SIMS Variance Decomposition. SIMS Variance Decomposition (Continued)

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1 SIMS Variance Decomposition The Second International Mathematics Study (SIMS; Muthén, 1991, JEM). National probability sample of school districts selected proportional to size; a probability sample of schools selected proportional to size within school district, and two classes randomly drawn within each school 3,724 students observed in 197 classes from 113 schools with class sizes varying from 2 to 38; typical class size of around 20 Eight variables corresponding to various areas of eighthgrade mathematics Same set of items administered as a pretest in the Fall of eighth grade and as a posttest in the Spring. 77 SIMS Variance Decomposition (Continued) Muthén (1991). Multilevel factor analysis of class and student achievement components. Journal of Educational Measurement, 28, Research questions: The substantive questions of interest in this article are the variance decomposition of the subscores with respect to within-class student variation and between-class variation and the change of this decomposition from pretest to posttest. In the SIMS such variance decomposition relates to the effects of tracking and differential curricula in eighth-grade math. On the one hand, one may hypothesize that effects of selection and instruction tend to increase between-class variation relative to within-class variation, assuming that the classes are homogeneous, have different performance levels to begin with, and show faster growth for higher initial performance level. On the other hand, one may hypothesize that eighth-grade exposure to new topics will increase individual differences among students within each class so that posttest within-class variation will be sizable relative to posttest between-class variation

2 SIMS Variance Decomposition (Continued) y rij = ν r + λ Br η Bj + ε Brj + λ wr η wij + ε wrij V(y rij ) = BF + BE + WF + WE Between reliability: BF / (BF + BE) BE often small (can be fixed at 0) Within reliability: WF / (WF + WE) sum of a small number of items gives a large WE Intraclass correlation: ICC = (BF + BE) / (BF + BE + WF+ WE) Large measurement error large WE small ICC True ICC = BF / (BF + WF) 79 Between fb_pre rpp_pre fract_pre eqexp_pre intnum_pre testi_pre aeravol_pre coorvis_pre pfigure_pre Within fw_pre fb_post rpp_post fract_post eqexp_post intnum_post testi_post aeravol_post coorvis_post fw_post pfigure_post 80 40

3 Table 4: Variance Decomposition of SIMS Achievement Scores (percentages of total variance in parenthesis) RPP FRACT Number of Items Between Within (34.0) (38.2) Pretest (66.0) (61.8) ANOVA Prop- Between Between Within (38.5) (40.8) Posttest Prop- Between (61.5) (59.2).41 % Increase In Variance FACTOR ANALYSIS Error-free Prop. Between Error-free % Increase In Variance Between Within Pre Post Between Within EQEXP (26.9) (73.1) (38.7) (61.3) INTNUM (25.2).358 (70.9) (30.6).442 (69.4) TESTI (33.3) (66.7) (34.5) (65.5) AREAVOL (17.2).451 (82.8) (24.1).490 (75.9) COORVIS (20.9).656 (79.1) (28.7).680 (68.3) PFIGURE (22.9) (77.1) (42.9) (67.1) Second-Generation JHU PIRC Trial Aggression Items Item Distributions for Cohort 3: Fall 1st Grade (n=362 males in 27 classrooms) Almost Never Rarely Sometimes Often Very Often Almost Always Stubborn (scored as 1) 42.5 (scored as 2) 21.3 (scored as 3) 18.5 (scored as 4) 7.2 (scored as 5) 6.4 (scored as 6) 4.1 Breaks Rules Harms Others Breaks Things Yells at Others Takes Others Property Fights Harms Property Lies Talks Back to Adults Teases Classmates Fights With Classmates Loses Temper

4 Hypothesized Aggressiveness Factors Verbal aggression Yells at others Talks back to adults Loses temper Stubborn Property aggression Breaks things Harms property Takes others property Harms others Person aggression Fights Fights with classmates Teases classmates 83 Two-Level Factor Analysis Within y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 fw1 fw2 fw3 Between y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 fb1 fb2 fb

5 Promax Rotated Loadings Stubborn Breaks Rules Harms Others Breaks Things Yells at Others Takes Others' Property Fights Harms Property Lies Talks Back to Adults Teases Classmates Fights With Classmates Loses Temper Within-Level Loadings Between-Level Loadings Two-Level Factor Analysis With Covariates 86 43

