An Introduction to
Carolyn Furlow
Hierarchical Linear Modeling (HLM)
Structural Equation Modeling (SEM)
Multilevel models or Hierarchical Linear
Models and Structural Equation Models are
both considered extensions of regression
 Both are frequently used with educational data
and are rapidly gaining in popularity.
When should HLM be used?
HLM is appropriate for use when we have
nested data structures which occurs frequently
with educational data.
 For example, when we have students who are
nested in classrooms, classrooms nested within
schools, etc…
 E.g., if we randomly sampled three classrooms
of students from 10 different schools and then
collected data from all these students.
Other HLM scenarios with nested data
Clients in groups for group therapy
 Employees in organizations
 School administrators in school districts
 Voters in voting precincts
 Homeowners in neighborhoods
Unit of Analysis
Researchers have difficulty deciding the
appropriate unit of analysis with educational
 Should the student be the unit of analysis or
the classroom mean, school mean, etc.?
 HLM simultaneously accounts for several
levels of data
HLM uses
We can simultaneously study the effects of
group level variables and individual level
variables with HLM
 There may be interactions across levels as well
that only HLM can account for.
 For example, the effect of student study time
may be related to teacher emphasis on
Why not just use multiple regression?
Students from Classroom A tend to be more
alike with each other than they would be with
students from Classroom B.
 Students within any one classroom, b/c they
were taught together tend to be similar in their
 As a result, they provide less information than
if the same number of students had been taught
separately by different teachers
Why not just use multiple regression?
Therefore the assumptions of constant variance
and independence of errors in multiple
regression are violated.
 Incorrect standard errors and tests of
significance for regression coefficients would
be given using MR when HLM should be
Example from Tate
Example of a policy analysis related to
ongoing school reform efforts in a hypothetical
 Set of instructional objectives for fifth grade
science were developed but individual schools
not required to use objectives in their
Example from Tate
Annual state-wide test was modified to reflect
the new objectives
 Evidence that individual schools vary with
respect to how consistent their science classes
are with objectives
Example from Tate
Policy makers have several research questions
 Question 1 (group level)
 Is
the average school achievement on the statewide science test, controlling for student aptitude,
related to the degree to which the school science
instruction is consistent with the state-wide
Example from Tate
Question 2 (individual level)
 Is
the relationship between individual science
achievement and individual aptitude within each
school related to the degree to which the school
science curriculum is consistent with the statewide objectives?
Hypothetical Study
Random sample of 20 schools from the state
 Collected measures of individual science
achievement and aptitude for all 580 students
in the 20 schools
 Each school has also been given a score on a
scale reflecting “Degree of Consistency of
School Science Instruction with State-Wide
Hypothetical Study
We can test at the group level how much the
school’s level of consistency affects the
variability of school’s scores on the
achievement test
 We can also test whether the relationship
between individual achievement and aptitude
is related to how consistent the curriculum is
with the objectives
Structural Equation Modeling (SEM)
Also seen as an extension of regression
 SEM attempts to analyze more complicated
causal models and can incorporate unobserved
(latent) variables and mediating variables as
well as observed (measured) variables
 SEM involves imposing a theoretical model on
a set of variables to explain their relationships.
Latent variables are unobserved/unobservable
variables such as self-esteem, marital happiness,
depression. These are sometimes called factors.
They are measured by indicators (observed
variables), often behaviors that can be observed such
as stated chance of getting divorced, number of fights
with spouse in the last week.
Standard SEM – consists of mediating
variables and latent variables
 Special Cases of SEM
 Path
analysis - all variables are observed but some
type of mediating variable exists
 Confirmatory factor analysis - where a latent
variable such as intelligence is measured by
several indicator variables
Obtain overall test of how well our data fits
with our proposed model
 Also obtain significance values for each of the
paths between variables
Example of SEM (path-analytic model)
Confirmatory Factor Analysis