Random Intercept Models

Transcription

1 Random Intercept Models Edps/Psych/Soc 589 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Spring 2019

2 Outline A very simple case of a random intercept model. When to use this model The model Empty model One explanatory variable Multiple explanatory variables Next: SAS and R overview followed by computer lab 1. Reading: Snijders & Bosker, pp C.J. Anderson (Illinois) Random Intercept Models Spring / 80

3 A Simple Model Situation: j = 1,...,N groups (e.g., schools). i = 1,...,n j individuals within the groups (e.g., students). Y ij a numerical response variable (e.g., math scores). x ij a possible explanatory variable (e.g., SES). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

4 Simple Level 1 Model Y ij = β 0j +β 1 x 1ij +R ij where β 1 is fixed in the population. R ij N(0,σ 2 ) and independent. The intercept, β 0j, depends on the group. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

5 Simple Level 2 Model The intercept, β 0j, can be broken down into two parts: An overall or average value of the intercept: γ 00 A group dependent part of the intercept: The Level 2 Model is U 0j β 0j = γ 00 +U 0j C.J. Anderson (Illinois) Random Intercept Models Spring / 80

6 The Linear Mixed Model Replace β 0j in the level 1 model by the level 2 model for β 0j : Y ij = γ 00 +U 0j +β 1 x 1ij +R ij = γ 00 +U 0j +γ 10 x 1ij +R ij where β 1 = γ 10 (to be consistent with later models). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

7 NELS88 Data What the model might look like applied to the NELS88 data (N = 10 schools): C.J. Anderson (Illinois) Random Intercept Models Spring / 80

8 Two cases of the model Y ij = γ 00 +γ 10 x 1ij +U 0j +R ij Case 1: ANCOVA If the U 0j s are fixed parameters in the population (i.e., The U 0j s are not random but are the group effects). Case 2: Random intercept model (i.e., HLM) If the U 0j s are random; that is, groups are a random sample in the population. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

9 Contrasting ANCOVA & HLM ANCOVA/ANOVA: Groups are unique and interest is only in the groups observed. A Group s observations provide information about U 0j, so with small n j, don t get precise estimates of group effects (large standard errors). InANCOVA, U 0j srepresentallthe differences between groups (no unexplained variability left). If U 0j and R ij are non-normal and/or dependent, random coefficient model may be unreliable. Random Coefficients Model: Groups are a sample from a (real or hypothetical) population and Group effects come from the same distribution, data from all groups is used= candowellwithsmalln j. Synonyms: random effects & unexplained variability i.e., U 0j N(0,τ0 2 ), groups are exchangeable. Can use a different distribution. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

10 Random Intercept Model: no x s The baseline/empty/null HLM (no explanatory variables). A regression model where the intercept is a random variable. Models for Clustered Data Level 1: Y ij = β 0j +R ij where R ij N(0,σ 2 ), and independent. The intercept, β 0j, is a random variable for which we specify a linear regression model (level 2, e.g., groups). Level 2: β 0j = γ 00 +U 0j where γ 00 is the intercept (fixed). U 0j N(0,τ0 2 ) and independent. U 0j and R ij are independent. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

11 Levels 1 and 2 Estimation of level 1 and level 2 models should be simultaneous. Substituting our equation for β 0j (level 2 model) into the equation for level 1, we get a composite or linear mixed model and the marginal model is Y ij = β oj +R ij = γ 00 }{{} +U 0j +R ij }{{} fixed random Y ij N(γ 00,(τ 2 0 +σ2 )) C.J. Anderson (Illinois) Random Intercept Models Spring / 80

12 Estimation Y ij = β oj +R ij = γ 00 }{{} +U 0j +R ij }{{} fixed random and Y ij N(γ 00,(τ 2 0 +σ 2 )) This (marginal model) is what is used to fit the model to data and in SAS/MIXED or R, we specify the linear mixed model. To use HLM7 software, you specify the component models (i.e., the level 1 model, and the level 2 models) and the program figures out what the marginal model is and fits it to the data. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

