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dissertation entitled

A MODEL FOR MULTILEVEL PATH ANALYSIS

presented by

Frank F. Jenkins

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A MODEL FOR MULTILEVEL PATH ANALYSIS

BY

Frank Ford Jenkins

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

1988

ABSTRACT
A MODEL FOR MULTILEVEL PATH ANALYSIS

BY

Frank Ford Jenkins

In the past couple of decades there have emerged two
particularly innovative trends in quantitative research
methodology in Education, path analysis and multilevel linear
models. In path analysis, networks of interrelationships
among variables are posited to represent the interconnected
relationships found in real life processes. In multilevel
linear models, analysis techniques have been devised to
represent real life processes as they naturally occur in
hierarchically' nested. contexts. Both. approaches seek; to
represent complexity in a way that corresponds to the
complexity of nature. There is developed in this thesis a
method which combines these two trends into a multilevel
path analysis. Such an approach combines the descriptive
power of both path analysis and multilevel linear models,
resulting in a single model which can define a complex network
of processes for numerous groups simultaneously. The
development of multilevel path models shows promise to
increase the descriptive power and theory building ability
of social science research.

The multilevel path modeling approach I have developed
is a direct extension of the recent developments in empirical

Bayes multilevel regression models. This is a methodology

by which to represent path analysis within numerous groups.
First of all a path model is stipulated for each group. It
is assumed that the path coefficients vary randomly from
group to group. This variation is modeled by a between-
group regression in which group-level variables predict the
path coefficients. Contextual variables at a higher level
of aggregation are introduced as predictors to explain why
processes vary from group to group. When the errors of the
within-group structural equations are assumed to be orthogonal,
estimates for the first-stage and second-stage parameters
are available via the EM algorithm.

Two hierarchical datasets are analyzed using this
technique. The results indicate that novel insights into
sociological processes can be gained by employing multilevel

path models.

This dissertation is dedicated to my mother and Robin and to

the world which I didn't help improve while I worked on it.

iv

ACKNOWLEDGMENTS

I wish to acknowledge the members of my committee:

Bill Schmidt, for his preserverence and longevity;

Joe Byers for asking the tough questions;

Richard Houang who was a creative sounding board during
the critical derivational phase of the work, and a helpful
friend during the whole project;

Steve Raudenbush who inspired the entire program
conceptually as well as personally. He encouraged my initial
wild speculation and helped me tame my wild prose.

I would also like to acknowledge my friends and family
who cajoled and endured. Finally I wish to remember the three

jokers in the Friday morning group.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

Chapter

I.

II.

Page
viii
ix
STRATEGIES FOR MULTILEVEL DATA 1
Introduction 1
Problems With Multilevel Contexts 6
Path Analysis and Multilevel Contexts . 9
Educational Research and Multilevel Contexts 10
The Hierarchical Bayesian Linear Model 14
Empirical Bayes Through the EM Algorithm 16
The Hierarchical Linear Model . . l8
Hierarchical Path Models 19
Demonstrating the Model 23
ESTIMATING MULTILEVEL PATH MODELS 26
Introduction 26
The General Bayesian Model 30
The Bayesian Mixed Model 34
Mixed Model Posterior Estimates 3S
Likelihood for the Hierarchical Case 38
Structure of the First- Stage Path System 39
Recursive Path Models 49
Change of Variable for the Probability
Density Function of Y . . . 43
The Whole- Group Likelihood . Q7
The Bayesian Likelihood for Many Groups 48
Transforming the Model Into the General
Bayesian Likelihood 50

The Matrix Structure of the Hierarchical Bayesian

Model .
The Likelihood of the Data

vi

51
S7

III. METHODS . . . . . . . . . . . . . . . . . . . 61

Introduction . . . . . . . . 61
Implementation of the EM Algorithm . . 61
EM Formula for Estimating the Second- Stage
Variance Matrix . . . . . 64
EM Formula for Estimating First- Stage
Variance Matrix . . . . . . . . . . . 66
Test Statistics . . . . 70
Statistical Test for Parameter Variances . 70
The Percent of Variance Accounted For by the
Second- Stage Model . . . . . . . . 72
2- Test for Second- Stage Regression
Coefficients. . . . . . 73
Accuracy Check of the Computer Path
Algorithm . . . . . . . . . . . 75
The Model and Data . . . . . . . . . 76
IV. USING THE MODEL . . . . . . . . . . . . . . . . 79
The Analysis of the High School and Beyond Data 79
The Sample and the Data . . . . . . . . . 86
Two Parallel Within- Class
Regression Models . . . . . . . . . 87
The Within- Group Path Model . . . . . . 91
Unconditional Between- Group Analysis . . . 91
Structured Between- Group Model . . . . . 98
The Analysis of Scottish Schools . . . . . . 111
First- Stage Model . . . . . . . . . . . 114
Baseline Analysis . . . . . 117
Second Run of the Scottish Data - Inclusion
of Second- Stage Predictors . . . . 121
V. CONCLUSION . . . . . . . . . . . . . . . . . . 130
Introduction . . . . . 130
Problems With the Multilevel Path Model . . . 132
Practicality . . . 132
Increased Burden of Proper Specification 133
Statistical Tests . . . . . . . . . . . 133
Limitations . . . . . . . . . . 134
Limited Model Definition . . . . . . . . 134
Lack of a Test of Fit . . . . . . . . . 134
Uncorrelated Disturbances . . . . . . . 135
Future Work . . . . . . . . . . . . . . . . . 140

vii

Defining Fixed Effects . . . 140
Adding Intercepts to the Within- Group Path

Model . . . . . . . . 141

Inclusion of a Measurement Model . . . . 143

Between- Group Measurement Model . . . . 146
Group-Level Path Model . . . . . . . . . 146

Full Blown Path Analysis . . . . . . . . 148

APPENDIX . . . . . . . . . . . . . . . . . . . . . 150
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . 152

viii

LIST OF TABLES

Table Page
1-A High School and Beyond Data Parameter Variance

of Paths Parallell Regression Model . . . . . . 89
1-B High School and Beyond Data Average Value of

Paths Parallel Regression Model . . . . . . . . 90
l-C High School and Beyond Data Parameter Variance

of Paths Unstructured Between-Group Model . . . 95
1-D High School and Beyond Data Average Value of

Paths Unstructured Between-Group Model . . . . . 97
2-A High School and Beyond Data Parameter Variance

of Paths Structured Between-Group Model . . . 106
2-B High School and Beyond Data Average Value of Paths

Structured Between-Group Model . . . . . . . . 107
2-C High School and Beyond Data Second Stage Regression

Coefficients . . . . . . . . . . . . . . . . . 108
3-A Scottish School Data Parameter Variance of Paths

Unstructured Between-Group Model . . . . . . . .119
3-B Scottish School Data Average Value of Paths

Unstructured Between-Group Model . . . . . . . .120
4-A Scottish School Data Parameter Variance of

Paths Structured Between-Group Model . . . . . .126
4-B Scottish School Data Average Value of Paths

Structured Between-Group Model . . . . . . . . .127
4-C Scottish School Data Second-Stage Regression

Coefficients . . . . . . . . . . . . . . . . . .128

ix

LIST OF FIGURES

Page
A Many-To-One Relationship . . . . . . . . . . 20
Path Diagram for a Two Equation System . . . . 39

Matrix Structure of a Two Equation Path System 40

Example of A Recursive Path System . . . . . . 40
Example of A Non-Recursive Path System . . . . 41
Example of A Full Recursive Path System . . . . 42
Individual Level Equation System -

Full Recursive Path Model . . . . . . . . . . . 42
Structure of Endogenous Paths in a Full Recursive
System . . . . . . . . . . . . . . . . . . . . 43
Augmented Single Equation Form of Path Model . 4S

Restructured Single Equation Form of Path Model 45
Path Diagram of Two Separate Regression Analyses 85
A Single Path Model Incorporating All Variables 85

Path Model Wilth Indirect Effect of Student
Background on Achievement . . . . . . . . . . . 86

High School and Beyond Data: Baseline Model . 93
High School and Beyond Data: Structured Model 101
Scottish School Data: Baseline Model . . . . . 116

Scottish School Data: Structured Model . . . . 123

CHAPTER I:

STRATEGIES FOR MULTILEVEL DATA

I r du 0

Educational researchers are faced with the task of
studying quite complex phenomena. Students in classrooms
possess a varied and unique personal histories. These
histories interact with a vast array of inherited traits to
form a matrix of propensities that the researcher must unravel.
In addition to the complexity of the individual, researchers
find that students function within a complicated hierarchy
of social institutions: students are grouped into classes,
classes are nested within schools, schools are nested within
communities and so on to national and international levels.

Instead of bracketing out the complexity of educational
contexts through tightly controlled '1aboratory’ experiments,
educational researchers have usually opted to capitalize on
the rich variability found in schools. By studying natural
settings researchers have hoped to address, and offer solutions
to, problems as they naturally occur in schools.

In the past couple of decades there have emerged two
particularly innovative trends in quantitative research
methodology in education that deal with complexity, path
analysis and multilevel linear models. In path analysis,
networks of interrelationships among variables are posited
to represent the interconnected relationships found in real

1

2

life processes. In multilevel linear models, analysis
techniques have been devised to represent real life processes
as they naturally occur in hierarchically nested contexts.
Both approaches seek to represent complexity in a way that
corresponds to the complexity of nature. As of yet, both
approaches have represented disparate lines of inquiry
informing and speaking to each other very little. In this
thesis there will be developed a method which combines these
two trends into a multilevel path analysis. Such an approach
combines the descriptive power of both path analysis and
multilevel linear models, resulting in a single model which
can define a complex network of processes for numerous groups
simultaneously. The development of multilevel path models;
shows promise to greatly increase the descriptive power and
theory-building capacity of social science research.

Path analysis has traditionally emphasized the need for
rich substantive theory. With origins in macro economics
(Theil, 1971) and sociology (Duncan & Featherman, 1973) it
has often been used to model large scale systems, e.g. the
economy, ignoring smaller subunits. For example, a national
analysis might not separately analyze each state economy.
The focus in path analysis is usually on the need to extract
a rich set of variables relevant to the processes being
modeled. In education, sociology and economics, path models
are applied as if one homogenous group were being studied.
The reality of groups imbedded in a hierarchy of social

structure is, for convenience, ignored. This oversight can

3

occur in two ways. In the first case what is regarded as a
single uniform group might actually be composed of numerous
dissimilar social units. For instance, students in a school
might be regarded as a homogenous group for the purposes of
a study while the fact is ignored that students are actually
grouped into classroom units within the school. The opposite
sort of oversight that can occur is the case in which the
researcher ignores the fact that the group under study is
one of numerous social units and each unit might exhibit
different relationships among educational processes. For
exampLe, a single classroom might be studied ignoring the
fact that effects estimated for that class mightiunzgeneralize
to other classes due to differences in the classroom context.
In both of these cases if group membership of subjects were
adequately taken into account, it would be possible to model
processes within groups and then explore the generalizibility
of effects across groups.

Educational researchers have long been concerned with
multilevel issues, But traditional research methods have not
provided adequate tools with which to analyzed data arising
in naturally occurring hierarchies. The paradigm of
educational research has been borrowed from the traditions
of agriculture and psychology in which subjects are randomly
assigned to each of several treatment conditions (Raudenbush
& Bryk, 1988). This assures that the expected effect of
confounding factors is zero. In addition the researcher, if

possible, administers treatment to each subject individually,

4
thus assuring that the responses of one subject is independent
of the responses of another subject (Raudenbush & Bryk, 1988).

Most educational research deviates from this paradigm
in both of its aspects. Usually students are not randomly
assigned to groups such as classrooms or schools and groups
are usually not randomly assigned to treatments. Unfortunately,
researchers find that they cannot control for confounding
factors occurring tunfll at the group and individual level,
”The problem is that the statistical methods the educational
researcher has inherited from experimental psychology provides
little guidance on how to implement such statistical controls"
(Raudenbush & Bryk, 1988).

The problem is further exacerbated by the fact that
usually the independent factor of interest, or the 'treatment',
is not administered individually to each student. Fbm'example,
school factors effect all students within the school at the
same time. In another example a classroom treatment often
is administered to all the students in a class simultaneously.
If students are affected as a group by independent factors,
they will to some extent have a common group history and
group experience. As a result, group responses will tend to
be correlated, not independent. This lack of independence of
responses violates statistical assumptions 1J1 traditional
linear models.

Because of the problems of analyzing multilevel data by

traditional methods a growing number of methodologists have

5
recognized the need to develop new research models which are
multilevel in character.

The roots of multilevel linear models go back to Lindley
and Smith (1972) and Smith (1973) and the definition of the
General Bayesian Linear Model, which defines a linear model
at multiple levels. Later researchers have capitalized on
this theme in research in multilevel contexts (Rubin, 1980;
Strenio, 1981; Morris, 1983; and others). .A single model
using an empirical Bayes approach for a wide range of
educational applications was reviewed by Raudenbush (1984).

The multilevel path modeling approach developed in this
thesis draws its inspiration from the notion of ”slopes as
outcomes" developed by Burstein and others (Burstein, Linn &
Capell, 1978) and is a direct extension of the empirical Bayes
estimation of a multilevel regression, model reviewed by
Raudenbush (1988). What is being proposed is a methodology
by which to represent path analysis within numerous groups.
First of all, a path model is stipulated for each group. It
is assumed that the path coefficients vary randomly from
group to group. In previous multilevel models it has been
assumed that processes are homogenous across groups (Wisenbaker
& Schmidt, 1979; Houang & Schmidt, 1981). The variation of
processes is modeled by a between-group regression in which
group-level variables predict the variance of the path
coefficients. This between-group regression is a way to
explicitly address heterogenous effects across groups. When

variables and the paths among them are properly specified,

6
group-level processes which vary from group to group provide
a source of explanation rather than constituting a violation

of a homogeneity assumption.

III PROBLEMS WITH MULTILEVEL CONTEXTS

.As was pointed out above, path analysis research and
traditional educational research have tended to ignore the
nesting of individuals within groups. In nested contexts
the lack of random assignment leads to confounded effects
and the group-wide administration of treatment leads to
correlated responses. As a result four major problems arise

in the traditional analysis of hierarchically nested data.

1. In analysis of variance studies, the assumption of
the independence of the errors of the units of analysis is
violated making statistical tests invalid. This will result
in an inflated actual alpha level, so that "The result
is an unacceptably high type I error rate." (Barcikowski,
1981). This occurs because the precision of the effect is
misestimated. Precision will be overestimated as a function
of' sample size and tntraclass correlation (Walsh, 1947).
Test statistics that are not corrected for this inflation
will have high type I error rates whenever the intraclass
correlation is greater than zero.

2. Aggregation bias can occur so that estimated
relationships at one level of aggregation may be quite

different from those at another level. This most commonly

7

occurs when the grouping variable is related to the outcome
(Cooley, Bond & Mac, 1981). For example, consider a regression
analysis predicting student achievement from student SES.
Suppose data is aggregated to class means and average
achievement is regressed on average SES. If students are
tracked into classes according to pretest achievement, the
regression coefficient estimated from class means will probably
be much larger than that estimated from individual scores.

Aggregation bias may also occur simply because processes
at one level are different from processes at another level
(Burstein, 1980: Cooley, Bond & Mao, 1981). This could be
because " Variables have different meanings at different
levels of analysis.” (Burstein, 1980) For example, a variable
may gauge a student's desire to work alone. At the student
level the variable may measure autonomy and motivation. But
aggregated to the class level the 'variable may indicate
group divisiveness.

Alternatively, there may be factors at one level of
aggregation that are absent at another level and which moderate
the process being studied. For example, a school district
might provide low achieving classes with tutors to coach
classes in the state achievement exam. Low achieving
classrooms would increase in mean achievement, thus attenuating
the SES/achievement relationship at the class mean level.
But within each classroom, student SES might still have a

large relationship with achievement.

8

3. Cross-level interactions alter individual level
relationships from group to group. Cronbach and Webb (1975)
realized that characteristics of the classroom interacted
with and altered processes occurring among individuals. He
saw this as a barrier to the formulation of scientific
theories. Cronbach believed that the interaction of the
setting with treatment made each study unique rendering it
impossible to generalize beyond each setting and blocking
the establishment of general theory.

4. Concentrating on one level of analysis loses
information at the other level, and ignores cross-level
interactions. Researchers can, for example, pool data within
groups which enables them to take out the mean group effect
and thus ignore group boundaries. This approach ignores
possible setting-by-treatment interactions and assumes all
groups have identical within-group processes. This of course
is only tenable when effects really do happen to be uniform
across groups. At the other extreme, researchers can aggregate
to group means, which leads to the problem described by
Page: "More rigorous investigators are apt to suppress most
of the richness within the classroom by using class means"
(1975 p.339). In this case possible cross-level interactions
are ignored as are problems of aggregation bias. Single-level
analyses must assume that effects are homogenous over groups.
It would be better not make such a restrictive assumption
and model the variation that occurs in effects from one

group to another by group-level variables.

S N U I V 0

Path analysis offers promise for establishing theories
in complex social contexts. .A network of variables tested
in a path analysis has an exact correspondence to a network
of processes proposed by substantive theory. lhuzpath analysis
has a history of largely neglecting the issue of hierarchical
nesting of subjects. One of the earliest attempts to address
this issue is found in Schmidt (1969) who articulated a
maximum likelihood technique for partitioning a covariance
matrix into orthogonal within-group and between-group parts.
This approach was later extended by Wisenbaker and Schmidt
(1979) to include structural models. A major limitation of
these techniques is that they required that the group sizes
be equal, constraining the practical applications.

Bianchi (1987) devised a Bayesian estimation technique
which employed the EM algorithm (Dempster, Laird Rubin,
1977) to provide maximum likelihood estimates of latent
random effects in the case of unequal group size. With the
unequal-n solution in hand, it is possible to apply a
partitioned covariance structure solution to natural settings.

Houang and Schmidt (1981) surveyed various methods of
partitioning estimates into orthogonal between-group and
within-group components in a context mostly pertinent to
regression applications. They devised an analytical model
which encompassed most of the partitioning approacheslnuzthey

constrained within-group effects to be uniform across groups..

