Mixed Effects Models - Level-2 Variables
- 1. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 2. Recap • Last week, we created a model of middle schoolers’ math performance that included a random intercept for Classroom • model1 <- lmer(FinalMathScore ~ 1 + TOI + (1|Classroom), data=math) Fixed effect of naive theory of intelligence Average intercept (averaged all classrooms) Variance in that intercept from one class to the next Residual (unexplained) variance at the child level
- 3. Notation • What is this model doing mathematically? • Let’s go back to our model of individual students (now slightly different): Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Fixed mindset γ10x1i(j)
- 4. Notation • What is this model doing mathematically? • Let’s go back to our model of individual students (now slightly different): Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Fixed mindset γ10x1i(j) What now determines the baseline that we should expect for students with fixed mindset=0?
- 5. Notation • What is this model doing mathematically? • Baseline (intercept) for a student in classroom j now depends on two things: • Let’s go back to our model of individual students (now slightly different): Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Fixed mindset γ10x1i(j) U0j = Intercept + Overall intercept across everyone B0j γ00 Teacher effect for this classroom (Error)
- 6. Notation • Essentially, we have two regression models • Hierarchical linear model • Model of classroom j: • Model of student i: Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Fixed mindset γ10x1i(j) U0j = Intercept + B0j γ00 Teacher effect for this classroom (Error) LEVEL-1 MODEL (Student) LEVEL-2 MODEL (Classroom) Overall intercept across everyone
- 7. Hierarchical Linear Model Student 1 Student 2 Student 3 Student 4 Level-1 model: Sampled STUDENTS Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class Level-2 model: Sampled CLASSROOMS • Level-2 model is for the superordinate level here, Level-1 model is for the subordinate level Variance of classroom intercept is the error variance at Level 2 Residual is the error variance at Level 1
- 8. Notation • Two models seems confusing. But we can simplify with some algebra… • Model of classroom j: • Model of student i: Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Fixed mindset γ10x1i(j) U0j = Intercept + B0j γ00 Teacher effect for this classroom (Error) LEVEL-1 MODEL (Student) LEVEL-2 MODEL (Classroom) Overall intercept across everyone
- 9. Notation • Substitution gives us a single model that combines level-1 and level-2 • Mixed effects model • Combined model: Student Error Ei(j) = End-of-year math exam score + + Yi(j) Fixed mindset γ10x1i(j) U0j + Overall intercept γ00 Teacher effect for this classroom (Error)
- 10. Notation • Just two slightly different ways of writing the same thing. Notation difference, not statistical! • Mixed effects model: • Hierarchical linear model: Ei(j) = + + Yi(j) γ10x1i(j) U0j + γ00 Ei(j) = Yi(j) B0j γ10x1i(j) U0j = + B0j γ00 + +
- 11. Notation • lme4 always uses the mixed-effects model notation • lmer( FinalMathScore ~ 1 + TOI + (1|Classroom) ) • (Level-1 error is always implied, don’t have to include) Student Error Ei(j) = End-of-year math exam score + + Yi(j) Fixed mindset γ10x1i(j) U0j + Overall intercept γ00 Teacher effect for this class (Error)
- 12. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 13. ! We’re continuing our study of naïve theories of intelligence & math performance ! We’ve now collected data at three different schools ! math1.csv from Jefferson Middle School ! math2.csv from Highland Middle School ! math3.csv from Hoover Middle School Combining Datasets in R
- 14. Combining Datasets in R ! Look at the math1, math2, math3 dataframes ! How are they similar? How are they different? ! TOI and final math score for each student
- 15. Combining Datasets in R ! Look at the math1, math2, math3 dataframes ! How are they similar? How are they different? ! TOI and final math score for each student Columns not always in same order
- 16. Combining Datasets in R ! Look at the math1, math2, math3 dataframes ! How are they similar? How are they different? ! TOI and final math score for each student Only Hoover has GPA reported
- 17. Combining Datasets in R ! Overall, this is similar information, so let’s combine it all ! Paste together the rows from two (or more) dataframes to create a new one: ! bind_rows(math1, math2, math3) -> math ! Useful when observations are spread across files ! Or, to create a dataframe that combines 2 filtered dataframes math1 math2 math3 math
- 18. bind_rows(): Results ! Resulting dataframe: ! nrow(math) is 720 – all three combined math1 math2 math3 math
- 19. ! Resulting dataframe: ! bind_rows() is smart! ! Not a problem that column order varies across dataframes ! Looks at the column names ! Not a problem that GPA column only existed in one of the original dataframes ! NA (missing data) for the students at the other schools bind_rows(): Results
- 20. bind_rows(): Results ! Resulting dataframe: ! You can also add the optional .id argument ! bind_rows(math1, math2, math3, .id='OriginalDataframe’) -> math ! Adds another column that tracks which of the original dataframes (by number) each observation came from
- 21. Other, Similar Functions ! bind_rows() pastes together every row from every dataframe, even if there are duplicates ! If you want to skip duplicates, use union() ! Same syntax as bind_rows(), just different function name ! Other related functions: ! intersect(): Keep only the rows that appear in all of the source dataframes ! setdiff(): Keep only the rows that appear in a single source dataframe—if duplicates, delete both copies