13 Example: Null Model High School and Beyond: A nationally representative sub-sample from the 1982 High School and Beyond Survey. It includes Information on n + = 7185 students. Students are nested within N = 160 schools: 90 public and 70 Catholic. Samples sizes averaged about n j 45 per school. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

14 Example: Level 1 Level 2 variables Level 1 Variables: Student level variables Outcome variable: Math achievement. Explanatory: Student socioeconomic status (SES) composite of parental education, occupation and income. This have been standardized such that average equals 0. Student ethnicity (1=minority, 0=not). Gender (1=female, 0=male). Level 2 Variables: School level variables Sector whether the school is Catholic (= 1) or public (= 0). Mean SES average SES of students w/in a school. School enrollment. Proportion of students in academic track. Disciplinary climate. Whether the percentage minority is 40% or > 40%. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

15 Example: the null HLM Y ij = math achievement. Groups are schools: j = 1,...,160. Diagram representing the situation:... math C.J. Anderson (Illinois) Random Intercept Models Spring / 80

16 The Hierarchical Null Model Level 1: Y ij = β 0j +R ij (math) ij = β 0j +R ij where R ij N(0,σ 2 ) and independent. Level 2: β 0j = γ 00 +U 0j where U 0j N(0,τ0 2 ) and independent....and R ij and U 0j are independent. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

17 The Marginal Model The Linear Mixed model: (math) ij = Y ij = γ 00 +U 0j +R ij where U 0j and R ij are independent and The Marginal Model: Y ij N(γ 00,(τ0 2 +σ 2 )). Implications for data. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

18 Results 1 Some descriptive statistics: The MEANS Procedure Variable Label Mean StdDev mathach Mathematics achievement ses standardized student ses Or in R, means(mathach), sd(mathach), where mathach is the name of math scores in hsb dataframe. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

19 Results 2 The Mixed Procedure Model Information Data Set SASDATA.HSBALL Dependent Variable mathach Covariance Structure Variance Components Subject Effect id Estimation Method ML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment C.J. Anderson (Illinois) Random Intercept Models Spring / 80

20 Results 3 Dimensions Covariance Parameters 2 Columns in X 1 Columns in Z Per Subject 1 Subjects 160 Max Obs Per Subject 67 Observations Used 7185 Observations Not Used 0 Total Observations 7185 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

21 Results 4 Iteration History Iteration Evaluations -2 Log Like Criterion Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z > t Intercept id <.0001 Residual <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

22 Results 5 Fit Statistics -2 Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

23 Summary of Results Null Model Parameter Value SE Parameter Value SE γ τ σ Estimate of the intra-class correlation, ˆρ I = ˆτ 2 ˆτ 2 + ˆσ 2 = = =.18 and ( math) ij = 12.64, the overall (total sample) mean, The overall variance of math scores is s 2 = ( ) = Note s = and s 2 = see descriptive statistics, p 18. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

24 Add SES as Level 1 Explanatory? C.J. Anderson (Illinois) Random Intercept Models Spring / 80

25 Some More Schools C.J. Anderson (Illinois) Random Intercept Models Spring / 80

26 Overlay Regressions for all Schools C.J. Anderson (Illinois) Random Intercept Models Spring / 80

27 Random Intercept: One x SES to help explain some of the variability of Y ij. The Hierarchical Model: Level 1: Y ij = β 0j +β 1j x ij +R ij where R ij N(0,σ 2 ) and independent. Level 2: β 0j = γ 00 +U 0j β 1j = γ 10 where U 0j N(0,τ0 2 ) and independent (and independent w/rt R ij ). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

28 Random Intercept: 1 x The appropriate diagram to represent this model?... SES ij math ij or more generally... x ij y ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