10

These approaches have remedied the first two problems
hierarchical data described above; the proper estimation of
precision and the avoidance of aggregation bias. However
the problem of cross-level interactions, was not addressed
by these models. Because all of these approaches partition
structural parameters into independent within—group and
between-group parts, the within-group partition is a single
set of parameters using information pooled from each of the
groups. This pooling of information is predicated upon the
assumption that the within-group parameters are identical
for all groups. Also, as they have been defined in the
literature, these models do not allow for group level variables

that are not also defined at the individual level.

V T ONA RESE CH AND U L V 0N EXTS

Investigators using quasi-experimental designs have become
increasingly aware of the problems for analysis posed by
multilevel data. As early as 1940 McNemar recognized the
problem of inflated alpha level for statistical tests, although
he did not articulate the causes of what has come to be
known as the "unit of analysis" problem. In educational
research the problem has taken on the aspect of a Devil's
bargain, as stated by Class and Stanley (1970, p.507), "The
researcher has two alternatives, though he is seldom aware
of the second one: (1) he can run a potentially illegitimate
analysis on the experiment by using the 'pupil' as the unit

of statistical analysis, or (2) he can run a legitimate

ll
analysis on the means of the classrooms, in which case he is
almost certain to obtain statistically nonsignificant results."
In my terms, the bargain is this: We can ignore groups and
get problems one and three, cited above, or we can aggregate
and get problems two, three and four.

More recently, researchers have begun to realize that
they need not accept this no-win approach to research.
Glass and Stanley framed the multilevel problem purely in
terms of problem 1, improper ANOVA estimates. Hopkins (1982)
sought to remedy the dilemma by devising a mixed model ANOVA
approach for individuals nested within groups, and groups
nested. within treatments. Barcikowski (1981) explicitly
defined the relationship between group size, intraclass
correlation, actual effect size, and power in the ANOVA
context. Cooley, Bond and Mao (1981) explain the origin of
several species of aggregation bias and suggest multilevel
structural equation modeling, of sorts, to remedy the
situation.

Most educational researchers have not addressed the issue
of cross-level interactions in multilevel contexts. Cronbach
and Webb (1975) recognized the salience (and magnitude) of this
issue. They contended that characteristics of the research
setting (or, group-level variables) interact with.within-group
treatment effects, destroying the external validity of most
quasi-experiments. Taking this lead, Burstein, Linn and
Capell (1978) addressed the problem of multilevel data in terms

of regression analysis. They suggested a model in which

12

regression parameters vary from group to group. In this way
the variability of processes could be defined in the model,
obviating the need to assume homogeneity across groups.
Moreover, Burstein et al, proposed the notion of "slopes as
outcomes", that is, using group characteristics as predictors
for the within-group slopes. In the simplest case, the
relationship between an outcome and a predictor is represented
by a within-group regression weight, or slope. The slopes
from all groups are then treated as outcomes for a second-
stage analysis. In the second-stage analysis group-level
variables predict the slopes in a multiple regression.
Hanushek (1974) proposed using slopes as outcomes as a means
of combining numerous regression studies, but his approach
required slopes tn) be statistically independent. .Although
Burstein, Linn and Capell realized the implication of this
approach for explicating cross-level interactions they did
not delineate the statistical properties of the least squares
estimates they proposed (Houang and Schmidt, 1981), leaving
issues of statistical assessment of estimators and variance
accounted for unresolved.

A problem with slopes-as-outcomes models which is even
more serious than the lack of an overall statistical framework
is the fact that slopes are very unreliable. The sampling
variance of beta weights is usually much larger than sampling
variance of ordinary outcomes, such as means. This

unreliability will often mask the effect of group-level

l3
predictors (Raudenbush and Bryk, 1986), washing out the
information to be gleaned from modeling the slopes.

Another statistical problem has to do with the fact
that the slopes vary in precision from group to group. For
the second-stage analysis the slopes are outcomes to be
analyzed by an ordinary least squares procedure that assumes
equal precision for each slope. As a result of the violation
of this assumption, the second-stage estimation procedure is
less efficient leading to less precise second-stage parameter
estimates. Because of this, it is more difficult to
demonstrate the relationship between group-level variables
and slopes (Raudenbush & Bryk, 1986).

A final problem has to do with the fact that the
variability of slopes can be partitioned into two components;
parameter variance, which represents the real differences in
the slope parameters from group to group, and sampling
variance, which is the error in slope estimates due to
sampling. Only the parameter variance can be explained by a
between-group model. For example, a between-group model
which explains only a small portion of the total variance
may in fact explain virtually all of the parameter variance.
Unless parameter and sampling variance can be distinguished
it will be very difficult to assess how well the between-
group model accounts for slopes. The slopes-as-outcomes
model does not provide means for partitioning slope variance

(Raudenbush & Bryk, 1986).

14
W

A statistical theory which had promise to flesh out the
slopes as outcomes approach was initiated when Lindley and
Smith (1972) used Bayesian theory to provide alternatives to
least squares estimates of the general linear model. The
result was a hierarchical Bayesian linear model (Smith,
1973) in which structural parameters could be estimated for
a two-stage hierarchy. These derivations assume that the
dispersions and structural coefficients of the prior
distributions are known.

In general, what the family of Bayesian linear models
provide is a scheme in which a model can be specified in two
stages. The first stage describes the data, given first stage
parameter vector, B;

Y - XB + R,
with "X" containing the fixed predictors and "R" containing
the random errors. The second stage describes the first
stage slope parameters, given second stage parameters 1;

B - W1 + U,
with "W" being fixed predictors and "U" being random errors.
A third stage defines the second stage parameters;

1 - AC + L.
This third stage simply defines our prior degree of certainty
about the value of 1 (Smith, 1973). The variance of the errors

are;

15
Var(R) - W,
Var(U) - T,
Var(L) - F

The goal is to get Bayesian or posterior estimates of the
first and second level parameters (B and 1), given Y and C.
Note that X, W, C, i, T and P are assumed to be known.

What makes these models peculiarly Bayesian in character
is the notion that the first and second stage parameters, B
and 1, have distributions. When the data are normal these
distributions can be described by linear models. The
motivation for shifting focus from classical estimators is
twofold. First, under most conditions the posterior estimates
of parameters have smaller expected means squared error than
classical estimators (Efron and Morris, 1977). Second, the
notion of parameters as random latent variables can be
conceptually appealing in multilevel contexts where within-
group effects are commonly seen to differ from one group to
another across a population of groups.

The hierarchical Bayes linear model proposed by Lindley
and Smith fits rather naturally into a notion like "slopes as
outcomes" where the first stage parameter 'vector, B, is
interpreted as the within-group slopes, and the second-stage
parameter vector, 1, is interpreted as the between-group
regression coefficients of "B" predicted by group-level
variables found in "W”.

An example of an application of the hierarchical Bayesian

linear model is found in Rubin (1981). This is an instance

16

of an empirical, Bayesian application of this model to
educational research. Empirical Bayes is ndifferent from
pure Bayes in that prior distributions of parameters are
estimated from the data, instead of being given. Although
Rubin assumed prior dispersions of the data to be known, he
used the data to estimate the prior location and dispersion
of the second-stage parameters. This study demonstrated how
Bayes and empirical Bayes techniques can give superior
estimates of treatment effects by combining information from
within-group and between-group sources.

Rubin (1981) had used a graphical method to get a maximum
likelihood estimate for second-stage dispersions, which was
an option available in the simplified model he employed.
Generally, though, there was at the time no uniform method
for estimating prior dispersions. Ina order to apply the
hierarchical Bayesian linear model one had to work out a

solution pertinent only to a specific, less complicated case.

V C L HROUG G

Widespread acceptance of the EM algorithm led to a
practical approach for estimating prior dispersions. Dempster,
Laird and Rubin (1977) outlined a general formulation of the
EM algorithm, an iterative computational method which yields
maximum likelihood estimates for a wide variety of estimation
problems. It was termed "EM" because each iteration consists
of an Expectation phase followed by a Maximization phase. The

power of this algorithm is that it will give estimates for

17

'incomplete' data when one specifies maximum likelihood
equations for 'complete' data. In linear models with normally
distributed data the 'incomplete' data consist of the observed
outcomes. The 'complete’ data include the observed data plus
certain latent variables, e.g. second-stage errors. If the
complete data were observed it would be a simple matter to
obtain maximum likelihood estimates for dispersions. By
acting 'as if' one had complete data one can greatly simplify
maximization equations. In the expectation phase dispersions
are treated. as known quantities, and. the "complete data
sufficient statistics" needed for the M step are estimated.

Generally, the algorithm works like this: In the "E"
step, parameter estimates from the previous iteration are
used to calculate the conditional expected value of the
"complete data" sufficient statistics, given the observed
(incomplete) data. So in this step, sufficient statistics
are derived as if parameters were known. In the "M" step,
the sufficient statistics from the "E" step are used to
calculate the maximum likelihood estimates of parameters.
So estimates of parameters are derived as if the complete
data had been observed.

By bouncing back and forth between the "E" and the "M"
step, the likelihood converges to a maximum. If the algorithm
is applied to data that is normally distributed, it should
converge to a global maximum and yield asymptotically efficient

ML estimates (Raudenbush, 1984).

18
W

The general formulation of the EM algorithm paved the way
for the widespread implementation of the empirical Bayesian
linear model. Strenio (1981) first implemented the EM
algorithm for this purpose. Raudenbush (1984) devised a
mixed model empirical Bayes approach he called the Hierarchical
Linear Model, or HLM. This is a very flexible model that
can be tailored to apply to three, heretofore disparate,
realms of research; School effects research with regression
modeling, meta-analysis with treatment effects, and growth
curve estimation for individual students. Other investigators
have used the EM algorithm in mixed linear models (see Laird
6: Ware, 1982; Strenio, Weisberg & Bryk, 1983; and Mason,
Wong, 5: Entwisle, 1984). Other approaches to dispersion
estimation have been proposed by Goldstein (1986), and Longford
(1985). Raudenbush (1988) reviews these developments.

In hierarchical linear models for school effects research,
there is a single regression model which is estimated in
numerous groups. This regression model is usually
characterized by there being one predictor of interest and
several covariates which control for the confounding effects
of student background characteristics.

In hierarchical linear models for meta-analysis there
is a set of separate studies all focusing on a similar
'treatment' issue. The effect from each group is usually a
standardized mean difference between treatment and non-

treatment groups .

19

The HLM of individual growth studies (see Bryk &
Raudenbush, 1987) look at student growth over time on an
outcome of interest, controlling for background characteristics
of the individuals.

Raudenbush (1988) demonstrates how the slopes as outcomes
interpretation of the hierarchical Bayesian linear model can
elucidate all of these research. contexts. This unified
approach to multilevel analysis is characterized by a)
heterogenous effects b) separate estimates of sampling variance
and parameter variance. of the first level parameters c)
posterior estimates which offer smaller expected mean squared
error than corresponding least squares estimates and d)
between-group predictor coefficients of the first level
parameters. This model speaks to all four problems of

multilevel analysis that have been outlined.

V C T OD S

There is an important limitation with the HLM approach:
it only uses one type of model (multiple regression) to
depict within-group processes. In the research paradigms where
the primary interest is with the relationship between several
independent variables and one dependent variable, multiple
regression is conceptually appealing. A multiple regression
depicts a 'many-to-one' type of relationship, as shown in

figure 1.1.

20

Figure 1.1

A Many-To-One Relationship

If' this relationship represents the .actual processes
according to substantive theory, a multiple regression defines
a structural equation. But if there are multiple interrelated
outcomes, multiple regression (and ANCOVA, which can be put
in terms of multiple regression) defines a prediction
relationship only and causal imputations can be extremely
misleading. For example, suppose that in actual classrooms,
enjoyment of reading contributes to reading comprehension,

and comprehension contributes to reading achievement:

 

Enjoyment > Comprehension >

 

Achievement

A regression analysis predicts achievement by enjoyment

and comprehension to give the following:

Achievement - B0 + B1 (Enjoyment) + B2 (Comprehension).

To impute a direct effect of enjoyment on achievement from

what could be a large regression coefficient would quite

distort the picture. Researchers often wish they could draw

21
structural conclusions from predictive equations, and not a
few succumb to the temptation to do so (if only in the
sanctuary of their own thoughts).

Muthen and Satorra (1987) surveyed numerous modeling
issues connected with multilevel structural models. They
broached the questions of l) heterogenous group parameters
and 2) correlated within-group responses. The discussion
ranges over a wide variety of issues to do with assumptions
of the nature of regressors, (fixed, random or latent) and
whether various parameters are homogenous or heterogenous
across groups. When they considered strategies for esti-
mating the models they defined, they were less than optimistic:

"It is clear that today's standard structural equation

modeling techniques and software cannot fully serve
the purposes of an appropriate multilevel analysis."

(13.19)

In this thesis the challenge will be taken to start
filling in the gap between theory and implementation for
multilevel structural models.

The model being proposed here deals with a particular
subclass of models considered by Muthen and Satorra. In the
first stage a path model is defined within each group. The
path model is the same for all groups but the path coefficients
can randomly vary from group to group. A single group model
for group j is given by;

Yj - Zij + RJ,

22

where R represents a vector of random errors for group j, 23
is a vector of fixed predictors which includes the endogenous
predictors found in Yj» and Bj is a vector of random path
coefficients.

The first-stage parameters are modeled at the second
stage by a between-group regression model of the form:

B - W1+ U,

Where B is the vector of path coefficients from all groups,
W is a set of fixed between-group predictors, 1 is a set of
fixed between-group coefficients and U is :1 set of random
errors.

In terms of Muthen and Satorra (1987) several features
define what subclass of possible hierarchical structural models
this is:

l) The within-group predictors, Zj, are fixed.

2) The within-group coefficients, BJ, are random and
heterogenous across groups.

3) The variance of the second-stage errors, Var(Uj) - r, is
homogenous across groups.

4) The second stage predictors, W, are fixed.

5) The second-stage parameters, 1, are fixed.

Other choices were possible for each of the five options
indicating that there are a large number of different
hierarchical models that can be devised.

An hidden feature of this model is the structure that
defines the path analysis, ZB. This appears to be identical

to a 'regression. model. But. the matrices are specially

23
constructed so that "2B" represents a series of structural
equations stacked on top of each other. Details of this

structure will be discussed in chapter two.

I MO E

In subsequent chapters the mathematical model for the
hierarchical path analysis will be defined. Then the computer
algorithm that was developed to implement the model will be
discussed. Finally, the efficacy of the model for explicating
educational research will be demonstrated by using the model
to analyze two actual data sets.

The efficacy of the model for explicating educational
research will be demonstrated by presenting the analysis that
has been done on two actual data sets. The first analysis was
drawn from a large scale research project called the High
School and Beyond study (Coleman, Hoffer & Kilgore, 1982).
This study measured variables at the student level and at
the school level in nearly a thousand schools across the
country. The students were measured on mathematics achievement
and various background variables. It has been found by
various researchers that the relationship between student
background (SES and race) and achievement is less strong in
Catholic high schools than in public high schools (Coleman,
Hoffer & Kilgore, 1982: Hoffer, Greeley & Coleman, 1985). A
conclusion that has been drawn is that Catholic schools are
more egalitarian than public schools since academic success

depends less on students' background in Catholic schools.

24

Data from a random sample of 158 schools will be analyzed.
The purpose of the analysis is to explain why Catholic schools
appear more egalitarian than public schools, or rather to
identify the school level characteristics account for the
discrepancy between public and private schools with respect
to the relationship of students' background to students'
achievement.

In a second demonstration of the multilevel path model
variables from a study by Willms (1987) will be reanalyzed.
This data consists of observations taken from 21 secondary
schools in one administrative division in Scotland. Measures
were taken on various student background variables. Students'
academic success was gauged by verbal reasoning score and a
score (”I a comprehensive achievement exam. School means
obtained by aggregating the data the group level were
introduced to measure school context. The original study by
Willms examined mean student achievement, controlled for by
student background and verbal reasoning.

In the present analysis the student level processes will
be construed as a network of causal relationships in which
student background affects academic achievement indirectly
through students verbal. ability. The within-school. path
coefficients will be modeled by a between-school regression
in which school context predicts variations in the processes
by which students achieve educational goals.

It is hoped that these two analyses will demonstrate that

hierarchical path models are feasible emu! that they can

25
significantly add to our understanding of educational processes

in their full multilevel contexts.

CHAPTER II:

ESTIMATING MULTILEVEL PATH MODELS

odu o
In this chapter estimators for parameters of the general
Bayesian linear model will be derived. The general Bayesian

model takes the individual form;

Y - A6 + R, (2.1)
where;
Y - is a K by 1 vector of outcomes for an individual,

with K - the number of outcomes;

A - is a K by 3 matrix of predictors, where s is the
number of structural parameters;

6 - is an S by 1 vector of parameters;
R - is a K by 1 vector of random errors.

This model is Bayesian because the parameters in 9 are
assumed to be random terms with a prior probability
distribution.

It will be assumed 6 has a normal prior distribution with mean
zero and dispersion matrix 0. In Bayes terms this represents
our prior belief about the location of the parameter vector,
9, and the precision of this prior belief (represented by

0’1). Estimation in the Bayesian context involves calculating
the posterior distribution of the parameters, that is finding
the posterior density function fKOIY). .After the formula

for the posterior distribution of 8 is derived, the mean

26

27
vector of this posterior distribution will be used as the
vector of point estimates of the vector parameter.

By an application of Bayes theorem to continuous
probability densities it can be shown that the posterior
density function is proportional to the product of two
independent density functions; f(6|Y) and f(9) (and a constant
term which drops out). This gives rise to the proportional
relationship, signified by or (see Hoel, Port and Stone,

1971, section 6.3);

new) a f(Y|9)f(9) . (2.2)

The first term, f(Y|6) represents the likelihood. of the
data, and the second term, f(9), represents the prior
distribution of the parameters. This division is fortuitous
because it enables one to develop the likelihood and the
prior distribution separately and bring the results together
in one expression. This greatly simplifies the exposition.
An expression for posterior density of 6 given Y can be
had rather straightforwardly by substituting the normal
probability density functions for f(Y|6) and f(6). Then by
multiplying, combining and simplifying terms a1 probability
density function, f(9|Y), results which is recognizably
normal. This expression will reproduce the standard Bayesian
results for the General Bayesian Linear Model. This general
solution is not hierarchical, i.e. it doesn't define parameters

at two levels of analysis.

