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Hierarchical Linear Modeling (HLM): A Conceptual Introduction Jessaca Spybrook Educational Leadership, Research, and Technology
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Slide 2 Overview What is hierarchical data? Why is it a problem for analysis? Example Modeling the hierarchical structure Example 1 student level predictor 1 student level predictor, 2 school level predictors Questions
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What is hierarchical (nested) data? Examples Kids in classrooms Kids in classrooms in schools Kids in classrooms in schools in districts Workers in firms Patients in doctors offices Repeated measures on individuals Other examples? Slide 3
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Why is it problematic? What is the relationship between SES and math achievement? Dependent variable: Math achievement Independent variable: Student SES Case 1: 1 School (school A) School A Mean achievement: SES achievement slope: Slide 4
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Why is it problematic? Case 2: 1 school (School B) School B Mean achievement: SES-achievement slope: Case 3: 160 schools 160 means, mean varies 160 SES-achievement slope parameters, slope varies Within school variation Slide 5
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Why is it problematic? Case 3: 160 schools Option A: Ignore nesting Violate assumptions for traditional linear model Standard errors too small Option B: Aggregate to school level Lose information Option C: Model the hierarchical structure Hierarchical linear models, multilevel models, mixed effects models, random effects models, random coefficient models Slide 6
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Modeling the hierarchical structure Advantages Improved estimation of individual (school effects) Test hypotheses for cross level effects Partition variance and covariance among levels Slide 7
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Example Results – what do they mean? Slide 8 Fixed EffectCoefficientStandard Error t-ratiop-value Overall mean achievement 12.640.2451.84<0.001 Mean SES-ach slope 2.190.1317.16<0.001 Random EffectsVarianceDfChi-squarep-value School means,u 0j 8.681591770.86<0.001 SES-ach slope, u 1j 0.68159213.440.003 Within school, r ij 36.70
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Example School-level predictors Do Catholic schools differ from public schools in terms of mean achievement (controlling for school mean ses)? Do Catholic schools differ from public schools in terms of strength of association between student SES and achievement (controlling for school mean ses)? Slide 9
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Example School level predictors Slide 10
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Example Results – what do they mean? Slide 11 Fixed EffectCoefficientStandard Error t-ratiop-value Model for school means Intercept12.090.1769.64<0.001 Catholic1.230.313.98<0.001 MEAN SES5.330.3315.94<0.001 Model for SES-ach slope Intercept2.940.1519.90<0.001 Catholic-1.640.24-6.91<0.001 MEAN SES1.030.333.110.002
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Example Visual Look Slide 12
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