29 Random Intercept: 1 x The Linear Mixed Model: Y ij = γ 00 +γ 10 x ij +U 0j +R ij where γ 00 is the intercept for the average group. γ 10 is regression coefficient for x ij (fixed). U 0j N(0,τ0 2 ) and independent. R ij N(0,σ 2 ) and independent. U 0j and R ij are independent. Notes: var(y ij x ij ) = var(u 0j )+var(r ij ) = τ 2 0 +σ2 cov(y ij,y i j x ij,x i j) = τ 2 0 Residual intra-class correlation, ρ I (Y X = x) = τ2 0 τ 2 0 +σ2 If τ0 2 = 0, then simply use ordinary linear regression (OLS). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

30 Marginal vs Hierarchical Model The Marginal Model: Y ij N(γ 00 +γ 10 x ij,(τ 2 o +σ 2 )) The hierarchical model implies the marginal model. The marginal model does not imply the hierarchical model. There are no random effects in the marginal model. The HLM is more restrictive than the marginal. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

31 Linear Mixed Model in Matrix Notation The linear mixed model, Y ij = γ 00 +γ 10 x ij +U 0j +R ij, In matrix notation, for marco unit (group) j, Y 1j Y 2j. Y njj = 1 x 1j 1 x 2j.. 1 x njj ) γ 10 + ( γ ( U 0j )+ R 1j R 2j. R njj Y j = X j Γ + Z j U j + R j C.J. Anderson (Illinois) Random Intercept Models Spring / 80

32 Marginal Model in Matrix Notation And the marginal model is Y j N(X j Γ,(Z j TZ j +σ 2 I)). Note that Z j TZ j is just (n j n j ) τ 2 τ 2... τ 2 1τ 2 1 τ 2 τ 2... τ 2 = τ 2 τ 2... τ 2 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

33 Example: HSB x ij = SES ij (student/micro level). Some descriptive statistics: The MEANS Procedure Variable N Mean Std Dev Mathematics achievement Standardized student ses C.J. Anderson (Illinois) Random Intercept Models Spring / 80

34 Edited output from SAS/MIXED The Mixed Procedure Model Information Data Set SASDATA.HSBALL Dependent Variable mathach Covariance Structure Unstructured Subject Effect id Estimation Method ML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment C.J. Anderson (Illinois) Random Intercept Models Spring / 80

35 Example: HSB Dimensions Covariance Parameters 2 Columns in X 2 Design for fixed Columns in Z Per Subject 14 Design for random Subjects 160 N Max Obs Per Subject 67 largest n j Observations Used 7185 n + Observations Not Used 0 Total Observations 7185 Convergence criteria met. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

36 Proc MIXED Output Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 Residual <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001 ses <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

37 Summary & Comparison Null Model Add SES Value SE Value SE Fixed effects γ γ Random effects τ σ Residual intra-class correlation, ˆρ I (math SES) = = =.11 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

38 Comparison Drop in between groups variance estimate ˆτ 2 0 : 4.73 =.55 or (1.56)100 = 45% descrease 8.55 Drop in within groups variance estimate ˆσ 2 : =.95 or (1.95)100 = 5% descrease Interpretation somewhat problematic because SES helps to explain both the between and within groups variance of Y ij : SES ij = SES j +(SES ij SES j ) C.J. Anderson (Illinois) Random Intercept Models Spring / 80

39 Problematic Interpretation... Between versus Within Group Regressions. Different processes at work at different levels. An example: C.J. Anderson (Illinois) Random Intercept Models Spring / 80

40 Problematic Interpretation (continued) Within individual: Child s reading ability improves with instruction. Between individuals: Children get tutoring/private instruction for different reasons. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

41 Returning to the HSB Example Groups differ with respect to mean SES (i.e., SES j ). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

42 Returning to the HSB Example Individual students within schools differ with respect to SES. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

43 Returning to the HSB Example Or for just one school, we have within school variability, Line is simple linear regression line. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