28
In order to define a hierarchical linear model we must

reparameterize, using the substitutions;

A - [ZW : Z] , (2.3)
and,
-1-
6 _ __, (2.4)
_U—

 

 

By substituting A and 6 into Equation 2.1 we get;

Y - 2W1 + Z0 + R . (2.5)

By introducing an identity for a new parameter, B, we can

decompose the model into two stages. The identity is,

B - W1 + U

Substituting this into Equation 2.5 leads to a two-stage
expression,

Y - ZB + R (2.6)
and,

B - w», + U . (2-7)

A convenient interpretation for this two-equation
expression is that forms a two-stage hierarchy in which Y -
28 + R represents a linear model within-groups and B - W1 +
U represents a between-groups linear model for the within-
group parameter vector, B (following Smith, 1973).

In this thesis the within-groups model represents a path

model defined within numerous groups in which set of paths for

29
each group j, Bj, differs from group to group. The between-
group model is a multiple regression in which group-level
variables, W, predict the group paths, Bj- The hierarchical
Bayes model enables us to model processes at the within-
group and between-group levels of analysis simultaneously,
which is the strength of the multilevel modeling approach.

In this chapter the Bayesian estimates of parameters will
first be derived for the general Bayesian linear model of
Equation 2.1. It will be shown that the general Bayesian
results are valid for the substituted or 'mixed model' case
represented by Equation 2.5. Finally by switching to the
two-stage model in Equations 2.6 and 2.7, it will be shown
that by stipulating a recursive path model at the within-
group level, we can justify the assumptions made in deriving
Bayesian estimates.

Throughout the derivation of the estimates it is assumed
that first and second stage variance matrices are known.
This is usually an untenable assumption. For this reason,
the EM algorithm, an empirical estimating routine, is used
to provide maximum likelihood estimates of the variance
terms. In the last section of this chapter I derive the
likelihood of the data, conditioned on the variance parameters.
It is this likelihood which is maximized by the EM algorithm.
The derivation of the formulas used in the EM algorithm will

be developed in chapter 3.

30

I G e e e
The general Bayesian linear model (Smith, 1973) as
depicted in Equation 2.1 was defined for one individual. We
will now define the model in terms of a whole group of N
individuals with K outcomes per individual,
Y - A6 + R,

where,
Y is a Kle vector of outcomes,

A is a KNxQ matrix of predictors,

9 is a Qxl vector of random structural coefficients,

R is a Kle vector of random errors,

N is the total number of individuals,

K is the number of outcomes observed for each
individual, and

Q is the number of parameters in the model.

The variance of the errors is,

Var(R) - W (2.8)
where,
W is a NK by NK variance matrix.
The sampling errors, R, are independent of the parameters
represented by 9. As a result 9 is assumed to have a normal

prior distribution,

6 ~ N (0’ 0) . (2.9)

where,

31

0 is a zero vector representing the prior mean and

O is the QxQ prior dispersion matrix.

The assumption of a zero prior mean can be made without loss
of generalizability (Raudenbush, 1984).

In order to find Bayesian point estimates of a model, one
must first derive an expression of the posterior distribution
of the parameters given the data and conditional on the prior
distribution of the parameters. In terms of the general
Bayesian linear model, if we have a prior normal distribution

of parameters, the posterior distribution has the form;

(a | Y,O) ~ N (9*,D9*), (2.10)
where 9* is the posterior mean and D9* is the posterior
variance matrix (Raudenbush, 1984). An explanation of how
to find a posterior distribution by employing Bayes theorem,
the heart of Bayesian estimation theory, is given in the
following section.

Recall from Equation 2.2 that the posterior density is
proportional to the product of the likelihood of the data

and the prior density of 6 or,

f(6|Y) a f(Y|9) f(6)

First we will focus on the likelihood, f(Y|9).
Conditional on 6, the errors of the observations will be
independent across individuals. Also, if we assume that we
have a well-specified, recursive structural equation system,
the errors for K outcomes observed for each individual will

also Ina independent (Land, 1973). This latter assumption

32
can be explicitly justified when we couch the model in
hierarchical terms in a later section. Under these assumptions
Land (1973) has shown that the ,joint 'normal probability
density can be depicted as the product of the densities of

the NK separate observations,

N K
f<Yil:-°~»YiKv --- :Yva---:YNK) 'inl knlf(Y1k) - (2-11)

When the errors between individuals and between measures are
uncorrelated, the variance of the errors, 9, is an NK by NK
diagonal matrix. When this is the case, the probability
density function depicted in Equation 2.11 takes the specific

form,

f(Y|6,W) -

(2«)'NK/2 lwl'l/Z exp{-l/2(Y - A9)’ 0'1 (Y - A6)}. (2.12)

This completes half the task» that. of (defining the
likelihood of YE The prior distribution of 6 follows from
the assumptions that 9 is normally distributed with zero

mean and dispersion matrix 0,
£(e) - (2«)'Q/2 lol'l/2 exp{-l/2(6’ 0'1 9)) , (2.13)

All that remains now is to combine f(Y|6) with f(6) to
get f(9|Y). We accomplish this by multiplying Equation 2.12

by 2.13 to get,

33
f(6|Y,i,O,A) «

(2«)-KN/2 (2,)-Q/2 |¢|-N/2 Inl-l/Z

expl-l/2(Y-A9)' 0'1 (Y-A9)} exp{-l/2 e' 0'1 e) .(2.14)

By expanding, combining terms, completing the square in
the quadratic term and combining constant terms (see appendix),
Equation 2.14 can be shown to be proportional to the following
density:

f(9|Y,i,O,A) «
exp {-1/2[e-(A'0'1A+o‘1)A'0'1Y]' (A'W'lA + 0'1)

[e-(A'0'1A+0‘1)A'w'1Y]) (2.15)

This is a normal density function with the mean and
covariance of 6 clearly visible. The posterior distribution

of 9 is therefore defined as follows;

(9|Y,0,o,A) ~ N(9*,D9*) , (2.16)

with, 9* - (A'W'IA + 0'1) A'w'lY , and

09* - (A'W'IA + 0'1)

This reproduces standard Bayesian.results (see Raudenbush,
1988). The estimate of 9 will simply be the mean of the
posterior distribution of 9 or, (A'it'lA + 0'1)A'0'1Y, from

Equation 2.16.

34

11 e e M el

We can write the mixed model form of the Bayesian model

by substituting Equation 2.7 into Equation 2.6 to yield;
Y - 2W1 + ZU + R

The first level parameter matrix, B, has disappeared so that
the parameters of this model are 1 and U. The parameters
have the prior normal distributions,

1 ~ N(O , F), and

U ~ N(0 , T).

The prior distribution for '1 serves the jpurpose of
representing our state of knowledge about 1 (Smith, 1973).
We assume 1 has an arbitrarily large variance matrix so that
the precision of 1, or P‘l, is near zero. This indicates a
complete lack of knowledge about the prior distribution of
1. Such a distribution has appropriately been termed a
vague prior (Smith, 1973). The mean of 1 is set to zero for
convenience, since with a vague prior the location of the
parameter is arbitrary (Raudenbush, 1984).

Because there is 1H) prior information about '7, it is
functionally equivalent to a fixed effect , while U is
considered to be a random effect (Dempster, Rubin & Tsutakawa,
1981). The combined model, then, loses the hierarchical
character and can be thought of as a one level mixed model.

Both of these conceptualizations, hierarchical and mixed

model, will be used in this chapter. It is important to

35
keep in mind that both of these conceptualizations are
equivalent.
The parameters of the mixed model, 1 and U, are assumed
to be independent so that the joint prior covariance of the

parameters equals,

 

 

 

 

‘91- r o
Var ---- - - O. (2.17)
-92- -0 T-
e ode e o a es
The estimating algorithm used in this thesis was designed
in terms of the Bayesian mixed model. This is because the

mixed model affords two statistical advantages over a
hierarchical model; it allows for groups 13) be analyzed
which have data matrices that are not full rank and it allows
for a flexible definition of which effects are fixed and
which effects are random. These issues will be taken up in
the concluding chapter.

A more general way to represent this model is to posit
the following definitions,

A1 - ZW; A2 - Z; 91 - 1; and 92 - U
Substituting these into Equation 2.5 we get the general form

for the mixed model,

Y - A191 + A292 + R, (2.18)

where 91 is the vector of fixed effects and 92 is the vector

of random effects.

36

A simple substitution show that there is a correspondence

between the mixed model and the General Bayesian model.

If
we set,
A - [A1 : A2] ,and (2.19)
-61-
6 _ __-- . (2.20)
-92-

 

 

and substitute into Equation 2.18,

we see that the mixed
model is just a partitioning of the General Bayesian model,
Y - A6 + R

Also note that the variance of 6, O,

is simply the joint

covariance of 61 and 62 as defined in Equation 2.18,

91
-92-

The formula for the posterior mean in Equation 2.16, calls

Var (6) - Var

 

 

 

for the inverse of O which is,

"r'1 o

 

 

Recall that the prior precision of 1 is F'l, which is virtually

zero. This leads to the results,

 

 

(2.21)

37
With these definitions in hand, we can now substitute
Equations 2.19, 2.20 and 2.21 into Equation 2.16 to get an
expression of the posterior distribution of parameters in terms
of the general mixed model.
By using various identities and simplifications it is
possible to derive the following mixed model definition of

the posterior distribution of parameters (see Raudenbush,

 

 

 

 

 

 

 

1988).
(2.22)
r w r __ __ l
l
_ _ _ *_

91 91 D11 I D12
---- (Y, ¢, 0, A1, A2) ~ N --;- : ...... | ......
-92- -92- I

D21 I D22
. , __ I -_
L. J

 

 

With:

91* - D11A1'(I- 0'1A20'1A2) 0'1Y

92* - C'1A2'W'1 (Y - A191*)

0 - A2'0'1A2 + 1'1

011 - (A1'0'1A1 - A10‘1A20'1A2'0'1A1)
D12 - D21' - - 0111‘11"I"11¥2C'1

D22 - C'1 + C'1A2'W'1A1D11A1'W'1A2C'1
The mixed model solution subsumes the special case in
which the model is strictly hierarchical as in Equation 2.10;

Y - 2W1 + ZU + R ,

38
where Z is a matrix of first-stage predictors and W is a matrix
of second-stage predictors. This is equivalent to the general
mixed model in Equation 2.18 only if ZW - A1, 2 - A2, 1 =
91, and U - 92. When these conditions are met ZW, Z, 1, and
U can be substituted into Equation 2.16 to yield simpler

equations for 91*, 92*, and D.
Simpler equations are desirable but there are important

cases in which A2 does not equal 2. For this reason we will

adhere to the more general mixed model approach.

IV e o for t e rarchica Case
In the derivation of posterior estimates for the general
Bayesian linear model, it was assumed that the K outcomes
observed for each individual had independent errors. This
assumption enabled us to devise a simple expression for the
likelihood of the data given parameters (Equation 2.12). In
order to justify this assumption. we must appeal to the
hierarchical form of the Bayesian model.
In the hierarchical path model a vector of first-stage
paths, B, is introduced as the set of parameters,
Y - ZB + R,
B - W1 + U
The probability distribution function of the data conditional
on B is f(Y|B), which is the likelihood of the data parallel
to f(Y|6). Since the only random terms that both 6 and B
contain are 1 and U, conditioning on B is equivalent to

conditioning on 9. For this reason the derivation in this

39
section of an expression for the likelihood f(Y|B) will be
pertinent to the expression for the general Bayesian likelihood
in Equation 2.12.
Conditional on first-stage paths, B, the model Y — ZB +
R represents a path model with fixed exogenous predictors in
Z and fixed paths, B. This defines an ordinary, non-Bayesian

path model.

Structure of the First-Stage Path System

In order elucidate the structure of the First-stage path
model we will examine a simple example in which the model
Y - 28 + R represents a two equation path system for an

individual, as illustrated by the path diagram in figure 2.1,

 

Figure 2.1
Path Diagram for a Two Equation System
In this example X is an exogenous variable because it
has no antecedents in the path system, while Y1 and Y2 are
endogenous variables because they both have causal antecedents
defined in the system.
The structure of the matrix equation which depicts this

path system is illustrated by figure 2.2.

40

Y - z B + R
_Y1- ‘x o 0‘ 73x17 _R1—
- Byl +
_Y2_ _0 Y1 X- -sz- _R2_

Figure 2.2

Matrix Structure of a Two Equation Path System

This figure shows that each row’ of Z contains the
predictors for one equation. Note that exogenous path are
subscripted with x and endogenous path are subscripted with
y. If we perform the matrix post-multiplication of 2 with B
and properly add elements we see that the matrix equation, Y
- ZB + R is a shorthand way of writing the two equations,

Equation 1 Y1 - XBx1 + R1

Equation 2 Y2 - Y1By1 + XBXZ + R2

_§cursive Path Models

 

Throughout this thesis focus will be restricted to the
subclass of path models which are recursive. A recursive
path model is one in which there are no causal loops. A
simple recursive path model is illustrated by the path diagram

in figure 2.3.

 

x Y3

Figure 2.3
Example of A Recursive Path System

41
The arrows stand for causal connections between the
processes represented by X, Y1, Y2, and Y3. In this case X
is an exogenous variable because no antecedents are defined
for it. The endogenous variables are the Y's because they
have causal antecedents defined within the model. Notice

that the causal flow is in one direction, from left to right.

A non-recursive path system is illustrated by figure
2.4. We arrive at this model by simply reversing the direction

of the arrow between Y1 and Y3.

 

Y1 :rYz

X Y3
Figure 2.4
Example of A Non-Recursive Path System
If you trace the causal flow from Y1 to Y2 to Y3, you will find
that you will loop back to Y1 again. Any such loop makes a
path model non-recursive.

The reason we are limiting ourselves to recursive path
models is that they define a class of models that have readily
interpretable parameters which are also easily estimated.
Fortunately, recursive path models alsotdescribe most processes
of interest in the social sciences. Hunter and Gerbing
(1980) states that in any non-recursive path system can be

transformed into a recursive one if it is represented

42
longitudinally, because causal paths cannot loop backward in
time.

Recursive path models have a certain general structure
which will be exploited in the derivation below. To see the
general structure we must define a £211 recursive model. If
we add some paths to figure 2.1 we will get the full path

model in figure 2.5.

 

 

X =TY3

Figure 2.5
Example of A Full Recursive Path System

This model is full in the sense that if any path is added, the
model becomes non-recursive. The individual level equation

system represented in figure 2.6 illustrates the structure.

Y1 - XBxl + R1

Y2 - YlByl + Xsz + R2

Y3 - Y1By2 + Y2By3 + XBX3 + R3

Figure 2.6

Individual Level Equation System - Full Recursive Path Model
All of the terms are scalars. The paths for the endogenous
and exogenous variables have been differentiated. The Bx's
are the exogenous path coefficients and the By's are the
endogenous path coefficients. Notice that the equations

compound in a stepwise fashion. Each equation has as

43
predictors a) the exogenous variable and b) all of the
endogenous variables in the equations above it. Every
recursive system can be ordered in such a way as to display
this structure. If we take the endogenous predictors and
put them in a matrix as they appear in the equation system,
we get the structure in figure 2.7.

0 (zero)
B - B 0
y Y1

By2 By3 0

Figure 2.7
Structure of Endogenous Paths in a Full Recursive System
From this we see that in recursive path system the
matrix of endogenous predictors will be a lower triangular
matrix. This is another way to define a recursive system.
This fact will come in handy when defining the likelihood
function of Y in the next section. Note that a less-full
recursive system would have some of the paths missing, so
some of the By's in figure 2.7 would be set to zero. The

general structure would be preserved, though.

ob u ct n of Y

A strong assumption of the individual path model is

that the errors of' the equations, Rk, are ‘uncorrelated.

This assumptions simplifies the probability density function

44

(PDF) of Y given B because the dispersion matrix of errors
for the individual, ¢. will be diagonal. In order to exploit
this fact, the PDF of Y must be expressed in terms of R,
which in calculus terms means there must be a change of
variables for the PDFS Standard calculus theory tells us
that with vector variables, in order to express a function
of one variable in terms of a function of another variable,
one must multiply by the determinant of the Jacobian of the
transformation (Hoel, Port & Stone, 1971). In terms of the
probability density functions in question we have,

f(YlB) - f(R|B) |6R / 6Y| (2.23)
where, f(YlB) is the density in terms of the error vector R
and 8R/8Y is the Jacobian of the transformation.

In order to express the density in terms of R we will
have to restructure the model somewhat. First we will put
the matrix equation in terms of R,

R - Y - ZB . (2.24)

As figure 2.1 shows each row of 2 contains the endogenous
and exogenous variables for one equation. For this reason,
some Yk's can appear in both the Y vector and the 2 matrix.
In order to differentiate (Y - ZB) with respect to Y (which
the Jacobian requires) we need to have all of the Yk's in
one vector. This can be accomplished by transforming the
right hand side of Equation 2.7 into a restructured but
equivalent form. To see how this could be done we will
rearrange the scalar terms from figure 2.6 and add some zero

terms,

45

Y - ZB + R Matrix Form
Y1 - 0Y1 + 0Y2 + 0Y3 + Bx1X + R1

Y2 - By1Y1 + 0Y2 + 0Y3 + szx + R1 Single Equation

Y1 - By2Y1 + By3Y2 + 0Y3 + BX3X + R1
Figure 2.8
Augmented Single Equation Form of Path Model
This suggests an alternative but equivalent structure
in which the endogenous paths are grouped together in one

matrix and the exogenous paths are grouped together in another

 

 

 

 

 

 

 

 

 

 

matrix;
Y - By Y + Bx x + R
_Y1- ‘0 o 0’ -Y1- -Bx1— —R1—
Y2 - By1 O 0 Y2 + sz X + R2
_Y2_ _By2 By3 0- _Y3_ _Bx3_ _R3_
Figure 2.9

Restructured Single Equation Form of Path Model

Notice that the endogenous path matrix, By, displays the lower
diagonal structure indicative of a recursive path system.

