Jessaca Spybrook Western Michigan University Multi-level Modeling (MLM) Refresher

2 MLM Refresher Goal of session  Brief review of multilevel models  Establish common language  Establish common notation

3 Hierarchical Data Individuals grouped into larger units Examples  Students in schools  Citizens in communities Focus Example  Africa Program for Education Impact Evaluation in the Gambia Students in schools Teachers in schools Classrooms in schools

4 Hierarchical Data Methods for analyzing data  Put everything at one level  Aggregate data up to level 2  Model both levels together Model both levels together  Improved estimation of individual effects  Questions related to cross-level effects  Partitioning of variance among levels

5 Hierarchical Data Modeling both levels together  Hierarchical linear models, multilevel models, mixed effects models, random effects models, random coefficient models

6 Hierarchical Data Scenario  The Gambia data (2008)  Students nested in schools  2,657 ->2,008 students (pupils data)  271 ->204 schools (head teacher data)

Hierarchical Data Variables  DV: Number of words read correctly in 60 seconds (reading fluency) [S2Q3_PP]  IVs: Treatment (L2) [trmt –WSC and Grant] Age (L1) [age_PP] Mean school age (L2) [age_PP_m] 7

Guiding Questions (A)  What is the mean reading fluency for all students?  How much variation in reading fluency is between schools? Within schools? 8

The Model Level 1 (students): Y ij is reading fluency for student i in school j is the mean reading fluency for school j r ij is the random error associated with student i in school j, Level 2 (schools): is the average school mean reading fluency across the schools u 0j is the random error associated with school means, 9

The Model Combined Model: - Demo in HLM - Fill in Table as we go 10

ICC Intraclass Correlation: is the between school variance is the within school variance 11

Guiding Question (B)  Is there a difference in reading fluency at baseline for those in the treatment condition compared to those in the control condition? 12

The Model Level 1 (students): Y ij is reading fluency for student i in school j is the mean reading fluency for school j r ij is the random error associated with student i in school j, Level 2 (schools): is the average school mean reading fluency for the control schools W j is the indicator for condition (1=treatment including WSC and Grant, 0=control) is the main effect of treatment, average difference in mean reading fluency for treatment and control schools is the random error associated with control school means, now a conditional variance, 13

The Model Combined Model: 14

Guiding Questions (C)  What is the relationship between students age and reading fluency? Consider 5 options 1 - Age is group mean centered at L1 2 - Age is uncentered at L1 3 - Age is grand mean centered at L1 4 - Age is group mean centered at L1, grand mean centered at L2 5 - Age is grand mean centered at L1, grand mean centered at L2 15

The Model-Option 1 Level 1 (students): Y ij is reading fluency for student i in school j is the average unadjusted mean reading fluency for school j is the average change in reading fluency for a 1 unit increase in student age in school j (within school age-reading fluency slope) r ij is the random error associated with student i in school j, Level 2 (schools): is the average school mean reading fluency across the schools is the average age-reading fluency slope within schools u 0j is the random error associated with school means, 16

The Model Combined Model: 17

Option 2 What if we left age uncentered?  Same models, age uncentered  Intercept is now average school mean reading fluency for schools when age = 0  Slope is now composite of within school age- reading fluency relationship and between- school age reading fluency relationship 18

Option 3 What if we grand mean centered age?  Same models, age grand mean centered  Intercept is now average adjusted school mean reading fluency for schools  Slope is now composite of within school age- reading fluency relationship and between- school age reading fluency relationship 19

Option 4 What if we group mean centered age at L1 and grand mean centered age at L2?  Need new model  Aggregate version of age for each school at L2 20

The Model – Option 4 Level 1 (students): Y ij is reading fluency for student i in school j is the average unadjusted mean reading fluency for school j is the average change in reading fluency for a 1 unit increase in student age in school j (within school age-reading fluency slope) r ij is the random error associated with student i in school j, Level 2 (schools): is the average school mean reading fluency across the schools adjusted for school mean age is the average change in school mean reading fluency for a 1 unit increase in school mean age across schools (between school mean age-reading fluency relationship) is the average age-reading fluency slope within schools u 0j is the random error associated with adjusted school means, now a conditional, 21

Option 5 What is we grand mean centered age at L1 and grand mean centered age at L2?  Same model, age grand mean centered at L1  Intercept is same but adjusted mean  is same  is the compositional effect of age, weighted composite of the within and between slopes, difference between 2 students with same age value but who attend schools that differ by one unit of school mean age  Note: 22

Next Steps Practice session in lab Questions/comments via video session 23