44 Solution to Problem Regarding different mean levels of SES between and within schools: Group mean centered variable, e.g., x ij = (SES ij SES j ) to model within group variability of Y ij w/rt SES. Group mean as a level 2 (school) variable z j = SES j C.J. Anderson (Illinois) Random Intercept Models Spring / 80

45 Solution to Problem (continued) Recall that overall mean SES = 0. First we ll just have a level 1 (student) variable for SES: x ij = (SES ij SES j ) A diagram for this model,... (SES ij SES j ) math ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

46 Add Centered SES at Level 1? C.J. Anderson (Illinois) Random Intercept Models Spring / 80

47 Some More Schools C.J. Anderson (Illinois) Random Intercept Models Spring / 80

48 Overlay Regressions for all Schools C.J. Anderson (Illinois) Random Intercept Models Spring / 80

49 Hierarchical Model with Centered SES Level 1: Y ij = β 0j +β 1j x ij +R ij math ij = β 0j +β 1j (SES ij SES j )+R ij where R ij N(0,σ 2 ) and independent. Level 2: β 0j = γ 00 +U 0j β 1j = γ 10 where U 0j N(0,τ0 2 ) and independent. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

50 Linear Mixed & Marginal Models The corresponding linear mixed model: math ij = γ 00 +γ 10 (SES ij SES j )+U 0j +R ij and Marginal Model: math ij N(γ 00 +γ 10 (SES ij SES j ),(τ 2 0 +σ2 )). C.J. Anderson (Illinois) Random Intercept Models Spring / 80

51 Edited SAS/MIXED Output Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 Residual <.0001 Solution for Fixed Effects Standard t Effect Estimate Error DF Value Pr > t Intercept <.0001 cses <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

52 Estimated Overall Regression Line math ij = (SES ij SES j ) C.J. Anderson (Illinois) Random Intercept Models Spring / 80

53 Sub-Sample of Groups Lines math ij = (SES ij SES j )+Ûj C.J. Anderson (Illinois) Random Intercept Models Spring / 80

54 Model Summary & Comparison Null Model Add SES Add cses Value SE Value SE Value SE Fixed effects γ γ Random effects τ σ C.J. Anderson (Illinois) Random Intercept Models Spring / 80

55 Conclusion Residual intra-class correlation, ˆρ I (math SES) = = =.19 It appears that SES account for some variability in math achievement, Y ij, within schools ( ) 100% = (1.945)100% = 5.5% Does z j = SES j help model between group variability? C.J. Anderson (Illinois) Random Intercept Models Spring / 80

56 Between School Variability & SES Using SES j as predictor for β oj. The Û0j s plotted below are from Y ij = (γ 00 +U 0j ) +γ 10(SES ij SES j )+R ij }{{} β 0j C.J. Anderson (Illinois) Random Intercept Models Spring / 80

57 HLM with x and z Level 1: math ij = β 0j +β 1j (SES ij SES j )+R ij where R ij N(0,σ 2 ). Level 2: Linear Mixed Model β 0j = γ 00 +γ 01 (SES j )+U 0j β 1j = γ 10 where U 0j N(0,τ0 2 ) and independent. math ij = γ 00 +γ 10 (SES ij SES j )+γ 01 (SES j )+U 0j +R ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

58 Guessing What Will Happen math ij = (SES ij SES j )+U 0j +R ij Put in regression of Û 0j on SES j into above: Y ij = (SES ij SES j )+( (SES j )+U 0j)+R ij = (SES ij SES j )+5.344(SES j )+U 0j +R ij How about a new τ 0? The R 2 between Û0j & SES j equaled.623 and previously estimated ˆτ 0 = ˆτ 0 ˆτ 0 ˆτ 0 R ˆτ ˆτ C.J. Anderson (Illinois) Random Intercept Models Spring / 80