If we multiply this matrix equation and combine the
proper terms we will get the same three equation system in
figure 2.6. This demonstrates that. the two .alternative

structures are equivalent,
ZB + R - ByY + BxX + R

This restructured form provides a convenient expression for

the model in terms of R,

46

R - Y-B Y-BXX

Y
Combining terms gives us,

R - (I-By)Y - BxX . (2.25)

This expression for R is also equivalent to the original

expression in Equation 2.23,
Y-ZB - (I-By)Y - BxX

Now Equations 2.24 and 2.25 can be substituted into Equation

2.23 to yield an expression of the PDF in terms of R,
f(YlB) - f(Y-ZBIB) |6[(I-By)Y - BxX]/6Y| . (2.26)

The simplification afforded by a recursive is apparent if we

focus on the Jacobian. Differentiating gives the result,
6[(I-By)Y-BXX]/6Y - I-By

The results, I - By, is a lower triangular matrix which

reflects the structure of recursive path systems,

 

 

1 0 0
I - By - -By1 1 o
_'By2 ”By3 1-

Because of this structure, the determinant of the matrix is
unity (Land, 1973). Substituting this result into Equation

2.26 yields,

a7
f(YlB) - f(Y-ZBIB) * 1

- f(Y-ZBIB) (2.27)

The normal probability density for one individual follows
straightforwardly from this form. It is assumed that 1)
there are R outcomes for an individual and 2) the error
vector for an individual is distributed, R ~ N(0,¢), where w
is a K by K diagonal matrix since the errors of the equations
are uncorrelated, 3) this is :1 well-specified recursive
path system which explains why the errors are uncorrelated
(Heise, 1975). The density according to Equation 2.27 and
these assumptions is,

f(YilB) - (2«>'K/2 I¢I'1/2 exp1-1/2<Y-B>' 0'1 <Y-B)}.
(2.28)

The Whole-Group Likelihood

Equation 2.28 gives us the density function for one
individual. Assuming that each individual's response is
independently and identically distributed, the whole-group
likelihood will simply be the product of each individual's
likelihood. If there are N individuals in the group this

leads to the total-N PDF

f(YIGJiJ’) -
(2«)'NK/2 |¢|'N/2 exp{-l/22(1) (Y-B)’ ¢'1 (Y-B)}. (2.29)
where, i is the index for individuals and,

2(1) is the summation over all individuals.

48

An important feature of this model is that, since the
dispersions are independent across equations, each equation
of the multiple equation system can be estimated by an
independent regression analysis. Also, the paths estimated
in such a model are full information maximum likelihood
estimates, as established by Land (1973). This will be
important when it comes to implementing estimates with the

EM algorithm, as we will see in chapter three.

W

The distinguishing feature of the hierarchical Bayesian
model is that the structural coefficients differ from group
to group in a way that is described by the second-stage
model. First let us introduce a structural coefficient
vector, Bj, that differs over group, j. The whole group
coefficient vector, B, now represents the parameters from

all groups stacked up vertically,

 

 

Conditional on B, individuals are independent within and
between groups. As a result the total-N likelihood is the
product of the individual likelihood functions within and

between groups. This results in the addition of a second

49
group summation to Equation 2.29. If N - an, is the total

N for all groups, the many group likelihood becomes,

f(YIB.¢) '
(210’va2 I16I'N/2 exei-l/Z X(j) 2(1) (Yij - ziij)' ¢’1
(Yij - 2138)}.

(2.30)

where 2(1) is the summation over individuals in a group and
2(j) is a summation over groups.

This formula can be rendered in a simpler form without
the summations. Since individuals are independent (conditioned
on B) the whole group error variance matrix, W,is a RN by KN
block diagonal with identical blocks equal to W- Since the
equations are uncorrelated each m is diagonal. So 9 is also
a diagonal matrix. For this reason the double summation in
the exponent of Equation 2.30 can be rewritten in terms of

whole group matrices,

f(YlB,W) -
(2«)'NK/2 |0|'1/2 exp{-l/2(Y - zs)' 0'1 (Y - ZB)}.
(2.31)
where Y is KN by 1, Z is KN by JP and B is JP by l. P is

the number of paths in the within-group model.

50

s e e t e Ge an
M
We have described the likelihood for Y conditional on
first-stage parameter matrix B. The likelihood we ultimately
seek is defined in terms of 6. Recall that 6 is a partitioned
matrix containing 1 and U. We can introduce these parameters
into the likelihood by recalling the identity for B found in
the second-stage of the hierarchical model, B - W1 + U
By substituting this identity into Equation 2.31, we
get the substituted form of the likelihood in terms of 1 and

U.

f(Yl'Ytuvw) '-

(2«)'NK/2 lwl'l/2 exp{-l/2(Y-ZW1-ZU)' 0'1 (Y-ZW1-ZU)).

Now use the identities;

A - [ZW : Z], and 6 -

 

 

Substituting into the last equation yields the General Bayesian

form,

f(Yle,w) -

(2«)'NK/2 lwl'l/z exp{-l/2(Y - A9)’ 0'1 (Y - A6)}. (2.32)

51
This is the likelihood that 'was ‘used tn) derive the
posterior estimates for the General Bayesian model. By
showing that the hierarchical likelihood leads to the general
Bayesian likelihood, we have proven that by assuming a
recursive first-stage path model one can derive the general

Bayesian estimates (Equation 2.16).

V e ture 0 he era h a e an M0 e1
The structure of the matrices has been illustrated only
for the single individual model. When considering all
individuals in many groups a special structure has been
devised for the matrices by Strenio (1981) to make elements
conformable in the hierarchical Bayesian linear model. The
whole group model can be constructed by stacking the individual
level matrices in an appropriate manner. For example, if
there are It outcomes and J groups and nj individuals in a
group, the whole group data vector, Y, is a Kinj by 1 vector
which results from vertically stacking the anK by 1 individual
outcome vectors. The sum, an, sums the number of all
individuals over all J groups and will be referred to as N.

The whole group data vector has the form;

52

Y11
Y21

 

 

The error vectors are stacked in a similar way to yield

a total-N error vector, R, with dimension N by 1;

R11
R21

R10
R2.)

 

 

The individual predictor matrices, zij: are stacked
vertically ‘within each group and. then each group's data

matrix is arranged in a block diagonal;

 

 

 

53

 

 

 

Z11
221
Zj - '
an
with,
__21 __
22
z -
__ ZJ__
Finally, the P by 1 coefficient vectors for each group,
Bj, can be stacked vertically to yield the JP by 1 whole
group vector, B;
__Bl __
B2
B - .
__BJ __
Combining these matrices gives us the first-stage whole
group model:
Y 2 B + R (2.33)
KN x KN x JP JP x 1 RN x l

The second-stage model is stacked by a similar process.

The P by 1 U5 error vectors are stacked just at the Bj vectors

are;

S4

 

 

__UJ __

Similarly, the p by s Wj group-level data matrices are
stacked vertically for each group to yield a Jp by 3 matrix

W:

 

 

__WJ __

But each group-level data matrix has a block diagonal

structure, with each block corresponding to each first-stage

 

 

coefficient, BPJ;

—_w1 __

w2 (zero)
WJ-
(zero)

__ wP...
with wp - [w1,w2,...,ws]. The diagonal block terms, wp, are
thus row vectors which can be of variable length. In this

way a different set of group-level predictors can predict

each path.

The structure of the 1 vector is no different in the

55
thole group case. Combining these terms leads to the whole
,group second-stage model;
B - w 7 + U (2.34)
Jle JPxSle Jle

The distributional assumptions are similar to the single

group case;
R ~ N(0, W) ,
U ~ N(0, T) ,
7 ~ N(0. I‘)

In addition it is assumed that R, U and 1 are mutually
independent.

As before, F'1 is assumed to be arbitrarily close to a
zero matrix. The structure of W and T differs from the
individual model. Since, conditional on B, individuals are
independent within and between groups, and since it assumed
that individual errors are identically distributed, W will
be a block diagonal matrix with each block equal to the

individual K by K variance matrix, ¢;

¢ (zero)
kxk W
W - kxk.
(zero) . W
__ kxk__

 

 

Nk x Nk

56

Due to the fact that errors are independent across

equations, each W is a diagonal matrix of the form,

 

 

a1 (zero)
02
¢ - .
(zero) 0K
K x K

The structural coefficient vector for a group, Bj, are
assumed to be independent and identically distributed across
groups. As a result the whole group variance matrix T is a
block diagonal with each block consisting of the identical

variance matrix for a groups structural parameters, Bj3

 

 

7 (zero)
T - pxl r
pxl
(zero) 7
__ pr
JP x 1

This completes the exposition of the hierarchical Bayesian

model.

57
VI 0 t

In the derivation for the mixed model posterior
distribution, it is assumed that variance matrices, W and r,
are known. By employing the EM algorithm it is possible to
get maximum likelihood estimates for Ifi and 1, but it is
necessary to have a criterion to judge whether the EM algorithm
has converged to the ML estimates. Therefore the relevant
likelihood of the data, f(Y|¢,r), must be monitored after
each iteration of the algorithm (Dempster, Rubin and Tsutakawa,
1981). In this section I give a derivation of this likelihood.

An expression for the probability density function of Y
given d and r can be obtained from the densities already
defined.

Let us consider several densities, all of which are
conditioned (”1 ¢ and r. Bayes theorem for ‘probability
densities gives us an expression for the posterior density
of 6 given Y, W and r (Hoel, Port & Stone, 1971),

f(9.Yl¢.f)
f(elY,W,T) -

 

f(Y|¢.r)

Solving for the PDF of Y given 9 and r,

f(9.Y|¢.T)
f(Y|r/:,t) - . (2.35)
f(elY.¢.f)

 

The likelihood we seek is f(Y|¢,r). We can cast the joint PDF,

f(9,Y|¢,r), into a more useful form by noting that,

58

f(9.Y|¢.f)
f(Y|9,¢,r) - . (2.36)
f(9|¢.r)

 

Solving for f(6,Y|¢,r) yields an alternative form of the
joint PDF,

f(ele¢07) - f(Y|99¢07) f(9|¢,r) - (2-37)

Substituting Equation 2.35 into Equation 2.33 gives,

f(Y|9.¢.f) f(9|¢.7)
f(Y|¢,r) - . (2.38)
f(GIY.¢.T)

 

This is the same as the form given by Demptser, Rubin and
Tsutakawa (1981).

The densities on the right hand side of Equation 2.38
have been defined in terms of normal density functions earlier
in this chapter in Equations 2.12, 2.13 and 2.15 respectively.
Substituting in to Equation 2.38 and eliminating some constant

terms reveals the following likelihood,

f(YlW,T) a

|W|’l/zexp{-l/2(Y-AO)'W'1(Y-A9)}IOI'1/2exp{-l/2(6'O'16)}

 

IDSI‘l/2 eXp{-1/2 (e-e*>' 03’1 (e-e*))

with, 03 - (A'W'1A+O‘1) and 9* - (A'W’1A+O‘1)A'W'1Y from

Equation 2.41. Combining terms leads to,

|0|‘1/2|o|'1/2|03|1/2exp(-1/2(Y-Ae)'0'1(Y-Ae)+e'o-1e

- (e-e*>03'1(e-e*>}

59
This expression holds for all 6, therefore a convenient
simplification can be had by evaluating the expression at
6-6* (see Dempster, Rubin & Tsutakawa, 1981). The exponential

term reduces to,
exp{-l/2(Y-A9)'W'1(Y-A9)+6*0'19*}

Using the fact that 9* - D3 A'W'lY this can be simplified to;
epr-l/ZYW'1(Y-A9)}

Now the log can taken to yield the final form,
f(Yl¢.r) “

—Lo 9 -Lo 0 Lo 0* -1/2 Y'W'1(Y-A9*) . (2.39)
s g s 9

This can be put in terms of the mixed model by

substituting the following terms into Equation 2.39;

 

 

79*" D I
* 1 * 11 I D12
A - [A1 : A2]. 9 - --;- , De- ...... | ......
-92- I
D21 : D22

 

 

As a result of these substitutions three terms in Equation 2.16

will change;

1) -Log|0| - - -Log |P| -Log |T|.

 

 

0 T
Because 1 is assumed to have a vague prior, its precision,
F‘l, goes to zero. This implies that I‘ is arbitrarily

large and fixed (Dempster, Rubin & Tsutakawa, 1981), so |F|

60
is treated as a constant and is taken out of the effective
part of the likelihood expression.
2) The term Leg|D3| can be reexpressed using a standard

method for taking the determinant of a partitioned matrix,

ID$| - I011| lDzz-021Dl1'1012I

By substituting the equivalencies for D22 and D21 from

Equation 2.22 and simplifying, The expression reduces to,

* -
IDeI - ID11| IC 1|.

where o‘1 - (A2'W'1A2+T‘1)‘1 (Raudenbush, 1987-B).

3) Finally, Y-AG has the mixed model form Y - A191 - A262

Substituting these changes into Equation 2.39 yields
the mixed model log likelihood:
Log P(Y|W,T) «
-1 -1 * *
-Log|W| - Log|T| + Log|D11| + Log|C | - YW (Y-A161-A292).

(2.40)

This last expression is used as the criterion for the EM

algorithm which will be discussed in chapter three.

CHAPTER III

METHODS

I d t o

This chapter will review some of the technical aspects
of implementing multilevel path analysis. First I review
the EM algorithm and explain its rationale. Then the specific
equations will be derived for implementing the EM algorithm
in a multilevel context for estimating variance components.
Next, statistical tests will be discussed. These will
encompass the chi-square test of parameter variance, the R2
statistic for assessing fit of the second-stage model and
the 2 test of second level parameters. Finally there will
be a discussion about the validation of the computer algorithm.
This will focus on the cross-referencing analysis that was
done with the multilevel path analysis program and on the
Hierarchical Linear Model program (Bryk, Raudenbush, Seltzer

& Congdon, 1986).

m t t o o t e E o t
The logic of the EM algorithm is to estimate parameters
for a hypothetically complete set of data from a sample
space which only contains incomplete data. Instead of using
the actual summary statistics found in the data, which are
by definition 'incomplete', the EM algorithm utilizes the

expected value of complete data summary statistics as a

61

 

 

62
substitute for having the 'complete data' statistics. The
advantage of this strategy is that maximum likelihood
estimators based on the assumption of complete data can be
quite simple to derive.

The EM algorithm is an iterative routine which cycles
through an expectation phase and a separate maximization
phase at each iteration. The maximization phase consists of
the calculation of maximum likelihood estimates for parameters
based on the assumption of complete data. Consider a simple
example of variance estimation. Let us assume a model for

individual i;
Y1 - XiB + 3i ,

where,

Y1 is a single outcome,

X1 is a matrix of fixed predictors,

B is a vector of regression coefficients and

e1 is the random error.

We want to estimate a given that Var(e) - 021, the

variance of the errors. If the complete data consists of
observations of Y1 as well as of e1, the maximum likelihood

estimator of 02 is simply defined as,

02 - Z eiz/N ,
where, Xeiz is the "complete data sufficient. statistic",

that is, the sufficient statistic needed to obtain the ML

estimate given that one has observed the complete data.

63
In actuality, though, we never observe the ei's. With
the EM algorithm we use the conditional expectation of the
complete data sufficient statistic instead of the actual

complete data sufficient statistic;

E(2612 I Y) .
where Y is the incomplete, i.e. observed, data. The expected
value of the sufficient statistics is calculated during the
Expectation phase of the EM algorithm.

A general schema for the EM algorithm is,
P1 - F{E9(Sufficient Statistics|P0, Incomplete Data))

with,
P0 - Vector of parameter estimates from the previous
iteration of the algorithm,
P1 - Vector of parameter estimates for the present
iteration,

FI } - The estimator of parameter vector P1 assuming
complete
data.

Ep - Expectation over all possible values of P, given the
complete data
Sufficient StatisticslPo, Incomplete Data - sufficient
statistics given previous estimates of parameters
and the observed, incomplete, data.
Note that sufficient statistics are conditioned on the
data. This means that parameter estimates involved in

calculating sufficient statistics will be the Bayesian

estimators derived in chapter two.

64
I will use the same model explicated in chapter two for

the EM derivations which has the hierarchical form,

Y - ZB + R

B - W1 + U
The substituted model is,
Y - ZW1 + ZU + R.
As before, by making the substitution of,
A1-ZW; A2-Z; 61-1; and 92-U,
we get the mixed model form, Y - A191 + A292 + R.

Also as before, Var(UJ)-r, and Var(Rij)-¢, where U3 is the p
by 1 vector of parameter errors for the paths of one group
and R11 is the k by 1 vector of sampling errors for person i
in group j. The purpose of the EM algorithm is to estimate r

and W.

V a a r
In. the context. of estimating 1' the 'complete data'
consists of the observed outcome data, Y, and the second-
stage errors, Uj- Assuming complete data the ML estimator

for r is simply,

XUJUJ'/J (Raudenbush, 1987-B),

with, U3 - The p x 1 vector of parameter errors associated
with the p paths in group j.

65
J - The number of groups.
With the EM approach we substitute E(UJUj'|Y,ro,¢o) for

UJUJ', at each iteration (Dempster, Rubin, Tsutakawa, 1981).

The expected sufficient statistics for a vector product
like UJUJ', comes out of the definition of variance in standard
statistics theory. The dispersion of Uj, Var(Uj), is defined

as,
Var(Uj) - E(UJUJ') - E(Uj)E(Uj)' (Searle, 1971).