6 Two-Level Factor Analysis With Covariates Within Between y1 y1 x1 fw1 y2 y2 y3 y4 w fb y3 y4 x2 fw2 y5 y5 y6 y6 87 Input For Two-Level Factor Analysis With Covariates TITLE: DATA: VARIABLE: ANALYSIS: MODEL: this is an example of a two-level CFA with continuous factor indicators with two factors on the within level and one factor on the between level FILE IS ex9.8.dat; NAMES ARE y1-y6 x1 x2 w clus; WITHIN = x1 x2; BETWEEN = w; CLUSTER IS clus; TYPE IS TWOLEVEL; %WITHIN% fw1 BY y1-y3; fw2 BY y4-y6; fw1 ON x1 x2; fw2 ON x1 x2; %BETWEEN% fb BY y1-y6; fb ON w; 88 44

7 Input For Monte Carlo Simulations For Two-Level Factor Analysis With Covariates TITLE: MONTECARLO: ANALYSIS: This is an example of a two-level CFA with continuous factor indicators with two factors on the within level and one factor on the between level NAMES ARE y1-y6 x1 x2 w; NOBSERVATIONS = 1000; NCSIZES = 3; CSIZES = 40 (5) 50 (10) 20 (15); SEED = 58459; NREPS = 1; SAVE = ex9.8.dat; WITHIN = x1 x2; BETWEEN = w; TYPE = TWOLEVEL; 89 Input For Monte Carlo Simulations For Two-Level Factor Analysis With Covariates (Continued) MODEL POPULATION: %WITHIN% x1-x2@1; fw1 BY y1@1 y2-y3*1; fw2 BY y4@1 y5-y6*1; fw1-fw2*1; y1-y6*1; fw1 ON x1*.5 x2*.7; fw2 ON x1*.7 x2*.5; %BETWEEN% [w@0]; w*1; fb BY y1@1 y2-y6*1; y1-y6*.3; fb*.5; fb ON w*1; 90 45

8 Input For Monte Carlo Simulations For Two-Level Factor Analysis With Covariates (Continued) MODEL: %WITHIN% OUTPUT: fw1 BY y2-y3*1; fw2 BY y5-y6*1; fw1-fw2*1; y1-y6*1; fw1 ON x1*.5 x2*.7; fw2 ON x1*.7 x2*.5; %BETWEEN% fb BY y2-y6*1; y1-y6*.3; fb*.5; fb ON w*1; TECH8 TECH9; 91 NELS Data The data National Education Longitudinal Study (NELS:88) Base year Grade 8 followed up in Grades 10 and 12 Students sampled within 1,035 schools approximately 26 students per school, n = 14,217 Variables reading, math, science, history-citizenshipgeography, and background variables Data for the analysis reading, math, science, historycitizenship-geography 92 46

9 NELS Two-Level Longitudinal Factor Analysis With Covariates Within Between r88 m88 s88 h88 r90 m90 s90 h90 r92 m92 s92 h92 r88 m88 s88 h88 r90 m90 s90 h90 r92 m92 s92 h92 fw1 fw2 fw3 fb1 fb2 fb3 female stud_ses per_adva private catholic mean_ses 93 Input For NELS Two-Level Longitudinal Factor Analysis With Covariates TITLE: DATA: VARIABLE: two-level factor analysis with covariates using the NELS data FILE = NELS.dat; FORMAT = 2f7.0 f f5.2 11f8.2; NAMES = id school f2pnlwt r88 m88 s88 h88 r90 m90 s90 h90 r92 m92 s92 h92 stud_ses female per_mino urban size rural private mean_ses catholic stu_teac per_adva;!variable Description!m88 = math IRT score in 1988!m90 = math IRT score in 1990!m92 = math IRT score in 1992!r88 = reading IRT score in 1988!r90 = reading IRT score in 1990!r92 = reading IRT score in

