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This is to certify that the

dissertation entitled

Access to Eighth—Grade Algebra:
A Bayesian, Multilevel Analysis

presented by

Yuk Fai Cheong

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Measurement and

 

Quantitative Methods

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MSU IeAn Affirmative Action/Equel Opportunity lnetituion
mm

 

ACCESS TO EIGHTH-GRADE ALGEBRA:
A BAYESIAN, MULTILEVEL ANALYSIS

By

Yuk Fai Cheong

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Counseling, Educational
Psychology and Special Education

1997

ABSTRACT

ACCESS TO EIGHTH-GRADE ALGEBRA:
A BAYESIAN, MULTILEVEL ANALYSIS

By

Yuk Fai Cheong

This dissertation addressed two research objectives: one substantive, the other
methodological. The first objective was to examine what school- and state-level factors
may influence a public school's decision to offer eighth-grade algebra for high school
credits. The analysis employed the data collected under the 1992 Trial State Assessment
Program (TSAP) in mathematics of the National Assessment of Educational Progress
(NAEP), and the state profile statistics compiled by the Council of Chief State School
Officers (CCSSO) (CCSSO, 1993). The second, related objective was to develop and
evaluate a fully Bayesian, multilevel approach that enables the study of public schools as
distributors of learning opportunities in advanced mathematics.

The Bayesian, multilevel approach developed by Zeger and Karim (1991) was
implemented via the Gibbs sampler (Gemen & Gemen, 1984; Gelfand & Smith, 1990).
The algorithm was coded using the Interactive Matrix Language of the Statistical
Analysis Systems (SAS/IML) (SAS Institute, Inc., 1989). The result of a small simulation
study, which generated and analyzed data sets having similar structure to that of the
TSAP, showed that the SAS/IML code performed well and the algorithm yielded
reasonable inferences.

The tested code was used to fit two models studying the distribution of learning

opportunities in mathematics. The results of the first unconditional model reveal

significant state-to-state variation in the likelihood of the offering of eighth-grade algebra
by public schools. The findings of the second model suggest that schools serving minority
students and students of low social-economic status (SES), small schools, schools located
in rural settings and states that spend less on education are comparatively less likely than
other schools to offer algebra. The implication is that size, composition, and location of a
school are linked to inequality in access to this educational resource. In these ways, the
schooling system reinforces social, ethnic and geographic inequalities regarding the

opportunities available for eighth-grade students to study algebra for high school credits.

This dissertation is dedicated to the

loving memory of my grandmom, Ah Mah.

iv

ACKNOWLEDGMENTS

This research was supported by a grant from the American Educational Research
Association which receives funding for its "AERA Grants Program" from the National
Science Foundation and the National Center for Education Statistics (U .S. Department of
Education) under NSF Grant #RED-9255437. I wish to thank Dr. Richard Shavelson and
Ms. Jeanie Oakes for their help in the grant application process and their continued
encouragement. Opinions expressed in this work reflect only those of the author and do
not necessarily reflect those of the granting agencies.

I wish to express my gratitude to my dissertation advisor, Dr. Stephen
Raudenbush, for his guidance and support from the initial stage to the completion of this
dissertation, for his mentoring and the many invaluable teaching and research
opportunities he has given me over the years. I wish to also express my appreciation to
my doctoral committee members, Dr. Betsy Becker, Dr. Aaron Pallas, and Dr. James
Stapleton, whose availability, support, and critical analysis were exceptional. I am
thankful to Dr. Michael Seltzer at the University of California, Los Angeles, for
generously sharing his expertise in Gibbs sampling.

I also appreciate the support of my colleagues involved with the Longitudinal and

Multilevel Methods Project. In particular, I am grateful to our project manager, Marcy

Wallace, for her encouragement. Special thanks to Dr. Randall Fotiu, Dr. Rafa Kasim,
and Mengli Yang for their assistance in programming, and to Todd D. Harcek for reading
and commenting on the manuscript.

Finally, I am especially appreciative of the love, understanding, and patience
given to me by my parents, sisters, brother, and in-laws throughout the pursuit of my

graduate education.

vi

TABLE OF CONTENTS

LIST OF TABLES ....................................................... x
LIST OF FIGURES ..................................................... xi
CHAPTER 1
INTRODUCTION ....................................................... 1
1.1 Objectives of the Study .......................................... 1
1.2 Substantive Goals ............................................... l
1.3 Methodological Goals ........................................... 3
1.4 Overview of the Study ........................................... 5
CHAPTER 2
BACKGROUND AND SIGNIFICANCE ..................................... 7
2.1 Introduction ................................................... 7
2.2 The Substantive Inquiry .......................................... 7
2.2.1 Background ............................................ 7
2.2.2 Relevant Literature ..................................... 11
2.2.3 Sources of School-to-School Variation in the Availability of Eighth-
Grade Algebra ....................................... 13
2.2.4 Sources of State-to-State Variation in the Availability of Eighth-
Grade Algebra ....................................... 16
2.2.5 Significance of the Substantive Inquiry ..................... 17
2.3 The Methodological Work ...................................... 19
2.3.1 Motivations ........................................... 19
2.3.2 An Illustrative Example ................................. 19
2.3.3 The Maximum Likelihood Approach ....................... 24
2.3.4 The Approximate Maximum Likelihood Approach\Penalized Quasi-
likelihood Approach ................................... 29
2.3.5 The Fully Bayesian Approach ............................ 32
2.3.6 Appropriateness of the Fully Bayesian approach .............. 37
2.3.7 Usefulness of the Bayesian Approach to Policy Research ....... 38

vii

CHAPTER 3
BAYESIAN ESTIMATION OF GENERALIZED LINEAR MODELS WITH RANDOM

EFFECTS VIA THE GIBBS SAMPLER .................................... 39
3.1 Introduction .................................................. 39
3.2 Logic of the Gibbs Sampler ...................................... 39
3.3 Implementation of the Gibbs Sampler .............................. 42
3.3.1 Full Conditional Distribution of y--p(y |T,u, y) ............... 43
3.3.2 Full Conditional Distribution of T--p(T| y,u, y) ............... 47
3.3.3 Full Conditional Distribution of u--p(u |y,T, y) .............. 48
3.3.4 Starting Values and Convergence Diagnostics ................ 50
3.4 A Simulation Study and Results .................................. 57
CHAPTER 4
BAYESIAN HIERARCHICAL ANALYSES OF
ACCESS TO EIGHTH-GRADE ALGEBRA ................................. 62
4.1 Introduction .................................................. 62
4.2 Data, Procedures, and Measures .................................. 62
4.2.1 Data Description ....................................... 62
4.2.2 Procedures ............................................ 63
4.2.3 Measures ............................................. 65
4.3 Results ...................................................... 73
4.3.1 Unconditional Model ................................... 73
4.3.2 Between-school/Within-state and Between-State Model ........ 80
4.4 Discussion and Implications ..................................... 87
CHAPTER 5
SUMMARY AND CONCLUSIONS ....................................... 92
5.1 Summary and Conclusions ...................................... 92
5.2 Future Research Needs ......................................... 93
5.2.1 Substantive Inquiry ..................................... 93
5.2.2 Methodological Development ............................. 95
APPENDIX A
SAS\IML Code for the Gibbs Sampler ...................................... 97

viii

APPENDIX B

Approval Letter from the US. Department of Education

for Accessing Individually Identifiable Survey Data Base ...................... 117
APPENDIX C

Approval Letter from the University Committee

on Research Involving Human Subjects .................................... 118
LIST OF REFERENCES ................................................ 119

ix

Table 2.1:

Table 3.1:

Table 3.2:

Table 3.3:

Table 4.1:

Table 4.2:

Table 4.3:

LIST OF TABLES

Defining Features for the Three Major Approaches .................... 36
CODA Output for Geweke Convergence Diagnostic ................... 52
CODA Output for Raftery and Lewis Convergence Diagnostic .......... 56
Results of Simulation Study ...................................... 6O
Descriptive Statistics ........................................... 72
Results of the Unconditional (No Predictors) Model ................... 74
Results of the Within- and Between-State Model ..................... 81

LIST OF FIGURES

Figure 3.1 Illustration of Rejection Sampling (Taken from Gelman et al., 1995, p. 304 )45
Figure 3.2 Geweke’s Convergence Diagnostic ................................ 54

Figure 4.1. Logits Plotted Against the Midpoints of the Categories of Percentages of
Minority Enrollment .............................................. 67

Figure 4.2: Plot of Logits Against Midpoints of Grouped Categories of Enrollment . . . 70

Figure 4.3: Marginal Posterior Distribution of r (The Unconditional Model) ........ 76
Figure 4.4a Distribution of Predicted Probabilities of Offering Algebra for the First Half
of Individual States ............................................... 78
Figure 4.4b Distribution of Predicted Probabilities of Offering Algebra for the Second
Half of Individual States ........................................... 79
Figure 4.5: Marginal Posterior Distribution of 1: (The Conditional Model) .......... 86

xi

CHAPTER 1
INTRODUCTION
1.1 Objectives of the Study
This dissertation has two related objectives, one substantive, and the other
methodological. The first objective is to describe and study unequal learning
opportunities in advanced mathematics in Grade 8 among schools across 42 states'. This
objective inspires the development and the evaluation of a fully Bayesian treatment of
random effects models with binary outcomes, which constitutes the second,

methodological objective of this dissertation.

1.2 Substantive Goals

The substantive purpose of the work is to investigate school and state influences
on public schools’ decisions to offer eighth-grade algebra for high school credits. Eighth-
grade algebra is ofien a "gatekeeping" course that allows students access to advanced high
school mathematics. Data collected under the 1992 TSAP of NAEP, and information
made available by CCSSO (CCSSO, 1993) are combined to relate school- and state-level
factors to the availability of an algebra course. Inquiry into the curriculum distinctions in
mathematics among schools across the forty-two states can help us understand the
interschool and interstate variation, and some of its sources, in the stratification of

learning opportunities in the middle grades mathematics curriculum. It can help us

 

1There were a total of 41 states and 1 territory in the sample. For brevity, they will be
referred to as states in this dissertation.

2
evaluate the effect of eighth-grade algebra on student achievement by offering a more
thorough understanding of the selection process. The analysis can inform discussion on
school- and state-level policies related to equity and funding. Furthermore, the likelihood
of course availability can perform as an opportunity-to-learn indicator (Oakes, 1989;
Pelgrum, Voogt, & Plomp, 1995; Porter, 1991) in monitoring the performance of
educational systems.

The study draws on perspectives primarily from Sorenson and Hallinan's (1977)
conception and Gamoran's (1987) model of opportunities for learning. Raudenbush,
Fotiu, Cheong and Ziazi's (1996) study on inequality in access to educational resources,
and Mullis, Jenkin, and Johnson's (1994) NAEP 1992 report on school effectiveness
pertaining to mathematics education, laid the groundwork for this analysis. Major
empirical work that guides the formulation of various research hypotheses includes
studies on course availability done by Becker (1990), MacIver and Epstein (1995), Monk
and Haller (1993), Oakes (1990), and Useem (1992).

At the school level, relationships between school composition, setting, size and
the likelihood of the offering of algebra for high school credits are examined.
Specifically, the analysis evaluates the hypotheses that schools 1) located in rural settings,
2) with greater percentages of African and Hispanic Americans, 3) of lower SES, and 5)
with smaller grade 8 enrollment are less likely to offer high school algebra. At the state
level, the study examines 1) whether the availability of the course varies with state
poverty levels, as indicated by the percent of children in poverty, 2) whether state

educational expenditures are related to how likely the algebra course is to be offered, and

3
3) whether state educational expenditures moderate the relationships between a school's
racial and SES composition and the likelihood that it offers algebra. Put differently, the
last question asks if there is an interaction effect between state educational spending and
school racial and SES composition on opportunities to learn. It is hypothesized that
schools located in states with higher poverty rates, on average, are less likely to offer the
advanced course. In addition, greater state educational investment efforts are associated
with a greater likelihood of offering algebra. The last state-level hypothesis postulates
that the association between school racial and social composition effects and the

probability of offering algebra depends on how much money states allocate per student.

1.3 Methodological Goals

The inquiry has motivated the development and evaluation of a fully Bayesian
analytical approach which accommodates 1) the binary nature of the outcome variable of
the offering of the algebra course, 2) the hierarchical or nested design of the TSAP in
mathematics, schools nested within states and territories, and 3) the need to incorporate
the uncertainty that arises about the variances at the state-level. By representing the state-
level variance as the investigator's uncertainty about the processes that produce it, the
hierarchical framework enables us to study state-level heterogeneity, the relationships
between state-level predictors and outcomes (Raudenbush, Cheong, & Fotiu, 1995), and
not to treat the states as fixed strata.

The fully Bayesian estimation approach has several advantages over two other

major approaches, the maximum likelihood (ML) (e. g., Karim, 1991) and the

4
approximate maximum likelihood (AML) (e.g., Goldstein, 1991) or the closely related
penalized quasi-likelihood (PQL) approaches (e.g., Breslow & Clayton, 1993). First,
relative to AML\PQL, Bayes estimates of the level-2 variances are less biased (Breslow &
Clayton, 1993; Rodriguez & Goldman, 1995). Second, relative to ML and AML\PQL,
Bayes inference about any parameter allows full assessment of uncertainty in other
parameters in the same model. Thus, unlike the other two approaches, Bayes estimates of
the regression parameters incorporate uncertainty of the estimate of variance. Lastly, the
Bayesian estimation supplies researchers with fuller inferential information by providing
the entire posterior distributions of estimates, rather than point and interval estimates
alone. This approach is useful: given the modest number of states, the posterior
distribution of the level-2 variance is likely to be skewed and inferences based solely on
the variance of the estimate can be misleading.

Zeger and Karim (1991) developed an algorithm for this approach and
implemented it via the Gibbs sampler (Gemen & Gemen, 1984; Gelfand & Smith, 1990).
They showed with a simulation that the algorithm yielded reasonable inferences in finite
sample cases in which the number of clusters was large relative to the number of
observations within each cluster (100 clusters, each with 7 observations). Rodriguez and
Goldman (1995) recommended the method as the most appealing option for more
complicated models in their assessment of estimation procedures for multilevel models
with binary responses, despite the intensive computation involved. However, Rodriguez
and Goldman noted that no computer program for implementing the algorithm is publicly

available. In this thesis, Zeger and Karim's algorithm is coded, evaluated and

5

implemented to study the distribution of learning opportunities. A small scale simulation
study is carried out to assess the accuracy of the code and to examine how well the
algorithm suits the present analysis, where the number of observations per cluster is larger
than the total number of clusters (42 clusters, each with 100 observations on average).

The approach can be applied to policy research that seeks to understand how
various policies and factors at the macro (e.g., nations, states, districts) and meso levels
(e.g., schools and classrooms) of the educational system(s) operate and interact with one
another, and whose design has relatively few self-selected macro units. For instance, the
approach can be applied to study how the probability of employment of adults is related
to one's educational attainment, social and ethnic background, literacy level, as well as
state policies on continuing education and retraining programs, using the 1992 National
Adult Literacy Study (Raudenbush, Kasim, Earnsukkawat & Miyazaki, 1996). It can be
applied to panel data to study trends as well. An example, which will be an extension of
the inquiry of this thesis, is to utilize the various waves of the TSAP in mathematics data
(I 990, 1992, 1996) to review the trends of the likelihood of offering algebra over time as

a function of state reform initiatives.

1.4 Overview of the Study

Chapter 2 describes the background and significance of the substantive inquiry
into learning opportunities as well as the development of a Gibbs sampler for Bayesian
hierarchical models with binary outcomes. Chapter 3 outlines the Gibbs sampler

technique developed by Zeger and Karim (1991), and presents the results of a simulation

6
study documenting the performance of the SAS\IML code written to implement the Gibbs
sampler. Chapter 4 reports and discusses the findings on school-to-school and state-to-
state variation in the likelihood of offering algebra, and the relationships between various
school- and state-level factors and a school's decision to include algebra in its middle
grades mathematics curriculum. Chapter 5 concludes the study and suggests future

research needs.

CHAPTER 2
BACKGROUND AND SIGNIFICANCE

2.1 Introduction

This chapter describes the theoretical frameworks and reviews relevant literature
for the substantive inquiry and methodological studies. It elaborates the research
objectives and hypotheses stated in Chapter 1. It is contended that the inquiry can add to
our understanding in how educational institutions function as distributors of learning
opportunities, and the methodological work done in this thesis can be beneficial to policy

research that addresses the multilevel structure of social and educational systems.

