Hierarchical Linear Modeling HLM A Conceptual Introduction Jessaca

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

Overview What is hierarchical data? n Why is it a problem for analysis? n ¨ Example Modeling the hierarchical structure n Example n ¨ 1 student level predictor, 2 school level predictors n Questions Slide 2

What is hierarchical (nested) data? n Examples ¨ Kids in classrooms in schools in districts ¨ Workers in firms ¨ Patients in doctors offices ¨ Repeated measures on individuals ¨ Other examples? Slide 3

Why is it problematic? n What is the relationship between SES and math achievement? ¨ Dependent variable: Math achievement ¨ Independent variable: Student SES n Case 1: 1 School (school A) ¨ School A Mean achievement: n SES achievement slope: n Slide 4

Why is it problematic? n Case 2: 1 school (School B) ¨ School B Mean achievement: n SES-achievement slope: n n Case 3: 160 schools ¨ 160 means, mean varies ¨ 160 SES-achievement slope parameters, slope varies ¨ Within school variation Slide 5

Why is it problematic? n Case 3: 160 schools ¨ Option A: Ignore nesting Violate assumptions for traditional linear model n Standard errors too small n ¨ Option n Lose information ¨ Option n B: Aggregate to school level C: Model the hierarchical structure Hierarchical linear models, multilevel models, mixed effects models, random coefficient models Slide 6

Modeling the hierarchical structure n Advantages ¨ Improved estimation of individual (school effects) ¨ Test hypotheses for cross level effects ¨ Partition variance and covariance among levels Slide 7

Example n Results – what do they mean? Fixed Effect Coefficient Standard Error t-ratio p-value Overall mean achievement 12. 64 0. 24 51. 84 <0. 001 Mean SES-ach slope 2. 19 0. 13 17. 16 <0. 001 Random Effects Variance Df Chi-square p-value School means, u 0 j 8. 68 159 1770. 86 <0. 001 SES-ach slope, u 1 j 0. 68 159 213. 44 0. 003 Within school, rij 36. 70 Slide 8

Example n 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

Example n School level predictors Slide 10

Example n Results – what do they mean? Fixed Effect Coefficient Standard Error t-ratio p-value Intercept 12. 09 0. 17 69. 64 <0. 001 Catholic 1. 23 0. 31 3. 98 <0. 001 MEAN SES 5. 33 0. 33 15. 94 <0. 001 Intercept 2. 94 0. 15 19. 90 <0. 001 Catholic -1. 64 0. 24 -6. 91 <0. 001 MEAN SES 1. 03 0. 33 3. 11 0. 002 Model for school means Model for SES-ach slope Slide 11

Example n Visual Look Slide 12