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Introduction to Hierarchical Models. Lluís Coromina (Universitat de Girona) Barcelona, 06/06/2005

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N=1371. Introduction

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Observed Variables M1T1M1T1 Frequency of contact / face-to-face M1T2M1T2 Feeling of closeness / face-to-face M1T3M1T3 Feeling of importance / face-to-face M1T4M1T4 Frequency of the alter upsetting to ego / face-to-face M2T1M2T1 Frequency of contact / telephone M2T2M2T2 Feeling of closeness / telephone M2T3M2T3 Feeling of importance / telephone M2T4M2T4 Frequency of the alter upsetting to ego / telephone Introduction

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1. How frequently are you in contact with this person (personally, by mail, telephone or Internet)? 1 Less than once a year. 2 Several times a year. 3 About once a month. 4 Several times a month. 5 Several times a week. 6 Every day. 2. How close do you feel to this person? Please describe how close you feel on a scale from1 to 5, where 1 means not close and 5 means very close Not Close Very Close 3. How important is this person in your life? Please describe how close you feel on a scale from 1 to 5, where 1 means not important and 5 means very important Not importantVery important 4. How often does this person upset you? 1 Never. 2 Rarely. 3 Sometimes. 4 Often. Introduction

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Model Y ij = t ij T i + e ij (1)) where: Y ij : response or measured variable i measured by method j. T i : unobserved variable of interest (trait). Related to validity. e ij : random error, which is related to lack of reliability. Model

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title: CLAS 2X4. TRAIT LOADS EQUAL. 1 nivell. RAW DATA FROM FILE dadesmodel.PSF LATENT VARIABLES T1 T2 T3 T4 RELATIONSHIPS M1T1 = 1*T1 M2T1 = T1 M1T2 = 1*T2 M2T2 = T2 M1T3 = 1*T3 M2T3 = T3 M1T4 = 1*T4 M2T4 = T4 SET THE ERROR VARIANCE OF M1T1 FREE SET THE ERROR VARIANCE OF M2T1 FREE SET THE ERROR VARIANCE OF M1T2 FREE SET THE ERROR VARIANCE OF M2T2 FREE SET THE ERROR VARIANCE OF M1T3 FREE SET THE ERROR VARIANCE OF M2T3 FREE SET THE ERROR VARIANCE OF M1T4 FREE SET THE ERROR VARIANCE OF M2T4 FREE SET THE VARIANCE OF T1 FREE SET THE VARIANCE OF T2 FREE SET THE VARIANCE OF T3 FREE SET THE VARIANCE OF T4 FREE T2 = T1 T4 T3 = T1 T4 LET T1 AND T4 CORRELATE LET T2 AND T3 CORRELATE LET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T2 -> M2T2 LET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T3 -> M2T3 LET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T4 -> M2T4 OPTIONS ND=3 sc RS PATH DIAGRAM END OF PROBLEM

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Model Figure I : Path diagram for the MTMM model

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Table I: Decomposition variance components T1M1T1M1 T2M1T2M1 T3M1T3M1 T4M1T4M1 T1M2T1M2 T2M2T2M2 T3M2T3M2 T4M2T4M2 trait variance87%79%83%74%87%82%85%78% error variance13%21%17%26%13%18%15%22% Model Structural Equations T2 = 0.376*T *T4, Errorvar.= 0.490, R² = (0.0245) (0.0322) (0.0244) T3 = 0.439*T *T4, Errorvar.= 0.566, R² = (0.0261) (0.0344) (0.0278) Error Covariance for T3 and T2 = (0.0242) Lisrel Output in latent growth curve Var (Y ij ) = t ij 2 Var (T i ) + Var (e ij ) (2)()

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The highest level: group level = egos = g The lowest level: individual level = alters = k Multilevel model Multilevel analysis. Two-level model.

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The mean centred individual scores for group g and individual k can be decomposed into: Between group component (3) Within group component (4) where: is the total average over all alters and egos. is the average of all alters of the g th ego. Y gk is the score on the name interpreter of the k th alter chosen by the g th ego. G is the total number of egos. n is the number of alters within each ego, constant. N=nG is the total number of alters. Multilevel model

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Sample covariance matrices: Multilevel model SW=SW=SB=SB= S T = S B + S W = (5)(6) (7) Population covariance matrices: T = B + W (8) Y ij = t Bij T Bi + e Bij + t wij T wi + e wij (9)() Y Bij Y Wij

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Härnqvist Method Separate analysis for S B and S W Group measures S w is the ML estimator of Σ W S B is the ML estimator of Σ W +cΣ B (10) Multilevel model Model estimated by Maximum Likelihood (ML).