2.2 The Substantive Inquiry
2.2.1 Background

With the philosophy that access to algebra in middle grades is critical to a broad
range of academic, and eventually occupational pursuits, and the mission to make algebra
available to all middle grades students, projects like the Algebra Project (Jetter, 1993;
Moses, 1994; Moses, Kamii, Swap, & Howard, 1989; Silva & Moses, 1990), and the
Urban School Science and Mathematics Program (Archer, 1993) have made serious
efforts to enhance mathematics and science education in the middle grades. Moses et al.
and Archer postulated that, given the sequential nature of the mathematics curriculum,
enrollment in algebra in middle grades is important in determining students' subsequent

access to college preparatory mathematics in high schools. Empirical support for their

8
premise was provided by Stevenson, Schiller and Schneider's (1994) research on
sequences of opportunities for learning mathematics from Grade 8 to Grade 10. They
found, afier controlling for achievement, coursework in eighth-grade algebra is associated
with a greater likelihood of enrollment in advanced high school mathematics.

The mission "algebra for all" was initiated in part by the concern for equity in
mathematics and science. A large number of inner-city, minority, and poor students are
denied access to prerequisite courses in rigorous academic programs. Differential access
to those courses is tied to race and class (Oakes, 1990, Raudenbush et al., 1996, Useem,
1992). The creators of the projects posed a challenge to the ability model, which decrees
only mathematically inclined students should be given access to algebra, and its
"institutional expressions" (Moses et al., p. 424)--curriculum tracking. They were
convinced that, with appropriate pedagogy, capable teachers, high expectations of
success, and strong support from school, parents and community, mastery of seventh- and
eighth-grade algebra is within the reach of every student. MacIver and Epstein (1995)
conducted an initial test of their belief that it is beneficial for public school students of all
academic tracks to be given access to algebra in the middle grades, notwithstanding their
past success in mathematics. Employing data from the base year of the National
Educational Longitudinal Study of 1988 (N ELS:88), MacIver and Epstein found that
students who had taken eighth-grade algebra scored .3 to .5 standard deviations higher on
a standardized mathematics test than those who did not, controlling for school
characteristics, such as social and ethnic composition, and students' track level, past

achievements in mathematics, and background variables. The increment in achievement,

9
moreover, was similar for students in different ability groups. Usiskin (1987) shared, to a
great extent, the views of Moses et al. and Archer. Citing the schooling experiences of
other countries and that of the University of Chicago School Mathematics Project, he
contended that algebra can be learned by average students at the eighth-grade level.

The proponents of these mathematics projects stressed the critical role of school
as a provider as well as a distributor of educational opportunity (MacIver & Epstein,
1995) in achieving equity as well as preparing the underrepresented non-Asian minorities
for careers in science (Kamii, 1990; Oakes, 1990). As Porter (1991) succinctly stated,
"Schools provide educational opportunity; they do not directly produce student learning"
(p.33). Components that constitute opportunities to learn include exposure to content,
pedagogy, instructional time, adequate institutional resources, and methods of assessment
(Porter, 1991, 1994).

Sorenson and Hallinan's (1977) model for the process of learning highlights the
impact schools can have through the instructional resources and learning environments
they provide. This model was further elaborated by Hallinan (1996), who collaborated
with Sorenson in its conceptualization. In the model, two individual attributes of the
students--level of ability and effort--together with opportunities for learning provided by
schools, constitute three primary determinants of student learning. The amount of
learning is a function of these three interdependent elements. While the three
determinants interact with one another in the learning process, the model, as Sorenson
and Hallinan stressed, is not an additive one. A high level of ability and effort does not

compensate for a lack of educational opportunity. Students cannot acquire the knowledge

10
from schooling if they are not exposed to the appropriate curriculum, no matter how able
or diligent the students are. Opportunities to learn can enhance and the lack thereof can
constrain a student's learning environment. As most low-income minority families cannot
afford to supplement school experiences with learning opportunities in the private sector,
such as hiring a tutor, the learning resources at school have a much larger impact on the
academic achievements of disadvantaged students (Gordon, 1967, Oakes, 1990).

Another important concern of the projects is the immediate need to tackle the
problem of the underparticipation of minorities in the fields of science, mathematics, and
engineering. With the demographic change of increasing proportion of minorities and
immigrants in the new labor force, Kamii (1990) argued that, in order to maintain the
competitiveness and leadership role of the United States in the global economy, enticing
the underrepresented groups into the "pipeline" leading to careers in science is
particularly urgent.

The present investigation compares public schools across 42 states with respect to
the opportunities they provide students to learn advanced mathematics in middle grades
through offering eighth-grade algebra. It seeks to understand how the schooling systems
in various states distribute to their students access to the critical gatekeeping algebra
course. The course availability measure (Monk, 1994) used here is whether a school
offers algebra for high school credits. The measure indexes the upper bounds of student
learning in advanced mathematics set by middle and junior high schools (Gamoran,
1987), as the absence of the course precludes most possibilities to learn algebra in the

eighth grade. Influences of various school- and state-level characteristics and policies on

11
this limit are considered. Past relevant studies, on which the formulation of research

hypotheses in this study are based, are reviewed in the following section.

2.2.2 Relevant Literature

Two studies laid the groundwork for this dissertation: Raudenbush, F otiu, et al.'s
(1996) work on inequality of access to educational opportunity, and Mullis et al.'s (1994)
report on effective schools in mathematics. Using the 1992 TSAP in mathematics of
NAEP, Raudenbush et al. investigated social and ethnic inequality in access to resources
for learning eighth-grade algebra. Of particular relevance here is the school resource--
course offerings that they considered. Using the proportions of students classified
according to ethnicity and level of parental education as predictors, the study modeled the
probability of attending a school that offers high school algebra for eighth graders. The
results indicates that the probability of attending a school with a rigorous mathematics
program is positively associated with the level of parental education and that modest
ethnicity gaps exist. In addition, there are substantial differences among states in the
overall probability of students' access to algebra in grade eight.

Mullis et al. (1994) carried out a school effectiveness study pertaining to
mathematics education using the 1992 NAEP 4th, 8th (not the TSAP in mathematics used
in this study), and 12th grade mathematics achievement data from 1500 public and
private schools. In the study, three series of analyses were carried out to classify the more
effective versus the less effective schools, to compare the characteristics of those groups

of schools, and to identify sets of variables associated with higher mathematics

12
achievement in the more effective schools.

In the first series of analyses, the researchers used the school mean mathematics
proficiency scores to classify the schools participating in the study into three categories:
highest-performing, medium-performing, and lowest-performing schools, and did a direct
comparison of the contexts for learning mathematics, e.g., students' course enrollment, in
the first and last groups of schools. In the second and third series of analyses, hierarchical
linear modeling techniques were employed to identify factors associated with school
effectiveness, controlling for home background of students and level of school SES.

The three series of analyses yielded congruent results. Their findings on school
effectiveness as related to opportunities to learn, socioeconomic characteristics of
students and schools, and school ethnic composition are reported here. With course taking
as an opportunity-to-learn measure, the study found that the top one-third of schools had
more students than their bottom one-third counterparts enrolled in advanced courses. For
example, 27 versus 13 percent of ninth graders in the two different groups of schools had
enrolled in algebra by eighth grade. Also, more economically disadvantaged, urban, and
minority students were in the lower-performing schools. After adjusting for
socioeconomic characteristics of students and schools, two of the eleven factors which
significantly differentiated the two groups of the most and least effective schools for
grade 8 were proportion of students enrolled in Algebra 1 and percentage of non-African
Americans. The more effective schools had more students enrolled in Algebra 1 and
fewer minority students (non-Afiican Americans). The last series of analyses investigated

predictors of mathematics achievement in more effective schools. In addition to the large

13
influences of socioeconomic factors, planning to enroll in or taking more advanced
courses was a powerful predictor of higher mathematics achievement at grades 8 and 12.
In sum, their study suggests that opportunities to learn, as measured by course enrollment,
are likely to be linked to school achievement and are socially and racially stratified
(Gamoran, 1987).

A few studies in addition to the two pieces of groundwork have implications for
the formulation of research hypotheses of this dissertation. The results of Education in the
Middle Grades: A National Survey of Practices and Trends (Becker, 1990) show
substantial between-school variability in the offering of algebra. The survey found that
out of a probability sample of 2,400 public schools that enrolled seventh-graders, 63% of
the middle-grade schools offered algebra courses in their curriculums in grade 7, and,
more typically, grade 8. More than one third of the schools in the sample did not offer
algebra. The finding, together with Raudenbush, Fotiu, et al. '5 (1996) report of
substantial interstate variation in the probability of the offering of algebra, indicates that
there is considerable variance at both the school and state level, and encourages an

examination of its sources.

2.2.3 Sources of School-to-School Variation in the Availability of Eighth-Grade
Algebra

At the school level, variability in the availability of the algebra course may reflect
differences in school composition, urbanicity, and size, as conceptualized in Gamoran's

(1987) model of opportunities for learning. Gamoran postulated that the location of a

14

school in an urban, suburban, or rural setting, and school composition may have
influences on school offerings, and lead to between-school variation and stratification of
learning opportunities. School offerings in turn may have an impact on students'
achievement. Gamoran labeled and tested these indirect "setting effects" on achievement
in his study. Oakes (1990), using data from the 1985-1986 National Survey of Science
and Mathematics Education (N SSME), studied the percentages of 1,200 junior high
schools of different SES (high poverty to high wealth) and Caucasian student populations
(different categories of different proportions) that had one or more sections of advanced
mathematics, i.e., eighth-grade algebra and ninth-grade geometry. An analysis of variance
revealed no significant differences in the percentages of schools that offered accelerated
mathematics classes among the SES or racial composition categories. She, however,
suggested that meaningful differences may exist. Specifically, low-SES and
predominantly minority schools are less likely to offer accelerated mathematics classes. In
another analysis, Oakes examined advanced mathematics sections per 100 students and
found that high-SE8 or all-white schools have significantly more of those advanced
sections. Useem (1992) reported that district parental education levels were related to the
striking district-by-district variability of student enrollment in eighth-grade algebra,
which ranged from 13% to 60%.

The present study cross-validates Oakes' findings of non-significant SES and
racial composition effects on course offering and evaluates the hypothesis that racial and
social stratification exist in the offering of algebra. Given the other related results and that

of Raudenbush, F otiu, et al. (1996), it is expected that schools with fewer economically

15
disadvantaged students and African and Hispanic minority students are more likely to
have rigorous mathematics programs.

Monk and Haller (1993) analde the 1980 High School and Beyond (HSB)
survey data with 1032 public and private schools in the US. and found a positive
significant interaction effect of urban setting and school size on the total number of
academic course credits. Monk and Haller argued as urban high schools face a less
restrictive local teacher labor market, which lifts the human resources constraint on
curriculum development, diversity in academic course offerings is favored there.
Therefore, all else being equal, the likelihood of offering algebra is hypothesized to be
higher in urban versus rural and, similarly, suburban versus rural schools.

Another source of variability among schools in the likelihood of the offering of
algebra may originate from a structural factor of schools, that of size. Monk and Haller
(1993) contended that greater school size increases efficiency through its potential to
generate economies, and their study showed a positive relationship between the
enrollment of a high school’s graduating class and how many different course credits
were offered. Furthermore, Useem (1992) in her study on variability in advanced
mathematics course placements across 26 districts, reported that in one K-8 system only
some of the larger elementary schools offered algebra. The two studies suggest that
greater school size is associated with the offering of a comprehensive and specialized
curriculum, as contended by Lee and Smith (1995). It is therefore hypothesized that
schools with larger grade 8 enrollments are more likely to offer algebra, net of the effects

of school racial and social composition and size.

16

2.2.4 Sources of State-to-State Variation in the Availability of Eighth-Grade Algebra

Two possible sources of state variation in the availability of algebra are state
investment efforts in public education and poverty levels. As financial support is related
to the quality of resource inputs, such as the qualifications of the mathematic teachers and
how updated the instructional materials are, the amount being invested in the educational
systems is likely to have an impact on the distribution of scarce educational resources.
This is particularly important for schools in economically disadvantaged settings where
there are typically fewer educational resources and less qualified mathematics teachers
(Archer, 1991; Tate, 1994). Indeed, Tate posited that without fiscal equity for those
schools, the chances for success of implementing the curricular reforms recommended by

the well-known , _ '

 

' ,_ (National
Council of Teachers of Mathematics (NCTM), 1989), are slim. His views echo those of
the proponents of systemic reforms (e. g., O'Day & Smith 1993).

Given that state educational spending accounts for nearly half of total K-12 public
school revenues (National Center for Education Statistics, 1994), the present inquiry tests
the exploratory hypothesis that different levels of financial resources at the state level are
associated with different likelihoods of offering of algebra. How much a state spends on
education may also influence the availability of the course through its impacts on the
relationships between school and ethnicity composition and the likelihood of the course.
In other words, state expenditures may interact with the effects of student social and
ethnic composition, if any, in influencing a school's decision to include algebra in its

mathematics curriculum. The exploratory hypothesis aims at investigating the indirect

17
effects of expenditures on opportunities to learn. Intervening processes, such as the
possibility that increased expenditures may help prepare and recruit better qualified
teachers and thus may supply the instructional staff necessary for the offering and
teaching of advanced courses, are not tested here. Another exploratory hypothesis
pertaining to the poverty level of the state is that schools in states with higher poverty

rates, on average, are less likely to offer algebra.

2.2.5 Significance of the Substantive Inquiry

While built upon the existing research, the inquiry of this thesis differs from, and
thus, adds to the literature on learning opportunities in several ways. First, unlike most
studies, except the one by Raudenbush, Fotiu, et al. (1996), it adopts a multilevel
framework and incorporates the clustering effects of states. It explicitly models the state
memberships shared by schools, an unprecedented opportunity presented by the TSAP in
mathematics. The framework allows one to study interstate and interschool variation
appropriately (see Raudenbush, 1988) and how they can be explained by state
characteristics and policies.

Whereas both Raudenbush, Fotiu, et al.'s (1996) and the present study investigate
access to algebra, they adopt different approaches. The former focused on person-to-
person variation and investigated the probability of placement of students in schools
which offered algebra as a function of the students' background characteristics. The
present work surveys what school- and state-level factors may affect a public school's

decision to offer high school algebra. The study focuses on one of the best "school-

18
controlled predictors of student achievement" (Porter, 1994, p.427)--content of
instruction.

Another major divergence from the previous research on course availability such
as Oakes' (1990) studies is the use of more comprehensive models in the present study.
The models assessed contain various school compositional, setting, and structural
predictor variables. It allows one to investigate the influences of individual variables net
those of the others. For instance, one may be interested in assessing the ethnicity bias
while holding constant the influences of the structural characteristic of a school. The
multilevel and better specified models in the study allow a better understanding of the
school decision-making process in delivering learning opportunities to their students, and
how the process may be related to state financial investment in education.

Furthermore, as state revenue makes up a significant portion of the total school
revenue, states can exert influences on how those monies are transformed into
educational opportunities. One way to do so, as argued by Monk (1994), is through the
explicit statements of the basis of the financial entitlement. A state, for instance, might
enumerate the sort of educational opportunities, or the "intended curriculum" (Pelgrum et
al., 1995), a district should provide and collect information on how well the district meets
the specified details. It is believed that the measure of course availability, operationalized
as the likelihood of the offering of algebra, can supply some of the relevant information
and act as a production and curriculum indicator (Oakes, 1989; Pelgrurn et al., 1995;

Porter, 1991).

19

2.3 The Methodological Work
2.3.1 Motivations

The substantive inquiry has motivated the development and evaluation of a fully
Bayesian analytical framework which accommodates l) the binary nature of the outcome
variable of the offering of the algebra course, 2) the hierarchical or nested design of the
TSAP in mathematics--schools nested within states and territories, and 3) the need to
incorporate the uncertainty about the variances at the state level. By representing the
state-level variance as the investigator's uncertainty about the processes that produce it,
the hierarchical framework enables us to study state-level heterogeneity, the relationships
between state-level predictors and outcomes (Raudenbush et al. 1995), and not to treat the
states as fixed strata. To illustrate the framework and to establish notation, I use an over-
simplified model for the availability of high-school algebra, with two predictor variables:
percentages of minorities (Hispanic and African American students) per school and state
educational expenditures per student. The fully Bayesian estimation approach, together

with the ML and AML\PQL strategies, is then outlined.