59 Theory and Proposition In terms of a diagram, our proposition (theory), SES j...ց... (SES ij SES j ) math ij Micro-micro: Higher a student s SES relative to the school mean, the higher his/her math score. Macro-micro: Higher average SES of school, higher a student s math score. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

60 Edited SAS/MIXED Output Dimensions Covariance Parameters 2 Columns in X 3 *** this changed Columns in Z Per Subject 1 Subjects 160 Observations Used 7185 Observations Not Used 0 Total Observations Convergence criteria met.. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

61 Edited SAS/MIXED Output Covariance Parameter Estimates Standard Z Cov Parm Sub Estimate Error Value Pr Z UN(1,1) id <.0001 Residual <.0001 Solution for Fixed Effects Standard t Effect Estimate Error DF Value Pr> t Intercept <.0001 cses <.0001 meanses <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

62 Summary Models Fit to HSB Null, empty, baseline, math ij = γ 00 +U 0j +R ij 1 uncentered variable, math ij = γ 00 +γ 10 (SES) ij +U 0j +R ij Group centered, math ij = γ 00 +γ 10 (SES ij SES j )+U 0j +R ij Group centered and mean, math ij = γ 00 +γ 10 (SES ij SES j )+γ 01 (SES j )+U 0j +R ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

63 Summary of Parameter Estimates Add cses ij Null Model Add (SES) ij Add cses ij & (SES) j Value SE Value SE Value SE Value SE Fixed effects γ γ γ Random effects τ0 2 (between) σ 2 (within) Note: cses ij = (SES) ij (SES) j SES j = (1/n j ) n j i=1 (SES) ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

64 Two more models Add all the student level variables that we have to see if we can account for more of the within groups residual variance. i.e., Add more x ij s to the model. SES ij SES j female (= 1 if female, = 0 if male) minority (= 1 if minority, = 0 if not) Add all the school level variables to see if we can account for more between groups residual variance. i.e., Add more z j s to the model. himinty = 40% minority, 0 =< 40% minority pracad = Proportion of students in academic track disclim = Disciplinary climate sector = 1 = Catholic, 0 = public size =school enrollment C.J. Anderson (Illinois) Random Intercept Models Spring / 80

65 HLM with Lots of Micro Variables Level1 : (math) ij = β 0j +β 1j (SES ij SES j ) where R ij N(0,σ 2 ) and independent. +β 2j (female) ij +β 3j (minority) ij +R ij Level2 : β 0j = γ 00 +γ 01 (SES) j +U 0j β 1j = γ 10 β 2j = γ 20 β 3j = γ 30 where U 0j N(0,τ0 2 ) and independent. R ij and U j are independent. Homework: What s the linear mixed model? What s the marginal model? C.J. Anderson (Illinois) Random Intercept Models Spring / 80

66 Edited SAS/MIXED output The Mixed Procedure Model Information Data Set WORK.HSBCENT Dependent Variable mathach Covariance Structure Unstructured Subject Effect id Estimation Method ML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment C.J. Anderson (Illinois) Random Intercept Models Spring / 80

67 Edited SAS/MIXED output Dimensions Covariance Parameters 2 Columns in X 7 Columns in Z Per Subject 1 Subjects 160 Max Obs Per Subject 67 Observations Used 7185 Observations Not Used 0 Total Observations 7185 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

68 Edited SAS/MIXED output Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 Residual <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

69 Edited SAS/MIXED output Solution for Fixed Effects Student student ethnicity gender (1=minority, (1=female, Standard Effect 0=not) 0=male) Estimate Error Intercept cses minority minority 1 0. female female 1 0. meanses C.J. Anderson (Illinois) Random Intercept Models Spring / 80

70 Note: cses ij = (SES) ij (SES) j SES j = (1/n j ) n j i=1 (SES) ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80 Simple Model Contrast ANCOVA & HLM Empty Model 1 Explanatory Multiple Explanatory Variables Summary Summary & Comparison Add cses ij Add all Null Model Add cses ij & (SES) j mirco level Value SE Value SE Value SE Value SE Fixed effects intercept centered SES male not minority mean SES Random effects τ0 2 (between) σ 2 (within)