Now we solve for the quantity we seek, the expected value of

the sufficient statistic, E(UJUJ');
E(UJUJ') - E(UJ)E(UJ)' + Var(Uj)

But the EM algorithm requires the expected sufficient
statistics given the 'incomplete' data, Y and parameter
estimates from the previous iteration, '0 and 100, so the

expectation is;

E(UJUJ|Y,10,¢0) - UJ*UJ*' + D*Uj ,

where, U * is the posterior parameter estimate of 92 given in
cgapter two in Equation 2.22, and

D*Uj is the posterior dispersion matrix for 92, also
given
in Equation 2.22.
The connection between.the posterior estimates given known

variances deve10ped in chapter two, and the EM estimating

routine is now explicit. At each iteration you plug in the

66
variance estimates from the previous iteration as the 'known'
variance and then you use the formulas for posterior estimates
developed in chapter two to calculate the expected sufficient
statistics.
The maximization phase for the estimation of r is
accomplished by the trivial operation of dividing the expected

sufficient statistics by j,
71 -Z(UJ*UJ*' + D*Uj)/J

This completes one iteration for estimating r.

orm s m ti F st-Sta e Variance Matrix

The first-stage variance term, ¢, is a K by K diagonal matrix
with off diagonals of zero. Because the first-stage errors
are uncorrelated, the variance terms can be estimated by K
separate EM estimation calculations. The K separate estimates
are then arranged along the diagonal of p to provide the
matrix estimate. Each of the K parallel estimates follows
the same format.

The quantity to be estimated in one EM calculation is
the k,k scaler diagonal element of W, 02k. The 'complete
data' in this case consist of the N by 1 observed outcome
vector, Yk, and the N by 1 first stage error vector, Rk.
The complete data maximum likelihood estimate for 02k is,

Rk'Rk -

X X Rzijk/N »
(3)”)

67
where, Rk is the N by 1 error vector for variable k,

Rijk is the error for person 1, group j, and outcome
(endogenous variable), k,

N - X(j)“j’ is the total number of individuals in all
groups, and

The double summation indicates summing the squared
errors
over persons and groups.
The derivation of the expected sufficient statistics in
the case of 02k is more complicated than for r. The term Rk

is the N by 1 error vector for outcome k. The mixed model

formula with Rk is,

Yk - Alkelk + A2k92k + Rk

We solve for Rk to get,

Rk - Yk - Alkelk - A2k92k

So the complete data sufficient statistics for 02k is,

XJXiRzijk -
(Yk - Alkelk - A2k92k)'(Yk - Alkelk - A2k92k>

This last formula can be made more tractable by putting the
mixed model into its simpler General Model form using the

substitution,
91k
A - [A1 : A2], and 9k - ----
92k

the sufficient statistic now has the form,

2321R213k '

(Yk - Akek)'(Yk - Akek)

68

The expected value for a scalar quadratic of the form Rk'Rk
is,

E(RkRk') - E(Rk)'E(Rk) + Tr{Var(Rk)},
were, TrIVar(Rk)} is the trace of the dispersion matrix
for
the N by 1 vector, Rk.

But for the EM algorithm we need the expectation given

$0, 70 and Y. In the first term we have,

E(Rk|¢0:'OsYk) - Yk - Ak9*k:

where, 9*k is the estimate of the posterior parameter mean for
equation k,found in Equation 2.22 of the last chapter.

The second term, Var(Rk), is the posterior variance of
RR which equals

Var(Yk - Ak9k|¢o,fo,Yk).
This is the same as, Var(-Ak6k|¢o,ro,Yk) which equals,

AkD*ekAk'
The variance matrix, D*9k, is the portion of the posterior
variance matrix, D*9, which is pertinent only to the outcome,
Yk-

The expression can be put in a computationally more
convenient form if we note that TrIAkD*9kAk]'<nu1be permutated

to yield,

TR(AkD*9kAk') - TR(Ak'AkD*ek)
Thus the conditional expected sufficient statistic for

2

a k can be expressed as,

E<RkRk'I¢o.ro.Y> -

(Yk - Akek)'(Y - Akek) + TR(Ak'Aka*9k)

69

This can be translated back to the mixed model form to
yield an equation similar to what was used in the multilevel

path program,

(Yk - Alkelk - A2k92k)'(Yk - A1191k - A2k92k) +

 

 

 

 

_A1'A1 A1'A2- _D*11 D*12_
TR * *
_A2'A1 A2'A2_ .0 21 D 22-
This involves rather large matrices. By multiplying the

partitioned matrices, expanding and simplifying terms this
can be broken down to a tractable computational formula in
terms of group-level variables.

As with the estimation of r, the maximization step is
relatively trivial. The estimated sufficient statistics is
simply divided by N to yield the maximum likelihood estimate
for 02k. The above steps are repeated for all K of the 02
terms and the diagonal terms of p are constructed from these
K estimates.

At this point we have r1 and $1 for one iteration of
the EM algorithm. These variance matrices are then used to
calculate all of the terms in the likelihood expression from

Equation 2.40,
L -l -l * *
- og|W| - Log|T| + Log|D11| + Log|C | - YW (Y - A191 - A292).

This is the criterion for convergence. The change in the
likelihood is positive from each iteration to the next as

the likelihood increases towards a maximum. If the positive

70
change in likelihood is less than .01%, then it is judged
that the algorithm has converged to the maximum likelihood

estimates.

11 est t

Three test statistics are utilized in the program. Two
provide tests for variances and one for second-stage regression
coefficients.

e V a

Recall that the second-stage model for the paths for
group j is,

Bj - W31 + Uj
The variance of Uj is r, which.is a P by P variance/covariance
matrix. The P diagonal elements of r, 'ppv are the parameter
variances of the paths. The larger this variance is the
more the structural parameter varies from group to group.
If the parameter variance is zero, the corresponding path is
considered to be the same for all groups, and is therefore a
fixed quantity. This has great implications for interpreting
an analysis, so it is useful to have a statistical test of
whether the parameter variance of a path is zero.

Such a test is a chi-square statistic which for group j
consists of the ratio of the estimated total variance of the
path (parameter variance + sampling variance) over the
parameter value of the sampling.variance. This ratio is

summed over all J groups;

71

g (Total Variance of Bp / Sampling Variance of Bp}

j-l
Since the numerator is the sum of parameter and sampling
variance, under the null hypothesis that parameter variance
is zero, the ratio should be small. Conversely, assuming
the parameter variance is not null, as the parameter variance
gets large so does the test statistic.

A statistic which is estimable in terms of the current

model is, according to Hedges (1982),

J A
j{31‘ij ‘ "197*p)2 / VPPJ’ '

where, ij is the least squares estimate for path p for
stone J.

ij is the second-stage predictor matrix for path p and
group J.

1*p is the posterior estimate of the s by 1, second-
stage
regression coefficient vector for path p,
V*p J is the sampling variance for path p. Its
estImate consists of the p,p element from the matrix,
(zj'9’1z1)‘1, where Z is the first stage predictor
matrix and d is the estimated variance of first -stage
errors. This is the familiar least squares estimate
for sampling variance of a regression weight.
This statistic has an asymptotic chi-square distribution
with J-S degrees of freedom with J equal the number of groups

and S equal the number of second-stage predictors for path
p.
Note that for this to be a true chi-square test it is

assumed variance term for each group, Vppj’ is a known

72
parameter. Raudenbush 6: Bryk (1986) point out that since
the sample size over all groups is usually large, this is
not a hazardous assumption. They point out a more serious
problem with the statistic, though. It may be sensitive to
departures in normality of Y and U. Raudenbush and Bryk
suggest that this statistic should be interpreted with caution

unless the probability is very small, e.g. in the .001 range.

T e cent Va iance Ac unted For b the
WW

As we will see in the analysis chapter, two models are
compared in a multilevel analysis, an unstructured between-
group model and a structured between-group model. The
unstructured model stipulates that first-stage paths vary
about a grand mean path;

Y - ZB + R

B - B-Mean + U;
where, B-Mean is the vector of grand mean paths. The motive
behind running this model is to get an estimate of the total
parameter variance of the paths, unconditional an a between-
group regression.

Typically we would proceed to specify a structured
between-group model in a subsequent run;

Y - ZB + R

B - W1 + U,
where the second-stage intercepts are incorporated into 1,

W is the matrix of between-group predictors, and

73

1 is the vector of regression coefficients for paths.
The variance of Uj, (7) now represents the residual
variance matrix of the second-stage model, conditional on
the second-stage predictors. In the ideal case where W1
predicts perfectly, the second-stage model would account for
100% of the parameter variance and 1 would be zero. A simple
test for the percent of variance accounted for by the between-
group model is given by Raudenbush and Bryk (1986);
{Var(ij) - Var(BJple)} / Var(ij) .
where, Var(Bj ) is the unconditional parameter variance of
path p. This is the p,p element from the r matrix
estimated in the unstructured between-group model,
Var(B |W ) is the conditional variance of path p.
This 1: t e parameter variance estimated in the
structured between-group model.
This statistic provides a useful criterion by which to
assess overall model performance and it also provides

information for modifying the between-group model for each

path.

- - e s 0
An ordinary 2 statistic can be used to test whether a

second-stage regression coefficient is zero. The standard 2

form is;

 

Standard Error(P - P)

74
where;
P is the estimated parameter value,
P is the hypothesized parameter value,
Standard Error (P - P) is the estimated standard error

of the difference score between the estimate and
the hypothesized parameter value.

In terms of the multilevel path analysis this becomes,

*

7 a

*
Dllégs

where, 1*s is the second-stage parameter coefficient, and

 

Dflé?s is the square root of the 5,8 diagonal
term from the posterior sampling variance of the fixed
effect, i.e. from D*91 in Equation 2.22.

This statistic has an asymptotic Z distribution.

Raudenbush & Bryk (1986) contended that statistical
tests of regression coefficients would be more robust to
such violations than chi-square tests of variances.

A note of caution has to do with the sheer number of 2-
tests that can occur. Each path can have numerous between-
group predictors so we could find ourselves performing myriad
non-independent Z-tests. The overall alpha level of the
entire set of Z-tests is unknown. It is therefore advised
that these tests only be use as a rule-of-thumb and not as

proof of the existence or nonexistence of particular effects.

75

The computer program to perform the multilevel path
analysis involves thousands of calculations over numerous
iterations of the EM algorithm. Checking the accuracy with
which the statistical formulae were translated into code by
hand calculations would be an unwieldy task, and would be
quite prone to error. It was therefore concluded that the
only reliably accurate way to check the equations and the
design of the program would be to compare it to an already
established estimating program.

The multilevel path model is an elaboration of the
Hierarchical Linear Model devised by Raudenbush (1984), so
the HLM program which estimates this model (Bryk, et al,
1986) is a natural choice for comparison. The difference
between the two models is that the multilevel path model
stipulates a multiple equation system, without intercepts at
the first-stage; while the HLM model stipulates a regression
model with intercepts. I therefore modified the multilevel
path program so that the first-level design could include
intercepts and could be restricted to one equation. With
these modifications the models for the two programs should
be the same. Also, under these conditions the Bayesian
estimating equations of the multilevel path model should
reduce to the HLM case.

To empirically test whether the algorithms were equivalent

76

in this case, both programs were used to analyze an identical
dataset stipulating an identical model for the data.

Wm

The data to be given the parallel analysis was drawn
from the High School and Beyond study (Coleman, Hoffer 6:
Kilgore, 1982). This study will be more fully described in
chapter four. For the purposes of exposition I will only
mention that in this analysis the data consisted of measures
on students in 94 schools. The dataset also included measures
at the school level but they were not included in the
validation run.

The within-school model is a standard regression with one
outcome and three predictors. For individual i and group j

this model is,

Math Achievementij - BOj + 813(M1nority Status)1j +
B2J(Gender)1j + B3J(SES)1J + Rij

where;
Math Achievement - a standardized math score,

BOj - is the mean math achievement for school j. The
student level predictors were mean deviated so
that the intercept was the group mean.

Minority Status - Whether the student was a minority,

Gender - Whether the student was male or female,

SES - Socioeconomic status index for student,

R11 - Sampling error.

The between-group model was unstructured,
Bj - B-Mean + Uj ,

where, Bj - The 4 by 1 vector of regression coefficients for

77
a group,
B-Mean - The 4 by 1 vector of grand mean paths,
Uj - The 4 by 1 vector of parameter errors.

This model was run on both analysis programs for 20
iterations of the EM algorithm.

The degree of agreement was quite high. The likelihood
function used to monitor the progress of the algorithm,
Equation 2.22, incorporates all the information of the
estimates. 'The value of the likelihood functions for the
two programs differed only from .001% to .02% over the 20
iterations. The estimates for the first-stage error variance
were identical at 1&498. Estimates for the elements of 1
were very close, with differences ranging from .002% to 11%
(see tables 5.1 and 5.2). The posterior estimates for the
second-stage intercepts, 10, were also very much in agreement,
with differences ranging from 0% to .07% (see table 5.3).

The two estimating programs produced virtually identical
parameter estimates when identical models were analyzed. What
little differences there were can be explained by the fact
that the programs were not written in the same programming
language and the form of the equations were not the same.
The HLM program was written in fortran with self-contained
matrix subroutines. The multilevel path model, on the other
hand, was written in SAS, using the Proc Matrix procedure
(SAS Institute, 1985). Rounding errors and the accuracy of

subroutines could differ between the two languages.

78
This parallel analysis has established that the multilevel
path program produces sensible results when compared to an
established estimating program for a restricted model. The
soundness of the algorithm has not been established for
other models and other datasets. The program will have to
be run under many conditions before the status of programming

bugs can be thoroughly assessed.

CHAPTER I“
USING THE MODEL

In chapter two the multilevel path model was defined, the
estimators derived and their statistical properties discussed.
In this chapter we ask if this model can be fruitfully applied
to educational research. We need to know a) if the model gives
estimates which have some meaningful correspondence to the
actual processes being studied, and b) if a multilevel path
analysis lends itself to an interpretation which enhances our
understanding about important educational questions. Ihiorder
to demonstrate the meaningfulness and interpretability of the

model I will analyze two educational data sets.

th S h e d a

The first data set is from the High School and Beyond
study (Coleman, et al, 1982). As mentioned in chapter one
this was a very large scale study in which a sample of 998
was taken nationwide. A major focus of the study was the
question, "What is the relationship between students'
characteristics, such as family background and ethnicity,
and academic success?" Previous studies have shown that the
relationship between students' SES and academic attainment
is substantial (Lee, 1986). This finding is of great concern
to many educators because it seems to undercut the ideal of
fairness, equality and equal access to opportunity. The
study by Coleman et al (1981) focused on the relationship

79

80

between student background and achievement in high schools of
two types, public and private Catholic. In this study,
background and achievement were examined in a broad context.
Within the schools, student background.information was gathered
on variables such as family SES and student's ethnicity.
Academic measures included number of math classes taken and
mathematics achievement score on a standardized math test.
One of the most controversial conclusions of this study was
that Catholic schools were found to be more egalitarian than
public schools. This claim was made on the basis of the
finding that the relationship between SES and achievement is
not as strong in Catholic schools as it is in public schools,
so that the disequalizing effect of SES on educational outcomes
is smaller in the Catholic sector. This has led to widespread
speculation that not only is much education inequitable in
this country, but that such inequity is concentrated in our
public institutions.

On the face of it, the appearance of inequity in public
schools is alarming and invites speculation about ”What is
wrong with our schools?" In order to gauge the seriousness
of the problem and to devise solutions, the mechanism behind
the inequity must be understood.

Multilevel linear models are particularly well suited for
exploring this question because the issue concerns processes
that arise at different levels of aggregation. The effect of
SES on achievement pertains to students within schools. The

influence that sector has on the SES to achievement

81

(SES—A>Ach) effect pertains to a school-level variable (sector)
and its effect on a student-level process. The mechanism which
explains how sector influences the SES-—4>Ach effect would
consist in school-level variables which characteriZe public
and Catholic schools and explain why the two types of schools
function differently. For example, it may be discovered
that certain policies and pmactices characterize Catholic
schools and explain why students of different SES backgrounds
achieve at the same level in these schools.

One effort to bring a multilevel approach to bear on this
issue was the reanalysis of Coleman, et al's study by
Raudenbush and Bryk (1986). They used an approach called
the Hierarchical Linear Model, or HLM, in which a single-
outcome, multiple regression is posited in numerous groups
as the within-group model” The 'variation in the group
parameters over groups is modeled at the between-group model
in such a way that characteristics of the group predict the
group's regression coefficients. Raudenbush and Bryk
demonstrated the existence of inequity within schools by
showing that there are schools which have a positive
SES——>Ach regression slope. Further, they demonstrated that
public schools were less equitable than Catholic schools by
showing that being, in the ‘public sector was positively
associated with a school having a larger SES——>Ach slope.
In other words, sector was introduced as a predictor in the
between-group model. Lee (1986) and Lee and Bryk (1986)

carried this logic further. Their goal was to demonstrate that

82
there was a mechanism which explained the greater equity of
Catholic schools. They added variables to the between-group
model which represented policies and practices of schools.
If the proper explanatory policy variables were introduced,
the estimated effect of sector on the SES——>Ach slope would
disappear.

It is noteworthy that Raudenbush and Bryk and Lee and
Bryk upheld the existence of a sector effect because Coleman
et a1. estimated this effect by performing separate student-
level regressions first for the public school students and
then for the Catholic school students. The classroom level
of analysis was ignored, making the results vulnerable to
aggregation bias.

Another possible source of bias is the presence of
confounding variables at both the within-class and the class
levels. Raudenbush and Bryk devised a model that controlled
for confounding variables at both levels of analysis. They
concluded that there remained a sector effect on the
SES/achievement relationship, even when aggregation bias and
confounding factors were controlled for.

In Lee and Bmyk's analysis two outcomes were used as
yardsticks of academic attainment: the number of math courses
taken in high school and math achievement. In the HLM approach
this means that two separate within-class models must be
analyzed. One had math achievement as the dependent variable,

predicted by student background. The other had number of

83
math classes as the dependent variable, also predicted by
student background.

The first within-class model had the form:

Ach - Bo + B1(Academic Background) + B2(Minority) + B3(SES)

In the second-stage model, the first-stage slopes, B2
(Minority-->Ach) and B3 (SES-->Ach) are predicted by school
context variables (i.e. average school SES and percent minority
enrollment in school) and school practice and climate variables
(e.g. number of math courses available in the school and level
of disciplinary problems in the school). The parameters, Bo,
the school mean, or B1, serve as statistical controls in this
analysis so the discussion here focuses on B2 and B3. After
school context and school climate variables were taken into
account in the second-stage model, the sector effect
disappeared. So the notion of "inequity" is explained away
and is replaced by the specific climate, policies and practices
of the school.

