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An Introduction to HLM and SEM 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 analyses. 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 data. 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 homework. 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 performance 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 used. Example from Tate Example of a policy analysis related to ongoing school reform efforts in a hypothetical state. Set of instructional objectives for fifth grade science were developed but individual schools not required to use objectives in their curriculum 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 objectives? 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 Objectives” 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 analysis. 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. SEM 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. SEM 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 SEM 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) Authoritative Parenting Style Family Stress Teacher Support Ethnic Identity Global Self-esteem Confirmatory Factor Analysis SEM