1. Hierarchical Linear Modeling (HLM): A Conceptual Introduction Jessaca Spybrook Educational Leadership, Research, and Technology

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Example School level predictors Slide 10

11. 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

12. Example Visual Look Slide 12