2.3.2 An Illustrative Example

Let y,j be an indicator outcome variable which takes on a value of 1 if high-school
algebra is offered, and 0 otherwise for school i in state j, where i = 1 to nj and j = 1 to J.
The goal of the analysis is to explore the correlates of the probability, pi}, that a particular
school offers eighth-grade algebra. At the school level, it is of interest to determine if

higher percentage of minorities in a school HiMin,j is associated with less likelihood of

20
the algebra course being offered, or if the learning opportunities for the gatekeeping
course to advanced high school mathematics are racially stratified (Gamoran, 1987).
HiMin,j is an indicator variable and is coded 1 for schools with more than 50% Hispanic
and African American students, and 0 otherwise. A discussion of the coding scheme of
this variable is given in Chapter 4. To constrain the probability to lie within a [0,1]
interval, pg. is transformed to flu using the logit link function. The transformed outcome is

now the log-odds of the offering of algebra. The model is expressed as:

Pr . .
log(—1—j—) = “U = Bo; + [lleszij , (2.1)

_q

or in vector form,

[5 .
“U = (i HiMin)U[B°’]. (2.2)
1]

Each coefficient defined in the school-level equation is modeled as an outcome at the
state level. The intercept of the school-level model, [50]., which is the average log-odds of
the offering of algebra for schools which are located in state j and have a relatively low
percentage of Hispanics and African American students, is estimated to see whether it is
related to fiscal equity. The intercept is modeled as a function of state educational

expenditure per student StateExpj‘ plus a random state effect according to the model

00. = 700 + YOIStateExp. '1' ac. , 2.3
1 1 1

where uoj ~ N( 0, r). In addition, whether or not educational expenditure may be related to

21
possible racial stratification in learning opportunities, as represented by flu, is assessed.

The following model expresses the relationship:

[31}. = 710 + ynStateExpj . (2.4)

In matrix notation, the combined state-level model can be expressed as

 

 

 

 

Y 00
[3 0/ l StateExpj 0 0 y or 1
= + [um], (2.5)
B I]. 0 0 1 StateExpj y 10 0
_Y 11

and the combined school- and state-level model can be stated as

 

 

11,-. -
J . ,
Yoo
701
(1 St‘ateExpj HiMinU. HiMinU*StateExpj) + (2.6)
710
5Y1].

[1] [“01] ’

which can be formulated as a more general two-level mixed model with a logit-normal

hierarchy (Searle, 1991),
71,-, = ngY + ngu, , (2.7)
where

x” is a p x 1 vector of predictors associated with the log-odds of the algebra course

being offered; for model (2.6), p = 4;

22
y is a p x 1 vector of regression coefficients for fixed effects;

z”, a subset of x.

,j, is a r x 1 vector of predictors associated with r random effect(s);

for model (2.6), r = 1; and

u] is a r x 1 vector of random effects and it is normally distributed with mean of
zero and variance of T. For (2.6), the variance is a scalar 1:.

The parameters of interest in model (2.7) are 7, u}, and T. The marginal likelihood

for the parameters 7 and T is proportional to

J I!

Km) « H II my, I u,)g(u,>du,. (2.3)

1:1 i=1

where

_ l-yt
1 y’ exp n” I
fty,-|u,) = _ _ . (2.9)
I 1 +exp "'1 l +exp "'1

 

 

and
_ 1 -
g(u,) « I T I "zexpl-E-uj'T 'u,]du,. (2.10)

Model (2.7) is a special case of the generalized linear model (GLM) (McCullagh
& Nelder, 1989, Nelder & Wedderbum, 1972) with random effects. GLM has unified
models with outcomes having a variety of different scales. Some examples are the linear
regression models for approximately Gaussian data (e.g., SAT scores), logit models for
dichotomous data (e.g., offering of algebra), log-linear models for count data (e. g., days

of absence from school), and proportional hazards models for survival time data (e.g.,

23
years taken to attain a doctoral degree). The unification has brought with it a common
theoretical framework and estimation procedure (see McCullagh & Nelder, 1989). Model
(2.7) is an extension of GLM with an incorporation of random terms uj to handle
clustered data. The conditional distribution of y”. given uj assumes an exponential family

distribution of the form

fiy,lu,> = exp<Ly,,e,., - c201,.) + bum/q») . (2.11)

where

6 ,1. is the natural parameter,

(p is the dispersion parameter, and

a(-) and b(-) are specific functions corresponding to the type of exponential
family.

The specific parameters of the conditional distribution are:

9,, = 11,, ,
_ (2.12)
a(e,,.) =log(1 + exp “"1.
and
126,.) = o .
(2.13)
(9 =1 .

Modeling the natural parameter, 0 directly as in (2.12) makes the link, 1],], a canonical

if,

one (McCullagh and Nelder, 1989). The conditional moments p,j =E(y,.j|u,) = a'(0 ,1.) and v,j

24

= var (yyluj)= a"(6,.j)<p are:

 

a’(0ij) -— _ =1)” ,
l + exp 11,,-
-. (2.14)
, _ ex " _
a'(e,.,.><p - p _ —p,.,.(1 -p,.,.).
(1 +exp "0)2

Three main estimation approaches for the model are 1) the ML approach (e. g.,
Anderson & Aitkin, 1985; Fahrmeir & Tutz, 1994; Gibbons & Hedeker, 1994; Hedeker &
Gibbons, in press; Karim, 1991), 2) the AML and the closely related PQL approach (e.g.,
Breslow & Clayton, 1993; Goldstein, 1991; McGilchrist, 1994; Schall, 1991; Stiratelli,
Laird, & Ward, 1984; Raudenbush, 1993; Wolfinger & O'Connell, 1993), and 3) the

Bayes approach (e.g., Dellaportas & Smith, 1993; Zeger & Karim, 1991).

2.3.3 The Maximum Likelihood Approach
The ML approach maximizes the marginal likelihood (2.8) or its logarithm with
respect to the fixed effects 7 and the variance components T. Let y} = (12”,...ynflf and (I) =

(y, T), the approach maximizes

J
log to) = 210g 10,14» . (2.15)
j=1

However, the marginal likelihood (2.8) does not have an analytic solution. Thus,

numerical or Monte Carlo integration methods are needed for approximations. Two

25
maximization strategies, direct and indirect, are available for parameter estimation
(F ahrrneir & Tutz, 1994). Whereas the direct strategy evaluates the marginal likelihood
(2.8) and its partial derivatives via numerical or Monte Carlo methods in parameter
estimation, the indirect one does not.
Fahrrneir and Tutz (1994) provided the algorithmic details for the direct
maximization strategy. The procedure starts with a reparameterization of the random

effects uj in the conditional mean E(y]|uj) as given in (2.14):

u1 = Tmaj , (2.16)

where T“2 is the left Cholesky factor of T and a] is a standardized random vector with zero
mean and the identity matrix as covariance matrix. The two—level mixed model (2.7) now

becomes

_ r T 1/2
r)”. —x” y + z” T a], (2.17)

and the marginal log-likelihood now becomes ((7,9), where Q = vec(T'/2),

n

J 1
101.0)“ H II f<y,,la,)g(a,>da . (2.18)

j=l i=1
Specifying g and maximizing [(7,0) with respect to the various parameters yield
maximum likelihood estimates for y and Q, and subsequently T. The maximization can
be accomplished by an iterative procedure such as Newton-Raphson or Fisher scoring
method, and the evaluation of the integral in (2.18) can be carried out by the Gauss-

Hermite quadrature technique when g is normal. The quadrature technique approximates

26
an integral by summing a certain number of quadrature points (1 for each dimension of the
integration. If the number of quadrature points d is large enough, the approximation can
be made increasingly precise and the approach could yield maximum likelihood estimates
that have the properties of consistency, asymptotic normality, and efficiency. Using
consistent estimators 1 and i, empirical Bayesian point estimator of a] can be computed,
which allows the recovery of estimates of u]. The algorithm allows likelihood-ratio tests
for comparisons of nested models differing in variance and covariance components in
addition to fixed effects.

A program called MIXOR (Hedeker and Gibbons, in press), which employs a
direct maximization strategy called marginal maximum likelihood (ML) for estimating
random-effects logit regression models, is available. The algorithm is very similar to the
one of Fahrmeir and Tutz (1994). The models are basically the same as model (2.7). In
their derivation, the models differ from (2.7) in their introduction of an unobservable or
latent continuous variable which "trigger[s] the discrete response" (Cramer, 1991, p.11).
A threshold value is assumed and a discrete response occurs if the value of the latent
continuous variable exceeds the threshold value. An example of a latent continuous
variable is a household's desire to own a car and there is a threshold value or certain
critical level of desire beyond which ownership may result (Crarner, 1991).

Karim (1991) developed an indirect maximization strategy to estimate the various
parameters using a Monte Carlo implementation of the EM algorithm (MCEM) (Wei &
Tanner, 1990, Dempster et al., 1977). In general, the EM algorithm is an iterative

procedure for finding maximum likelihood estimates in the presence of unobserved or

27
missing data in probability models. The idea is to augment the observed data with latent
data in order to replace one complicated maximization by an iterative series of simple
maxirnizations (Tanner, 1990). Let y = (y,, ..., yj)T and u = (u ,, ..., u,)T, for model (2.7), y
is the observed or incomplete data, u is the latent data , and (u, y) is considered to be the
complete data. The complete data log-likelihood is given by log flu, y|¢) and the
conditional predictive distribution of the unobserved or missing data is fluly,¢("), where
4)") is a current guess of the parameters. The iterative series of the simplified
maxirnizations consists of two steps: the E-step (for expectation) and the M-step (for
maximization). In the E-step the expected value of the complete data log-likelihood with

respect to the conditional predictive distribution is determined, i.e.,

Q(¢ 14>“) = ang minnow» , (2.19)

which can be obtained by evaluating

Q(¢I¢"’) = flog flurlM/(ulmmwu . (2.20)

In the M-step, Q((|)|¢( i)) is maximized by finding the solutions to

fian‘") = 0 . (2.21)

and d)” 1) is thus determined. This completes one cycle and the iterative series continues

until convergence. Iterations between E- and M-steps lead ultimately to the maximum

28

likelihood values of the parameters under mild regularity conditions (Dempster et al.,
1977; Wu, 1983). The algorithm does not yield any variances and covariances of the
estimators, but they can be obtained from the Hessian of the score functions of the
various parameters.

For model (2.7), however, no closed form exists for the E-step in (2.20). Karim
(1991) adopted a Monte Carlo implementation of the E-step (Wei & Tanner, 1990) as a
result. The general scheme of Monte Carlo integration involves generation of random
variates, which adds another step to the EM iterative series. The modified EM series
begins with the generation of a sample of latent data, u”), ..., am), where m = l to M from
the current approximation to the conditional predictive distribution flu [y ,d)‘ '7). The
expected value of the complete data log-likelihood is then obtained as a mixture of the

complete data log-likelihood over the generated latent data, i.e.,

M
Q(,.,,(<i>.¢“’) . 11?: log lbw ("”1 «1) . (2.22)
m=l

Then the M-step maximizes Q(,,1)(¢|¢( ") and the new maximizer is used to update the
conditional predictive distribution.

Karim (1991) used an indirect sampling method called importance sampling
(Ripley, 1981) to generate latent data from flu ly ,(b‘ '7). He approximated the posterior
with a Gaussian distribution with mean and variance equal to the posterior mode and the
posterior curvatures of the conditional predictive distribution. Then random variates were

sampled from the approximated Gaussian distribution. To adjust for the discrepancy

29
between the true posterior and the approximating distribution, each simulated draw was
assigned a weight. Let g(-) be the Gaussian function, f (-) be the conditional predictive
function, and let g(u‘) be a draw from g(u), the adjustment weight was computed as
flu‘)/g(u‘).

Karim (1991) noted that with MCEM, the estimates fluctuate around the true
maximum even after convergence. He recommended various tests including visual
inspections of the traces of the estimates and assessment of variability in consecutive
iterations. He implemented a simulation study and showed that the algorithm yielded

reasonable inferences. No software is publicly available for this algorithm.

2.3.4 The Approximate Maximum Likelihood Approach\Penalized Quasi-likelihood
Approach

Another main approach to parameter estimation for model (2.7) is the AML (e. g.,
Goldstein, 1991; McGilchrist, 1994; Schall, 1991; Stiratelli et al., 1984; Wolfinger &
O'Connell, 1993 ), or the closely related PQL approach (e.g., Breslow & Clayton, 1993;
Raudenbush, 1993). The essence of the approach is to derive an approximation to the
marginal likelihood (2.8) and its partial derivatives, which allows repeated application of
normal theory (Kuk, 1995). The approximation can be achieved by the linearization of the
model (e. g., Goldstein, 1991), or by the use of penalized quasi-likelihood (e. g., Breslow
& Clayton, 1993).

The linearization can be implemented by a Taylor series expansion of p,j in (2.14)

around the 1‘" current guess of ‘y = y") and u] = uj ( '1. p,j is approximated in the region of a

30
point in n ,1"). The procedure, which can be implemented by the Newton-Raphson
algorithm, yields a set of linearized dependent variables y,j *'s and weights wy's for each
data point (McCullagh & Nelder, 1989; Raudenbush, 1993; Schall, 1991), where
(r)

. 6p. .
_ (l) ' - ()
ygj* — “I! + ( :0) ‘(yq‘ — pi], )’ (2.23)

1‘1
with

= 10') Ta)
””111 ”aui’

(i) _ ‘11") -1
1" _ (l + 6X 0) 9
p’ p (2.24)

6p (1')
if _ (i) _ (1') _ (i)
— — wrj _ pi} (1 pi} ) 9

Lindstrom and Bates (1990) pointed out that this Taylor series expansion can be
conceptualized as a step for creating psuedo-data.

Alternatively, the approximation can be achieved by replacing f(y,j In!) in (2.9) by
the exponential of the quasi-likelihood that denotes the deviance measure of fit (Breslow
& Clayton, 1993; McCullagh & Nelder, 1989), which depends only on the first two
moments of the model. Laplace’s method for integral approximation is then applied to the
marginal likelihood with the substituted quasi-likelihood. The approximation then allows
the use of normal theory to estimate the parameters.

Iterative algorithms such as generalized least squares (e.g., Goldstein, 1991),

Fisher scoring (e.g., Breslow & Clayton, 1993), and EM-type algorithm (e. g.,

31
Raudenbush, 1993) can be applied to the approximated likelihood to obtain approximate
maximum likelihood estimates. The EM-type algorithm is related to MCEM in that it is
obtained when M in (2.22) is equal to one (Wei & Tanner, 1990). The iterative algorithm
consists of forming and maximizing the function log f(u]=a},y*| (1)) over (I), where a, is
the posterior mode of f(u,[y,¢"’), to obtain (NM) and update a, This EM-type algorithm
which relies on posterior modal analysis, coupled with the Newton-Raphson algorithm
used for creating psuedo-data, are employed by Stiratelli et al. (1984) and Raudenbush
(1993) to estimate model (2.7).

The Taylor expansion for the approximation can be either about u] = 11]") or uj = 0.
The former type of model corresponds to the PQL model and the latter marginal quasi-
likelihood (MQL) model in Breslow and Clayton's (1993) classification of models with
and without the incorporation of the random effects terms zfu, in the linear predictor.
Goldstein and Rabash (1996) showed that the PQL estimates with a second order Taylor
expansion are less biased than those of the first order MQL.

An option to estimate restricted approximate maximum likelihood estimates for
degree-of-freedom adjustment (Breslow & Clayton, 1993) is also available in this
approach. In cases where the outcomes are normal, such adjustment can alleviate the
downward bias of the maximum likelihood estimates of T. To implement the estimation,
Stiratelli et al. (1984) and Raudenbush (1993) handled y as random effects, whose
variances tend to infinity, in the EM-type routine in their algorithms.

Several software packages for the approximate maximum and penalized quasi-

likelihood estimation are available. HLM2/3 (Bryk, Raudenbush, & Congdon, 1996),

32
ML3 (Prosser, Rasbash, & Goldstein, 1991), and VARCL (Longford, 1988) are some
examples. Several simulation studies were done to assess the estimators of this approach.
Rodriguez and Goldman (1995) and Breslow and Clayton (1993) found that the MQL
estimators display considerable bias, especially with reference to T and when the
binomial denominators are small. Kuk (1995) and Goldstein and Rabash (1996) applied
iterative boostrap bias correction and Kuk showed that the procedure yields

asymptotically consistent and unbiased parameter estimates.

