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# Multilevel Analysis

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1. Multilevel Analysis By Zach Andersen Jon Durrant Jayson Talakai

2. OUTLINE • Jon – What is Multilevel Regression • Jayson – The Model • Zach – R code applications / examples

3. WHAT IS MULTILEVEL REGRESSION • Regression models at multiple levels, because of dependencies in nested data • Not two stage, this occurs all at once

4. EXAMPLES • Students in schools • Individuals by area • Employees in organizations • Firms in various industries • Repeated observations on a person https://www.youtube.com/watch?v=wom6uPdI-P4

5. WHEN TO USE A MULTILEVEL MODEL? • Individual units (often people), with group indicators (e.g. Schools, area). • Dependent variable (level 1) • More than one person per group • Generally we need at least 5 groups, preferably more. (Ugly rule of thumb) https://www.youtube.com/watch?v=wom6uPdI-P4

6. WHEN TO USE A MULTILEVEL MODEL? • Use a multilevel model whenever your data is grouped (or nested) into categories (or clusters) • Allows for the study of effects that vary by group • Regular regression ignores the average variation between groups and may lack the ability to generalize http://www.princeton.edu/~otorres/Multilevel101.pdf

7. DATA STRUCTURE AND DEPENDENCE • Independence makes sense sometimes and keeps statistical theory relatively simple. • Eg; standard error(sample average) = s/n requires that the n observations are independent • But data often have structure, and observations have things in common; same area, same school, repeated observations on the same person • Observations usually cannot be regarded as independent https://www.youtube.com/watch?v=wom6uPdI-P4

9. PROBLEMS CAUSED BY CORRELATION • Imprecise parameter estimates • Incorrect standard errors

10. A SIMPLE 2-LEVEL HIERARCHY School 1 School 2 Student 1 Student 2 Student 3 Student 1 Student 2 Student 3 https://www.youtube.com/watch?v=wom6uPdI-P4

11. A SIMPLE 2-LEVEL HIERARCHY School 1 School 2 Level 2 Student 1 Student 2 Student 3 Student 1 Student 2 Student 3 Level1 https://www.youtube.com/watch?v=wom6uPdI-P4

12. PEOPLE ARE AT LEVEL 1?? • The first level of a hierarchy is not necessarily a person https://www.youtube.com/watch?v=wom6uPdI-P4

13. A SIMPLE 2-LEVEL HIERARCHY Industry 1 Industry 2 Level 2 Firm 1 Firm 2 Firm 3 Firm 1 Firm 2 Firm 3 Level1 https://www.youtube.com/watch?v=wom6uPdI-P4

14. A SIMPLE 2-LEVEL HIERARCHY Person 1 Person 2 Level 2 Event 1 Event 2 Event 3 Event 1 Event 2 Event 3 Level1 https://www.youtube.com/watch?v=wom6uPdI-P4

15. BRIEF HISTORY • Problems of single level analysis, cross level inferences and ecological fallacy https://www.youtube.com/watch?v=wom6uPdI-P4

16. DISCUSSION AS TO WHY A NORMAL REGRESSION CAN BE A POOR MODEL • Because Reality might not conform to the assumptions of linear regression (Independence) • Because in nature observation tend to cluster • A random person in Lubbock is more likely to be a student then a random person in another city (clustering of populations/not independent) • Different clusters react differently https://www.youtube.com/watch?v=wom6uPdI-P4

17. EXTENSIONS • Focus was initially on hierarchical structures and especially students in schools • Also longitudinal, geographical studies • More recently moved to non hierarchical situations such as cross-classified models. (single level is part of more than one group)

18. INTRACLASS CORRELATION • Level 1 variance explained by the group (level 2) • ICC is the proportion of group-level variance to the total variance • Formula for ICC: • Variance in group • Overall variance http://en.wikipedia.org/wiki/Intraclass_correlation

19. MULTILEVEL MODELING • Random or Fixed Effects • What are random and fixed effects? • When should you use random and fixed effects? • Types of random effects models • The Model • Assumptions of the model • Building a multilevel model

20. **Anytime that you see the word “population” substitute it with the word “processes.” Fixed vs random effects http://www2.sas.com/proceedings/forum2008/374-2008.pdf

21. INTRODUCING THE MODEL

22. Types of Models: Random Intercepts Model • Intercepts are allowed to vary: • The scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups.  • http://en.wikipedia.org/wiki/Multilevel_model

23. Types of Models: Random Slopes Model • Slopes are different across groups. • This model assumes that intercepts are fixed (the same across different contexts).  • http://en.wikipedia.org/wiki/Multilevel_model http://www.strath.ac.uk/aer/materials/5furtherquantitativeresearchdesignandanalysis/unit4/randomslopemodelling/

24. Types of Models: Random intercepts and slopes model • Includes both random intercepts and random slopes • Is likely the most realistic type of model, although it is also the most complex. • http://en.wikipedia.org/wiki/Multilevel_model

25. Assumptions for Multilevel Models • Modification of assumptions • Linearity and normality assumptions are retained • Homoscedasticity and independence of observations need to be adjusted. • Observations within a group are more similar to observations in different groups. • Groups are independent from other groups, but observations within a group are not. http://en.wikipedia.org/wiki/Multilevel_model