Explore the concept of Hierarchical Linear Modeling (HLM) in educational research, including the challenges of analyzing hierarchical data structures, nested data examples, and the problematic nature of ignoring nesting. Learn how modeling the hierarchical structure can lead to improved estimation of individual effects and hypothesis testing for cross-level effects.
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Hierarchical Linear Modeling (HLM): A Conceptual Introduction Jessaca Spybrook Educational Leadership, Research, and Technology
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 Slide 2
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
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: = + + 2 ( ) ~ , 0 ( N ) Y SES r r 0 1 i i i i ^ = . 9 73 0 ^ = . 2 51 1 Slide 4
Why is it problematic? Case 2: 1 school (School B) School B Mean achievement: SES-achievement slope: = + + 2 ( ) ~ , 0 ( N ) Y SES r r 0 1 i i ^ i i = 13 51 . 0 ^ = . 3 26 1 Case 3: 160 schools 160 means, mean varies 160 SES-achievement slope parameters, slope varies Within school variation Slide 5
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
Modeling the hierarchical structure Advantages Improved estimation of individual (school effects) Test hypotheses for cross level effects Partition variance and covariance among levels = + + 2 : 1 ( ) ~ , 0 ( ) Level Y SES r r N 0 1 ij j j ij ij ij = + : 2 Level u 0 00 0 j j u 0 j 00 01 = + ~ u MVN 1 10 1 j j u 10 11 ij = + + + + : ( ) ( ) Combined Y SES u u SES r 00 10 0 1 ij ij j j ij ij Slide 7
Example Results what do they mean? Fixed Effect Coefficient Standard Error 0.24 t-ratio p-value Overall mean achievement Mean SES-ach slope 10 12.64 51.84 <0.001 00 2.19 0.13 17.16 <0.001 Random Effects School means,u0j SES-ach slope, u1j Within school, rij Variance 8.68 0.68 36.70 Df 159 159 Chi-square 1770.86 213.44 p-value <0.001 0.003 Slide 8
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
Example School level predictors = + + : 1 ( ) Level Y SES r 0 1 ij j j ij ij = + + + : 2 Catholic ( ) ( ) Level MeanSES u 0 00 01 02 0 j j j ) j = + + + Catholic ( ) ( MeanSES u 1 10 11 12 1 j j j j = + + + : Catholic ( ) ( ) ( ) Combined + 11 Y MeanSES SES 00 01 02 10 ij j j ij ) + + + + Catholic ( )( ) ( )( ) ( SES MeanSES SES u u SES r 12 0 1 j ij j ij j j ij ij Slide 10
Example Results what do they mean? Fixed Effect Coefficient Standard Error t-ratio p-value Model for school means Intercept Catholic MEAN SES Model for SES-ach slope Intercept Catholic MEAN SES 12.09 1.23 5.33 0.17 0.31 0.33 69.64 3.98 15.94 <0.001 <0.001 <0.001 00 01 02 2.94 -1.64 1.03 0.15 0.24 0.33 19.90 -6.91 3.11 <0.001 <0.001 0.002 10 11 12 Slide 11