2.3.5 The Fully Bayesian Approach

The third major approach to parameter estimation is the fully Bayesian approach.
Unlike the previous two approaches, which estimate T and/or y as fixed parameters,
Bayesian inference treats all parameters as random quantities with prior distributions.
Prior distributions have two basic interpretations, the population and the state of
knowledge (Gelman, Carlin, Stern & Rubin, 1995). In the first interpretation, "the prior
distribution represents a population of possible parameter values, from which the
[parameter] of current interest has been drawn" (Gelman et al., 1995). In the second
interpretation, which is more subjective, "the guiding principle is that we must express
our knowledge (and uncertainty) about the [parameter] as if its value could be thought
of as a random realization from the prior distribution" (Gelman et al., 1995). The main
inferential goal of Bayesian analysis can be considered to be updating the prior
knowledge of the random quantities in light of the observations. The analysis requires

computation of the joint posterior distribution, p(y,T,u[y), which is proportional to the

33

product of the likelihood function for the parameters and their prior distributions:

J";

p(r.T.u M = k "11 111201,)!Y.u,)p(u,lT)p(Y.T). (2-25)
j=li=l

where

J "j

k = II p0,,~lY.u,)p(u,lT)p(r T)6u,616T . (226)
1:1 i=1

In the joint distribution (2.25), the joint prior for T and y is p(T,y), and the prior for u,- is

the product of the J distributions p(uj |T). Through the likelihood function for the

parameters T, y, and u, which is

J";

I<y.T.u) = II Hp<y,lv.u,)p(u,lT) . (2.27)

j=li=l

the prior distributions are modified and updated in Bayesian posterior analysis (Box &
Tiao, 1973). When inferences on single parameters are required, the joint posterior
distribution is to be integrated with respect to the other or what is called nuisance

parameters (Gelfand, Hills, Racine-Poon & Smith, 1991) to obtain the marginal

posteriorspw 11'). NH» and p(u|y).

m Iy)=ffp(v.T.u Iy)auaT . (2.2s)

p(T Iy) =ffp(r.T,u 0061161! . (2.29)

34

and

Ma ly1=ffp(Y.T.u|y)816T . (2.30)

Characteristics of the joint and marginal posterior characteristics such as means,
modes, and variances are legitimate for Bayesian inference. The integrals above do not
have analytic solutions. Thus Bayesian inference requires numerical evaluation. The task
may become formidable when the model increases in complexity. An alternative to avoid
the difficulty is to employ the Gibbs sampling technique (Gemen & Gemen, 1984;
Gelfand & Smith, 1990) to simulate the posteriors, therefore leading to draws of the
parameters. The next chapter describes the application of the Gibbs sampler to random
effects generalized linear models.

A simulation study done by Zeger and Karim (1991) showed that the algorithm
yielded reasonable inferences in finite sample cases in which the number of clusters was
large relative to the number of observations within each cluster (e.g., 100 clusters, each
with 7 observations). When compared to AML or PQL, the algorithm yielded less biased
estimates. No computer program for implementing the algorithm developed and tested by
Zeger and Karim is publicly available.

How the three estimation approaches compare and contrast with one another in
their inferential goals and strategies are summarized in Table 2.1. In the AML/PQL and
ML approaches, the major objective is to maximize or find the maximizer of the
likelihood functions and the posterior distributions. In the Bayesian approach, the main

goal is to seek the marginal posteriors of the various parameters (Tanner, 1993). Unlike

35

ML and Bayes, data affect inference via the quasi-likelihood fimction in AML/PQL.
Finally, T is treated as random only in the Bayesian approach. The advantages offered by

the distinguishing features are given in the next section.

36

Table 2.1: Defining Features for the Three Major Approaches

 

 

 

 

 

 

 

 

Features/ AML\PQL ML Bayes

Approach

Objectives Approximate Maximize the Find the marginal
likelihood (2.8) and marginal posterior distributions
its partial likelihood (2.8) of the parameters:
derivatives and with respect to p(y | y), p(Tl y) and
maximize the y and T, then p(uj | y)
approximated predict u], given
likelihood with y and i
respect to y and T,
then predict u},
given 9 and i

Data affect No Yes Yes

inference via

the likelihood

All No No Yes

parameters

are treated as

random

quantities

 

 

37

2.3.6 Appropriateness of the Fully Bayesian approach

Among the three main approaches, the Bayesian approach is deemed more
appropriate for the inquiry studying the distribution of learning opportunities for the
following reasons. First, relative to AML\PQL, Bayes estimates of the level-2 variances
are less biased (Breslow & Clayton, 1993; Rodriguez & Goldman, 1995). This is because
in the Bayesian approach, the data affect inference via the likelihood instead of the
approximate or quasi-likelihood. Second, relative to ML and AML\PQL, Bayes
inferences about any parameter firlly take into account uncertainty about other parameters
in the same model. In the AML\PQL procedures for restricted approximate maximum
likelihood estimation, for instance, T is treated as fixed at its point estimates. Estimates of
all regression coefficients and their standard errors are then conditioned on those point
estimates. This poses two problems. First, Zeger and Karim (1991) pointed out that the
estimates of the fixed effects 7 and the variance T are asymptotically correlated and
inferences about the fixed effects have to incorporate the uncertainty in the estimate of T.
In addition, as there are only 42 states in the data, the variance and co-variance
components are likely to be estimated with moderate or poor precision. By treating T as
random in the fully Bayesian approach, uncertainty arising from the state-to-state
heterogeneity can be incorporated into the estimates of the regression coefficients. The
incorporation of uncertainty is particularly important with regard to inferences about
relationships between state-level predictors and outcomes, which is of great importance in
this study. Lastly, the Bayesian estimation provides researchers with richer inferential

information by outputting the entire posterior distributions of estimates, rather than point

38
and interval estimates alone. The Bayesian estimation approach is useful here, given the
modest number of states and the posterior distribution of the level-2 variance is likely to

be skewed. Inferences based on the variance of the estimates alone can be misleading.

2.3.7 Usefulness of the Bayesian Approach to Policy Research

The approach has much to offer to policy research that seeks to understand how
various policies and factors at the macro (e.g., nations, states, districts) and meso levels
(e. g., schools and classrooms) of the educational system(s) operate and interact with one
another, and whose design has a relatively small number of self-selected macro units. For
instance, it can be applied to study how the probability of employment of adults is related
to one's educational attainment, social and ethnic background, literacy level, as well as
state policies on continued education and retraining programs, using the 1992 National
Adult Literacy Study (Raudenbush et al., 1996). It can be applied to panel data studying
trends as well. An example, which will be an extension of the inquiry of this thesis, is to
utilize the various waves of the TSAP in mathematics data (1990, 1992, 1996) to study
the trends of the likelihood of offering algebra over time as a function of state reform

initiatives.

CHAPTER 3
BAYESIAN ESTIMATION OF GENERALIZED LINEAR MODELS
WITH RANDOM EFFECTS VIA THE GIBBS SAMPLER

3.1 Introduction

This chapter describes the Gibbs sampling algorithm that Karim (1991) and Zeger
and Karim (1991) developed for Bayesian estimation in random effects generalized linear
models. It first explains the logic of the algorithm. It then describes how to implement the
technique to obtain Bayesian inferences on y, T, and a, which involves the use of an
indirect sampling method called rejection sampling (Ripley, 1987) to generate variates
from non-standard distributions. In addition, how the convergence of the Gibbs chains is
assessed using Geweke (1992) and Raftery and Lewis' (1992) approaches is illustrated.
The final section reports the results of a simulation study, which documents the
performance of the SAS\IML code written to implement the Gibbs sampler and how well
the algorithm works in analyses where the number of clusters, J, is less than the number
of observations per cluster, nj. The results are compared to those of Zeger and Karim,
who established the validity of the algorithm in applications where J is large relative to

the nj's.

3.2 Logic of the Gibbs Sampler
Built on the work of Metropolis, Rosenbluth, Rosenbluth, and Teller (1953) and

Hastings (1970), Gemen and Gemen (1984) employed Gibbs sampling in image

39

40
restoration. Later Gelfand and Smith (1990) and Gelfand et al. (1990) introduced this
technique to the statistical community as a tool for fitting statistical models. Casella and
George (1992) defined Gibbs sampling as "a technique for generating random variables
from a distribution indirectly, without having to calculate the density" (p. 167). Its
success lies in its ability to reduce "the problem of dealing simultaneously with a large
number of intricately related unknown parameters into a much simpler problem of
dealing with one unknown quantity at a time, sampling each from its fill] conditional
distribution" (Gelman et al., 1995 p. 39). A full conditional distribution of a parameter is
defined as its distribution conditional on the data and on the current values of all the other
parameters in the model (Gilks, Best, & Tan, 1995). To obtain Bayesian inference in
random effects generalized linear models, the technique reduces the complex tasks of
computing the joint posterior distribution p(y,T,u|y) and the marginal posteriors p(y | y),
p(TI y) and p(u | y) to a relatively straightforward task of sampling from the three full
conditionals: p(y |T,u, y), p(Tl y,u, y) and p(uly,T, y). The strategy as well as the
elegance of the technique can be revealed by re-expressing the joint posterior distribution
as consisting of two components, a full conditional and a joint marginal distribution, in

three different ways:

p(Y.T.uly) =p(r|T.u.y)p(T.uly) , (3-1)

p(Y.T.u|y) = p(Tlv.u.v)p(v.uly). (3.2)

41

and

p(Y,T.u|y) =p(u|Y.T.v)p(v.le) . (3.3)

It is difficult to compute the joint marginal distributions above. For instance, the posterior

distribution p(y,T| y) is given by

J "j

H II}D(V,,-|1.14,).v(u,|T)1D(1.T)r3uj
1:1 i=1

 

p(r .T ly) = - (3.4)

J "j

H 11190,)!Y.u,)p(u,|T)p(v.T)au,81/8T
j=l i=1

However, as will be seen in the next sections, it is relatively easy to simulate from each of
the three full conditionals above. The Gibbs sampler exploits this advantage. It employs
the full conditional of each parameter to construct and drive a Markov chain which has,
upon convergence, the joint distribution as its stationary or invariant distribution (Besag,
Green, Higdon, & Mengersen, 1995). A property of Markov chain exploited here is that
the probability of an event is conditionally dependent on a previous state. Let k = 0, 1,...,K
denote the k‘h iteration in the Markov Chain simulation scheme. The iterative simulation
scheme begins with some starting values y‘o’, T10), and "(0). Given T10) and “(0) draw y“)
from p(le‘o),u (my), next draw T“) from p(Tl y“),u°’, y), and then conclude an iteration
cycle by drawing u“) from p(uly‘”,T‘”,y). Gemen and Gemen showed that after the
sampler has been nm for enough iterations to achieve convergence, 7“), T1"), and u“) can
be regarded as random variates from p(y,T,u|y), regardless of the starting values chosen.

The joint distribution or any lower dimensional marginal distribution can be

42
approximated by the empirical distribution of r subsequent successive draws obtained
from the Gibbs chain after convergence, where r = l to R is chosen to provide sufficient
precision to the empirical distributions of interest (Karim, 1991). If a conditional
distribution has a closed form, such as T in random effects generalized linear models, the
empirical distribution can also be estimated by

R
p(T|y) = %EP(TIYM»UIM) . (3.6)

r=1

The next section describes how the Gibbs sampler is applied to obtaining

Bayesian inferences on the various parameters of interest.

3.3 Implementation of the Gibbs Sampler
The implementation of the Gibbs sampler requires first the derivation of each of

the conditional distributions from the joint posterior distribution p(y,T,uLy):

 

J "1 “'1: 1 ”’11
1 ’1: ex ’
p(Y.T.u|y) 1* 1111 _ p _
1:“:11+exp n.) l+exp "‘1 (3_7)

_ 1 -
I T I meXpl'EufT '11,] °p(Y.T) -

The strategy is to pick out the term(s) in the joint density (3.7) which involve the relevant

parameter (Gilks, 1996).

43
3.3.1 Full Conditional Distribution of y-p(y |T,u, y)
Assuming the priors p(y) or constant and p(T) or constant, and that the two priors

are independent, i.e.,

p(r.T) =p(Y)p(T) . (3.8)

p(y |T,u, y) is independent of T. The conditional distribution can thus be stated as p(y | u,
y). Given the values u“) for u at iteration k, with reference to the joint density (3.7),

p(y | u“), y) is proportional to the likelihood function

J "j
(k) a
p(vlu .v) ENE. 11‘, + m»,
J 1 +exp ”7 z‘“
r (k) l_y!/ (3'9)

1'
“(qu + 2,)“, )

1 ”1

 

exp

 

T T (k)
+
l + exp-0“” 2".” )

As no standard algorithm exists for generating random variates from the non-standard
conditional distribution (3.9), Zeger and Karim (1991) adopt the rejection\acceptance
sampling algorithm (Ripley, 1987). To illustrate the logic of the technique, the procedure
for generating variates having a non-standard probability density function (p.d.f.)f(6) and
a cumulative distribution function (c.d.f.) F (0) is outlined here.

To begin with, a different p.d.fg(0) known as an envelope function with c.d.f.
0(6), which resembles f(B) and is easy to sample from, is chosen. Then random variates
are generated from g(0). If the random variates drawn pass an acceptance\rej ection test,
which will soon be illustrated, the variates will be accepted as belonging to 1(0). The

algorithm requires that the envelope function, g(0), dominate f(0). To ensure its coverage

44
over f(B), g(0) is always multiplied by a constant c. The more closely g(0) imitates f(B),
the lower the rejection rate will be. Figure 3.1, taken from Gelman et al. (1995, p.304),
shows how cg(0) enve10pesf(0). The vertical line indicates that a random variate 0, is

sampled from g(0), with a realization 00.

45

Figure 3.] Illustration of Rejection Sampling (Taken from Gelman et al., 1995, p. 304 )

 

08(9)

f(e)

46
The ratio, r(0), of the height of the lower curve f(Bo) to the height of the upper

curve cg(60) gives the probability that 60 will behave according to 1(6). To test if 00 is to
be accepted, a number U is drawn at random from the unit interval [0,1]. If U 3 1(6), 00 is
accepted. If U > r(0), 00 is rejected. If the random variate is rejected, another draw from
g(0) and the unit distribution will be made, followed by another acceptance/rejection test.
The validity of the algorithm lies in the equivalence of 6(00) conditional on U s r(0) to
F (00), as shown by Dagpunar (1988).

The envelope function chosen by Zeger and Karim to cover the full conditional
distribution (3.9) is the Gaussian density N(?"",2- V,""), where W maximizes p(y | u“), y)
and V7“) is the inverse Fisher information at the k‘" iteration of the Gibbs sampler. Given
the values of u“), the task of estimating 9“) and V7“) is simplified because 7.), Tuf") has
become a vector of offsets, or explanatory variables with known coefficients (Gelman et
al., 1995). Model (2.7) therefore reduces to a fixed effects generalized linear model,
which can be estimated via iteratively weighted least squares with y,""* being the
linearized dependent variable and w”. being the weight. How they can computed is given
in the equations (2.23) and (2.24). Let yj’w‘): (y ,1. *‘k’ *. ..., ynfl’"), X]: (ijT, , wa), Zj=

(zUT, , znfiT), W11") = diag{p,.j("’(1-p,j“))}, then at the klh iteration of the Gibbs chain,

.1 J
A0) = T (k) -1 T (k) 00'
7 (2X1 W1 X1 (2X1 W1 y, )’
j=I 1:1
(3.10)

J
V9 (1):; X] W] X!) .

47
To ensure that the envelope fimction covers the full conditional (3.9), Zeger and Karim
(1991) inflate the variance K“) by a factor of 2 and multiply the envelope function with a
constant c, which matches the mode of the envelope function and the full conditional
(3.9). Let the full conditional (3.9) be denoted by fly) and the envelope function
N(?"",2°V,"") by g(y), c is then computed as the ratio fl?"")/g(?“)). Using a Cholesky
factorization of V,"", a random variate y* can be generated. Then a rejection\acceptance
test involves evaluating the full conditional (3.9) and the envelope function at y“ by
computing the ratio f(y* )/(c-g(y*)) is implemented. The decision rule is that if the ratio is
greater than or equal to a random variate U generated from the uniform distribution [0,1],
y* is accepted as a random variate belonging to the full conditional (3.9) with a
probability equal to the ratio and 7“”) = y*. Otherwise, draw another variate from the

envelope function and subject it to the rej ection\acceptance test again.

