HIERARCHICAL LINEAR MODELS

1. HIERARCHICAL LINEAR MODELS • USED WITH NESTED DESIGNS IN EDUCATION, PSYCHOLOGY • USES RANDOM FACTORS EXPECTED MEAN SQUARE THEORY • COMBINES INFORMATION ACROSS LEVELS TO ESTIMATE VARIANCES, STANDARD ERRORS • PRODUCES CORRECT STANDARD ERRORS FOR F-TESTS

2. TWO LEVEL ANOVA Student score = Class mean + Indiv. Error yij = Ci + pij Class mean = School mean + Class error Ci = S + eij yij = ( S + eij ) + pij Thus, student score has class variation in it, the effect of being in, and influencing a particular class.

3. TWO LEVEL ANOVA Correction to standard errors: SEcor = SE [ 1 + (nclus -1)  ] , nclus = # students per class where = Intraclass correlation : var(class)/[var(class) + var(student)]

4. TWO LEVEL ANOVA SEcor = SE [ 1 + (nclus -1)  ] , nclus = # students per class where = Intraclass correlation : Average (rclass) > rschool Example: SE = SD/ SQRT(nclus) = 2.236 let = .1, SD= 10, nclus = 20 SEcor = {10/4.472}[ 1+ (20-1)*.1] = 6.485

5. Y raverage= .5 School regression rschool = 0 X

6. ANCOVA and ATI for HLM • Dep. Variable: Popularity • Fixed effects: Pupil gender • Random effects: Teacher experience • Model: yij = b0j + b1j Gen + eij b0j = g00 + g01 Exp + u0j Intercepts b1j = g10 + g11 Exp + u1j Slopes

7. boys Popular I ty girls Class 1 Class 2 Teacher Experience

8. GLM ANALYSIS OF PREFERENCE AND AGGRESSION R-Square Coeff Var Root MSE mrat1 Mean 0.171843 20.13344 0.658391 3.270139 Source DF Type I SS Mean Square F Value Pr > F ovg 1 292.0628803 292.0628803 673.76 <.0001 agmean 1 0.2496626 0.2496626 0.58 0.4480 ovg*agmean 1 0.0163565 0.0163565 0.04 0.8460 Source DF Type III SS Mean Square F Value Pr > F ovg 1 21.83289074 21.83289074 50.37 <.0001 agmean 1 0.24986698 0.24986698 0.58 0.4478 ovg*agmean 1 0.01635649 0.01635649 0.04 0.8460 Standard Parameter Estimate Error t Value Pr > |t| Intercept 3.293216737 0.03268211 100.77 <.0001 ovg -1.521097692 0.21433132 -7.10 <.0001 agmean -0.124159266 0.16353429 -0.76 0.4478 ovg*agmean -0.199813164 1.02863958 -0.19 0.8460

9. HLM ANALYSIS OF PREFERENCE AND AGGRESSION Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) cls 0.09404 0.01264 7.44 <.0001 INTERCEPT OVG UN(2,1) cls 0.03315 0.02609 1.27 0.2038 COV INT-SLOPE UN(2,2) cls 0.4096 0.1071 3.82 <.0001 SLOPE OVG Residual 0.3328 0.008725 38.15 <.0001 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F OVG 1 163 37.39 <.0001 OVGMEAN(CLASS) 1 2920 5.06 0.0246 OVG*OVGMEAN 1 2920 0.01 0.9068

10. CLASSROOM 1 PREFERENCE CLASSROOM 2 CLASSROOM 5 CLASSROOM4 CLASSROOM 3 CLASSROOM MEAN AGGRESSION