- 22. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 23. Multiple Random Effects • Schools could differ in math achievement—let’s add School to the model to control for that • Is SCHOOL a fixed effect or a random effect? • These schools are just a sample of possible schools of interest " Random effect. School 1 School 2 Sampled SCHOOLS Sampled CLASSROOMS Sampled STUDENTS LEVEL 3 LEVEL 2 LEVEL 1 Student 1 Student 2 Student 3 Student 4 Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class
- 24. Multiple Random Effects • No problem to have more than 1 random effect in the model! Let’s a random intercept for School. School 1 School 2 Sampled SCHOOLS Sampled CLASSROOMS Sampled STUDENTS LEVEL 3 LEVEL 2 LEVEL 1 Student 1 Student 2 Student 3 Student 4 Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class
- 25. Multiple Random Effects • model2 <- lmer(FinalMathScore ~ 1 + TOI + (1|Classroom) + (1|School), data=math) School 1 School 2 Sampled SCHOOLS Sampled CLASSROOMS Sampled STUDENTS LEVEL 3 LEVEL 2 LEVEL 1 Student 1 Student 2 Student 3 Student 4 Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class
- 26. Multiple Random Effects • model2 <- lmer(FinalMathScore ~ 1 + TOI + (1|Classroom) + (1|School), data=math) • Less variability across schools than classrooms in a school
- 27. Multiple Random Effects • This is an example of nested random effects. • Each classroom is always in the same school. • We’ll look at crossed random effects next week School 1 School 2 Sampled SCHOOLS Sampled CLASSROOMS Sampled STUDENTS LEVEL 3 LEVEL 2 LEVEL 1 Student 1 Student 2 Student 3 Student 4 Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class
- 28. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 29. Level-2 Variables • So far, all our model says about classrooms is that they’re different • Some classrooms have a large intercept • Some classrooms have a small intercept • But, we might also have some interesting variables that characterize classrooms • They might even be our main research interest! • How about teacher theories of intelligence? • Might affect how they interact with & teach students
- 30. Level-2 Variables Student 1 Student 2 Student 3 Student 4 Sampled STUDENTS Mr. Wagner’s Class Ms. Fulton’s Class Ms. Green’s Class Ms. Cornell’s Class Sampled CLASSROOMS • TeacherTheory characterizes Level 2 • All students in the same classroom will experience the same TeacherTheory LEVEL 2 LEVEL 1 TeacherTheory TOI
- 31. Level-2 Variables • Is TeacherTheory a fixed effect or random effect? • Teacher mindset is a fixed-effect variable • We ARE interested in the effects of teacher mindset on student math achievement … a research question, not just something to control for • Even if we ran this with a new random sample of 30 teachers, we WOULD hope to replicate whatever regression slope for teacher mindset we observe (whereas we wouldn’t get the same 30 teachers back)
- 32. Level-2 Variables • This becomes another variable in the level-2 model of classroom differences • Tells us what we can expect this classroom to be like Student Error Ei(j) = End-of-year math exam score + + Baseline Yi(j) B0j Growth mindset γ10x1i(j) U0j = Intercept + Overall intercept B0j γ00 Teacher effect for this classroom (Error) LEVEL-1 MODEL (Student) LEVEL-2 MODEL (Classroom) Teacher mindset + γ20x20j
- 33. Level-2 Variables • Since R uses mixed effects notation, we don’t have to do anything special to add a level-2 variable to the model • model3 <- lmer(FinalMathScore ~ 1 + TOI + TeacherTheory + (1|Classroom) + (1|School), data=math) • R automatically figures out TeacherTheory is a level-2 variable because it’s invariant for each classroom • We keep the random intercept for Classroom because we don’t expect TeacherTheory will explain all of the classroom differences. Intercept captures residual differences.
- 34. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 35. What Changes? What Doesn’t? • Random classroom & school variance is reduced. • Teacher theories of intelligence accounts for some of the variance among classrooms (and among the schools those classrooms are in). • TeacherTheory explains some of the “Class j” effect we’re substituting into the level 1 equation. No longer just a random intercept. WITHOUT TEACHERTHEORY WITH TEACHERTHEORY
- 36. What Changes? What Doesn’t? • Residual error at level 1 essentially unchanged. • Describes how students vary from the class average • Divergence from the class average cannot be explained by teacher • Regardless of what explains the “Class j” effect, you’re still substituting it into the same Lv 1 model WITHOUT TEACHERTHEORY WITH TEACHERTHEORY
- 37. What Changes? What Doesn’t? • Similarly, our level-1 fixed effect is essentially unchanged • Explaining where level-2 variation comes from does not change our level-1 model • Note that average student TOI and TeacherTheory are very slightly correlated (due to random chance); otherwise, there’d be no change. WITHOUT TEACHERTHEORY WITH TEACHERTHEORY
- 38. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab
- 39. Cross-Level Interactions • Because R uses mixed effects notation, it’s also very easy to add interactions between level-1 and level-2 variables • model4 <- lmer(FinalMathScore ~ 1 + TOI + TeacherTheory + TOI:TeacherTheory + (1|Classroom) + (1|School), data=math) • Does the effect of a student’s theory of intelligence depend on what the teacher’s theory is? • e.g., maybe matching theories is beneficial
- 40. Cross-Level Interactions • Because R uses mixed effects notation, it’s also very easy to add interactions between level-1 and level-2 variables • In this case, the interaction is not significant
- 41. Week 5.1: Level-2 Variables ! Notation ! Multiple Random Effects ! Combining Datasets in R ! Modeling ! Level-2 Variables ! Including Level-2 Variables in R ! Modeling Consequences ! Cross-Level Interactions ! Lab