3.3.2 Full Conditional Distribution of T--p(T| y,u, y)

Assuming uj's are independent normal with mean 0 and variance T and a uniform
prior for T, i.e., p(T) °< constant, the full conditional distribution is independent of y and
y. It thus can be expressed as p(Tl um), which is proportional to

J
p(T lam) e ,E,‘ T l ‘mexvl "%u,“'TT "ufk’i . (3.11)
Another choice of prior for p(T), which Zeger and Karim use, is the Jeffrey's prior
|T|*"+')’2. More recent work on Bayesian analysis for hierarchical models (e. g., Seltzer,

Wong, & Bryk, 1996), however, has found that the choice may cause a problem. The

48
problem can be illustrated with the case when the variance term, 1:, is a scalar, that is q =
l. The corresponding Jeffrey's prior is p(t) o< 1/1:, 1: > 0. When 1: is close to zero, 1/1:
becomes infinitely large and "a spike at 1: = 0" (Seltzer et al., 1996, p. 139) in the
posterior distribution for the parameter may occur. It will result in a standstill in the
updating scheme of the Gibbs sampler. I encountered this problem in the simulation runs
and the problem went away when I adopted the uniform prior, i.e., p(T) o< constant.
To obtain random draws from (3.11), it is convenient to work with the conditional
posterior distribution of '1”, given u, which is a Wishart distribution with parameters
J
8‘“ =2 1.100.110“ , (3.12)

[=1
and J-q-l degrees of freedom, where q is equal to the dimension of T. As one can use a
standard algorithm for sampling from the full conditional of '1" here, rejection sampling
is not needed. Tu”) can then be obtained by inverting the values of T' sampled.
Specifically, '1“ is equal to H“) TWIH‘” , where S“) " = H“) TH“) and W1 is a
standardardized Wishart variate with J-q-l degrees of freedom (Odell and Feiveson,

1966).

3.3.3 Full Conditional Distribution of u—p(u |y,T, y)
Given the values 7"" and T“), the full conditional distribution p(u | y“),T"",y) is

proportional to

49

 

J "j 1 y”
p(u I y”),T("),y) a, II II .
j=11=l 1 + exp- (quYfl) + zUTul)
“(xTY(k)+zTu) 1'?”
exp ” " ’ (3.13)

 

- Tour
xv 214)
1+exp(" ”’

_ I -1
(it) 1/2 __ (k)
| T | exp[ 2ulT u 1] .

(3.13) is a non-standard distribution, so Zeger and Karim (1991) again use the rejection
sampling algorithm. For each cluster j, the envelope function employed is N(0,‘*),2- Val‘“).
Using iterative weighted least squares, at the k‘h iteration of the Gibbs sampler, the mode

11,“) for cluster j

~(k) = T (k) + -1""
u (Z, W] Z] T

1 )"ZITWI(")(yj*("’ —le“") , (3.14)

and its curvature

_ (k) _
ij’ = (ZITWIU‘VI + T ‘ ) ' , (3.15)

can be estimated. To ensure coverage, the variance of the envelope function is inflated by
a factor of 2, and the function itself is multiplied by a constant C}, which matches the
modes of the full conditional and the envelope function for cluster j. For each cluster, a
random variate “1* is drawn from the envelope function. A rejection\acceptance test that
computes and compares the ratio j(u,"‘)/(cj-g(uj *)), where flu!) is the full conditional for
cluster j and g(uj) is the envelope function, to a random variate U generated from the
uniform distribution [0,1]. If U 2 j'(uj"‘)/(cj-g(uj *)), uj‘m’ = u,*, otherwise repeat the

sampling procedure.

50
3.3.4 Starting Values and Convergence Diagnostics

The starting values of the Gibbs sampler for uj's and T are set to be zeros and the
identity matrix respectively by Karim (1991). No starting values for y are needed as the
Gibbs sampler starts with sampling from the full conditional distribution of y. To assess
convergence at iteration k, Zeger and Karim (1991) employ Q-Q plots in which the last
specified number of draws, say m, are plotted against the preceding m draws. In the
present investigation, the convergence diagnostics developed by Geweke (1992) and
Raftery and Lewis (1992) and automated in the software Convergence Diagnosis and
Output Analysis Software for Gibbs Sampling Output (CODA) (Best et al., 1995;
Cowles, 1994) are used to monitor the convergence of the Gibbs chains. Besides the
availability of the computer code, the two diagnostics are chosen because 1) the methods
are theoretically motivated, 2) each of them requires one sampler run, and 3) the
conclusion drawn from each diagnostic could be used to compare with one another to
check for the reliability of the diagnostic results.

To briefly explain and illustrate the two approaches, I apply the diagnostics to a
chain of 800 simulated values of 1: generated to monitor the convergence of the chain.
The parameter value for ‘i.’ is 0.250.

The first diagnostic used is Geweke's (1992) convergence diagnostic, which is
based on a standard time-series method. For each variable, the chain is divided into two
"windows" containing different fractions of the chain, for example, the first 10% and the
last 50%. If the whole chain is stationary, the means of the values early and late in the

sequence should be similar. Geweke's approach involves computing a convergence

51
diagnostic Z, which is the difference between the 2 means in the two different windows
divided by the asymptotic standard error of their difference. As the length of the chain K
-' co, the sampling distribution of Z -+ N(0,1) if the chain has converged. Thus values of Z
which fall in the extreme tails of a standard normal distribution may imply that the chain
has not fully converged. Table 3.1 gives the CODA output for Geweke's convergence

diagnostic.

52
Table 3.1: CODA Output for Geweke Convergence Diagnostic

GEWEKE CONVERGENCE DIAGNOSTIC (Z-score):

 

Iterations used = 1:800
Thinning interval = 1
Sample size per chain = 800

Fraction in 1st window = 0.1
Fraction in 2nd window = 0.5

 

Variable Convergence Diagnostic Z

13 -0.594

 

 

 

 

 

53

As Table 3.1 shows, the number of iterates used is 800. The diagnostic uses every
one of the 800 iterates, thus the thinning interval is equal to 1. Sometimes instead of
using every iteration, one may "thin" the chain by choosing to save and use every t‘"
iteration, where t > 1. The option is useful in runs when there is high correlation between
consecutive iterations (Raftery & Lewis, 1996). Here I follow Zeger and Karim (1991) in
using every iteration and a single chain. The two "windows" contain the first 10% and the
last 50% of the iterates respectively. Given that the Z-score is only -0.594, no evidence
against convergence is established. According to Geweke (1992), a score in excess of 4
indicates problems. One can also use plots to illustrate the diagnostic. In essence, the
approach computes and plots different Z-scores for different segments of the chain. A
large portion of Z-scores falling outside the 95% confidence interval for a N(0,1)
distribution will suggest possible convergence failure. Figure 3.2 gives the plot of the
analysis and the 95% confidence interval. As shown in the figure, most of the scores are
within or cluster near the confidence interval (the two broken lines), therefore, no

convergence failure is suggested.

54

Figure 3.2 Geweke's Convergence Diagnostic

Z-score

 

 

 

 

m4
x x
x x
x x
xx x X X
x xx x
X x X
04 xx x xx x x x
x x x x
x x
....... x’--------------------x-T---x.---------'--"----.x-'----x'--"x---'--'--‘
xx
x
x
T I I I
O 200 400 600

First iteration in segment

55
Raftery and Lewis' (1992) approach, based on two-state Markov chain theory, as
well as standard sample size formulas involving binomial variance, is an alternative
convergence diagnostic. The method detects convergence as well as provides a way of
bounding the variance of the estimates of quantiles of functions of parameters. Table 3.2
gives a slightly edited CODA output of the analysis of the same chain of 800 simulated
values used earlier. Several letter symbols of the original output were changed to avoid

confusions with the ones already employed in the thesis.

56

Table 3.2: CODA Output for Raftery and Lewis Convergence Diagnostic

RAFTERY AND LEWIS CONVERGENCE DIAGNOSTIC:

 

Sample size per chain = 800

Quantile = 0.025
Accuracy = +/- 0.02
Probability = 0.9

Chain: c:/coda/diagnose

 

 

 

 

 

 

 

 

VARIABLE Thin Bum-in Total Lower bound Dependence
(t) (B) (N) (N min) factor (D)
1: 1 5 25 3 165 l .53

 

 

57

In the output, the sample size per chain indicates the number of iterates used,
which is again 800. This diagnostic specifies the convergence criteria to be estimating the
0.025 quantile of the posterior distribution of 1: with a precision of $0.02 with probability
0.9. The burn-in value indicates how many initial iterations can be discarded. The small
burn-in value given in Table 3.2, B = 5, suggests the chain has converged almost
immediately to t. In order to estimate the 2.5th percentile of the posterior distribution to
the specified accuracy and probability, i.e., $0.02 and 0.9 respectively, the result
recommends that an estimated 254 iterations (with a minimum of 165) out of the 800
iterations are needed. The last column reports the dependence factor, which is the ratio of
N to Nmin. Values of D much greater than 1.0 indicate high within-chain correlations and
probable convergence failure (Raftery and Lewis (1996) suggest that D > 5.0 often

indicates problems). The result here (D = 1.53) indicates convergence is achieved.

3.4 A Simulation Study and Results

A small simulation study was done to assess the accuracy of the SAS/IML code
and how well it would suit the present analysis. The structure of the data sets generated
basically followed that of the simulations carried out by Rodriguez and Goldman (1993)
and Yang (1995), in which 1: was a scalar. The model specifications of the simulated data
were similar to the over-simplified model (2.6) discussed earlier. The model had one
predictor at each level, denoted by llepred,.j and lleprealj for the level-1 and level-2

predictors. However, here only the level-1 intercept but not the regression coefficient was

58

modeled using Ilepreab. The model could be expressed as:

Yoo
"if = (l llepredj llepredij) Yo] + [11140)]. (3.16)

 

 

LY 10‘

Two kinds of data sets, each had J = 40 and nj = 100, were generated. The number of
clusters and observations per cluster was chosen to be similar to those of the TSAP in
mathematics, which had 42 states and 100 schools per state on average. In both data sets,
yoo = -.796, yo, = 7,0 = 1. They differed in their value of 1:. One was equal to 0.250 and
the other was equal to l. The two different values were chosen to examine how well the
algorithm works with moderate as well as large between-cluster variance. As the
algorithm is very computationally intensive, the number of replications was chosen to be
50, half of what Zeger and Karim (1991) ran. It is important to note that the goals of this
simulation, as stated in the beginning of this section, was different from Zeger and
Karim's, which was to establish the statistical properties of the Gibbs sampler algorithm.
The Gibbs sampler was run for 2,800 to 5,000 iterations until convergence. Table

3.3 lists the values of the true parameter 0, and reports the mean and standard deviation
of 6‘, the square root of the average variance of the estimates, and the coverage of the
nominal 90% Bayes interval from the 5th to 95th percentile based on 200 additional

simulated values generated after convergence. 200 iterates were used as it would facilitate

the comparison of the results of this simulation with those of Zeger and Karim (1991),

59
who used the same number of iterates to compute the posterior characteristics of the

various parameters.

Table 3.3: Results of Simulation Study

60

 

 

9 = Yoo to. Y... r
Easel
6 (true value) 0796 1.000 1.000 0.250
mean(9I )9 -0.763 1.042 0.991 0.287 (mode = 0.250)
std(9| y) 0.187 0.233 0.152 0.104
(mean(var((6| y))))'/’ 0.163 0.201 0.154 0.096
Coverage of nominal 84 82 94 88
90% interval
£25.12,
0 (true value) -0.796 1.000 1.000 1.000
mean(9|y) -0.778 1.042 1.016 1.211 (mode = 1.070)
std(9l y) 0.302 0.372 0.144 0.359
(mean(var((0| y))))'/’ 0.270 0.334 0.159 0.349
Coverage of nominal 86 84 96 82

90% interval

 

The results are similar to those of Zeger and Karim (1991). In their simulation,

they reported a slight bias in the intercept estimate while the other fixed effects were

approximately unbiased. Here all fixed effects are approximately unbiased. Zeger and

61
Karim reported that the posterior means for the random effects variances were positively
biased by 20%-3 0%. They attributed the bias to the long tail of the marginal distribution,
which was alleviated when posterior modes were used instead of means to indicate
central tendency. Here the positive bias ranges from 15% to 22% and the bias is alleviated
by using posterior modes. The coverage probabilities of the nominal 90% Bayes posterior
intervals ranged from 80% to 100% in Zeger and Karim's simulation study and they range
from 82% to 96% here. Given the number of trials was 50, the standard deviation of the
proportion of coverage is sqrt(.9(l-.9)/50) = .04. Thus two standard deviations below and
above the 90% coverage ranged from .82 to .98. The results obtained here, 82% to 96%,
gave another indication that the code and the algorithm were performing well.

Both simulations show the algorithm yields reasonable inferences, which suggests
that the algorithm works well in analyses where the number of clusters, J, is less than the
number of observations per cluster, nj as well and can be applied to analyze the TSAP in
mathematics data to study the distribution of learning opportunities. The next chapter
reports a use of the tested code to perform Bayesian analysis of models studying access to

opportunities to learn.

CHAPTER 4
BAYESIAN HIERARCHICAL ANALYSES OF
ACCESS TO EIGHTH-GRADE ALGEBRA

4.1 Introduction

In this chapter, the estimation approach and algorithm from Chapter 3 are
implemented to analyze the data collected under the 1992 TSAP in mathematics and state
background characteristics data made available by CCSSO. The purpose of the analysis is
to study the distribution of learning opportunities in advanced mathematics in Grade 8
among schools across 42 US. states. I first provide a description of the data sets and the
procedures used. Then the selection and the construction of relevant variables and
measures are presented in conjunction with the various research hypotheses. The final

sections describe and discuss the findings and their implications.

4.2 Data, Procedures, and Measures
4.2.1 Data Description

In TSAP, 44 states volunteered to participate. Within each state, on average, 100
public schools were selected at random. Within each school, approximately 30 eighth
graders were then sampled. More specifically, eligible schools were first stratified and
selected according to grade 8 enrollment, urbanicity, percentage of African and Hispanic
American students, and median household income (for a description of the sample design

and selection, see Mohadjer, Rust, Smith and Severynse, 1993). A systematic sample of

62

63
students was then drawn from each selected school. In the survey, the students were
administered a test to assess their mathematics proficiency. Personal and school
background information was solicited from the students, their teachers, and their school
principals or administrators.

The present investigation employed primarily the school response data on school
characteristics and policies collected from the principals or other administrators. After
listwise deletion of cases with missing values, complete data were available for 3,525
schools in 42 states. About 5% of schools in the total sample (3,725) and two territories
were removed as a result. The two territories, each with six schools, were excluded as
they had no data on the availability of algebra. A comparison of the descriptive statistics
of the total sample and the actual sample used in the analyses revealed little difference.
Another data set used in conjunction with TSAP was data on state profile statistics
compiled by CCSSO, for which there were no missing data (CCSSO, 1993). In the

analysis, the two data files were merged.

4.2.2 Procedures

Using the TSAP and state background data, I constructed measures of the central
concepts of this study for the selected schools and participating states in the sample.
Reliability analyses and the less computationally intensive penalized quasi-likelihood
estimation approach implemented via the EM algorithm (Raudenbush, 1993) were used in
scale construction.

With the log odds of the offering of high school algebra as the outcome, 1 first

64
formulated an unconditional model that included 1) a within-states model in which the
outcome for a given school in a given state was seen as varying around the mean of the
outcome for that state and 2) a between-states model in which the mean for the state on
that outcome was seen as varying randomly around the grand mean for the outcome. The
model estimated the unconditional variance in the outcome that lies between states.

To test hypotheses, I then expanded the model to include school- and state-level
predictors, which consisted of school composition, setting, eighth-grade enrollment, state
poverty level and educational expenditure. In the analysis, all coefficients that represented
the effects of school-level predictor variables were assumed not to randomly vary among
the 42 states. Systematic state-to-state variation of the racial and social composition
effects on the offering of algebra associated with different levels of public educational
investment was postulated. The model specification regarding between-state variation of
the effects of the various school-level predictors was guided by the principle of parsimony
and based on the reasoning that the effects on the outcome due to school size and
urbanicity are similar across the 42 states. And after controlling for the state educational
investment and poverty level, it is likely that there will be no significant unique ethnicity
and social composition effects associated with individual states. An exploratory run using
HLM2/ 3 (Bryk et al., 1996) found non—significant random variation in the ethnicity and
social composition effects, controlling for the two state-level predictors.

In addition, all predictors in the within- and between-state model were centered
around their means. As a result, the mean logits of offering algebra for individual states

were adjusted for the various predictors.

65
4.2.3 Measures

A central focus of this research is to describe and study the availability of high
school algebra in the 42 states and the extent to which the availability depends on various
demographic composition, setting, structural, and state economic and educational
financial factors. To measure the availability of the course, an indicator variable was
used. Schools received a score of 1 if the course was offered and a 0 otherwise. The mean
for the variable was .76, indicating that 76% of the 3,525 schools offered the course (the
standard deviation was .43). 846 schools in the sample did not include algebra in their
eighth-grade mathematics curriculums.