In the second model, the number of math classes is the

outcome for the within-class regression:

# Classes - Bo + B1(Academic Background)

+ B2(Minority) + B3(SES)

In the subsequent analysis, the between-group predictors

for the B2 (Minority-->Classes) slope and the B3

84
(SES-->Classes) slope were school climate variables (i.e.
minority enrollment and school SES). This analysis was much
less conclusive than the previous one. The sector effect was
never explained away.

A limitation of the Raudenbush and Bryk analysis is that
it only modeled one outcome. In Lee's analysis it is contended
that the two outcomes, number of classes taken and achievement,
are both important outcomes which bear on the equity issue.
But the main limitation of Lee's approach is that the two
outcomes of interest must be assessed by two separate analyses.
AS Lee asserts (1986), it is reasonable to assume that the
number of math classes taken has a strong effect on math
achievement. This implies that a properly specified model
would include the effect of Number of Classes on achievement.
Such an analysis requires path modeling and is outside the
scope of the HLM model. The analysis presented in this
chapter employs this sort of within-groups path model.

The issue can be illustrated by path diagrams. A model
similar to the two parallel within-groups regression models

used by Lee would have the form:

 

 

Minority 311 >—Classes
B21

SES 322 > Ach
Figure 4.1

Path Diagram of Two Separate Regression Analyses

A separate regression analysis is run for each outcome.
The slope estimates of one analysis does not effect the slope
estimates of the other analysis. But what if we connect the

separate models by drawing an arrow between the outcomes:

 

 

Minority 311 :7 Classes
B22 B21
/B12
SES 323 >7Ach
Figure 4.2

A Single Path Model Incorporating All Variables

Instead of two regression models we have one path model.
Such a model is different from separate regression models in
two ways 1) an additional relationship, the Classes/Ach effect,
is estimated and 2) some of the previous relationships may be

estimated to have very different values under this model. For

86
example, suppose minority status and student SES affect
achievement only by affecting the number of classes taken, i.e.

suppose the actual model is;

 

 

Minority Bll ; Classes
7'
B21
B12
/ 1‘
SES Ach
Figure 4.3

Path Model With Indirect Effect of
Student Background on Achievement
If this represents the state of affairs of the world,
when the Classes/Ach path is added to the model, the estimates
of Minority/Ach and the SES/Ach path will tend towards zero.
Contingencies such as these can only be explored by a path

analysis.

The sample and the Data

In the analysis I performed on the High School and Beyond
data a random sample of 158 schools out of the original base
of 998 schools was employed. In this sample there were 68
Catholic schools and 90 public schools. Four within-group
variables were used in the present analysis:
Minority Status (Minority) - Whether or not the student was

a minority. O-White,
l-Minority.

87
SES - A composite index of student's
social class.

Number of Classes (C1asses)- Number of advanced math classes
taken in high school.

Math Achievement (Ach) - Senior year math achievement.

wo a W - ss el

In order to demonstrate the explanatory power of a within-
groups path analysis, we will first estimate the two parallel
but separate regression models illustrated by figure 4.1.
We will then compare these results to an analysis employing
a within-group path analysis as depicted by figure 4.2.

The two regression models have the single-equation form

for person i and group j,

Classesij - B113(minority) + B12J(SES) + R131

Achievementij - B211(Minority) + B22J(SES) + Rij2

The between-groups model is unstructured, i.e. there are
no group-level predictors so that regression parameters vary
about a grand mean,

The results of the parallel regression run can be found

in table 1. In table l-A, the estimated parameter variances
are listed. For each estimated parameter variance there is
a corresponding chi-square test. This chi-square statistic

tests the null hypothesis that the parameter variance is
zero (see chapter 3). A larger chi-square statistic indicates
a less probable result given the null hypothesis. If the

probability of the chi-square test is below some a priori

88
critical level, that is grounds for inferring that the
parameter variance is not zero. The exact value of the
critical probability is of course arbitrary, but the customary
.05 value will be assumed.

In table l-A the chi-square tests indicate that all the
parameter variances are significant. In other words every
regression coefficient varies from group to group.

Table l-B shows the weighted average of the coefficients,
with the coefficients from each group weighted by the precision
of the group estimate. This gives us some idea ofzni'average'
regression model from which all the groups deviate. In the
special case in which the coefficient is inferred to have zero
parameter variance the average coefficient represents the
structural relationship for all groups. The 2 test indicates
whether the average coefficient is significantly different from
zero. As we see, all coefficients are different from zero.

This analysis is dramatically more interesting when

compared to the within-group path model in the next section.

B11

B12

B21

B22

 

mus.
W
a e Va e th
MW
Chi— Parameter
Parameter Square To Total

MWMM

.324 232.616 (.0001 .285
.059 219.700 <.0001 .235
6.692 211.456 <.0001 .270

.688 160.966 .034 .156

 

 

 

ch d e

v a e V ue o

 

 

WW
Average 2
Path Value Statisgig Pgobabiligy
B11 (Minority->
Classes) -.279 -3.620 .0003
B12 (SES->
Classes) .343 10.420 <.0001
B21 (Minority->
Ach) -2.690 -7.318 <.0001
B22 (SES->
Ach) 1.326 9.189 <.0001

 

 

 

 

91

W1.

In contrast to the parallel regression analyses we now
propose a path model which 1) relates background variables
to both Classes and achievement, but also relates Classes to
achievement as depicted by figure 4.2. The within-group
path model is a two equation system which has the individual

form (for individual i and group j):

Classesij - B11j(minority) + 3121(SES) + R131

Achievementij - B213(Classes) + B22J(Minority)
+ B23j(SES) + R1j2

o etw en- u n l s

The first computer run that is performed with a multilevel
path analysis specifies an 'unconditional' between-group model,
one that stipulates no between-group predictors, as was the
case with the parallel regression analyses. Since the
parameter variance estimate is not conditioned on between-
group predictors it is at its maximum possible value.
Unconditional estimates of parameter variance provide baseline
estimates of the total variance, if any, that may be explained
by future runs which include group-level variables. This
baseline run will also yield the mean slopes across groups,
giving us an idea of what the central tendencies of the
paths are. The unconditional second-stage model takes the

simple form:

92

B11 - B11 + U11
B12 - B12 + U12
B21 - B21 + U21
B22 - B22 + U22
B23 - B23 + U23

The ka terms are the average, or pooled-within—group,
estimates of the paths. If there is no parameter variance,
i.e. if Var(Ukp)-0, then the pooled within group path is
characteristic of all groups. Figure 4.4 is a multilevel
path diagram of the baseline model. The bold arrows represent
the first level. (within-group) 'paths. The finely' etched
arrows represent predictive relationships between the second
level (between-group) variables and the first level paths.
In this model only the second-stage errors (Ukp) impinge on

the first-stage paths.

93

U11
MINORITY

\\\\‘ B11
/

U22 2 321

;/////U21

4>—CLASSES

 

U
::::>,312

355 323 a» ACH

/

U23

 

Figure 4.4
High School and Beyond Data: Baseline Model

94
We can see from table l-C that parameter variance is
only a small part of the total variance of the path estimates.

Column five lists the ratio of the parameter variance to the

total variance. The numerator is from column one, the
parameter variance. The denominator is :1 statistic which
represents the sum of parameter and sampling variance. This

ratio, then, is the percentage of total variance represented
by the parameter variance. The estimated parameter variance
of the first three paths accounts for only 28%, 24%, and 22%
of total variance, respectively. The relatively small group
sizes could account for why the sampling variance is large
when compared to parameter variance.

Column four lists the probabilities of the chi-square
tests. From this we see that the first three parameter
variances are significantly different from zero. The parameter
variance for B22 represents only 13% of the total variance,
even though the chi-square test still indicates that this is
a non-zero quantity (p-.007). The last path, B23, has a
parameter variance which is only 9% of total, and indeed the
chi-square indicates this is ‘not significantly different
from zero (p-.95). With virtually no systematic variance to
be explained, it is unlikely that any between-group variables

will predict the B23 path.

95

 

 

001 a d e 0 ate
a e e V e a
Unagragguged Between-Gggap Mgdel
Chi- Parameter
Parameter Square To Total
Raga Variaace Statistic Eggbability Vagiance
B11
.319 322.601 <.0001 .281
B12 '
.059 219.546 <.0001 .236
B21
.335 210.709 <.0001 .221
B22
2.173 172.674 .0073 .130
B23
.198 104.037 .955 .087

 

 

 

 

96

This model offers a sharp contrast: to the parallel
regression analyses. First of all let us compare the average
coefficients from the two analyses in tables 1-B and l-D. The
first. two coefficients, Minority/Classes and SES/Classes,
arevirtually unchanged. But the last two coefficients,
Minority/Ach and SES/Ach, have become much smaller. The
average minority/Ach effect went from -2.69 to -l.928, while
the average SES/Ach effect went from 1.326 to .391. From
this we can conclude that on the average, much of the effect
of students' background on achievement is through the number
of classes taken. In other words, student background
determines achievement largely by determining how many classes
the student will take.

It might be argued that we have set up a straw man, that
the HLM approach could have modeled the same two equations as
the multilevel path model, in two separate runs. This is a
viable option but it allows for less satisfactory modeling at
the between-group stage. With the multilevel path analysis
the paths from all equations can be modeled by group variables
as one set. Since the covariances among paths across equations
are accounted for, the simultaneous approach can be expected

to yield more appropriate results for the second-stage model.

Path

 

811 (Minority->

B12

B21

B22

B23

Classes)

(SES->
Classes)

(Classes->
Ach)

(Minority-
Ach)

(SES->
Ach)

Ifihl2_l;2
o d e
MW
Un uct e Be w e -
Average Z
Value Sgatiatia
-.278 ~3.620
.342 10.414
2.951 38.029
>
-1.928 -7.411
.391 3.622

 

 

del

Egobabiiity
.0003

<.0001

<.0001

<.0001

.0003

 

 

98

Another striking contrast occurs if we compare the
regression model parameter variances, table l-A, with duzpath
model parameter variances, table l-C. As before, the values
for the first two coefficients are virtually unchanged between
the two models. But the variances for the last two
coefficients have diminished by two-thirds from the regression
to the path models. In fact, the path model variance for
the SES/Ach effect is not significantly different from zero.
The SES/Ach effect is constant over groups when achievement
is controlled for the number of classes taken. Almost all
of the variation in equity is accounted for by variation in
how classes are distributed. The situation is like the path
model depicted in figure 4.3 where SES affects achievement
through Classes. In order to explain apparent differences
in the SES/Ach relationship from school to school, we must
find school-level variables which explain tflua SES/Classes
effect and the Classes/Ach effect. This is the issue taken

up in the structured between-group analysis.

c u wee - ou Anal sis
A second statistical analysis is now presented in which
group-level predictors have been included in the second-stage
model. The three group-level variables that were used in this

analysis are defined as follows:

99

Sector - Whether the school belonged to
the public school or the
Catholic sector.
0-Public, l-Catholic.

Ave-SES - Average social class of all
students in the school.

Sd-SES - Standard deviation of students'
social class in a school.

Several models were estimated to explore different
combinations of second-stage predictors. The multilevel path
analysis is highly sensitive to changes in the model. Because
second-stage predictors can be multicollinear and because the
estimation procedure is full information maximum likelihood,
the estimate for one parameter affects estimates for all other
parameters. Note that the first-stage model does not change.
After some exploration a second-stage model of the following

form was settled upon:

B11(minority-->classes) - E11 + 1111(sector) + U11

B12(SES-->Classes) - B12 + 1121(Sector) + 1122(Ave-SES)
+ U12

B21(Classes-->Ach) - B21 + 1211(Sector) + U21
B22(Minority-->Ach) - B22 + 1221(Sd-SES) + U22
B23(SES-->Ach) - 323 + 023

As with the baseline analysis, the intercepts of the

between-group regression are slope averages. This is because

all between-group predictors were mean deviated. A pictorial

100
depiction of this multilevel path model is given by figure
4.5. As with figure 4.4, the bold arrows represent the first-
level model and the finely etched arrows representaipredictive
relationship between group-level variables and paths. As

before, the U's are parameter errors.

101

U11 sector
——> CLASSES

MINORITY \\\\\‘ 311\\\\\‘
sector

U22 /////' B22 2}/////
U21
sd-ses ,/////

sector

ave-se::::\\‘

0
12\\\ B12

/

SES 323 >>ACH

/

U23

 

Figure 4.5
High School and Beyond Data: Baseline Model

102

The striking feature of this model is that sector does
not enter into the relationship between student's background
(i.e. minority status and SES) and achievement. Raudenbush
& Bryk (1986) and Lee (1986) both found such relationships.
We see that sector helps determine B11, the Minority-->Classes
path, B12, the SES-->Classes path and B21, the
Classes-->Achievement path. The implication would seem to
be that once the effect of Classes on Achievement is taken into
account for students within schools, sector no longer
determines the relationship between background and achievement.
Such a conclusion would not be apparent without a path model
at the within-group level.

Sector La important for mediating the relationship between
background and number of classes. The conclusion that we draw
is that sector mediates equity but ag£,by directly influencing
the relationship between students' background and achievement.
Rather, sector influences the indirect relationship between
background and achievement. In other words, Catholic schools
accomplish greater equity in achievement by promoting equity
in classes. This is demonstrated by the sector influence on
the Minority/Classes path and on the Classes/Ach path.

Lee found that this relationship was resistent to being
explained away by school context and school climate factors.
Thus a mechanism which explains how Catholic schools function
differently from public schools, was not found.

Table 2-A gives information about the parameter variances

and helps us to assess how well the between-group model fit

103
the data. If between-group variables in the model perfectly
predicted a path, the parameter variance would fall to zero.
Although no model achieves this ideal the last column of
table 2-A, the R2 column, helps us assess how close the
model came to the ideal. R2 is the proportion of parameter
variance accounted for when compared with the baseline analysis

(Raudenbush & Bryk, 1986). The formula is:

Var(B) - Var(B|W)

 

R2 -
Var(B)
Var(B) is the parameter variance for a path from the
baseline model. Var(B|W) is the parameter variance from a

model in which the parameter variance is conditioned on group
level predictors, W. The R2 for B11 is .32, so Sector accounts
for 32% of the total parameter variance for this path.

The estimated second-stage regression coefficients are
given in table 2-C. In. column three. we find that the
coefficient for Sector predicting the B11 path is .62. The
2 test of whether this coefficient is different from zero
has a probability less than .0001, which is convincingly
significant.

The test of the usefulness of the model is in what it can
say about school processes. From table 2-B we see that the
average value of 811 is -34, which is significant at
probability"< .0001. This indicates that (”I the average

being a minority leads to having fewer math classes. As we

104
have seen, the second-stage regression coefficient of Sector
predicting B11 is .62. This means that going to a Catholic
school will have a positive effect on the B11 path (i.e.
makes the slope less negative by an increment of .62). The
effect of being in a Catholic school (W-l) is demonstrated

by the second-stage regression equation:

A

B11 - -.34 + (l) .62

Being in a Catholic school flips the sign of the
Minority/Classes path from -.34 to; -.34 + .62 - .28. In
public schools being a minority is a disadvantage for taking
math classes, while in Catholic schools it is an advantage.
This defines a disordinal interaction between sector and the
Minority-->Classes path.

Now let us focus attention on the other background
variable, SES. Table 2-B indicates that the average

SES-->Classes path is .35. On average, higher SES students
take more math classes. This B12 path is predicted by two
group level variables:

1) Sector, which has the regression coefficient of -.25. In
Catholic schools the B12 path is smaller indicating that there
is a weaker relationship between SES and number of classes
taken. Catholic schools seem more egalitarian by this
criterion.

2) Ave-SES has a .15 coefficient for predicting B12. In
schools with higher average SES, the student's SES is more

important for determining number of classes taken, i.e higher

105

SES schools are less egalitarian.

Path.
B11

B12

B21

B22

B23

 

 

106

 

V c s
t u u e wee - o e
Chi- Parameter
Parameter Square To Total
Varianga Sgagiagic Probabilitv YQL13322___

.216 204.312 <.0001 .203
.048 205.549 <.0001 .198
.318 207.220 <.0001 .213
1.849 166.351 .0149 .109
.187 103.918 .955 .082

 

.32

.19

.05

.15

 

 

 

Tabla Z-B
o o d
Averaga Valaa 9f Ragga
tu e etwee -G on el
Average 2
Path Vaiue Statiagia o ab it _
B11 (Minority->
Classes) -.343 -4.705 <.0001
B12 (SES->
Classes) .353 11.010 <.0001
B21 (Classes->
Ach) 2.953 38.323 <.0001
B22 (Minority—>
Ach) -1.868 -7.319 <.0001
B23 (SES->Ach)
.371 3.451 .0006

 

 

 

 

B11

B12

B21

1322

B23

Path

ed

108

ab 2-
e ata
o t e e cie ts

Second
Second Stage
Stage Regression Z

MMMW

Sector .617 4.34 <.0001
Sector -.246 -3.52 .0004
Ave-SES .149 1.80 .0722
Sector .278 1.82 .069
Sd-SES -7.02 -2.72 .007

 

 

 

 

109

The direction of all the second-stage effects is
coincident with previous research and substantive theory.
The R2 for the B12 slope indicates that these three second-
stage predictors (Ave-SES, Sector and Ave-Classes), account
for 22% of the total parameter variance of the path.

The B21 path, representing the relationship of the number
of math classes a student takes to math achievement, has an
average value of 2.95 (as indicated by table 2-B). Taking more
classes is strongly related to higher achievement. This
relationship is quite variable across schools, with an easily
significant parameter variance indicated in table 2-A. It is
helpful to look at the "parameter to total variance” ratio in
column 4. Twenty one percent of the total variance is
parameter variance even after being conditioned on the second-
stage model.

The single between-group variable that predicts the B21
(Classes-->Ach) path is Sector. The regression coefficient
of B21 on Sector is .28 (re table 2-C) which is only marginally
significant (P - .07). The interpretation that can be given
this is that Catholic schools evidence a somewhat stronger
positive relationship between number of classes and achievement
than public schools. It could be said that classes are more
efficient in Catholic schools, i.e. taking a math class
Creates a greater gain in math achievement in Catholic schools.
TC> find a mechanism for this influence we might inquire into

(”lrricular differences between Catholic and private schools.

