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Hierarchical Linear Modeling: An Introduction & Applications in Organizational Research Michael C. Rodriguez

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Common Problems The “unit of analysis” problem – misestimated precision Testing hypotheses about effects occurring at each level and across levels Problems related to measurement of change or growth

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What’s in a name… Sociology: Multilevel Models Biometrics: Mixed-Effects Models or Random-Effects Models Econometrics: Random-Coefficient Regression Models Statistics: Covariance Components Models

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A way to think about HLM A regression is computed for each school, given the students and student level characteristics within each school At a second level, the resulting regression equations are regressed across schools, given the schools and school level characteristics

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An Example Is there a relationship between socio- economic status and achievement? Is the relationship between SES and achievement the same in public schools and catholic schools?

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Regression: Achievement & SES Is the level of achievement of student i in school j related to the student’s Socio-Economic Status? If SES scores are “centered” within each school so that the average SES for each school is zero, the intercept of the regression becomes the mean achievement level of school j

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Means as Outcomes Since the SES variable is centered within each school, we need to account for potential differences in the average SES between schools. Is the mean achievement level of school j related to the Sector of the school (public versus catholic)?

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Slopes as Outcomes Does Sector predict the within-school slopes? Is the relationship between SES and Achievement related to Sector?

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Advantages of HLM Estimation of parameters requires some distributional assumptions. One requires the error term (the part of the outcome that is not explained by observed factors) to be independent and identically distributed. This is in contrast with the idea that people exist within meaningful relationships in organizations. Frank, K. (1998). Quantitative methods for studying social contexts. Review of Research in Education, 23,

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Advantages of HLM Adjustment for nonindependence of error terms – when subjects are nested in meaningful or relevant organizations Larger framework for real-life problems; when organizational contexts matter Unbalanced designs and missing data are accommodated

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What do we gain through HLM? Improved estimation of effects within individual units. Example: Developing an improved estimate of a regression model for an individual school by borrowing strength from the fact that similar relationships exist for other schools.

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What do we gain through HLM? Formulation and testing of hypotheses about cross-level effects. Example: How school size might be related to the magnitude of the relationship between social class and academic achievement within schools.

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What do we gain through HLM? Partitioning variance and covariance components among levels. Example: Decomposing the correlation among a set of student-level variables into within- and between-school components. How much of the variance is within or between schools?

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A Classroom Assessment Study Is achievement in mathematics related to mathematics self-efficacy and effort (about 1 hour of homework per day)? Does the relationship between student self- efficacy, student effort, and achievement depend on classroom assessment practices?

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The Model

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The Results

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