Two measures of the demographic composition were the minority enrollment and
the SES of a school. Schools in which Afiican and Hispanic American students made up
more than 50% of their enrollments were represented by an indicator variable and coded
as high minority enrollment schools. The variable was constructed from the continuous
variable of percentage of minority students in a school. The decision to adopt the dummy
coding scheme was based on an examination of various plots of predicted logits of the
probability of offering algebra against the midpoints of grouped percentages of minority
enrollment, as suggested by DeMaris (1992). In addition, an exploratory model was fit
with an alternative coding scheme for the independent percent minority variable.

Eleven groups of percentages (0, 1-10, 11-20, 21-30, 31-40, 41-50, 51-60, 61-70,
71-80, 81-90, and 91-100) were formed by breaking the range of the percentages of
minority students in schools. All but the no minority enrollment group, which was chosen

to be the reference group, were used to model the log odds of offering algebra using

66
HLM2\3 (Bryk et al., 1996). Thus the model had ten dummy predictor variables. After the
run, I plotted the estimated logits for all eleven categories of percentages against the
eleven midpoints of the groups. The estimated logits for all but the reference group were
computed by adding the regression coefficient of each group to the intercept, with the
intercept being the estimated logit for the no minority group. Figure 4.1 gives the plot.
The result shows that higher percentage of minority enrollment has a generally negative
relationship to the logits of offering algebra after the 40th to 50th percentile category. In
addition, I plotted the estimated logits against the logarithm and the logits of the
midpoints of the various groups. It was found that the two transformations of the
independent minority enrollment variable did not result in linearity in the logit of offering
algebra. To explore an alternative coding scheme, I ran a model with percentage of
minority enrollment and its quadratic term as predictors. The result of the run showed that
the quadratic term did not achieve a conventional level of significance, and no quadratic
relationship was suggested. Therefore the analysis used an indicator variable indicating
schools with high minority enrollment, which allowed a straightforward interpretation as

well.

67

Figure 4.1. Logits Plotted Against the Midpoints of the Categories of Percentages of
Minority Enrollment

0.85

. R,
0.80 r/ \M

0.75

.-____ M _—l

 

LOGITS

_T___--

0.70

__-___.- , _[__

0.50 -. 1 1 1 I _I
O 20 4O 50 80 100

PERCENTAGE OF MINORITY ENROLLMENT

 

68

The measure of the SES of schools was a 3-item scale. Two items in the scale
covered the home background of students of a school, and the third was the percentage of
students who were in the free lunch program. The home background items were based on
students' reports about the educational level of their parents and their access to reading
materials (receiving a newspaper regularly, an encyclopedia in the home, more than 25
books in the home, and receiving regular magazines). To create the SES scale, I first
averaged the responses of the students within each school to obtain the school means for
the two measures on home background. Z scores were then created for the two
aggregated home background measures and the percentages of students receiving free
lunches. Finally, the scale was constructed as the average of the three standardized scores.
The scale had a Cronbach's alpha of 0.76.

Grade 8 enrollment was measured by the actual number (in hundreds) of students
enrolled. Past research suggests that the positive relationship between school size and
program comprehensiveness is nonlinear (e.g., Monk & Haller, 1993). The relationship
weakens as the size of the school increases. The finding warrants a check of the tenability
of the assumption of linearity in the logit of the outcome. To proceed, I broke the range of
the enrollment variable into groups, following the approach of DeMaris (1992) and
Hosmer and Lemeshow (1987). The values of the variable ranged from 0.01 to 16.07.
Grouping based on quartiles of the distribution of the variable yielded four categories
with midpoints at 0.585, 1.615, 2.525, and 9.25. I then created three indicator variables to
represent three of the four categories and entered them as predictors to model the log odds

of offering algebra. To compute the estimated logits for each category, I added

69
successively the regression coefficient for each indicator variable to the intercept, which
was the estimated logit for the reference group. Finally, I plotted them against the
rrridpoints of the four categories. Figure 4.2 displays the plot of logits against the various
midpoints. It shows obvious departure from a straight line and reveals nonlinearity in the

relationship. As a result, a quadratic term was created to appropriately model the trend.

70

Figure 4.2: Plot of Logits Against Midpoints of Grouped Categories of Enrollment

LOGIT

 

 

O l l I I J
0 200 400 000 800 1000

GRADE 8 ENROLLMENT

71

The measure of the setting of the school was represented by three indicator
variables showing whether or not a school was located in an urban, suburban, or rural
setting. In the sample, 23% were urban schools, 53% were suburban schools, and 24%
were rural schools.

At the state level, two measures employed were the percent of children in poverty
and the state educational expenditure per pupil. CCSSO compiled the two measures for
each state for 1990, which was relatively close in time to 1992, the year the data on
algebra offering were gathered (CCSSO, 1993). Percent of children in poverty is defined
as "the percentage of related children under age 18 who live in families with incomes
below the US. poverty threshold" (The Annie E. Casey Foundation, 1994, p. 159). The
state educational expenditure measures how much money (in hundreds) states allocate for
each student, and indexes state efforts in funding public education. Percent of children in
poverty is negatively correlated with state educational expenditure per pupil (r = -.41).

Table 4.1 shows the means, standard deviations, ranges, and variable names for

each of the variables in this analysis.

Table 4.1: Descriptive Statistics

72

 

 

Variables Range Mean SD

School-level Data

(3,525 schools)

Offering 8th Grade 0 = no 0.76 0.43

Algebra for High School 1 = yes

Credits

Schools with Minority 0 = no 0.14 0.35

(Hispanic and African 1 = yes

Americans) in Excess of

50%

School SES (-3.42, 2.08) 0.00 0.82

Schools in Urban Setting 0 = no 0.23 0.43
1 = yes

Schools in Suburban 0 = no 0.53 0.50

Setting 1 = yes

Grade 8 Enrollment (0.01, 16.07) 2.17 1.39

(in hundreds)

t - v ta

(42 states)

Percent of Children in (7, 33.50) 17.72 5.92

Poverty

Educational Expenditure (25.47, 78.27) 45.22 12.35

per Pupil (in hundreds)

 

73

4.3 Results
4.3.1 Unconditional Model

Table 4.2 gives the posterior characteristics of the marginal distributions of the
parameters in the unconditional model. Listed are the means, the standard deviations of
the intercept or the grand mean log odds of offering algebra and between-state variance,
the 90% credibility interval (C.I.) for each parameter, and the mode for the variance
estimate. These statistics were computed from 1000 simulated values obtained after
convergence. The results show that on average the logit of offering algebra is 1.312 and
there is statistically significant state-to-state variation in the log-odds of offering algebra.
The predicted odds ratio is exp{ 1 .312} = 3.714. Schools are more likely to offer the
course than not on average. The predicted probability is (1+exp{- 1.312))" = 0.789. Note
that this differs from the mean outcome across the whole sample, which is equal to 0.76.
The difference arises because the grand mean estimated here is based on the conditional
distribution of the outcome given the random effects are null, rather than on the marginal

distribution of the outcome (Zeger, Liang & Albert, 1988; Raudenbush, 1993).

74

Table 4.2: Results of the Unconditional (N o Predictors) Model

 

Posterior Characteristics

 

 

Explanatory Variables Mean Std. Dev. 90% CI.
From To
Intercept
Intercept 1.312 0.145 0.886 1.810

State-to-state Heterogeneity
1.’ (mode = 0.730) 0.806 0.223 0.492 1.222

 

75
Figure 4.3, a histogram that approximates the posterior distribution of the
between-state variance of the log odds based on 1000 sampled values of 1:, portrays the
variability. As Figure 4.3 implies, values of 1: as small as 0.30 and as large as 1.70 are
plausible, whereas the 90 01. covers from 0.49 to 1.22. A sense of the magnitude of
state-to-state differences in the likelihood of offering algebra can be conveyed by
computing a plausible range of predicted probabilities with the random effects varying

from -1 .5 to 1.5 standard deviation units, i.e., p,j = 1/(1+exp{-Boj}) for

13,, e (v i 1.5%).

where [30]. is the average log odds of offering algebra for schools in state j (Raudenbush,
1993). When 1: is equal to the mode of the posterior, 0.806, the predicted probabilities
range from 0.51 to 0.93; when 1: is at the 10th percentile, 0.48, they range from 0.57 to
0.91; and when 1: is at the 90th percentile, the range is from 0.41 to 0.95. The ranges

indicate substantial state differences in the predicted probabilities of offering algebra.

76

Figure 4.3: Marginal Posterior Distribution of 1: (The Unconditional Model)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.15 _ - 150
m '_I
at) 010 .. j — 100
E13 /;I\r
a Z __
Z
9 i \
'—
5
CL 005 — 72 Br -50
O ' V
E i
j l I
0.30 0.85 1.40 1.95

STATE-LEVEL VARIANCE

lNROO

77

For individual states, Figures 4.4a and 4.4b display box plots describing the
distribution of the predicted probabilities computed using the marginal posterior of u.
Each box plot summarizes 1,000 probabilities computed from 1,000 corresponding
simulated values of mean logits. Inside the box are the middle 75 percent of the values of
the probabilities, while the ends of the whiskers delimit the top and bottom 1 percent of
those values. Thus, the plots give 75 percent and 98 percent intervals for the predicted
probabilities of offering algebra for each state. The figures offer another graphical
assessment of between-state variability. For instance, there are quite a few non-
overlapping distributions. PA, for example, is shown to differ from OK and TN in its

predicted probabilities of offering algebra.

78

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80
4.3.2 Between-school/Within-state and Between-State Model
The second model consisted of between-school/within-state and between-state
predictors. Table 4.3 summarizes the result and displays posterior characteristics of the
marginal distributions of the parameters, based on successive sets of 1000 sampled values
collected after convergence, for the school- and state-level predictors and the state-level

variance.

81

Table 4.3: Results of the Within- and Between-State Model

 

Posterior Characteristics

 

 

Explanatory Variables Mean Std. Dev. 90% CI.
From To

Intercept

Intercept 1.540 0.121 1.349 1.747

State Expenditure Per Pupil 0.044 0.012 0.023 0.063

Percent of Student in Poverty 0.016 0.023 -0.018 0.053
Minority Enrollment

Intercept -0.664 0.161 -0.939 -0.401

State Expenditure Per Pupil -0.046 0.013 -0.067 -0.024
SES

Intercept 0.383 0.068 0.267 0.496

State Expenditure Per Pupil 0.008 0.006 -0.001 0.017
Urban

Intercept 0.398 0.159 0.131 0.651
Suburban

Intercept 0.230 0.117 0.105 0.486
Grade 8 Enrollment

Intercept 0.645 0.050 0.562 0.725
Grade 8 Enrollment (Quadratic
Term)

Intercept -0.073 0.013 -0.097 -0.053
State-to-state Heterogeneity

1: (mode = 0.437 ) 0.477 0.147 0.286 0.736

 

82

At the school level, the log odds of offering algebra is negatively related to
percent of minority enrollment and positively related to school SES, urban and suburban
setting, and size. A sense of the magnitude of the effects of specific predictors can be
conveyed by computing the scale-invariant measure of odds ratio, exp{y}, which yields
the ratio of predicted odds of offering algebra, or by obtaining the predicted probabilities
of the existence of the course for schools which differ with respect to particular
characteristics.

I first examine the effects of the percent of minority enrollment on the likelihood
of offering algebra. The estimated ratio of the odds of offering algebra for schools which
have minority enrollment greater than or equal to 50% versus schools which do not is
exp(-0.664) = 0.515, after controlling for the effects of other variables in the model. It
indicates on average that the odds of offering the course among typical schools with high
minority enrollment are about half of those with less minority enrollment.

Another way to convey the effect of minority enrollment is to compute the
predicted probabilities for two typical schools which differ in their racial composition.
The probabilities can be obtained by holding all of the grand-mean centered predictors
constant at zero and allowing only the minority enrollment indicator to vary. As Table 4.1
shows, the mean of high minority enrolhnent is 0.14, indicating that 14% of the sample
had minority enrollment in excess of 50%. The grand-mean centered version of minority
enrollment takes on a value of 1 - 0.14 = 0.86 for schools with high minority enrollment
and 0 - 0.14 = -0.14 otherwise. Given random state effects of zero (“0) = 0), the predicted

probabilities of offering algebra are (1 + exp{-[1.540 + -0.664 * 0.86]})'1 = 0.725 for a

83
typical school with high minority enrollment and (1 + exp{-[1.540 + -0.664 * -0.14]})’I =
0.837 for a typical school with fewer minority students, other things being equal.

As Table 4.3 implies, school SES composition is associated with the log odds of
offering algebra, net of the effects of other variables. The likelihood of offering the
advanced mathematics course is higher among schools with higher SES. To gauge the
effect, the odds ratio for two typical schools which are two standard deviations apart in
their SES composition (2*0.82) = 1.64 is computed. The odds ratio estimated for the
higher SES versus lower SES schools is 1.874 (exp(0.383* 1.64)). Given random state
effects are null, the predicted probabilities for the two typical schools to offer algebra are
(1 + exp{-[1.540 + 0.383 * 0.82]})'1 = 0.865 and (1 + exp{-(1 + exp{-[1.540 - 0.383 * -
0.82]})‘1 = 0.773.

Rural schools, when compared with their urban and suburban counterparts, are in
general less likely to include 8th grade algebra in their mathematics curricula, when
controlling for the school demographic composition, size, state educational expenditure,
and percent of children in poverty. The estimated ratios of odds for an urban versus a
rural school and a suburban versus a rural school in offering the advanced mathematics
course are 1.488 and 1.259 respectively. The predicted probabilities for typical urban,
suburban, and rural schools to offer algebra are 0.864 ((1 + exp{-[1.540 + 0.398 * 0.77]})‘
'), 0.839 ((1 + exp{-[1.540 + 0.23 * 0.47]})") , and 0.784 ((1 + exp{-[1.502 + 0.398 * -
0.23 + 0.23*-0.53]})").

The partial effect of school size on the logits of offering algebra, net of those of all

other predictors, is positive and statistically significant. The relative odds for schools with

84
eighth-grade enrollments of 356 (one standard deviation above the average) to schools
with average eighth-grade enrollments are exp(0.645* l .39 + -0.073*1.39*1.39) = 2.129,
indicating the odds that the former schools offer algebra are twice as likely as the latter.
The predicted probability of offering algebra for a typical school with 356 eighth-grade
students is 0.909, whereas it is 0.824 for a typical one with average eighth-grade
enrollment. Note that 0.824 here differs from 0.789 estimated in the unconditional model
as it is conditioned on the random effect after controlling for state educational
expenditure and percent of children in poverty.

At the state level, the log odds of offering algebra, adjusted for school
composition, urbanicity, and size, are associated with state educational expenditure per
pupil but not percent of children in poverty. The estimated odds ratio for schools in states
with educational expenditure per pupil one standard deviation above the average versus
their counterparts in states with average educational expenditure is exp(.044*12.35) =
1.72, after controlling for the percent of student in poverty and other school-level
predictors. The predicted probabilities of offering algebra for two typical schools, one
located in a state with an educational expenditure one standard deviation above average,
the other in a state with average educational expenditure, are 0.824 and 0.889,
respectively.

The relationship between minority enrollment and the likelihood of offering
algebra depends on state educational expenditure per pupil. The gap in the log odds of
offering algebra between predominantly minority schools and schools with fewer

minority students is widened as state educational expenditure per pupil increases. The

85
estimated odds ratio for high minority schools in states with educational expenditure per
pupil one standard deviation above the average versus their counterparts in states with
average educational expenditure is exp(-.046*12.35) = 0.568. The predicted probabilities
of offering algebra for two typical schools located in a state with an educational
expenditure one standard deviation above average, with and without high minority
enrollment, are 0.670 and 0.725 respectively.

As the result in Table 4.3 suggests, the relationship between school SES and how
likely a school is to offer algebra does not depend on state per pupil educational
expenditures. Regarding the state-to-state heterogeneity, substantial variance remains to
be accounted for after putting the school- and state-level predictors into the model. There
is about a reduction of 35.6% in the mode of the variance. Figure 4.5 gives a histogram of
the posterior distribution of the conditional variance for the second model. Note that the
90% CI. (0.286, 0.736) of the adjusted or residual variance is much narrower than that of
the unconditional model 90% C1. (0.492, 1.222). Allowing the random effect to vary
from -1.5 to 1.5 standard deviation units and holding all the variables constant at their
grand means, the adjusted predicted probabilities for individual states range from 0.63 to

0.93 when 1: is equal to its mode, 0.437.