110
As table 2-A shows, the R2 for B21 is only .05, i.e. only 5%
of the parameter variance is explained by Sector. Sector is
not very important for mediating the Classes to Ach effect,
and there are other factors, not represented in this analysis,
which would explain the path.

The final two paths represent the relationship between
background variables and achievement. Looking at table 2-A
we see that the parameter variance of both paths is a small
percentage of total, account for only 11% and 8% of total
variance. Once the number of classes is controlled for there
is little variation from school to school in the relationship
between student's background and achievement.

The Minority to Achievement path, B22, has an average
value of -1.87 (table 2-B), indicating that being a minority
has a negative effect on achievement. This path is predicted
by the standard deviation of SES for a school (Sd-SES).
From table 2-0 we see that the coefficient for Sd-SES is
-7.0 (significant at a P - .007). This second-stage
coefficient implies that as a school gets more heterogenous
in its social mix, a student's minority status is a bigger
determinant of achievement. The precise interpretation to
give this relationship in terms of school processes would be
difficult to determine without more information about how
the schools functioned. The last column of table 2-A indicates
that Sd-SES accounted for only 15% of the parameter variance

in B22 (Minority-->Ach). Given that the total parameter

lll
variance for B22 was initially quite small, the import of
the Sd-SES prediction is minimal.

The final path is SES to achievement, B23. It has an
average value of .37, which is significantly different from
zero (p-.0006, from table 2-B). Since there is virtually no
parameter variance in this path, the average value represents
the relationship for every school. As with number of classes
taken, higher SES is associated. with. higher achievement
scores. This holds equally true for the public and the
Catholic sector.

The multilevel path model indicates that the processes
in the schools that are responsible for making the Catholic
seem more 'egalitarian' than public schools pertain to how
math classes are distributed to students. Once we account
for the number of math courses students take, the relationship
between student background and achievement is quite constant

across schools.

11 e na 8 s Sc tt ools

The second dataset that is to be analyzed was first
interpreted by Willms (1985). The dataset I have access to
comes from 20 secondary schools in one administrative division
in Scotland. The total number of students in the dataset is
1292, so on average 65 students were sampled per school. The
original intent of gathering the data was to estimate the

effectiveness of each school based on the school mean on an

112
achievement index. In one study, "effectiveness” was
controlled for student level socioeconomic background, and
student level academic background (Willms, 1987). School
”effectiveness“ was also controlled for school level Nunuext"
factors consisting of aggregated SES and academic background.

In the 1987 analysis by Willms, a technique devised by
Longford (1985) was employed for obtaining maximum likelihood
estimates of covariance components in a multilevel mixed
model. Using this technique, Willms was able to estimate
school mean achievement controlled for by variables at two
levels of analysis, i.e. the individual student level and
the school level. In the present analysis the dataset will
be used for a quite different purpose than originally intended.
The present analysis will a) define a path model at the
within-school level, b) will ascertain if the paths vary
from school to school, and c) will explore the possibility
of accounting for path variability with a between-school
model which incorporates school context factors as predictors.
Note that school means, which were the focus of previous
analyses, do not appear in the present model at all.

It will be of technical interest to see how well the
multilevel path analysis performs when there is a small number
of groups, 20 in this case. In contrast, in the previous
analysis of the High School and Beyond data there were 158
schools. Since the Scottish data has a small set of schools
taken from. a Icontiguous geographical area, the range of

variation of the within school processes might be severely

113
restricted. This could result in small parameter variance
for paths, and attenuated second-stage regression estimates.
There are five variables which make up the within-class
data set:
Education of Mother (Edmoth) - Educational level of
student's mother.
Occupation of Father (Occfath) - Occupational status of
father. A sociological
index of occupational

status .

Number of Siblings (Numsib) - Number of brothers and
sisters.

Verbal Reasoning Quotient (VRQ)- A verbal IQ battery,
intended to represent
general academic skills.

Achievement (Ach) - An index of measures
covering the last three
years of secondary school.

The first three variables are intended to measure a
student's socioeconomic status. The verbal reasoning score
is intended to capture the student's academic background,
i.e. the academic skills the student enters secondary schools
with (Willms, 1987).

The three school-level variables consist of aggregated
student-level measures and represent school context. It is
often believed that aggregated individual level variables
represent more than simply the average impact of the
individuals' values. For example, if average verbal reasoning

is high, a school might have a more interesting and creative

curriculum, contributing to a positive learning environment,

114

even for those students with low verbal reasoning skills.

This is an example of a variable changing its meaning from

one level of analysis to another (Burstein, 1980).

The school level variables are:

Average SES (Ave-SES) - An average socioeconomic
background score. The SES
score for a student was a
weighted combination of
education of mother,
father's occupation and
number of siblings, where
the weights were derived
from principle components
analysis (Willms, 1987).

Average Occupational Status

of Father (Ave-Occfath) - Average of the student's

occupational status index.

Average Verbal Reasoning - Average of the students'
(Ave-VRQ) VRQ score.

F s -S e del

The path model devised on this data set was a two-equation
system similar to the model posited for the High School and
Beyond data. Although it would have been preferable to
demonstrate the multilevel path approach with a very different
model, (e.g. a four equation system with numerous endogenous
predictors) a certain similarity between the two sets of
data constrained the choice of sensible models. The two
equation system has the following form for individual 1,

within group j:

115

VRQiJ - B113(Edmoth)1j + 8121(Occfath)1j + B13j(Numsigs;1gl

Achij - B21j(Edmoth)ij + B22j(0ccfath)1j + B23J(Numsibs)1
+ 3241(VRQ) + Rij2

The path diagram depicted in figure 4.6 is more
descriptive. This is parallel to the High School and Beyond
model in general definition” In both cases there are two
equations in the system and in the first equation student
social background ‘variables are antecedents for academic
background. In the second equation social background and
academic background (as an endogenous predictor) are
antecedents for achievement. The parallel is further extended
by the fact that in ‘both studies the SES and academic
background variables were aggregated to the school level to
define school context, but more of this when the second-
stage model is described. The primary reason for the striking
parallel between the two data sets is the fact that they were
compiled for similar reasons, to estimate academic outcomes
which are controlled for factors at two levels of aggregation.

Parallel purpose led to parallel structure.

116

U11

\\\\ U24

EDMOTH 311

 

U21_______‘ .
-21

 

U12-1\“ B12

OCCFATH B22

U22*””"

U13‘-‘—’Bl3

NUMSIBS .132

/

U23

ACH

V

Figure 4.6
Scottish School Data: Baseline Model

117
Baseline_Anelxsis
As before, an initial baseline analysis is performed with
an unstructured second-stage model, i.e no school level
variables are stipulated, so the paths vary around the grand

mean:

B11 - B11 + U11
B12 - B12 + U12
B13 - E13 + U13
B21 - E21 + U21
B22 - E22 + U22
B23 - E23 + U23

324 - 324 + 024

Combining this with the first-stage model gives rise to
the multilevel path diagram in figure 3. As before, the
bold arrows represent paths of the first-stage model, and
the finely etched arrows pointing to the paths represent the
impact of school level factors (in the baseline model, random
parameter error) on the paths.

Table 3-A lists the estimated parameter variances of the
paths. By inspecting the chi-square probabilities (column
4) it. is apparent that three paths have no significant
parameter variance. The paths B12, B13 and B24 have chi-square
probabilities of .67, .84 .34 respectivelyu As a result, these
paths will not be modelled with school level predictors.

Table 3-B gives the 'average' betas. These are the

intercepts of the second-stage regressions:

118

B11, B12, ... , B24

The average betas coincide with commonsense expectations. 811-
Ave and B12-Ave are positive indicating that a higher level
of mother's education and a higher level of father's
occupational status is associated with higher verbal reasoning
skills. B13 is negative indicating that verbal reasoning
skills are inversely related to size of family. All things
being equal, having a larger family is probably associated
with a generally lower socioeconomic status since more children
means a greater financial ‘burden. The same pattern of
relationship between SES variables and outcome is found in
the second equation. Taken together the paths, B21 (Edmoth-
->Ach) , B22 (Occfath-->Ach) and B23 (Numsibs-->Ach), indicate
that higher SES is associated with greater academic
achievement. The final path, B24, indicates a strong positive

relationship between academic background and achievement.

 

 

Chi-
Path Parameter Square
Variaaga Statistic oba
B11
.02286 51.66 .0001
B12
.00298 16.79 .67
B13
.00395 13.82 .84
B21
.00494 33.33 .03
B22
.00689 35.23 .02
B23
.01102 45.84 .0008
B24 .00234 22.05 .34

 

 

 

Parameter
To Total

Vagiance

.60

.23

.39

.35

.45

.57

.25

 

Path

B11 (Edmoth->

B12

B13

B21

B22

B23

324

VRQ)

(Occfath->
VRQ)

(Numsibs->
VRQ)

(Edmoth->
Ach)

(Occfath->
Ach)

(Numsibs->
Ach)

(VRQ->Ach)

 

Average

.086

.236

-.l70

.093

.111

-.060

.639

120

 

Z

l.

27.

99

.96

.72

.69

.01

.95

61

del

Vel22____u_E£a£1§£1£___u_fiighahili£1__

.046

<.0001

<.0001

.0002

<.0001

.050

<.0001

 

 

121

a e t

A number of exploratory runs were made to determineIflfich
of the school level variables predict each path. .As was
mentioned earlier, since this is a full information maximum
likelihood procedure and since predictors are multicollinear,
inclusion. or exclusion of 13 single predictor alters the
entire solution. It is therefore necessary run numerous
trials, testing whole sets of second-stage predictors. The
end product of this exploratory phase is a quite modest

model which has the form:

B11 - B11 + 1111(Ave-VRQ) + 011

B12 - B12 + U12

B13 - B13 + U13

B21 - i521 + U21

B22 - E22 + U22

B23 - 323 + 1231(Ave-SES) + 1232(Ave-Occfath) + 023

324 - 324 + 024

Only two paths have predictors. Table 3-A, for the
baseline model, indicated that B12, B13 and B24 had virtually
no parameter variance and so were not susceptible to
prediction. Two other paths (B21 and B22) although having
significant parameter variance, evidenced no relationship
with the available set of school level predictors. This

raises the possibility that important school level processes

122
are not represented and that the model may be misspecified
at the second stage.

Another issue is purely statistical. The degrees of
freedom are quite small in relationship to the number of
parameters being estimated. There are ten fixed effects and
only twenty schools. With more groups a more predictive

between-group model may have been possible.

123

ave-vrq

\ \

EDMOTH 811

 

U21—_______.
B21

 

U12~\1\\‘ BI2
”’/”

OCCFATH 822

“22””’/'

 

U13 B13

NUMSIBS B23 _,ACH

/

023 ave-see ave-occfath

Figure 4.7
Scottish School Data: Structured Model

124

The B11 (Edmoth-->VRQ) path is explained by average school
VRQ. The second-stage regression coefficient (table 4-C) is
.25, implying that in schools with a high average VRQ, mother's
education is more predictive of a student's verbal reasoning
than in schools with a low average VRQ. Why this is the case
is a matter of speculation. Perhaps average school VRQ is
indicative of a facilitative school learning atmosphere
where a mother's contribution to the general education of
her children will be reinforced rather than drowned out.
More in-depth information on the nature of the school processes
would be required to illuminate this question. Whatever the
underlying mechanism, Ave-VRQ explained 46 percent of the
total parameter variance, as is indicated by table 4-A.
Also, the chi-square probability of the conditional parameter
variance is .06, which is non-significant by a strict
criterion. 80 the B11 path has been substantially explained
by the model.

A rather different result was found in the second-stage
prediction model for B23 (Numsibs-->Ach). Table 4-C shows that
average school SES has a second-stage regression coefficient
of .61 for predicting B23. Since the average B23 path is
negative (-.06) higher school SES would tend to make this
path less negative, or more positive. School SES has a
large enough standard. deviation that 111 the ‘highest SES
schools the B23 path would flip around and become positive.

Perhaps this is a result of a threshold effect. If the

125
family is financially well off having siblings increases
opportunities for a child to learn. But below a certain
economic threshold, a bigger family means greater financial
burden and greater deprivation for the student. Again, only
process information about schools can begin to answer these
questions.

Another school level variable predicts the B23
(Numsibs-->Ach) path, namely the average occupational status
of father. Surprisingly, this has a negative predictive
coefficient for B23 which equals -.04. The 2 test for this
coefficient is significant at the .01 level. Why an SES-
related variable would predict with an opposite sign as
average SES is mysterious. This suggests that the model is
not fully specified. If other relevant school level variables
could have been added to the model, such an anomaly might
disappear. Table 4-A indicates that the conditional parameter
variance of this path is significant 80, although the present
second-stage model accounted for 38 percent of the parameter
variance, there is more that can be explained.

In sum, table 4-A shows us that three paths (312, B13 and
324) had virtually no between-group (parameter) variation.
These paths can be regarded as constant over schools. One
path, B11, had virtually all of its between-group variation
explained. Another path, B23, had only part of its parameter
variance accounted for. Two paths, B21 and B22, had a
significant amount of parameter variance but none of it was

explained in a second-stage model.

B11

B12

B13

B21

17‘22

B23

324

 

Parameter

@MM

.01023

.00401

.00272

.00580

.00743

.00683

.00287

 

126

 

I3hl£_&;fi.
W
V t s
w de
Chi- Parameter
Square To Total
bab v Variance
29.43 .06 .43
16.81 .67 .31
13.86 .84 .27
33.48 .03 .42
35.40 .02 .49
29.96 .04 .52
22.16 .33 .31

 

 

 

.38

Path

811 (Edmoth->

B12

B13

B21

B22

323

324

VRQ)

(0ccfath->
VRQ)

(Numsibs->
VRQ)

(Edmoth->
Ach)

(Occfath->
Ach)

(Numsibs->
Ach)

(VRQ->Ach)

 

127

1311124143.
co t
Average Eglgg 9f Bathe
tu e tw -G
Average Z

.087 2.48 .013
.232 7.59 <.0001
-.l68 -5.85 <.0001
.095 3.61 .0003
.112 3.99 <.0001
-.059 -2.06 .027
.639 27.02 <.0001

 

VLMW

 

 

IEhJE 4-g
Sc tt 0
S d-St e o effic e ts
Second
Second Stage
Path Stage Regression Z
MMMMW
B11
Ave-VRQ .025 4.21 <.0001
B12
B13
B21
B22
B23
Ave-SES .607 3.36 .0008
Ave-Occfath -.040 -2.48 .013
324 ——

 

 

 

 

129

Although the inability of the analysis to explain much
of the between-group variation.in.paths indicatesznuincomplete
model, .a baseline model is valuable in a multilevel path
analysis. If our interest lies getting precise estimates of
a path model for each group, a multilevel path analysis yields
posterior estimates of paths which have the smallest possible
means square error. If our interest lies in explaining paths
rather than estimating them, a rich and correctly specified
second-stage model is a necessity.

The analyses of both datasets has illustrated the
usefulness in stipulating path models at the between-group
level. In both cases the path model made substantive sense
and yielded sensible results after the analysis was performed.
The multilevel path analysis was less successful in explaining
the between-group variance of the paths. This difficulty is
symptomatic of the fact that the illustrations offered here
were borrowed from datasets that were designed for other
purposes. If the possibilities of multilevel path analysis
are going to be fully realized in the future, studies will
have to be designed for the purpose of explicating a path
model in numerous groups. This means that a rich mix of
process related variables has to be gathered at all levels
of analysis. When there is a more thorough matching of
statistical modeling and research design, substantive theory

will be better informed.

CHAPTER V:
CONCLUSION
I. Introduction

In the preceding chapters it has been argued that a
multilevel path model would merge the statistical traditions
of multilevel analysis and path analysis into a single powerful
analytic tool. In chapter two a statistical model was derived
which represented one type of multilevel path model. Chapter
three reviewed issues associated with the production of a
computer algorithm which would create estimates derived from
the model. Finally analyses of actual datasets were presented
in chapter four utilizing a computer program written according
to the principals outlined in chapter three. The analysis
section demonstrated that the multilevel path model gives
interpretable results when applied to the sorts of datasets
that occur in large-scale educational studies.

In principle, the multilevel path approach would be
pertinent whenever information is presented to the researcher
at two levels of analysis and the researcher is interested
in deducing causal processes at the 'lower' level. The

most obvious example of this is a study of students nested

within numerous schools, the situation in both datasets
analyzed in chapter four. In this instance we are interested
in modelling processes within each of numerous schools. A

less obvious example would be to study a set of individuals
observed at numerous time points. In the previous example a

130

131
group was represented by observations on numerous individuals
within a school. In this example the 'group' is represented
by observations at numerous time points within an individual.
If we were studying what contributes to the development of
math skill in early elementary students, we might collect data
on computational skills, understanding of concepts and general
math ability at ten time points. A within-student path model

might take the following form:

 

Time s—>fiath Achievement

Computation

Concepts

From such a model we could determine which, if any, math
skills are important for the development of math ability.
This application of the multilevel path model parallels the
applicathmn of the Hierarchical Linear Model to individual
growth curves, as outlined by Bryk and Raudenbush (1987).

These examples suggest that there is 21 wide range of
applicability for the multilevel path models. Nevertheless,
such models are especially useful if certain conditions are
met:

1. There are large datasets. Experience with the related
HLM has shown that data from tens of groups is required in

order to give precise estimates (Raudenbush, 1984). In

132
fact, for a given total sample size, it is better to have
many small groups than a few large groups.

2. Similar processes occur in all groups. It is assumed
in the multilevel path model that the same variables are
related by the same causal network in all groups. In a
sense, each set of within-group paths represents a replication
of path model experiment.

3. If there is information available about the nature
of the groups, this information must explain why processes are
different from one group to another. This is because such
variables serve as predictors in a between-group model which

models variation in the within-group paths.