10 Input For NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued)!s88 = science IRT score in 1988!s90 = science IRT score in 1990!s92 = science IRT score in 1992!h88 = history IRT score in 1988!h90 = history IRT score in 1990!h92 = history IRT score in 1992!female = scored 1 vs 0!stud_ses = student family ses in 1990 (f1ses)!per_adva = percent teachers with an MA or higher!private = private school (scored 1 vs 0)!catholic = catholic school (scored 1 vs 0)!private = 0, catholic = 0 implies public school MISSING = BLANK; CLUSTER = school; USEV = r88 m88 s88 h88 r90 m90 s90 h90 r92 m92 s92 h92 female stud_ses per_adva private catholic mean_ses; WITHIN = female stud_ses; BETWEEN = per_adva private catholic mean_ses; 95 Input For NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) ANALYSIS: MODEL: OUTPUT: TYPE = TWOLEVEL MISSING; %WITHIN% fw1 BY r88-h88; fw2 BY r90-h90; fw3 BY r92-h92; r88 WITH r90; r90 WITH r92; r88 WITH r92; m88 WITH m90; m90 WITH m92; m88 WITH m92; s88 WITH s90; s90 WITH s92; h88 WITH h90; h90 WITH h92; fw1-fw3 ON female stud_ses; %BETWEEN% fb1 BY r88-h88; fb2 BY r90-h90; fb3 BY r92-h92; fb1-fb3 ON per_adva private catholic mean_ses; SAMPSTAT STANDARDIZED TECH1 TECH8 MODINDICES; 96 48

11 Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates Summary Of Data Number of patterns 15 Number of clusters 913 Average cluster size Estimated Intraclass Correlations for the Y Variables Variable Intraclass Correlation Variable Intraclass Correlation Variable Intraclass Correlation R88 H88 S90 M M88 R90 H90 S S88 M90 R92 H Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Tests Of Model Fit Chi-Square Test of Model Fit Value Degrees of Freedom P-Value Scaling Correction Factor for MLR * Chi-Square Test of Model Fit for the Baseline Model Value Degrees of Freedom 202 P-Value 0 CFI/TLI CFI TLI Loglikelihood H0 Value H1 Value

12 Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Information Criteria Number of Free Parameters Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate SRMR (Standardized Root Mean Square Residual Value for Between Value for Within Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Model Results Estimates S.E. Est./S.E. Std StdYX Within Level FW1 BY R88 M88 S88 H88 FW2 BY R90 M90 S90 H

13 Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) FW3 BY R92 M92 S92 H92 FW1 ON FEMALE STUD_SES FW2 ON FEMALE STUD_SES FW3 ON FEMALE STUD_SES Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Residual Variances R88 M88 S88 H88 R90 M90 S90 H90 R92 M92 S92 H92 FW1 FW2 FW

14 Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Between Level FB1 BY R88 M88 S88 H88 FB2 BY R90 M90 S90 H90 FB3 BY R92 M92 S92 H Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Between Level FB1 ON PER_ADVA PRIVATE CATHOLIC MEAN_SES FB2 ON PER_ADVA PRIVATE CATHOLIC MEAN_SES FB3 ON PER_ADVA PRIVATE CATHOLIC MEAN_SES

15 Output Excerpts NELS Two-Level Longitudinal Factor Analysis With Covariates (Continued) Residual Variances R88 M88 S88 H88 R90 M90 S90 H90 R92 M92 S92 H92 FB1 FB2 FB Multiple-Group, Two-Level Factor Analysis With Covariates

16 NELS Data The data National Education Longitudinal Study (NELS:88) Base year Grade 8 followed up in Grades 10 and 12 Students sampled within 1,035 schools approximately 26 students per school Variables reading, math, science, history-citizenshipgeography, and background variables Data for the analysis reading, math, science, historycitizenship-geography, gender, individual SES, school SES, and minority status, n = 14,217 with 913 schools (clusters) 107 ses minority Between generalb mathb y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 generalw math sc hcg ses gender Within

17 Input For NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools TITLE: DATA: VARIABLE: DEFINE: ANALYSIS: NELS:88 with listwise deletion disaggregated model for two groups, public and catholic schools FILE IS EX831.DAT;; NAMES = ses y1-y16 gender cluster minority group; CLUSTER = cluster; WITHIN = gender; BETWEEN = minority; GROUPING = group(1=public 2=catholic); minority = minority/5; TYPE = TWOLEVEL; H1ITER = 2500; MITER = 1000; 109 Input For NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) MODEL: %WITHIN% generalw BY y1* y2-y6 y8-y16 y7@1; mathw BY y6* y8* y9* y11 y7@1; scw BY y10 y11*.5 y12*.3 y13*.2; hcgw BY y14*.7 y16*2 y15@1; generalw WITH mathw-hcgw@0; mathw WITH scw-hcgw@0; scw WITH hcgw@0; generalw mathw scw hcgw ON gender ses; %BETWEEN% generalb BY y1* y2-y6 y8-y16 y7@1; mathb BY y6* y8 y9 y11 y7@1; y1-y16@0; generalb WITH mathb@0; generalb mathb ON ses minority;

