hierarchical - 6 Hierarchical Modeling 6.1 Introduction...

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6 Hierarchical Modeling 6.1 Introduction: Learning About OGT Success Rates The Ohio Graduation Test (OGT) is a test administered to all high school students in the state of Ohio. To graduate from high school, a student must de- velop a proficiency in reading, science, mathematics, social studies and writing as measured by his/her performance on the OGT. This test is used to assess the quality of schools in the state and the State of Ohio releases summary OGT scores for all public and private schools in the state. A student will receive a grade on each of the five sections of the OGT. The raw score for each section is categorized as Advanced, Accelerated, Proficient, Basic, or Limited. A student needs to have a grade of Proficient, Accelerated, or Advanced on all sections to pass the OGT. We focus on the performance of the students from the nine public school districts in Wood County. For each school district, we collect the number of students taking the exam, and the number who received an Advanced score (the highest category) on the Writing section of the OGT. School.District N ADVANCED Proportion Bowling Green City Sd 259 5 0.019 Eastwood Local Sd 145 10 0.069 Elmwood Local Sd 95 3 0.032 Lake Local Sd 138 2 0.014 North Baltimore Local Sd 53 0 0.000 Northwood Local Sd 82 4 0.049 Otsego Local Sd 140 2 0.014 Perrysburg Ex Vill Sd 355 16 0.045 Rossford Ex Vill Sd 139 4 0.029 Overall, only 3% of the students received an Advanced score on the Writing test for this county, so clearly this was a challenging test. One can compare
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2 6 Hierarchical Modeling schools by computing the proportion of Advanced that are given in the last column of the table. Several questions arise when one tries to make sense of these proportions. First, we note that none of the North Baltimore Local students received an Advanced score. Does this mean that the population proportion of students from North Baltimore Local who received this score is equal to zero? Although it is unlikely for these students to receive this grade, one would expect at least a few students to get “Advanced” in future years. Second, we note that Eastwood’s proportion of Advanced (0.069) is higher than Perrysburg (0.045). Does this mean that Eastwood’s Advanced population proportion is higher than Perrysburg? Since the actual success counts are small, it is possible that there is no difference in the quality of the Eastwood and Perrysburg schools and we are simply observing sampling variability. 6.2 To Pool or Not to Pool? In the OGT testing example, the general problem is to estimate the propor- tions p 1 , ..., p 9 , where p j represents the proportion of students from the j th school district who score at an Advanced level on the writing section of the OGT. We observe ( y j , n j ), where y j is the number of Advanced students in a sample of size n j from the j th district. The typical estimate of the proportion p j is the sample proportion ˆ p j = y j n j and the sample proportions are displayed in Table ???. As discussed in the introduction, there are some problems with these estimates. Due to the rela-
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