I W ve h Model

W

Although the results in chapter four demonstrated that
application of the multilevel path model can lead to
interesting results, this methodology may not always be
feasible because the algorithm is computationally intensive.
The EM algorithm requires many passes through the data before
it converges, so without a fast mainframe computer and a
healthy computer budget, such techniques may be impractical.
Ironically, one way around this problem may be the 'low
tech' approach. If a version of the estimating program
could be written to work on a micro computer, expense would

not be a factor. The analysis could be set into motion and

133
the computer could be left on its own for however long is
required. This could be cumbersome from a time standpoint

though.

WW

Another problem with this type of analysis is a
conceptual, rather than a practical, one. Multilevel path
models will give proper estimates only if the models are
properly specified at both levels of analysis. This imposes
a heavy a priori burden on theory and emphasizes that this
technique is not particularly appropriate for exploratory

purposes.

t al est

A third problem that comes to light is statistical in
nature. The chi-square test for parameter variances is only
approximate. One reason for this is because the statistic
is a function of an estimated regression coefficient that is
the least squares estimate of paths and involves the inversion
of the data matrix for each equation, for each group. There
will often be groups that do not have full rank predictor
matrices. For example, if gender is a predictor and a group
is composed. of all females, the 'variance and covariance
terms for gender will be zero. 4At present, non-full-rank
cases are simply excluded from the calculation of the
statistic, but not from the Bayesian estimates (which do not

require the inversion of data matrices). So the estimates of

134
parameters and the test statistics for parameter variances
may be based on two sets of groups.
Another reason that test statistics are approximate is
because they treat dispersion parameters as known quantities.
The error of the dispersion estimates cannot be estimated by

the present program.

III m at ons

Limited H9481 definition

The main limitation of this version of the multilevel
path model is that it represents only one of many feasible
configurations. Muthen and Satorra (1987) define the
conceptual possibilities for defining multilevel structural
models which include: 1) measurement models at the first
and/or second stage 2) random predictors at the first and/or
second stage 3) A path model at the second stage. These
possibilities define 32 different configurations which would
be possible for multilevel path model, of which the present

model is one.

W

In any path analysis it is very useful to have some
criterion by which to assess how well the model fits the
data. In a LISREL model one can test the fit of the model

by employing an omnibus chi-square test based on a likelihood

135
ratio test (Joreskog, 1973). An analogous test has not been
developed for the multilevel path analysis. Perhaps a
likelihood ratio test based on the log likelihood of the

data (derived in chapter two) could be devised.

d u anc
The next issue may not be a limitation as such, but a
strong assumption. In the model devised in this thesis, the
first-stage disturbances (or errors) are uncorrelated, meaning
that e is diagonal. This means that the RR are uncorrelated

for the within-group equation system (individual i and group

J);

Y - Z B + R -
equation 1 Y1 - X1 b(x)11 + x2 b(x)l2 + °'° + xq b(x)1q
+ R1
equation 2 Y2 - Y1 b(y)21 + x1 b(x)21 + ~-- + xq b(x)2q
+ R2
equation K YK - Y1 b(y)1<1 + Y2 b(y)K2 +
+ Y(K-1) b<y>K<K-1) + x1 b<x>1<1
+ ... + xq b(x)Kq + Rx

This is a fortuitous assumption because it enables us to
use the separate-equation regression approach to estimate the

paths (Land, 1973). The assumption of uncorrelated

136

disturbances is perfectly reasonable assuming that all
pertinent variables are in the model and the configuration
of paths is correct. It is commonly believed that if there
is a variable missing from the path model and that this
variable has a causal influence on two or more endogenous
(outcome) variables, the disturbances will be correlated.
This sentiment is echoed by Hanushek and Jackson (1977) "If
the same explanatory factor is excluded from more than one
equation, the effect of that factor will be present in more
than one error term and will cause the error terms to be
somewhat correlated" (p. 230). In Joreskog's LISREL model
(Joreskog & Sorbom, 1978) one can allow the error terms, Rk,
to be correlated. The motivation behind doing so is to make
up for such missing, confounding predictors (confounding in
that the missing variable is related to at least two endogenous
variables). It is assumed that since gm variables are
almost always left out of any model, correlated error terms
are a way to represent the effect of these missing variables
and the result will be a model that fits the data well.

Hunter and Gerbing have disputed this claim and have
devised two counter examples to disprove it (Hunter & Gerbing,
1981). The first counterexample illustrates a situation in
which a confounding variable is left out of the model but
the resultant path model has uncorrelated errors. The model
can remain well specified even in the face of missing
confounding variables and uncorrelated disturbances if paths

are added to the model which make the connections which

137
would have been made if the missing variables had been in
the model. These connections will be indirect causal paths
because they would have been mediated by a missing variable.
As an example I will give a simplified version of Hunter and

Gerbing's illustration. Suppose the complete causal system

 

pictured below; The values are the causal paths for the
population:
8
/\
A C
3 .3 3
D ::E
.3

Now suppose the path model that is specified leaves out
factor 'D'. The usual custom is to leave out all direct paths

associated with 'D'. This leads to the following estimate:

This is a poor fitting model but it could be 'fixed up'

by allowing the disturbances to be correlated. But another

138
model could have been specified which would not necessitate
correlated disturbances by simply putting the indirect paths

which would have been mediated by 'D':

One often expects there to be missing intervening
variables, but these can be accommodated by the correct
specification of paths. In a second counterexample Hunter
and Gerbing illustrate a situation in which a misspecified
model is made to display apparent good fit by incorrectly
allowing disturbances of the equations to be correlated.

First we have the actual model;

X d1::Y1"””,¢rr'
\Y

3‘k\\\‘d3

 

Where d1, d2 and d3 are uncorrelated.

A misspecified model will be defined if we leave out the

path from Y2 to Y3;

139

X #Y 1

Hunter and Gerbing claim that a LISREL analysis on such
a misspecified model will not fit the data well in the sense
that it will not reproduce observed correlation matrix.

But what if we further misspecify the model by stipulating
that the disturbance terms for Y2 and Y3 are correlated (as

is indicated by a curved arrow);

Y
3\63

In this case they claim that a LISREL analysis fits the data
almost perfectly. The moral is that specifying correlated
disturbances does not make up for missing variables. It may
instead cover up misspecified paths.

The upshot of these examples is that there is no analytic
substitute for a properly specified path model; one that has
properly defined paths as well as variables. As a result, in
multilevel path models even more onus is plated.on proper model

specification.

140
In the less ideal world of actual data analysis the
equations may in fact be correlated to some extent. It
would be quite useful in the future to do monte carlo studies
to determine how results are effected by mild departures

from the assumption of uncorrelated errors.

IV u u Wo k

e n ec

The multilevel path model defined in this thesis is an
instance of a mixed model, i.e. both fixed and random effects
are present in the combined model (Braun, Rubin and Thayer,

1983). From chapter two the mixed model is given as:
Y - A191 + A292 + R , (5

where 1 is the fixed effect and U is the random effect.

The mixed model is expressed in the combined-level form by,
Y - 2W7 + ZU + R . (S
This in turn can be separated into its two-stage
hierarchical form by equations:

Y-ZB-I-R, (5

B - W1 + U . (5

.1)

.2)

.3)

.4)

By substituting 5.4 into 5.3 we get Equation 5.2. In the

combined form it is apparent that the vector U ,containing the

141
random effects, constitutes the parameter differences across
groups and the vector 1, containing the fixed effects,
constitutes the second-stage regression coefficients. If
the chi-square test indicates that one of the parameter
variances is zero, this implies that the parameter error,
Upj: corresponding to path BPJ’ is zero in every group j.
In other words, if a path has zero parameter variance, the
random component constituting the path is null so the second-

stage model for path p is simply:
Bp-Wy

The advantage of the mixed-model formulation is that a
first-stage parameter can be modeled as a fixed effect by
eliminating the corresponding Upj in every group. If J is the
number of groups this can be accomplished by deleting the J
Upj elements from U, and suitably reducing the column dimension
of A2 (or Z) by J. When some paths in fact have no variance,
estimates assuming that these paths are fixed are more valid
than estimates assuming that these paths are random. The
programming needed to fix effects will be the next feature
added to the multilevel mixed model. The analyses presented
in this thesis will be redone with the appropriate paths

defined as fixed effects.

e e W -G u at ode
Another elaboration that can readily be added to

multilevel path analysis is the addition of intercepts to

142
the first-stage model. If predictors are also mean deviated,
the intercepts will be group means which are interesting
from a substantive point of view. Presently, the multilevel
path program has the option to add group-mean intercepts to
the path model. This elaboration wasn't presented in the
current analysis because paths were the main focus of the
thesis. The path model for the High School and Beyond defined

for intercepts would have the form (for individual i and

group J):

Classes - Classes + B11(Minority-Minority)

+ B12(SES-SES) + R1

 

Achievement - Achievement + 821(Classes-Classes) +

322(Minority-Minority) + 323(SES-SES) + R2

The advantage of adding group means is that they can be
analyzed in a second-stage model so that we can define the
antecedents of group effects as well as group processes. For
example, it would be interesting to know if Catholic schools
were more equitable (smaller SES->Ach path) but at the expense
of average achievement for the school. The data analysis
presented in this thesis will be rerun with group means in

the near future.

143

Wade].

One of the possibilities summarized by Muthen and Satorra
(1987) involved adding a measurement model to the within-group
model. Ignoring measurement error at the first stage tends
to bias estimates of paths and inflate the estimates of
sampling error. The notion of a measurement model in a
multilevel context reiterates many of the issues that arose
with the concept of a within-group path modelling. Do we
assume that the factor structure is the same for all groups?
Do we further assume that the factor loadings constant across
groups? Parallel to the path model formulation, it is my
judgement that the factor structure will be constant across
groups but the actual loadings will vary. This implies that
a measurement model would be formulated by running separate
confirmatory factor analyses which test the same factor
structure, in all groups.

Another central question is how would one best enact a
measurement model simultaneously defined over numerous groups.
The LISREL program currently has the capability of estimating
a measurement model. But executing a separate LISREL analysis
in each of over a hundred groups would be computationally
prohibitive. Also, LISREL is based on large sample estimating
theory which might be inappropriate for the often small
group size to be found in multi-group data bases.

There is another, more conceptual, objection to a LISREL

type approach. LISREL simultaneously estimates measurement

144

coefficients and path coefficients. Since LISREL is based on
full information maximum likelihood (as is the multilevel path
program) misspecification errors will tend to bias the path
estimates and visa versa. Heise says of full information
maximum likelihood (FIML) estimation ”All the observed
variances and covariances simultaneously contribute to the
estimation of all the parameters...FIML methods are quite
sensitive to specification error" (Heise, 1975). This is an
especially acute problem if the definition of the measurement
model. is in, an exploratory ‘phase where different factor
structures are being piloted or items are being assessed for
feasibility of inclusion in scales. Imbedding the measurement
part of the model within a larger two-stage path model would
mean that an independent assessment of the quality of the
measurement instruments could not be had.

Gerbing and Hunter site an example where a deliberately
misspecified path model gives incorrect estimates of factor
correlations, "Even though the error was in the causal mode,
LISREL placed aflJ. of the error into the estimated factor
correlations and maintained perfect consistency between factor
correlations and the incorrect path coefficients" (Gerbing &
Hunter, 1980). They conclude that the simultaneous analysis
of measurement and causal models may be suited for correctly
specified models but "There is no a priori reason why a
researcher would induce the additional complexity of

simultaneous analysis of untested. measurement and causal

145
models except that the necessary machinery exits for such an
analysis" (p19).

This problem can be avoided if measurement model
definition is a wholly separate phase of the analysis. This
gives the researcher an opportunity to troubleshoot scales
through numerous confirmatory runs. When a valid measurement
structure is confirmed, the latent covariances of factors
are input into the path analysis. This approach was
successfully employed by the author (Jenkins, 1985) in defining
a large scale measurement model for later input into LISREL.
The approach in that study was to first define the measurement
structure through a least squares confirmatory factor analysis
procedure. This structure was then validated by a confirmatory
factor analysis using LISREL. Interestingly the least squares
and LISREL runs gave similar factor loadings.

A measurement model as a separate stage of analysis would
be relatively straightforward to implement with the present
estimation program. A separate routine could be written to
implement a least square confirmatory factor analysis in all
groups simultaneously (see Hunter 6: Gerbing, 1979). Some
summary statistics to represent fit of the model would have
tn) be devised (e.g. average residual correlations; means,
maximum and minimum values of loadings, average factor/factor
correlations, etc.). Variables will be added to and deleted
till a good-fitting factor structure has been defined. Then
the estimated factor/factor correlations for each group

would be fed into the multilevel path program as it exists now.

146

w e - e

The issues involved with defining a between-group
measurement model are simpler because only one model has to
be devised, rather than one for every group. If a separate
measurement model analysis were planned, it could be done
via LISREL or a least squares confirmatory package (Hunter &
Gerbing, 1979) and the resulting factor scores could be fed
into the existing multilevel path program.

An attempt to simultaneously define a measurement and a
path model would be part of an effort to define a group-level

path model. This will be discussed in the next section.

It is not immediately apparent that defining a path model
at the group level would be useful. For a group-level path
model first-stage paths would be the second-stage endogenous
variables. I cannot think of a sensible interpretation for
a situation where one within-group path causes another such
path” .A between-group path model would be sensible under
two conditions a) we want to represent a network of
relationships among the group-level variables and b) intercepts
(which are group means) are included in the model as predictors
in the second-stage path analysis. These sorts of intercepts

are the same as group-level variables.

147

The easiest way to devise a between-group path model would
be to repeat the same general approach used for the within-
group path model. This would involve defining a simultaneous
equation system in which paths were outcomes and group level
variables (or intercepts) are predictors or outcomes. As
with the first-level model, the errors of the equations
would be uncorrelated

For example consider again the first-stage path model for
the High School and Beyond data. This model, defined with
intercepts and assuming mean-deviated predictors takes the

following form for individual i and group j:

Classes - Classes + B11(Minority) + B12(SES) + R1

Ach - Ach + B21(Classes) + 322(minority) + B23(SES) + R2

There are four variables defined at the group level;
Sector, Ave-SES, Sd-SES, and Ame-Classes. Note that Ave-
Classes and the Classes intercept would be the same variable.

A Possible second-stage path model might be defined as follows:

Sector - 110(Ave-SES) + 112(ET;§§Z§) + 113(Sd-SES) + U11
B11 - 120 + 121(Sector) + U12

B12 - 730 + 131(Sector) + 132(Ave-SES) + U12

821 - 740 + 141(Sector) + U21

B22 - 150 + 151(Sd-SES) + U22

B23 - 760 + U23

Here we see that the model previously defined for the

paths is as it was before, but now there are relationships

148
among the group-level variables and intercepts. Specifically,
all the group-level variables which characterized Sector are
explicitly introduced. as exogenous predictors (the first
equation).

The complexity has increased quite a bit over the
previously defined multilevel path model. Possibly the
greatest problem with these models will be to keep them
simple enough.

This sort of scheme would fit into a second-stage model

which, in matrix terms, looks the same as before:

B*-W1+U

But unlike the previous formulation the outcome vector,
B*, contains more than just paths, it can contain paths,
intercepts and group-level variables. As before for group
j, Var(UJ) - T. But now T is defined as a diagonal matrix,
(i.e. errors are not correlated) for the same reason that ¢
is diagonal; disturbances are independent in a: properly
specified path system. This simple scheme for a group-level

path model would fit into the present estimating program

with little renovation.

ow 3
One may ask, "Why not just do a LISREL model at the group
level?" This is a theoretical possibility; but a between-group

LISREL model would depart from the statistical assumptions

149

of the present estimation theory. First of all in the LISREL
model it is assumed that exogenous variables are random. In
the General Bayesian Linear Model exogenous variables are
fixed. Also, with LISREL path coefficients are defined in
two rectangular matrices of fixed parameters. In contrast,
with the Bayesian approach second-stage path coefficients
are defined as a vector of random coefficients with a vague
prior distribution. At present the LISREL model would not
fit into the context of the General Bayesian Linear Model.
The assumption of random exogenous variables at the first or
second stage of the hierarchy would necessitate a reformulation
of the multilevel path model, possibly along lines other
than those developed in chapter two.

This work represents a beginning of the actual estimation
of multilevel path models. This final chapter has suggested
a few ways the present statistical approach could be extended.
There are doubtless other approaches which would further expand
the scope of these models.

Regardless of the analytic form such approaches take in
the future, I hope to have demonstrated that multilevel path
models have promise to be a powerful tool for illuminating

important issues in social science research.

APPENDIX

150

V UN ON 9

From Equation 2.14 the posterior density of 9 is:

f(6|Y,i,O,A) -

(2«)'KN/2 (2«)'t/2 |w|'1/2 Ifll'1/2

exp{-l/2(Y-A9)' w'l (Y-A6)} exp{-1/2e'0'19} (A.1)

The exponential component is the quadratic Q
Q - (Y-Ae)'w'1(Y-Ae) + e'n'le (A.2)

Expanding terms we get:

Q - Y'w'lY - Y'W'IAG - e'A'w‘lY + e'A'w'lAe + e'n‘le
(A.3)

The first term, Y'W'IY,is not a function of 9 and so is
a constant with respect to the density. The corresponding
term, exp{-l/2Y'¢'1Y}, is taken out of the exponent and put
into the constant term. The remaining terms are combined

and arranged in descending powers of 8 to yield:

Q - 9'(A'w‘1A+0‘1)e - 2Y'W'1A9 (A.4)

151

Q now has the general quadradic form:

Q - X'MX - 2B'X , with : (A.5)
l) X - 9

2) M - (A'w'1A+n'1)

3) B - A'n’ly

The square of the quadratic can be completed to put Q into

the algebraically equivalent form:
Q - (X-MB)'M(X-MB) - B'MB ‘ (A.6)
Substituting for X, M and B we get:

Q - [e - (A'w'1A+n'1)A'w'1Y]'(A'w'1A+o'1) (A.7)

[e - (A'0'1A+0'1)A'W'1Y] - Y'w'1A(A'w'1A+n'1)A'W'1Y

The last term, not being a function of 6, can be taken

out of the quadratic and put into the constant to yield:

Q - [e - (A'W'1A+0'1)A'W'1Y]'(A'W'1A+0'1) (A.8)

[e - (A'w'1A+n'1)A'W'1Y]}

Which is the result required for Equation 2.15 in the text.

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