18 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools Summary Of Data Group PUBLIC Number of clusters 195 Size (s) Cluster ID with Size s Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued)

19 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Group PUBLIC Number of clusters 195 Average cluster size Estimated Intraclass Correlations for the Y Variables Variable Intraclass Correlation Variable Intraclass Correlation Variable Intraclass Correlation Y1.111 Y7.100 Y Y2.105 Y8.124 Y Y3.213 Y9.069 Y Y4.160 Y Y Y5.081 Y Y Y

20 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Group CATHOLIC Number of clusters 40 Average cluster size Estimated Intraclass Correlations for the Y Variables Variable Intraclass Correlation Variable Intraclass Correlation Variable Intraclass Correlation Y1 Y2 Y3 Y4 Y5 Y Y7 Y8 Y9 Y10 Y Y12 Y13 Y14 Y15 Y Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Tests Of Model Fit Loglikelihood Value Degrees of Freedom P-Value Scaling Correction Factor for MLR Chi-Square Test of Model Value Degrees of Freedom P-Value * CFI/TLI CFI TLI Loglikelihood H0 Value H1 Value

21 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Estimates S.E. Est./S.E. Std StdYX Group Public Within Level GENERALW ON GENDER SES MATHW ON GENDER SES SCW ON GENDER SES HCGW ON GENDER SES Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Estimates S.E. Est./S.E. Std StdYX Group Catholic Within Level GENERALW ON GENDER SES MATHW ON GENDER SES SCW ON GENDER SES HCGW ON GENDER SES

22 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Group Public Between Level Estimates S.E. Est./S.E. Std StdYX GENERALB ON SES MINORITY MATHB ON SES MINORITY GENERALB WITH MATHB Intercepts GENERALB MATHB 119 Output Excerpts NELS:88 Two-Group, Two-Level Model For Public And Catholic Schools (Continued) Estimates S.E. Est./S.E. Std StdYX Group Catholic Between Level GENERALB ON SES MINORITY MATHB ON SES MINORITY GENERALB WITH MATHB Intercepts GENERALB MATHB

23 Further Readings On Two-Level Factor Analysis Harnqvist, K., Gustafsson, J.E., Muthén, B, & Nelson, G. (1994). Hierarchical models of ability at class and individual levels. Intelligence, 18, (#53) Hox, J. (2002). Multilevel analysis. Techniques and applications. Mahwah, NJ: Lawrence Erlbaum Longford, N. T., & Muthén, B. (1992). Factor analysis for clustered observations. Psychometrika, 57, (#41) Muthén, B. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, (#24) Muthén, B. (1990). Mean and covariance structure analysis of hierarchical data. Paper presented at the Psychometric Society meeting in Princeton, NJ, June UCLA Statistics Series 62. (#32) Muthén, B. (1991). Multilevel factor analysis of class and student achievement components. Journal of Educational Measurement, 28, (#37) Muthén, B. (1994). Multilevel covariance structure analysis. In J. Hox & I. Kreft (eds.), Multilevel Modeling, a special issue of Sociological Methods & Research, 22, (#55) 121 Two-Level SEM: Random Slopes For Regressions Among Factors

24 y1 y5 Within y2 f1w s f2w y6 y3 y7 y4 y8 y1 y5 Between y2 y6 f1b f2b y3 y7 y4 y8 x s 123 Input For A Two-Level SEM With A Random Slope TITLE: DATA: VARIABLE: ANALYSIS: a twolevel SEM with a random slope FILE = etaeta3.dat; NAMES ARE y1-y8 x clus; CLUSTER = clus; BETWEEN = x; TYPE = TWOLEVEL RANDOM MISSING; ALGORITHM = INTEGRATION;

25 Input For A Two-Level SEM With A Random Slope (Continued) MODEL: OUTPUT: %WITHIN% f1w BY y1@1 y2 (1) y3 (2) y4 (3); f2w BY y5@1 y6 (4) y7 (5) y8 (6); s f2w ON f1w; %BETWEEN% f1b BY y1@1 y2 (1) y3 (2) y4 (3); f2b BY y5@1 y6 (4) y7 (5) y8 (6); f2b ON f1b; s ON x; TECH1 TECH8; 125 Output Excerpts Two-Level SEM With A Random Slope Tests Of Model Fit Loglikelihood H0 Value Information Criteria Number of Free Parameters Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + 2) / 24)

