comparison criteria, DIC and its component, p D, RPS, LogS, and DSS, under each fitted model. Numbers in italics indicate best model under the respective criterion.
25 Posterior summary of the coefficients β 1 lit (logit θ tij) Level 1 Level 2 Level 3 Intercept HDI ADM LAB LIB NEL
26 Posterior summary of the coefficients β 2 lit (µ tij) Level 1 Level 2 Level 3 Intercept HDI ADM LAB LIB BOYS NEL
27 Posterior summary of the coefficients β 3 lit (log φ tij) Intercept Level 1 Level 2 Level 3 Overall effect NSTUDENT
28 Sign of the random effects in the probability of presence Negative Effect Administrative Educational Level Level Total Federal State Municipal Total Null Effect Educational Level Total Positive Effect Educational Level Total Sign of the random effects in the mean Negative Effect Administrative Educational Level Level Total Federal State Municipal Total Null Effect Educational Level Total Positive Effect Educational Level Total
29 Posterior summary of the coefficients α 1 lt Intercept HDI 6 4 ADM LAB 1.5 LIB 1.0 NEL
30 Posterior summary of the coefficients α 2 lt 0.55 Intercept 0.05 HDI 0.20 ADM 0.05 LAB LIB 0.05 BOYS 0.05 NEL
31 Posterior summary of the fitted values (solid lines) and probability of presence (gray) for some schools School 52 School 128 School Y θ School 221 School 750 School Y θ School 963 School 1292 School Y θ
32 Posterior summary of the predictive distribution for 2013 of the scores of 90 schools that took part in the 2nd phase and were left out from the inference procedure for predictive purposes (rows=educ. levels) Federal State Municipal Score Score Score School School School Score Score Score School School School Score Score Score School School School
33 As different exams are given in different years, we propose an ad hoc standardization, per level and year, of the schools mean scores such that they lie in the interval (0, 1). We propose a hierarchical mixture dynamic regression model that Estimates the probability of presence of a school in the 2nd phase of OBMEP Estimates a school s score even if it has not taken part in the 2nd phase The coefficients of the covariates vary per level and year Inference procedure is performed under the Bayesian paradigm Model comparison criteria suggest model M6 (normal response with school random effect in the mean and probability of presence)
34 As different exams are given in different years, we propose an ad hoc standardization, per level and year, of the schools mean scores such that they lie in the interval (0, 1). We propose a hierarchical mixture dynamic regression model that Estimates the probability of presence of a school in the 2nd phase of OBMEP Estimates a school s score even if it has not taken part in the 2nd phase The coefficients of the covariates vary per level and year Inference procedure is performed under the Bayesian paradigm Model comparison criteria suggest model M6 (normal response with school random effect in the mean and probability of presence)
35 As different exams are given in different years, we propose an ad hoc standardization, per level and year, of the schools mean scores such that they lie in the interval (0, 1). We propose a hierarchical mixture dynamic regression model that Estimates the probability of presence of a school in the 2nd phase of OBMEP Estimates a school s score even if it has not taken part in the 2nd phase The coefficients of the covariates vary per level and year Inference procedure is performed under the Bayesian paradigm Model comparison criteria suggest model M6 (normal response with school random effect in the mean and probability of presence)
36 As different exams are given in different years, we propose an ad hoc standardization, per level and year, of the schools mean scores such that they lie in the interval (0, 1). We propose a hierarchical mixture dynamic regression model that Estimates the probability of presence of a school in the 2nd phase of OBMEP Estimates a school s score even if it has not taken part in the 2nd phase The coefficients of the covariates vary per level and year Inference procedure is performed under the Bayesian paradigm Model comparison criteria suggest model M6 (normal response with school random effect in the mean and probability of presence)
37 Normal and Beta distributions (for the response variable) provided very similar results Overall, the mean performance of schools tend to be small Federal schools perform better than state or municipal ones The HDI has a positive effect on the mean performance of the school for most of the educational levels and years The proportion of boys has a (very small) positive effect on the mean performance for some years and educational levels
38 Normal and Beta distributions (for the response variable) provided very similar results Overall, the mean performance of schools tend to be small Federal schools perform better than state or municipal ones The HDI has a positive effect on the mean performance of the school for most of the educational levels and years The proportion of boys has a (very small) positive effect on the mean performance for some years and educational levels
39 Normal and Beta distributions (for the response variable) provided very similar results Overall, the mean performance of schools tend to be small Federal schools perform better than state or municipal ones The HDI has a positive effect on the mean performance of the school for most of the educational levels and years The proportion of boys has a (very small) positive effect on the mean performance for some years and educational levels
40 Normal and Beta distributions (for the response variable) provided very similar results Overall, the mean performance of schools tend to be small Federal schools perform better than state or municipal ones The HDI has a positive effect on the mean performance of the school for most of the educational levels and years The proportion of boys has a (very small) positive effect on the mean performance for some years and educational levels
41 Normal and Beta distributions (for the response variable) provided very similar results Overall, the mean performance of schools tend to be small Federal schools perform better than state or municipal ones The HDI has a positive effect on the mean performance of the school for most of the educational levels and years The proportion of boys has a (very small) positive effect on the mean performance for some years and educational levels
42 Next steps Organizers of OBMEP should use some tool to standardize the level of difficulty of the exams across years Our current interest is to investigate what kind of impact the OBMEP has on the ian educational system. We plan to focus on students in the last year of high school. Considering different years we plan to use causal inference and propensity score methods (Hirano and Imbens, 2004) to investigate the effect of the OBMEP on the performance of students in different editions of the High School ian National Exam (Exame Nacional do Ensino Médio, ENEM).
43 Next steps Organizers of OBMEP should use some tool to standardize the level of difficulty of the exams across years Our current interest is to investigate what kind of impact the OBMEP has on the ian educational system. We plan to focus on students in the last year of high school. Considering different years we plan to use causal inference and propensity score methods (Hirano and Imbens, 2004) to investigate the effect of the OBMEP on the performance of students in different editions of the High School ian National Exam (Exame Nacional do Ensino Médio, ENEM).
44 Thanks to Hedibert F. Lopes, for the invitation CNPq, CAPES, and FAPERJ,, for the financial support Last, but not least, Mike West, for influencing many generations of ian Statisticians Muito obrigada!
45 Thanks to Hedibert F. Lopes, for the invitation CNPq, CAPES, and FAPERJ,, for the financial support Last, but not least, Mike West, for influencing many generations of ian Statisticians Muito obrigada!
46 Thanks to Hedibert F. Lopes, for the invitation CNPq, CAPES, and FAPERJ,, for the financial support Last, but not least, Mike West, for influencing many generations of ian Statisticians Muito obrigada!
47 Thanks to Hedibert F. Lopes, for the invitation CNPq, CAPES, and FAPERJ,, for the financial support Last, but not least, Mike West, for influencing many generations of ian Statisticians Muito obrigada!
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