G89 2247 Lecture 6 Multilevel Notation Level 1 and Level 2 Equations Multilevel Models with Random Intercept and Slope Models Incorporating Cross Level Interactions Estimation Issues G89 2247 Lecture 6 1 Bar Examination Study Design Examinee 1 1 1 1 1 1 1 1 2 2 1 2 2 100 100 100 100 100 Day 0 1 2 3 4 29 30 31 Distress 1 034 1 892 2 022 2 496 2 942 5 122 4 382 4 627 0 1 29 30 31 27 28 29 30 31 1 619 1 508 5 122 2 324 0 941 3 719 3 957 3 659 3 314 3 191 Exam Day Exam Day Exam Day Exam Day Exam Day Exam Day G89 2247 Lecture 6 2 Level 1 Equation for one individual Y i 0 1X i ri 1 Y 0 Intercept X G89 2247 Lecture 6 3 Level 1 Equation for many individuals Y ij 0j 1jX ij rij Y X G89 2247 Lecture 6 4 Level 2 Equations 0j 00 u0j 1j 10 u1j G89 2247 Lecture 6 5 L evel 1 E quation for many individuals Y ij 0j 1j X ij r ij Y Average Regression Line 10 00 X G89 2247 Lecture 6 6 Level 1 Equation Yij 0j 1jXij rij Level 2 Equations 0j 00 u0j 1j 10 u1j Combined Equation Yij 00 10Xij u0j u1jXij rij Fixed Random Effects Effects G89 2247 Lecture 6 7 Assumptions About Error Terms Level 1 The rij are independently and identically distributed iid normal random variables with mean 0 and variance 2 Level 2 The u0j are iid normal random variables with mean 0 and variance 00 Level 2 The u1j are iid normal random variables with mean 0 and variance 11 Level 2 The u0j and u1j can be correlated G89 2247 Lecture 6 8 Covariance Matrix of the Level 2 Random Effects u0 j 00 01 T Var u 1 j 10 11 G89 2247 Lecture 6 9 Dependency in Yij Due to Sampling Level 2 Units Level 1 Equation Yij 0j rij Level 2 Equation 0j 00 u0j Combined Equation Yij 00 u0j rij Fixed Random Effect Effects G89 2247 Lecture 6 10 Dependency in Yij Due to Sampling Level 2 Units Var rij 2 Within person variation in Y Var u0j 00 Between person variation in Y Total variation in Yij 2 00 Intraclass correlation proportion of total variance that is between person variance 2 00 00 Intraclass correlation the correlation between level 1 G89 2247 11 units due to being groupe d inLecture the6same level 2 unit Models With Cross Level Interactions G89 2247 Lecture 6 12 Level 1 Equation for many individuals Y ij 0j 1jX ij rij Y X G89 2247 Lecture 6 13 Level 2 Equations 0j 00 01Zj u0j 1j 10 11Zj u1j G89 2247 Lecture 6 14 Level 1 Equation Yij 0j 1jXij rij Level 2 Equations 0j 00 01Zj u0j 1j 10 11Zj u1j Combined Equation Yij 00 10Xij 01Zj 11XijZj u0j u1jXij rij Fixed Random Effects Effects G89 2247 Lecture 6 15 Y ij 00 10 X ij 01 Z j 11 X ij Z j u 0j u 1j X ij r ij 10 11 u 1j Slope ZW 1 1 10 11 00 01 u 0j Slope 00 01 Y 00 10 S lo p e WZ 0 0 X G89 2247 Lecture 6 16 Covariance Matrix of the Residual Level 2 Random Effects u0 j 00 01 T Var u 1 j 10 11 G89 2247 Lecture 6 17 Estimation Issues Unlike standard regression methods random regression must take into account two kinds of variability Within macro level within person Between person random effects The estimation process requires both estimates of the regression coefficients fixed effects and the the variance terms random effects G89 2247 Lecture 6 18 ML and REML Overview Proc Mixed and similar programs use iterative methods Step 1 estimate fixed effects assuming a first guess for the variances Step 2 use residuals to estimate variances Step 3 use estimated variances to estimate fixed effects again Step 4 use residuals to estimate variances again Continue until values don t change G89 2247 Lecture 6 19 Estimating Variances Maximum likelihood estimation gives consistent estimates of variances Estimates are biased however For sample variances we fix bias by using N 1 rather than N Restricted Maximum likelihood is a general approach to fix the bias Gives same point estimates of fixed effects but gives better estimates of the variances G89 2247 Lecture 6 20