3 HLM High School and Beyond HLM6 HLM6 C: Program Files HLM6S 2 C: Program MATHACH Files HLM6S Examples AppendxA school SECTOR Socio-Economic

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1 S. W. Raudenbush HLM6 student edition SAS/STAT MIXED R 2 HLM6 HLM HLM Hierarchical Linear A. S. Bryk S. W. Raudenbush Models HLM SSI *1 HLM6 student edition *2 student edition HLM6 (1) GUI (2) (3) [email protected] *1 *2

2 3 HLM High School and Beyond HLM6 HLM6 C: Program Files HLM6S 2 C: Program MATHACH Files HLM6S Examples AppendxA school SECTOR Socio-Economic Status: SES MEANSES MATHACH SES SECTOR MEANSES 2 Raudenbush and Bryk 1 (2002) MATHACH SECTOR SES MEANSES SES 1: 7185 i j SES SES.j j MATHACH SES MAT HACH ij = β 0j + β 1j (SES ij SES.j ) + r ij (1) 2: { SECTOR 1 (Catholic) SECT OR = 0 (public) MEANSES β 0j = γ 00 + γ 01 (SECT OR j ) + γ 02 (MEANSES j ) + u 0j (2) β 1j = γ 10 + γ 11 (SECT OR j ) + γ 12 (MEANSES j ) + u 1j (3) 2 2 u 0j, u 1j 1 2 SECTOR, MEANSES

3 HLM MDMT.mdmt MDM MDM.mdm STS.sts 3. HLM.hlm MLM.mlm HLM HLM ASCII 2. SPSS, SAS, SYSTAT, STATA ASCII HLM6 ID ID ID ID ID 3

4 3.3 ID ID 1. 1 (HSB1.DAT) 8 2 (HSB2.DAT) SPSS SPSS.sav 1 ASCII MDM ASCII ASCII 4

5 3.4 MDM 1 (HSB1.SAV) 2 (HSB2.SAV) ASCII 3.4 MDM HLM6 MDM Multivariate Data Matrix.mdm MDM MDM MDM HLM5 SSM Sufficient MDM Statistics Matrix 6 ASCII.dat SPSS.sav MDM 5

3.4 MDM 3.4.1 HLM6 HLM6 HLM6 HLM6S WHLMS.exe WHLMS.exe [ ] [ ] HLM6

6 3.4 MDM HLM6 HLM6 HLM6 HLM6S WHLMS.exe WHLMS.exe [ ] [ ] HLM ASCII MDM ASCII MDM MDM 1 1 [File]-[Make new MDM file]-[ascii input] 6

7 3.4 MDM [OK] 2 HLM2 Make MDM - HLM2 MDM MDM File Name MDM hsb-gd.mdm Nesting of input data persons within groups Level-1 Specification 1 Level-1 File Name: 1 C: Program Files HLM6S Examples Browse AppendxA hsb1.dat 7

3.4 MDM Number of Variables: ID 4 MDM / Missing Data: Data Format: (A4,4F12.3) ( ) A ID A ID Labels: F 4F12.3 3 4 ID F [ ].[ ] hsb1.

8 3.4 MDM Number of Variables: ID 4 MDM / Missing Data: Data Format: (A4,4F12.3) ( ) A ID A ID Labels: F 4F ID F [ ].[ ] hsb1.dat X / 2 8X 8, ACH Level-2 Specification MINORITY, FEMALE, SES, MATH- 2 1 C: Program Files HLM6S Examples 2 AppendxA hsb2.dat 6 Data Format (A4,6F12.3) Save mdmt file [Save mdmt file] MDMT MDM Template.mdmt SIZE SECTOR PRA- CAD DISCLIM HIM- MDMT MDM *3 INTY MEANSES *4 *3 MDM *4 MDMT MDM [File]-[Make new MDM from old MDM template (.mdmt) file] MDMT MDM 8

9 3.4 MDM Make MDM [Make MDM] MDM MDM Check Stats [Check Stats] STS hlm2mdm.sts STS Done [Done] MDM MDM [File]-[Display MDM stats] MDM 9

10 3.4 MDM SPSS MDM SPSS.sav MDM [File]-[Make new MDM file]-[stat package input] [OK] ASCII 2 HLM2 Make MDM - HLM2 MDM 10

11 3.4 MDM MDM File Name MDM.mdm hsb-gd.mdm Input File Type SPSS/Windows Nesting of input data persons within groups Level-1 Specification 1 Browse: [Browse] Level-1 File C: Program Files HLM6S Examples Name: AppendxA hsb1.sav Choose Variabes: [Choose Variables] 1 SPSS ID MDM : Missing Data? Yes No Yes hsb1.sav MDM / Level-2 Specification 2 Browse: [Browse] 2 C: Program Files HLM6S Examples AppendxA hsb2.sav Choose Variables: [Choose Variables] 2 11

12 3.4 MDM Save mdmt file [Save mdmt file] MDM hsb-gd.mdmt Make MDM [Make MDM] MDM MDM Check Stats [Check Stats] STS hlm2mdm.sts STS 12