86

Figure 4.5: Marginal Posterior Distribution of 1: (The Conditional Model)

PROPORTION PER BAR

0.15

0.10

0.05

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ - 150
7'1
- SK — 100
~ 51 ~ 50
I I’T
0.15 0.70 1.25 1.80

STAT E—LEVEL VARIANCE

lNITOO

87
4.4 Discussion and Implications

Given that the results of the analyses confirmed most of the hypotheses stated in
the study, a question that arises is whether or not the statistically significant partial effects
of the school- and state-level predictors are practically significant as well. Translating the
changes in odds or predicted probabilities to the opportunities to learn at the student level
allows us to assess the practical significance and the magnitudes of effects of the
predictors. It enables us to examine the extent to which factors at one level of the
multilevel educational system influence, alter, or limit the events at lower levels.

To proceed, I use the predicted probabilities of offering algebra obtained for two
typical schools with and without high minority enrollment. All else held constant, the two
predicted probabilities are 0.725 and 0.83 7, respectively. They differ by 0.112. Thus, for
100 typical schools with high minority enrollment, it is predicted that there will be 11
fewer schools offering algebra than if the probabilities were equal. As seen in Table 4.1,
the average grade eight enrollment is 217. Approximately 2,400 students will attend
schools that do not offer algebra for high school credits as a result. This indicates that the
practical significance of the racial composition effect is substantial. By the same token,
differences in the predicted probabilities of offering algebra among typical schools with
different SES composition, size, and in different settings (rural versus others) could have
a significant impact on students' access to the course.

The correlational effect of the denial of access to algebra initiated at a higher level
of the educational system can also be illustrated using a unit above schools--the state. For

example, all else being equal, a difference of 0.065 in the predicted probabilities of

88
offering algebra brought about by a one standard deviation decrease in state educational
expenditure per student will be translated to the absence of the algebra course in hundreds
of typical schools within a state and, on average, to the preclusion of possibilities to learn
algebra for 217 students in each school.

Whereas the partial effects allow us to gauge the influence of a particular factor
net those of the other factors, it may be useful and even important to examine how the
outcome varies with more than one variable. One reason is that a school's disadvantage in
its capacity to offer algebra may be compounded by other factors. For instance, high
minority schools on average have lower SES than low minority schools, and rural schools
usually have fewer students than their urban and suburban counterparts. In the 1992
TSAP sample, high minority schools have a mean SES of -0.982, and low minority
schools have a mean of 0.163. The two means are more than one standard deviation
apart. Thus, other factors remaining equal, the predicted probability of offering the course
for a typical high minority school whose SES is equal to -0.982 is (1+exp{-[l .540 + -
0.664*0.86 + 0.383*-0.982]}) = 0.644. It is 0.081 less than that for a high minority school
with average SES, i.e., 0.

After assessing the practical significance of most predictors, I now offer and
discuss some tentative explanations as well as the implications of the findings. The
analyses indicate racial and social stratification in the opportunities to learn algebra
among schools in the 42 states examined. The fact that high minority schools are more
likely to have lower SES suggests there is a clustering of poor and minority children in

schools, which leads to a compounding of school's disadvantage. The clustering, as

89
suggested by Hawley (1993) and Carter (1995), could have been brought about by
residential segregation, policies of neighborhood school assignment, and school district
configurations.

Regarding school size, the results support the assertion that greater grade
enrollmentencourages greater specialization and differentiation of the curriculum (Lee &
Smith, 1995). The association, however, is a nonlinear one and it weakens as school size
increases. One possible reason that rural schools are less likely to offer the advanced
mathematics course, among other things, may be the limited availability of highly-
qualified teachers in the rural sector due to the more restrictive markets in certain
demographic regions (Monk & Haller, 1993). There may be an inequitable distribution of
well-trained teachers across the different sectors.

The association of increased levels of financial investment in public education
with a greater access to the course lend some support to the conclusions of Hedges, Laine,
and Greenwald's (1994) meta-analysis of studies of the effects of differential school
inputs on student outcomes. They concluded that financial resource input is likely to be
related to school outcomes. The result of the present investigation shows that increased
state spending is associated with the offering of a course and in turn is associated with
higher mathematics achievement (MacIver & Epstein, 1995) and a greater likelihood of
enrollment in advanced high school mathematics (Stevenson et al., 1984). It is important
to note here, however, as no educational expenditure at the district level was entered as a
predictor, district-to-district variability in funding was not modeled. In order to model the

relationship between the two educational inputs (district educational spending and the

90
availability of the course), two indices of educational investment efforts would be needed.
These indices are the educational expenditure per student as well as the level of special
education funding. TSAP does not provide the relevant data for the inclusion of these
indices in the analysis. Hence what was examined here was the general effect of overall
state public education investment efforts on the availability of the advanced mathematics
course.

Whereas higher levels of public education investment efforts in public education
are associated with a greater likelihood of the offering of the course, they also widen the
racial composition gap in access to algebra. This seems to suggest that increased state
educational expenditures benefit low-minority schools more than high-minority schools.

In sum, the results suggest that schools serving minority students and low SES
students, small schools, and schools located in states spending less on education are
comparatively less likely than other schools to offer algebra. The implication of the
finding is that size, composition, and location of a school are linked to inequality in
access to this particular educational resource. In these ways, the schooling system
reinforces social, ethnic, and geographic inequality in the opportunity to study high-
school algebra in the middle grades.

Finally, the analyses show that in addition to allowing one to analyze the specific
effects of variables of interest, while holding other factors constant, the fitted or predicted
probabilities could act as a production and curriculum indicator (Oakes, 1989; Pelgrum et
al., 1995; Porter, 1991) at the state and the school level. They inform us about the

"implemented curriculum" (Pelgrum et al., 1995) at those levels and how they are related

91
to "curriculum antecedents" (Pelgrum et al., 1995) such as the SES composition of the
school. The fitted probabilities could perform as an indicator as well as a diagnostic of
how well the educational systems perform. For example, they can indicate the presence of
the stratification of learning opportunities due to race, SES, and location in terms of

student attendance at a school that offers algebra.

CHAPTER 5

SUMMARY AND CONCLUSIONS

5.1 Summary and Conclusions

In this work, I coded Zeger and Karim’s (1991) Bayesian posterior analysis for
generalized linear models with random effects via the Gibbs sampler. The SAS/IML code
performed well and the algorithm gave reasonable inferences. This was documented with
the help of a simulated study with datasets having structure similar to the TSAP in
mathematics data. Using the algorithm, 1 implemented a Bayesian, multilevel analysis of
access to eighth-grade algebra. The analysis accommodated the hierarchical or nested
design of the TSAP in mathematics and the need to incorporate the uncertainty that arose
about the variances at the state level.

The results suggest that schools serving minority students and low SES students,
small schools, schools located in the rural setting, and states spending less on education
are comparatively less likely than other schools to offer algebra. The implication is that
size, composition, and location of a school are linked to inequality in access to this
educational resource. In these ways, the schooling system reinforces social, ethnic and
geographic inequality in the opportunity to study advanced mathematics. The analyses
demonstrated that the fitted probabilities could be used as an indicator for certain
academic opportunities in schools with different demographics, enrollment, setting, and
levels of state financial investment efforts. In addition, the approach can be applied to

policy research addressing the multilevel structure of the educational and social systems.

92

93
5.2 Future Research Needs
5.2.1 Substantive Inquiry

This study provided a snapshot of the different availability of eighth-grade algebra
among public schools in 42 states in 1992-93. The focus of the inquiry is on an essential
component of the opportunities to learn algebra in middle grades, which is content of
instruction that schools make accessible to students (Porter, 1994). With the current
reforms in mathematics that stress a high-quality curriculum for all students (Porter,
1994; Wheelock, 1996), efforts to effect changes in the content of and access to algebra in
the mathematics curriculum for the middle grades have been initiated over the past few
years (Edwards, 1994; Olson, 1994).

Two main and related curricular changes regarding algebra instruction are
advocated by the widely-circulated WW
Mathematics (NCTM, 1989), often abbreviated as the Standards. First, as Jack Price, the
president of NCTM, stated, algebra should be seen as "a strand throughout the K-12
curriculum" (Olson, 1994, p.11 ), and not as an isolated instructional topic. Indeed it is
recommended not to segregate basic algebraic ideas into a one-year course (N CTM,
1993). Second, algebraic thinking that emphasizes understanding, reasoning, problem-
solving, and real-world applications is to be introduced gradually to all students in a
broad and integrated curriculum for the middle grades. Other topics included in the
recommended curriculum are measurement, geometry, probability and statistics. An
example of an integrated middle grades mathematics curriculum that reflects the

Standards is the Connected Mathematics Project Curriculum (Connected Mathematics

94
Project, 1996).

According to the proponents of the Standads, extensive curriculum development
is required to guarantee access to algebraic competence for all students (Silver, 1995). For
supporters, the traditional first-year course in algebra is not the "right algebra" for
everyone (Chambers, 1994), in part because of the course puts undue emphasis on rules
and algorithms.

The requirement of eighth-grade algebra in various school districts such as
Cambridge, Massachusetts (Olson, 1994) has brought greater access to algebra to
students. The district requirement shares the goals of the Standards to engage more
students in learning important mathematical ideas and promote their proficiency in
mathematics (Silver, 1995).

Further research can be carried out to study the impacts of current reforms on
access to algebra. Of particular interest is to examine the availability of algebra
instruction in the middle grades amidst the perceived incompatibility between the
advocacy of a "problem-based" curriculum suggested by the NCTM and the more
conventional "scope-and-sequence" approach (Schoenfeld, 1994). In addition, the great
amount of instructional and financial support and community involvement called for by
the standard-based reforms, for example, extensive curriculum development and teacher
education, add additional difficulties in the reform process to guarantee access and
opportunity for all students (Bruckerhoff, 1995; Silver, 1996; Tate, 1994). An important
research question one may ask is what state- and school-level factors may influence a

school's decision to implement curricular reform and offer "problem-based" algebra

95
instruction in the middle grades.

The present inquiry focused on the existence of an algebra course for high school
credits in the middle grades. Further research can be carried out to review its "breadth" or
the percentage of students within a school who enroll in the course. Becker (1990) found
substantial within-school variability in access to algebra. He found that out of the 63% of
the 2,400 middle-grade schools which offered algebra courses in their curriculums in
grade 7, and, more typically, grade 8, only 35% provided access to algebra to at least one-
quarter of their students. A future study of the student- and school-level correlates of
enrollment in the algebra course can complement the present investigation by offering an
understanding of the selection process at another level of the hierarchical educational
system, that of the student. Using Pelgrum et al.'s (1995) term, the study will allow one to
study the "attained curriculum" (p. 81) at the micro, or student level, and will complement
the present investigation of the "implemented curriculum" (p.81) at the meso, or school

and state levels, in monitoring the performance of the multilevel educational system.

5.2.2 Methodological Development

The Gibbs sampler was coded in SAS/IML and its execution was very time
consuming. In the simulation, for instance, it took approximately 8 to 16 hours to fit a
model. Coding in a lower or intermediate-level language such as C will greatly reduce the
run times. A prototype written in C is currently being developed. My early prediction is
that it will take approximately half an hour to forty five minutes to execute the same

analysis mentioned previously.

96

The simulations and analyses of the present study focused on cases where 1: is a
scalar. With the development of a faster computer code, simulations to document the
performance of the algorithm for cases where T is a matrix can be carried out.
Satisfactory results will allow the code to be applied to analyze random coeffrcient
models. A slight modification of the over-simplified model (2.6) described in Chapter 2
gives us an example of a random coefficient model. By assuming the racial composition

effect to be randomly varying across the 42 states after controlling for StateExpj, i.e.,

BU. = 710 + YHStateExpj + “11 , (51)

model (2.6) becomes a random coefficient model with u] being a 2 x 1 vector of random

effects with mean of zero and variance T.

APPENDICES

APPENDIX A

APPENDIX A

SAS\IML Code for the Gibbs Sampler

/*******************************************************************/

/*
Program name: bayessas

Purpose and Modules: The code implements the Gibbs sampler developed
by Karim (1991) and Zeger and Karim (1991). Its main program consists of
the following modules:

1) import -- reads in the data set,

2) initial -- enters various parameter values and sets up vectors and
matrices,

2) glimgam -- performs GLIM to obtain 9 and V,

3) gbgarn -- generates y

4) gbtau -— generates T

5) ridgeuj -- performs ridged regression to obtain a, and V0,

6) gbb' -- generates ul

7) output -- outputs simulated values of y, T and uj

*/

/********************************************************************/

/******** formats and directories ******/
Options pagesize= 100 linesize = 80 nocenter number date;

Libname gibbs 'c:\sas\data';
Libname output 'c:\sas\output';

Proc iml;
reset nolog;

/********* formats and directories ******/

97

98

/********* mOdUIe import ************/

start import (yij,lev2var,lev1var,nj); /* begin module import: get raw data */

/* yij = outcome variable
lev2var = a level 2 variable
levlvar = a level 1 variable
nj = number of level 1 units in cluster j */

use gibbs.rawdata;
read all var{yij lev2varlev1var};
close metgibbsrawdata;

use gibbsnj;
read all var {nx} into nj;
close metgibbsnj;

finish;

99

/*********pan 2 ************/

start initial (yij,lev2var,levlvar,nj,xij,zij,ngroup,N,nf,nq,inititer,
niter,nsample,gam,uj,Tau,seed1 ,outgam,outuj ,outTau);

/* xij = predictors for fixed effects

zij = predictors for random effects

N = total sample size

ngroup = number of level-2 units

nf = number of fixed effects --- beta's
nq = number of random effects --- z's
inititer = counter of iteration

niter = number of maximum iteration
nsample = sample size of the posterior dist.
gam = fixed effects

uj = random effects

Tau = variance of uj's

seedl = seedl number

outgam = output beta 0 = no, 1 = yes
outuj = output bj 0 = no, 1 = yes
outTau = output Tau 0 = no, 1 = yes */
ngroup = 40;

nf = ;

nq = ;

inititer = 10;

niter = 5000;

nsample = 2000;

seedl = 12378942;

outgam = ;

outbj = ;

outTau = ;

bj = j(ngroup*nq,l,0);

yij = yij;
N = nrow(yij);
x1 =j(N,1,1);

xij = x1 ||lev2var||lev1 var;

zl =j(N,nq,l);
zij = 21;
finish;

100

/******** mOdUIe glimgam ********/

start glimgam (yij,nj,nf,nq,ngroup,N,xij,zij,gam,uj,vgam,inititer,det,
loggamx,ofstuj);

crit=1 ;

lo=0;
iterval=j(20,2,0);
loggamx = 0;

ofstuj = j(N,1,0); /* offset zijuj */
a = ;

c = ;

e = ;

f= ,

g = nq;

Do i = l to ngroup;
ijth = a -1+ nj[c:e];
ofstuj[a:ijth] = zij[a:ijth,1:nq]*uj[f:g];
a = a + nj[c:e];

c=c+1;

e=e+l;

f=f+nq;

g=g+n¢
End;

Do it=1 to 20 while(crit> 1.0e-8);
etaij =j(N,l,0);

loglhood=0;

xwx = j(nf,nf,0);

xwystar = j(nf,1,0);

ystarij =j(N.l.0);

wij =j(N.1,0);

uij =j(N.1,0);

a— ,
c=1;
e= ,
f= .