26 Output Excerpts Two-Level SEM With A Random Slope (Continued) Model Results Within Level Estimates S.E. Est./S.E. F1W Y1 Y2 Y3 Y4 F2W Y5 Y6 Y7 Y8 F1W F2W BY BY WITH Output Excerpts Two-Level SEM With A Random Slope (Continued) Estimates Variances F1W F2W Residual Variances Y Y Y Y Y Y Y Y S.E Est./S.E

27 Output Excerpts Two-Level SEM With A Random Slope (Continued) Estimates S.E. Est./S.E. Between Level F1B Y1 Y2 Y3 Y4 F2B Y5 Y6 Y7 Y8 F2B F1B BY BY ON Output Excerpts Two-Level SEM With A Random Slope (Continued) Estimates S ON X Intercepts Y Y Y Y Y Y Y Y S Variances F1B Residual Variances F2B S S.E Est./S.E

28 Multilevel Estimation, Testing, Modification, And Identification Estimators Muthén s limited information estimator (MUML) random intercepts ESTIMATOR = MUML Muthén s limited information estimator for unbalanced data Maximum likelihood for balanced data Full-information maximum likelihood (FIML) random intercepts and random slopes ESTIMATOR = ML, MLR, MLF Full-information maximum likelihood for balanced and unbalanced data Robust maximum likelihood estimator MAR missing data Asparouhov and Muthén 131 Multilevel Estimation, Testing, Modification, And Identification (Continued) Tests of Model Fit MUML chi-square, robust chi-square, CFI, TLI, RMSEA, and SRMR FIML chi-square, robust chi-square, CFI, TLI, RMSEA, and SRMR FIML with random slopes no tests of model fit Model Modification MUML modification indices not available FIML modification indices available Model identification is the same as for CFA for both the between and within parts of the model

29 Practical Issues Related To The Analysis Of Multilevel Data Size Of The Intraclass Correlation Small intraclass correlations can be ignored but important information about between-level variability may be missed by conventional analysis The importance of the size of an intraclass correlation depends on the size of the clusters Intraclass correlations are attenuated by individual-level measurement error Effects of clustering not always seen in intraclass correlations 133 Practical Issues Related To The Analysis Of Multilevel Data (Continued) Within-Level And Between-Level Variables Variables measured on the individual level can be used in both the between and within parts of the model or only in the within part of the model (WITHIN=) Variables measured on the between level can be used only in the between part of the model (BETWEEN=) Sample Size There should be at least between-level units (clusters) Clusters with only one observation are allowed

30 Steps In SEM Multilevel Analysis For Continuous Outcomes 1) Explore SEM model using the sample covariance matrix from the total sample 2) Estimate the SEM model using the pooled-within sample covariance matrix with sample size n - G 3) Investigate the size of the intraclass correlations and DEFF s 4) Explore the between structure using the estimated between covariance matrix with sample size G 5) Estimate and modify the two-level model suggested by the previous steps Muthén, B. (1994). Multilevel covariance structure analysis. In J. Hox & I. Kreft (eds.), Multilevel Modeling, a special issue of Sociological Methods & Research, 22, (#55) 135 Technical Aspects Of Multilevel Modeling

31 Numerical Integration With A Normal Latent Variable Distribution Weight Points Fixed weights and points 137 Non-Parametric Estimation Of The Random Effect Distribution Weight Weight Points Points Estimated weights and points (class probabilities and class means)

32 Numerical Integration Numerical integration is needed with maximum likelihood estimation when the posterior distribution for the latent variables does not have a closed form expression. This occurs for models with categorical outcomes that are influenced by continuous latent variables, for models with interactions involving continuous latent variables, and for certain models with random slopes such as multilevel mixture models. When the posterior distribution does not have a closed form, it is necessary to integrate over the density of the latent variables multiplied by the conditional distribution of the outcomes given the latent variables. Numerical integration approximates this integration by using a weighted sum over a set of integration points (quadrature nodes) representing values of the latent variable. 139 Numerical Integration (Continued) Numerical integration is computationally heavy and thereby timeconsuming because the integration must be done at each iteration, both when computing the function value and when computing the derivative values. The computational burden increases as a function of the number of integration points, increases linearly as a function of the number of observations, and increases exponentially as a function of the dimension of integration, that is, the number of latent variables for which numerical integration is needed

