Hierarchical Linear Models (HLM) Using R Package nlme. Interpretation. 2 = ( x 2) u 0j. e ij
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1 Hierarchical Linear Models (HLM) Using R Package nlme Interpretation I. The Null Model Level 1 (student level) model is mathach ij = β 0j + e ij Level 2 (school level) model is β 0j = γ 00 + u 0j Combined model is mathach ij = γ 00 + u 0j + e ij Selected output below: Linear mixed-effects model fit by REML AIC Data: hlm AIC BIC loglik = ( x 2) Fixed effects: math ~ 1 Value Std.Error DF t-value p-value (Intercept) γ 00 > VarCorr(model1) school = pdlogchol(1) Variance StdDev u 0j (Intercept) Residual e ij As shown in Table 1 below, the grand mean math achievement estimate is with a standard error of The intra class correlation (ICC) describes the proportion of variance associated with differences between schools, where 00 is the school level variance and 2 is the student level variance: 1 P age = 00 / ( ) = 8.61/ ( ) =.18,
2 which indicates that about 18% of the math achievement variance is between schools. Table 1. Results from the Null Model Fixed Effect Coefficient Standard Error p Value Average school mean, γ <.001 Random Effect Variance Component School mean, u 0j 8.61 Level 1 effect, e ij Model Fit AIC P age
3 II. Means as Outcomes Model Level 1 (student level) model is mathach ij = β 0j + e ij Level 2 (school level) model is β 0j = γ 00 + γ 01 (meanses) j + u 0j Combined model is mathach ij = γ 00 + γ 01 (meanses) j + u 0j + e ij Selected output below: Linear mixed-effects model fit by REML AIC Data: hlm AIC BIC loglik = ( x 2) Fixed effects: math ~ meanses γ 00 Value Std.Error DF t-value p-value (Intercept) γ 01 meanses > VarCorr(model2) school = pdlogchol(1) Variance StdDev u 0j (Intercept) Residual e ij As shown in Table 2 below, there is a statistically significant positive relationship between school mean socioeconomic status and mean math achievement (γ 01 = 5.86, t = 16.22, p <.001). Additionally, school achievement means vary significantly when school mean socioeconomic status is controlled. 3 P age
4 Table 2. Results of the Means as Outcomes Model Fixed Effect Coefficient Standard Error p Value Intercept, γ <0.001 Mean SES, γ <0.001 Random Effect Variance Component School mean, u 0j 2.64 Level 1 effect, e ij Model Fit AIC P age
5 III. Random Coefficient Model Level 1 (student level) model is mathach ij = β 0j + β 1j (cses) j + e ij Level 2 (school level) model is β 0j = γ 00 + u 0j β 1j = γ 10 + u 1j Combined model is mathach ij = γ 00 + γ 10 (cses) j + u 0j + u 1j (cses) j + e ij Selected output below: Linear mixed-effects model fit by REML AIC Data: hlm AIC BIC loglik = ( x 2) Fixed effects: math ~ cses γ 00 Value Std.Error DF t-value p-value (Intercept) γ 01 cses > VarCorr(model2) school = pdlogchol(1) Variance StdDev (Intercept) Residual > VarCorr(model3) school = pdlogchol(cses) Variance StdDev Corr 5 P age
6 u 0j Information Research and Analysis (IRA) Lab (Intercept) (Intr) u 1j cses Residual e ij The random coefficient model analyzes the student level socioeconomic status and math achievement relationship within the 160 schools. The intercept represents the mean math achievement score of students and it was statistically significant, γ 00 = 12.65, t = 51.73, p <.001. Socioeconomic status was statistically significant (γ 01 = 2.19, t = 17.10, p <.001), indicating that students with higher socioeconomic status had higher math achievement scores. Furthermore, variances of the random effects were statistically significant as well. Table 3. Results of the Random Coefficients Model Fixed Effect Coefficient Standard Error p Value Intercept, γ <0.001 Mean SES achievement slope, γ <0.001 Random Effect Variance Component School mean, u 0j 8.68 SES achievement slope, u 1j 0.69 Level 1 effect, e ij Model Fit AIC P age
7 IV. Intercepts and Slopes as Outcomes Model Level 1 (student level) model is mathach ij = β 0j + β 1j (cses) j + e ij Level 2 (school level) model is β 0j = γ 00 + γ 01 (mses) j + γ 02 (sector) j + u 0j β 1j = γ 10 + γ 11 (mses) j + γ 12 (sector) j + u 1j Combined model is mathach ij = γ 00 + γ 01 (mses) j + γ 02 (sector) j + γ 10 (cses) j + γ 11 (mses) j *(cses) j + γ 12 (sector) j *(cses) j + u 0j + u 1j *(cses) j + e ij Selected output below: Linear mixed-effects model fit by REML AIC Data: hlm AIC BIC loglik Random effects: Formula: ~cses school 2 = ( x 2) Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) (Intr) cses Residual Fixed effects: math ~ meanses * cses + sector * cses γ 00 Value Std.Error DF t-value p-value γ 01 γ 10 γ 02 (Intercept) e+00 meanses e+00 cses e+00 sector e-04 meanses:cses e-04 γ 11 7 P age
8 γ 12 cses:sector e+00 > VarCorr(model4) school = pdlogchol(cses) Variance StdDev Corr u 0j (Intercept) (Intr) u 1j cses Residual e ij The intercepts and slopes as outcomes model is an explanatory model that accounts for variability across schools. Table 4 displays the results. The mean math achievement score was and statistically significant at the.05 alpha level. The regression coefficients relating the schools socioeconomic status and school sector are positive and statistically significant, γ 01 =5.34 and γ 02 = 1.22, respectively at the.05 alpha level. This signifies that as school s socioeconomic status increase, so does math achievement scores. Additionally, Catholic schools have higher mean math achievement scores than public schools, controlling for individual and school mean socioeconomic status. In reference to the slopes, individuals with high socioeconomic status have steeper slopes than do individuals with lower socioeconomic status. Also, Catholic schools have statistically significant weaker socioeconomic slopes than do public schools. Table 4. Results of Intercepts and Slopes as Outcomes Model Fixed Effect Coefficient Standard Error p Value Intercept, γ <0.001 School level Mean SES, γ <0.001 Sector, γ <0.001 Student level SES, γ <0.001 Student level SES*Mean SES Achievement Slope, γ Student level SES*Sector Achievement Slope, γ <0.001 Random Effect Variance Component School mean, u 0j 2.38 SES achievement slope, u 1j 0.10 Level 1 effect, e ij Model Fit AIC P age
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