101

Do i = 1 to ngroup;

end;

ijth = a - 1 + nj[c:e];

etaij[a:ijth] = xij[a:ijth,l :nf]*gam + ofstuj[a:ijth];
nunit=ijth-a+1;

if inititer <= 10 then

do;

if it <= 3 then

do;

h = a;

do k = 1 to nunit;

if .95 <= uij[h:h] then

do;
uij[h:h] = .95;
* print 'large probability';
end;
h = h+l;
end;
end;
end;

wij[a:ijth] = uij[a:ijth]#(1-uij[arijth]);

ystarij[a:ijth] = etaij[a:ijth] +
((yijlaiiith] - uij[aiijthlywijlaiijflll);

xwx = xwx + (xij[a:ijth,l:nf]#wij[a:ijth])‘*xij[a:ijth,1:nf];

xwystar = xwystar + ((xij[a:ijth,1:nf]#wij[a:ijth])‘*(ystarij[a:ijth]-
ofstuj[a:ijth]));

loglhood = loglhood + sum(yij[azijth]#log(uij[a:ijth]) +
(1-yijlaiijth])#10g(1-uij[aiijth]));

a=a+nj[c:e];
c=c+1;
e=e+1;
f=f+nq;
g=g+nm

gam = solve(xwx,xwystar);
vgam = inv(xwx);
crit=abs(loglhood-lo);
iterval[it,]=it||loglhood;

10 = loglhood;

end;

102
loggamx = loglhood;
finish;

/***** module gimgam ****/

103

/********* module gbgam ******/

start gbgam (yij,nj,nf,nq,ngroup,N,xij,zij,gam,uj,vgam,seed1,loggamx,ofstuj,inititer);

/* ygam = envelope function (multivariate normal distribution)
xgam = actual function

c1 = mode-matching constant
c2 = variance-inflation constant
loggamx = pi(x) from glimgam */
ygam = 0;

xgam = 0;

gamstarl = j(nf,1,0);

c1 = 0;

c2 = 2;

etaij =j(N,1,0);

uij =j(N.1,0);

wij =j(N.1,0);
nordev = j(nf,1,0);
ratio = 0;

/* compute the log of ordinate of ygam--the envelope function at the mode */
loggamy = - ((nf/2) * log (02 * 3.1416))

- ((nf/2) * log (det(c2 * vgam»);
/* compute c1 */

c1 = loggamx - loggamy;
print cl;

/* generate gamstar from ygam */
u = 1;

gamcnt = 0;
Do while (ratio < u);

a=1;
b=1;
doi=ltonf;

nordev[a:b] = rannor(seed 1 );
a = a + 1;

104

b=b+ 1;
end;

CholeskU = j(nf,nf,0);
CholeskU = root(c2*vgam);
gamstar= gam+CholeskU‘ *nordev;

/* compute the ordinate */
loggamy = - ((nf/2) * log (c2 * 3.17))
- ((nf/2) * log (det(c2 "‘ vgam)))
- ((gamstar-gam)‘ *inv(c2*vgam)*(gamstar-gam))/2;

/* compute the ordinate of xgam */

a=.
c=.
e=,
f=,
g=nq,

Do i = 1 to ngroup;
ijth = a - l + nj[c:e];
nijth = 0;
nijth=ijth-a+1;
etaij[a:ijth] = xij[a:ijth,l :nf]*gamstar + ofstuj[a:ijth];
uij[azijth] = 1/(1+exp(-l*etaij[a:ijth]));
loggamx = loggamx + sum(yij[a:ijth]#log(uij[a:ijth]) +
(1-yij[a:ijth])#log(l-uij[a:ijth]));
a = a + nj[c:e];
c = c + 1;
e = e + 1;
f = f + nq;
g=g+n¢
end;

/* calculate the ratio fgam/ggam */
gamcnt = gamcnt + 1;

ratio = exp(loggamx - loggamy - c1);
u = uniform(seedl);

end;
/* updating gam */

gam = gamstar;
end;

finish;

/******* mOdUIe gbgam *********/

105

106

/******* mOdUIC ngau ***************/
start ngau (uj,Tau,nq,ngroup);

seedl = 344456;
3 = j(nq.nq.0);

c=1;
e=nq;

Do i = 1 to ngroup;

S = S + (uj[c:e]*uj[c:e]‘);
c=c+nm

e=e+nm

end;

do i = 1 to nq;
doj = 1 to nq;
if(i <j) then S[i,j] = S[j,i];
end;
end;

inVS =j(nq.n<L0);
invS = inv(S);

do i = 1 to nq;
do j = 1 to nq;
if (i < j) then invS[i,j] = invS[j,i];
end;
end;

H = root(invS);
W = j(nq.nq,0);
do i = 1 to nq;
do j = 1 to nq;
if (i <j) then do;
W[i,j] = rannor(seedl);
end;
if (i = j) then do;
df= ngroup - nq - 1;
* df=ngroup-j+1;

107

if (mod(df,2) = 0) then
do;
df = int(df/2);
W[i,j] = sqrt(2.0 * rangam(seedl,df));
end;
else do;
df = int(df/2);
W[i,j] = sqrt(2.0 * rangam(seed1,df) + rannor(seed1)**2);
end;
end;
end;
end;

Tau = inv(H‘ * W‘ * W * H);
do i = 1 to nq;
do j = 1 to nq;
if (i < j) then Tau[i,j] = Tau[j,i];

end;
end;

finish;

/******* module ngau **************/

108

/****** module ridgCUj **************/

start ridgeuj (yij,xij,zij,nj,N,ngroup,nf,nq,gam,Tau,uj,varuj,inititer,
logujx,ofstgam);

crit=1 ;
lo=0;
iterval=j(20,2,0);

ofstgam = j(N,1,0); /* offset xijgam */

a=;
c=;
c=;

Do i = 1 to ngroup;

ijth = a — 1 + nj[c:e];

ofstgam[a:ijth] = xij[a:ijth,l :nf]*gam;
a = a + nj[c:e];

c=c+1;
e=e+1;
End;

ystarij =j(N,1.0);

wii =j(N,1,0);

uij =j(N,1.0);

etaij =j(N,1.0);

varur' = j(ngIOUP*nq.nq.0);

3: 9
c=l;
e=1,
f: 9
g=nq;
ijth=0;

loglhdpl =j(N.1,0);
logUJ'X =j(ngIOUP,1,0);

109
Do i = 1 to ngroup;

ijth = a - l + nj[c:e];
nijth= ijth - a+ 1;
zwzian = j(nq,nq,0);
zwystaro = j(nq.1,0);
crit = 1;

it = 0;

lo = 0;

Do it = l to 20 while (crit> 1.0e-8);
loglhood = 0;
etaij [azijth] = ofstgam[a:ijth]+
+ zij[a:ijth,l :nq]*uj[f:g];
uij[a:ijth] = 1/(1+exp(-etaij[a:ijth]));
nunit=ijth-a+ 1;
if inititer <= 50 then
do;
if it <= 20 then
do;
h = a;
do k = 1 to nunit;
if uij[h:h] <= .01 then

do;
uij[h:h] = .01;
end;
else if .99 <= uij[h:h] then
do;
uij[h:h] = .99;
end;
h =h+1;
end;
end;
end;

wij[a:ijth] = uij[a:ijth]#(1-uij[a:ijth]);

ystarij[a:ijth] = etaij[a:ijth] +
(yij[a:ijth] - uij[a:ijth])/wij[a:ijth];
zwzian = (zij[a:ijth,1 :nq]#wij [a:ijth])‘ *zij[a:ijth,1 :nq]+inv(Tau);
zwystaro = (zij[a:ijth,1:nq]#wij[a:ijth])‘*(ystarij[a:ijth]-
ofstgam[a:ijth]);

loglhdp1[a:ijth] = (yij [azijth]#log(uij[a:ijth]))

110

+ ((1-yii[aiijth])#log(1-uij[aiijth]));
loglhood = sum(loglhdp1[a:ijth]) - (uj[f:g]*inv(D)*uj[f:g])/2;
crit = abs(loglhood-lo);
iterval[it,]=it||loglhood;
10 = loglhood;
uj[f:g] = solve(zwzian, zwystaro);
end;

varuj [fz g,1 :nq]= inv(zwzian);
logujx[c:e] = loglhood;

a = a + nj[c:e];

c = c + 1;

e = e + 1;

f=f+nq;

g=g+nm
end;

finish;

/******* module ridgeuj ***********/

lll

/******** mOdUIe gij ************/

start gbuj (yij,xij,zij,nj,N,ngroup,nf,nq,gam,Tau,uj,varuj,seedl ,
logujx,accuj,ofstgam,inititer);

CholeskU = j(ngroup*nq,nq,0);
u = j(ngroup*nq,1 ,0);

logujy = j(ngroup,1,0); /* (y --- the envelope/proposal function) */

ujstar = j(ngroup*nq,1 ,0);
c1 = j(ngroup,1,0);

c2 = 4;

etaij =j(N.1,0);

uij =j(N,1,0);

wij =j(N,1.0);

logur'Xpl =j(N.1,0);
ujcount = j(ngroup,1,0);

/* calculate c1 */

c=1,
e= .
f= .
g=nq;

Do i = 1 to ngroup;
/* compute the ordinate of ggam --- the envelope function at the mode */

logujy[c:e] = - ((nq/2) * log (2 * 3.14))
- ((nq/2) * log (det(c2 * varuj[f:g,1:nq])));

112
/* compute the ratio c1 */
c1[c:e] = logujx[c:e] - logujy[c:e];
c = c + l;
e = e + 1;
f= f+ nq;
g=g+nm

end;

/* generate ujstar from yuj --- the envelope function */

a=.
c=.
e=,
f= .
8:110;
nijth=0,

Do i = 1 to ngroup;

ratio = 0;

u = 1;

ijth = a - 1 + nj[c:e]; /* size index */
nijth= ijth- a+ 1;

CholeskU[f:g,l :nq] = root(c2*varuj [f:g,1 :nq]);

Do while (ratio < u);

nordev = j(nq,1,0);
x = 1;

do k = 1 to nq;
nordev[xzx] = rannor(seed l );
x = x + 1;

end;

uj star[f: g] = uj [f: g] + CholeskU[f:g,1 :nq]‘ *nordev[l :nq];

113
/* compute the ordinate of uj star using guj ---- the envelope function */

logujy[c:e] = - ((nq/2) * log (2 * 3.14))
- ((nq/2) * log (det(c2 * varuj[f:g,1:nq])))
- ((UJ'StarlfIgl-ujlfigl)‘
*inv(c2*varuj[f:g,1:nq])*(ujstar[f:g]-uj[f:g]))l2;

/* the ordinate of ujstar using fuj ---- the true function */

etaij[a:ijth] = ofstgam[a:ijth]+ zij [a:ijth,1 :nq]*ujstar[f: g];
uij[a:ijth] = 1/(1+exp(-etaij[a:ijth]));
logujxp1[a:ijth] = ((yij [azijth]#log(uij [azijth]))

+ ((1-yijlaiijth])#10g(1-uij[a=ijth])));
logujx[c:e] = sum(logujxpl [azijth])

- (uj star[f:g] *inv(Tau)*ujstar[f:g])/2;

/* compute the ratio */

ujcount[c:e] = ujcount[c:e] + 1;
ratio = exp(logujx[c:e] - cl [c:e] - logujy[c:e]);
u = uniform(seedl);

end;

ujlfig1= ujstarlftg];
a = a + nj[c:e];

c = c + 1;

e = e + 1;

f=f+nq;
g=g+nm

end;
totujcnt = 0;
totujcnt = sum(ujcount);

print totuj cnt;

end;
finish;

/****** mOdUIe gbuj ****************/

114

/****** module output ***************/

start output (itcount,firstout,ngroup,nf,nq,niter,nsample,
outgam,outTau,outuj,gam,Tau,uj);

gamt = j(1,nf,0); /* gain */
gamt = gatn‘;

uniqTau = nq * (nq + 1)/2; /* unique elements of Tau */
elemTau = j( 1 ,uniqTau,0);

ujt = j(1,ngroup*nq,0); /* uj */
ujt = uj‘;

if (firstout = itcount) then do;
if (outgam = 1) then create output. gampost from gamt;
if (outTau = 1) then create output.Taupost from elemTau;
if (outuj = 1) then create outputujpost from ujt;

end;

/* Write out the posterior distribution of the parameters */

if (outgam = 1) then do;
setout output. gampost;
append from gamt;
end;

if (outTau = 1) then do;

index = 1;
do i = 1 to nq;
do j = 1 to nq;
if (i <=j) then do;
e1emTau[1,index] = Tau[i,j];
index = index + 1;
end;
end;
end;

setout output.Taupost;
append from elemTau;
end;

if (outuj = 1) then do;
setout outputuj post;
append from ujt;
end;

finish;

115

116

/******* main program **********/
itcount= 1;

run import (yij,lev2var,lev1var,nj);

run initial (yij,lev2var,lev1var,nj,xij,zij,ngroup,N,nf,nq,
inititer,niter,nsample,
gam,uj,Tau,seed1 ,outgam,outuj ;outTau);

Tau = 1;
111' =j(ngr0up*nq.1,0);

firstout = niter - nsample + 1;

Do while (itcount <= niter);

run glimgam (yij,nj,nf,nq,ngroup,N,xij,zij,gam,uj,vgam,inititer,det,
loggamx,ofstuj);

run gbgam (yij,nj,nf,nq,ngroup,N,xij,zij,gam,uj,vgam,gengam,seed1,
loggamx,c2gam,cntgam,accgam,ofstuj,inititer);

run ridgeuj (yij,xij,zij,nj,N,ngroup,nf,nq,gam,Tau,uj,varuj,inititer,
logujx,ofstgam);

run gbuj (yij,xij,zij,nj,N,ngroup,nf,nq,gam,Tau,uj,varuj,genuj,seedl ,
logujx,o2uj,cntuj,accuj,ofstgam,inititer);

run ngau (uj,Tau,nq,ngroup);

If (itcount >= firstout) then do;

run output (itcount,firstout,ngroup,nf,nq,niter,
nsample,outgam,outTau,outuj,gam,Tau,uj);

end;

print itcount;

itcount = itcount + 1;

end;

quit;

APPENDIX B

APPENDIX B

Approval Letter from the US. Department of Education
for Accessing Individually Identifiable Survey Data Base

US. DEPARTMENT OF EDUCATION
spruce CF soucmowu RESEARCH mo MPROVEMENT

 

NATIONAL CENTER FOR EDUCATION STATISTICS

Yuk Fai Cheong

Department. of Counseling ..
College of Education SEP 5 1994

Michigan State University
East Lansing, Michigan 48824-1034

Dear Mr. Cheong:

IampleuedtomformyouthstthoDepsrunentofEducatiomMichiganStats
University has met the requirements for accessing the individually identifiable
survey data base entitled: "1992 NAEP Trial State Assessment".

The following information is enclosed for your use:

0 A signed copy of the license and 1 cop(ies) of the Affidavit of Non-
Disclosure, for yourself and three project stsfl';

o Onecopyofthedambaseyourequescadonfour9-track
tapes numbered: W-1565, W-01520, W-01483 and W05297; and

o Disk containing documentation.

Please keep the single copy ofthe W914, GEPA. and the N088
Wm enclosed with your initial licensing application. with the
executed license for reference by you and those project staff who will be accessing
the data. Also retain a copy of your approved data W with the examined
license. Violations of any of the licensing provisions by any member of your
research project stafl' could result in cancellation.

MMmonlmeLEWmmwaa
period not to exceed 5 years commencing with the date of the NCES Commissioner’s
signature on the license. You have been assigned license number: 940830148.
Reference this number in all future correspondence.

If you have any questions. you may call me at (202) 219-1920.

Alan W. oorehead
Data Security Officer

Enclosures

.‘JASHINGTON DC 22235--

117

APPENDIX C

APPENDIX C

Approval Letter from the University Committee
on Research Involving Human Subjects

MICHIGAN STATE

UNIVERSITY

November 20. 1996

 

To: Ste hen w. Raudenbush
Col ego of Education
461 Erickson hall

R3: IRES: 90-‘13

TITLE: CORRELATES OF ST!!! VIRIATION IN TR: SOCIRL
DISTRIBUTION OF ACE : A RAYSERN
ANALYSI : METHODOLOGICAL ALTERNIIIV‘S 18 TH:
ANALYSIS OF Dflxh FROM NAEP

REVISION REQUESTED: 11/05/96

CATEGORY: 1'3

APPROVAL DATK: 09/13/96

 

The University Committee on Research Involving Human Sub eots'IUCIIISI
review of this project is complete. I am pleased to a so that the
rights and welfare of the human subjects appear to be adequately
rotected and methods to obtain informed consent are a ropriate.

refore. the UCIIBS approved this project and any rev sions listed
above.

hllllhns UCIIHS approval is valid for one calendar year. beginning with
the approval date shown above. Investigators planning to
continue a project be one year must use the green renewal
form (enclosed with t original a rovel letter or when a
ptOject is renewed) to seek u date certification. There is a
maximum of four such expedite renewals ossible. Investigators
wishing to continue a reject beyond the time need to submit it
again or complete re ew.

asvrsross: UCIIBS must review :3! changes in rocedures involving human
subjects. rior to tiation of t change. If this is done at
the time o renewal. please use the green renewal form. To
revise an approved protocol at an other time during the year.
send your written request to the CRIBS Chair. requesting revised
approval and referencing the project's IRS # and title. Include
in your request a description of the change and any revised
ins ruments. consent forms or advertisements that are applicable.
paostxxs/ .
ell-088: Should either of the following arise during the course of the
work. investigators must noti UCRIES promptly: (l) roblems
(unexpected side effects, comp aints. etc.) involving uman
subjects or (2) changes in the research environment or new
information indicating greater risk to the human sub ects than
existed when the protocol was previously reviewed approved.

 

STUDIES If we can be of any future help. lease do not hesitate to contact us
at (517)355-2180 or PA! (Sl7Is 2- 171.

Mumsldlmm Sincere

...,...‘W (Jr

David E. Wright. Ph. .
UCRIRS Chair

DEN : bed

 
 
   
 

 

sums-2180
FAX: SIT/£3241"

cc: Yuk Fai Cheong

mums-mm
memes-m
fan‘s-m

“worm

118

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