33 Practical Aspects Of Numerical Integration Types of numerical integration available in Mplus with or without adaptive quadrature Standard (rectangular, trapezoid) default with 15 integration points per dimension Gauss-Hermite Monte Carlo Computational burden for latent variables that need numerical integration One or two latent variables Light Three to five latent variables Heavy Over five latent variables Very heavy 141 Practical Aspects Of Numerical Integration (Continued) Suggestions for using numerical integration Start with a model with a small number of random effects and add more one at a time Start with an analysis with TECH8 and MITERATIONS=1 to obtain information from the screen printing on the dimensions of integration and the time required for one iteration and with TECH1 to check model specifications With more than 3 dimensions, reduce the number of integration points to 5 or 10 or use Monte Carlo integration with the default of 500 integration points If the TECH8 output shows large negative values in the column labeled ABS CHANGE, increase the number of integration points to improve the precision of the numerical integration and resolve convergence problems

34 Technical Aspects Of Numerical Integration Maximum likelihood estimation using the EM algorithm computes in each iteration the posterior distribution for normally distributed latent variables f, [ f y ] = [ f ] [ y f ] / [ y ], (97) where the marginal density for [y] is expressed by integration [ y ] = [ f ] [ y f ] df. (98) Numerical integration is not needed for normally distributed y - the posterior distribution is normal 143 Technical Aspects Of Numerical Integration (Continued) Numerical integration needed for: Categorical outcomes u influenced by continuous latent variables f, because [u] has no closed form Latent variable interactions f x x, f x y, f 1 x f 2, where [y] has no closed form, for example [ y ] = [ f 1, f 2 ] [ y f 1, f 2, f 1 f 2 ] df 1 df 2 (99) Random slopes, e.g. with two-level mixture modeling Numerical integration approximates the integral by a sum [ y ] = [ f ] [ y f ] df = w k [ y f k ] (100) Κ k=

35 Multivariate Approach To Multilevel Modeling 145 Multivariate Modeling Of Family Members Multilevel modeling: clusters independent, model for between- and within-cluster variation, units within a cluster statistically equivalent Multivariate approach: clusters independent, model for all variables for each cluster unit, different parameters for different cluster units. Used in latent variable growth modeling where the cluster units are the repeated measures over time Allows for different cluster sizes by missing data techniques More flexible than the multilevel approach, but computationally convenient only for applications with small cluster sizes (e.g. twins, spouses)

36 Figure 1. A Longitudinal Growth Model of Heavy Drinking for Two-Sibling Families O 18 O 19 O 20 O 21 O 22 O 30 O 31 O 32 Older Sibling Variables Male ES HSDrp S21 O LRate O QRate O Black Family Variables Hisp FH123 FH1 FH23 Younger Sibling Variables Male ES HSDrp S21 Y LRate Y QRate Y Y 18 Y 19 Y 20 Y 21 Y 22 Y 30 Y 31 Y 32 Source: Khoo, S.T. & Muthen, B. (2000). Longitudinal data on families: Growth modeling alternatives. Multivariate Applications in Substance Use Research, J. Rose, L. Chassin, C. Presson & J. Sherman (eds.), Hillsdale, N.J.: Erlbaum, pp Three-Level Modeling As Single-Level Analysis Doubly multivariate: Repeated measures in wide, multivariate form Siblings in wide, multivariate form It is possible to do four-level by TYPE = TWOLEVEL, for instance families within geographical segments

37 Input For Multivariate Modeling Of Family Data TITLE: DATA: VARIABLE: MODEL: Multivariate modeling of family data one observation per family FILE IS multi.dat; NAMES ARE o18-o32 y18-y32 omale oes ohsdrop ymale yoes yhsdrop black hisp fh123 fh1 fh23; s21o lrateo qrateo s21y lratey qratey s12o ON omale oes ohsdrop black hisp fh123 fh1 fh23; 221y ON ymale yes yhsdrop black hisp fh123 fh1 fh23; s21y ON s21o; lratey ON s21o lrateo; qratey ON s21o lrateo qrateo; 149 Twin Modeling

38 Twin1 y1 Twin2 y2 a c e a c e A1 C1 E1 A2 C2 E2 1.0 for MZ for DZ Neale & Cardon (1992) Prescott (2004) 151 Multilevel Growth Models