13 3.5 Done [Done] MDM 3.5 MDM MDM [File]-[Create HLM6 (1) (3) a new model using an existing GUI HLM6 MDM file] MDM (1) MATHACH SES [Level-1] 1 1 MATHACH Outcome variable MATHACH 13

14 3.5 SES add variable group centered β 0 2 SES MEANSES HLM6 1 SES Delete variable from model 2 [Level-2] 2 1 β 0 (2) 14

15 3.6 SECTOR MEANSES uncenterd β 1 (3) SECTOR MEANSES uncenterd 2 u 1 u Basic Settings [Basic Settings] [Basic Setting] [Level-1] [Outcome] 15

16 3.6 [Level-1 Residual File] 16

17 3.6 Variables in residual file Residual File Type SPSS resfil1.sav Free Format.txt [OK] CSV.csv 2 [Level-2 Residual File] SPSS Excel Variables in residual file 1 Residual File Type SPSS [OK] resfil1.sav resfil2.sav 2 Title no title Output file name HSB-GD Graph file name [OK] 17

3.7 HLM 3.6.2 Other Settings 3.7 HLM HLM.

18 3.7 HLM Other Settings 3.7 HLM HLM.hlm HMLM.mlm [File]-[Save As] HLM 3.8 hsb-gd.hlm [Run Analysis] HLM 3.9 [File]-[View Output] [Basing Settings] hlm2.txt 18

19 3.9 SPECIFICATIONS FOR THIS HLM2 RUN Problem Title: HSB-GD MDM The data source for this run = hsb-gd.mdm.hlm The command file for this run = C:\Program Files\HLM6S\Examples\AppendxA\hsb-gd.hlm Output file name = C:\Program Files\HLM6S\Examples\AppendxA\hlm2.txt The maximum number of level-1 units = 7185 The maximum number of level-2 units = 160 The maximum number of iterations = 100 Method of estimation: restricted maximum likelihood Weighting Specification Weight Variable Weighting? Name Normalized? Level 1 no Level 2 no Precision no MATHACH The outcome variable is MATHACH The model specified for the fixed effects was: Level-1 Level-2 Coefficients Predictors INTRCPT1, B0 INTRCPT2, G00 SECTOR, G01 MEANSES, G02 * SES slope, B1 INTRCPT2, G10 SECTOR, G11 MEANSES, G12 19

20 3.9 SES * - This level-1 predictor has been centered around its group mean. The model specified for the covariance components was: Sigma squared (constant across level-2 units) Tau dimensions INTRCPT1 SES slope Summary of the model specified (in equation format) Level-1 Model Y = B0 + B1*(SES) + R Level-2 Model B0 = G00 + G01*(SECTOR) + G02*(MEANSES) + U0 B1 = G10 + G11*(SECTOR) + G12*(MEANSES) + U1 Iterations stopped due to small change in likelihood function ******* ITERATION 61 ******* 1 Sigma_squared = Tau INTRCPT1,B SES,B Tau (as correlations) INTRCPT1,B SES,B (Reliability) Random level-1 coefficient Reliability estimate INTRCPT1, B SES, B

21 3.9 2 The value of the likelihood function at iteration 61 = E The outcome variable is MATHACH Final estimation of fixed effects: Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value For INTRCPT1, B0 INTRCPT2, G SECTOR, G MEANSES, G For SES slope, B1 INTRCPT2, G SECTOR, G MEANSES, G The outcome variable is MATHACH Final estimation of fixed effects (with robust standard errors) Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value For INTRCPT1, B0 INTRCPT2, G SECTOR, G MEANSES, G For SES slope, B1 INTRCPT2, G SECTOR, G MEANSES, G Final estimation of variance components: Random Effect Standard Variance df Chi-square P-value Deviation Component INTRCPT1, U SES slope, U level-1, R

22 3.10 Deviance Deviance Statistics for current covariance components model Deviance = Number of estimated parameters = HLM6 (1) 1 SECTOR [File]-[Graph Equations]-[Level-1 equation graphing] 1 22

23 3.10 X-focus Level-1: Number of groups: 1 SES groups All Z-focus SECTOR [OK] Cathoric SECTOR=1 SECTOR SECTOR INTRCPT1, B SES slope, B

24 HLM6 1 MDM l1resid 1 fitval 2 mathach 24

3.11.2 2 2 eb : ebintrcp 1 β 0j 1 empirical Bayes u 0j ebses 2 1 β 1j u 1j ol 2 ol : olintrcp olses 1 β 0j β 1j 2 ordinary least-squares u 0j u

25 eb : ebintrcp 1 β 0j 1 empirical Bayes u 0j ebses 2 1 β 1j u 1j ol 2 ol : olintrcp olses 1 β 0j β 1j 2 ordinary least-squares u 0j u 1j ebintrcp olintrcp eb- eb ses olses fv : 1 2 ec : 1 ( fv ) ( eb 4 SAS PROC MIXED β 0j β 1j ) 23 HLM6 HLM6 1 SAS MIXED 29 R SAS PROC MIXED HLM6 25