71 Lots of Macro-level Variables Add all of the school (macro-level) variables: himinty = 40% minority, 0 =< 40% minority pracad = Proportion of students in academic track disclim = Disciplinary climate sector = 1 = Catholic, 0 = public size =school enrollment C.J. Anderson (Illinois) Random Intercept Models Spring / 80

72 HLM for Lots of Macro-level Level 1: (math) ij = β 0j +β 1j (SES ij SES j )+β 2j (female) ij +β 3j (minority) ij +R ij where R ij N(0,σ 2 ) and independent. Level 2: β 0j = γ 00 +γ 01 (SES) j +γ 02 (himinty) j +γ 03 (pracad) j β 1j = γ 10 β 2j = γ 20 β 3j = γ 30 +γ 04 (disclim) j +γ 05 (sector) j +γ 06 (size) j +U 0j where U 0j N(0,τ0 2 ) and independent, and R ij and U j are independent. C.J. Anderson (Illinois) Random Intercept Models Spring / 80

73 Linear Mixed w/ Lots of Macro-level (math) ij = γ 00 +γ 10 (SES ij SES j )+γ 20 (female) ij +γ 30 (minority) ij +γ 01 (SES) j +γ 02 (himinty) j +γ 03 (pracad) j +γ 04 (disclim) j +γ 05 (sector) j +γ 06 (size) j +U 0j +R ij What s the marginal model? C.J. Anderson (Illinois) Random Intercept Models Spring / 80

74 Edited SAS/MIXED output The Mixed Procedure Dimensions Covariance Parameters 2 Columns in X 13 Columns in Z Per Subject 1 Subjects 160 Max Obs Per Subject 67 Observations Used 7185 Observations Not Used 0 Total Observations 7185 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

75 Edited SAS/MIXED output Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 Residual <.0001 C.J. Anderson (Illinois) Random Intercept Models Spring / 80

76 Edited SAS/MIXED output: Fixed Effects Student student Sector Std Effect ethnicity gender Est. Error Intercept cses minority not minority minority 0. female male female female 0. meanses himinty pracad disclim sector public sector Catholic 0. size C.J. Anderson (Illinois) Random Intercept Models Spring / 80

77 Edited SAS/MIXED output Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F cses <.0001 minority <.0001 female <.0001 meanses <.0001 himinty pracad disclim sector size C.J. Anderson (Illinois) Random Intercept Models Spring / 80

78 SES j = n j i=1 (SES) ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80 Simple Model Contrast ANCOVA & HLM Empty Model 1 Explanatory Multiple Explanatory Variables Summary Summary & Comparison: Fixed Effects Add cses ij Add all All micro Null Model & (SES) j mirco level and macro Value SE Value SE Value SE Value SE intercept centered SES male not minority mean SES himinty pracd disclim sector size Note: cses ij = (SES) ij (SES) j

79 Summary & Comparison: Random Add cses ij Add all All micro Null Model & (SES) j mirco level and macro Value SE Value SE Value SE Value SE Random effects τ0 2 (between) σ 2 (within) Note: cses ij = (SES) ij (SES) j SES j = n j i=1 (SES) ij C.J. Anderson (Illinois) Random Intercept Models Spring / 80

80 Summary: Random Intercept Models Concepts covered: Within group dependency accounted for by random effect (i.e., R ij ). Between group differences accounted for by random intercept (i.e., U 0j ). Central role of variance between and within groups. Effect of adding Level 1 and Level 2 predictors. Mean centering of level 1 predictors and adding the mean back in at Level 2. Intra-class correlation or ICC. HLMs with only random intercepts imply homoscedasticity. C.J. Anderson (Illinois) Random Intercept Models Spring / 80