MULTILEVEL MODELING Multilevel: what does it mean? Consider the following graph: LIKINGLIKING AGGRESSION LO HI

MULTILEVEL MODELING Each oval represents a classroom Each regression line has a b weight For a classroom, y i(j) = b 0j +b 1j x i(j) + e i(j) but the regression weights also have a structure, for slopes: b 1j =  10 +  11 Z j + u 1j

MULTILEVEL MODELING y i(j) = b 0j +b 1j x i(j) + e i(j) And for intercepts, b 0j =  00 +  01 Z j + u 0j

MULTILEVEL MODELING So that by combining, y i(j) = [  00 +  10 X i(j) +  01 Z j + +  11 X i(j) Z j ] + [ u 1j X i(j) + u 0j + e i(j) ] DETERMINISTIC or FIXED EFFECTS STOCHASTIC or RANDOM EFFECTS

MULTILEVEL MODELING where  00 =common intercept,  10 X i(j) = 1 st level aggression effect pooled  01 Z j = second level slope (liking-aggression for classes)  11 X i(j) Z j = cross level interaction (classroom slope by level of classroom aggression) u 1j X i(j) = agression effect error u 0j = slope error, e i(j) = individual error

MUTHEN PSEUDOBALANCED EQUAL SAMPLE SIZES ASSUMED  T =  B +  w S wp = pooled within groups S B = S wp + c S B C = weight factor = group sample size (or weighted average sample size)

MULTILEVEL ANALYSIS PROC MIXED IN SAS –ANALYZES ALL REGRESSION-GLS BASED MODELS –CANNOT ANALYZE CORRECTLY SEM POSSIBLE TO ANALYZE IN AMOS OR LISREL BUT NOT STRAIGHTFORWARD

ANALYSIS MUTHEN PSEUDOBALANCED: AMOS OR LISREL: TREAT B AND W COVARIANCE MATRICES AS SEPARATE GROUPS USE DEGREES OF FREEDOM ASSOCIATED WITH EACH

ANALYSIS SAS:two group modeling AMOS: two group modeling MPLUS: single group data with second level variables included

Aggression Liking Mean Aggression- b n Mean Liking n

Aggression Liking Mean Aggression- b 0/n Mean Liking 0/n TWO GROUP MODELING bw bB