26 4.1 SAS (2) (3) (1) MAT HACH ij = [γ 00 + γ 01 SECT OR j + γ 02 MEANSES j + γ 10 (SES ij SES.j )+ γ 11 SECT OR j (SES ij SES.j )+ γ 12 MEANSES j (SES ij SES.j )]+ [u 0j + u 1j (SES ij SES.j ) + r ij ] (4) [ ] 2 [ ] HLM6 SAS R (4) SAS PROC MIXED Singer (1998) HSB HLM6 4.1 SAS ***************************; * SAS HSB ; * By Taichi OKUMURA ; ***************************; OPTIONS ls = 80; /* */ DATA hsb; INFILE C:\Program Files\SAS Institute\SAS\V8\hsb-sas.dat ; INPUT id school ses mathach meanses cses sector; RUN; /* proc mixed */ PROC MIXED DATA=hsb NOCLPRINT COVTEST; CLASS school; MODEL mathach = sector meanses cses sector*cses meanses*cses / SOLUTION DDFM=bw NOTEST; RANDOM intercept cses / TYPE = un SUB=school; RUN; QUIT; MIXED PROC MIXED NOCLPRINT: COVTEST: school 26

27 4.2 SAS CLASS school MODEL SOLUTION: DDFM=bw: NOTEST: RANDOM TYPE = un: sub=school: school 4.2 SAS / / SAS The Mixed Procedure Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.HSB mathach Unstructured school REML /* REML */ Profile Model-Based Between-Within Dimensions Covariance Parameters 4 Columns in X 6 Columns in Z Per Subject 2 Subjects 160 Max Obs Per Subject 67 Observations Used 7185 Observations Not Used 0 Total Observations

28 4.2 SAS Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z /* 2 */ UN(1,1) school <.0001 UN(2,1) school UN(2,2) school /* 1 */ Residual <.0001 Fit Statistics -2 Res Log Likelihood /* deviance */ AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.0001 /* 2 */ Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001 /* G00 */ sector <.0001 /* G01 */ meanses <.0001 /* G02 */ cses <.0001 /* G10 */ sector*cses <.0001 /* G11 */ meanses*cses /* G12 */ 28

29 5 R R (R Development Core Team, 2005) nmle lme *5 nlme HSB 5.1 SAS HSB 10 school ses mathach meanses cses sector Public Public Public Public Public Public Public Public Public Public 5.2 R R HSB By Taichi OKUMURA nlme library(nlme) 1 data(mathachieve) MathAchieve[1:10,] 2 data(mathachschool) MathAchSchool[1:10,] SES mses attach(mathachieve) *5 29

30 5.3 R mses <- tapply(ses, School, mean) detach(mathachieve) HSB HSB <- as.data.frame(mathachieve[, c("school", "SES", "MathAch")]) names(hsb) <- c("school", "ses", "mathach") SES mses HSB$meanses <- mses[as.character(hsb$school)] HSB$cses <- HSB$ses - HSB$meanses sector HSB sector <- MathAchSchool$Sector names(sector) <- row.names(mathachschool) HSB$sector <- sector[as.character(hsb$school)] HSB$sector <- factor(hsb$sector, levels=c( Public, Catholic )) result.hsb result.hsb <- lme(mathach ~ sector + meanses + cses + sector*cses + meanses*cses, random = ~ cses school, data = HSB) summary(result.hsb) 5.3 R / / Linear mixed-effects model fit by REML /* REML */ Data: HSB AIC BIC loglik /* LogLik -2 deviance */ Random effects: Formula: ~cses school Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) (Intr) /* 2 */ cses /* 2 */ Residual /* 1 */ /* 2 */ Fixed effects: mathach ~ sector + meanses + cses + sector * cses + meanses * cses Value Std.Error DF t-value p-value (Intercept) e+00 /* G00 */ sectorcatholic e-04 /* G01 */ meanses e+00 /* G02 */ cses e+00 /* G10 */ sectorcatholic:cses e+00 /* G11 */ meanses:cses e-04 /* G12 */ Correlation: (Intr) sctrct meanss cses sctrc: sectorcatholic meanses cses

31 REFERENCES sectorcatholic:cses meanses:cses Standardized Within-Group Residuals: Min Q1 Med Q3 Max Number of Observations: 7185 /* */ Number of Groups: 160 /* */ References R Development Core Team. (2005). R: A language and environment for statistical computing. Vienna, Austria. (ISBN ) Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Sage. Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 24,

: (EQS) /EQUATIONS V1 = 30V F1 + E1; V2 = 25V F1 + E2; V3 = 16V F1 + E3; V4 = 10V F2 + E4; V5 = 19*V99

: (EQS) /EQUATIONS V1 = 30*V F1 + E1; V2 = 25*V *F1 + E2; V3 = 16*V *F1 + E3; V4 = 10*V F2 + E4; V5 = 19*V99 218 6 219 6.11: (EQS) /EQUATIONS V1 = 30*V999 + 1F1 + E1; V2 = 25*V999 +.54*F1 + E2; V3 = 16*V999 + 1.46*F1 + E3; V4 = 10*V999 + 1F2 + E4; V5 = 19*V999 + 1.29*F2 + E5; V6 = 17*V999 + 2.22*F2 + E6; CALIS.