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title: CLAS 2X4. TRAIT LOADS EQUAL. BETWEEN SIMPLIFICAT GROUP 1: BETWEEN RAW DATA FROM FILE dadesmodel.PSF $CLUSTER EGO LATENT VARIABLES T1 T2 T3 T4 RELATIONSHIPS M1T1 = 1*T1 M2T1 = 1*T1 M1T2 = 1*T2 M2T2 = 1*T2 M1T3 = 1*T3 M2T3 = 1*T3 M1T4 = 1*T4 M2T4 = 1*T4 SET THE ERROR VARIANCE OF M1T1 FREE SET THE ERROR VARIANCE OF M2T1 FREE SET THE ERROR VARIANCE OF M1T2 FREE SET THE ERROR VARIANCE OF M2T2 FREE SET THE ERROR VARIANCE OF M1T3 FREE SET THE ERROR VARIANCE OF M2T3 FREE SET THE ERROR VARIANCE OF M1T4 TO SET THE ERROR VARIANCE OF M2T4 FREE SET THE VARIANCE OF T1 FREE SET THE VARIANCE OF T2 FREE SET THE VARIANCE OF T3 FREE SET THE VARIANCE OF T4 FREE T2 = T1 T4 T3 = T1 T4 LET T1 AND T4 CORRELATE LET T2 AND T3 CORRELATE... Multilevel model

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GROUP 2: WITHIN RAW DATA FROM FILE dadesmodel.PSF LATENT VARIABLES T1 T2 T3 T4 RELATIONSHIPS M1T1 = 1*T1 M2T1 = T1 M1T2 = 1*T2 M2T2 = T2 M1T3 = 1*T3 M2T3 = T3 M1T4 = 1*T4 M2T4 = T4... END OF PROBLEM Multilevel model

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CLAS 2X4. TRAIT LOADS EQUAL. BETWEEN SIMPLIFICAT GROUP 1: BETWEEN LISREL Estimates (Maximum Likelihood) Measurement Equations M1T2 = 1.000*T2, Errorvar.= , R² = M1T3 = 1.000*T3, Errorvar.= , R² = M2T2 = 1.000*T2, Errorvar.= , R² = M2T3 = 1.000*T3, Errorvar.= , R² = M1T1 = 1.000*T1, Errorvar.= , R² = M1T4 = 1.000*T4, Errorvar.= 0.000, R² = 1.00 M2T1 = 1.000*T1, Errorvar.= , R² = M2T4 = 1.000*T4, Errorvar.= , R² = Structural Equations T2 = *T *T4, Errorvar.= , R² = T3 = 0.152*T *T4, Errorvar.= 0.103, R² = Error Covariance for T3 and T2 = (0.0) Multilevel model Lisrel Output in latent growth curve

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GROUP 2: WITHIN LISREL Estimates (Maximum Likelihood) Measurement Equations M1T2 = 1.000*T2, Errorvar.= 0.184, R² = M1T3 = 1.000*T3, Errorvar.= 0.193, R² = M2T2 = 0.950*T2, Errorvar.= 0.151, R² = M2T3 = 0.950*T3, Errorvar.= 0.151, R² = M1T1 = 1.000*T1, Errorvar.= 0.154, R² = M1T4 = 1.000*T4, Errorvar.= 0.225, R² = M2T1 = 0.950*T1, Errorvar.= 0.167, R² = M2T4 = 0.950*T4, Errorvar.= 0.181, R² = Structural Equations T2 = 0.474*T *T4, Errorvar.= 0.386, R² = T3 = 0.502*T *T4, Errorvar.= 0.448, R² = Error Covariance for T3 and T2 = (0.0) Multilevel model

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Interpretation: To analyse each component separately: Y ij = t Bij T Bi + e Bij + t wij T wi + e wij (11) () Y Bij Y Wij Decompose the variance: Var (Y ij ) =t ij 2 w Var (T iW ) + t ij 2 B Var (T iB ) + (12)() Var (e ijw ) + Var (e ijB ) Multilevel model

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Table II: Decomposition into 4 variance components. T1M1T1M1 T2M1T2M1 T3M1T3M1 T4M1T4M1 T1M2T1M2 T2M2T2M2 T3M2T3M2 T4M2T4M2 trait variance within error variance within trait variance between error variance between* * Boldfaced for small non-significant variances constrained to zero. Results and interpretation

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Table III: Percentages of decomposition into 4 variance components* T1M1T1M1 T2M1T2M1 T3M1T3M1 T4M1T4M1 T1M2T1M2 T2M2T2M2 T3M2T3M2 T4M2T4M2 trait variance within error variance within trait variance between error variance between* * Boldfaced for small non-significant variances constrained to zero. Results and interpretation

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t ij 2 w Var(T iw )/ [t ij 2 w Var(T iw ) + t ij 2 B Var(T iB )] T1M1T1M1 T2M1T2M1 T3M1T3M1 T4M1T4M1 T1M2T1M2 T2M2T2M2 T3M2T3M2 T4M2T4M Table IVTable IV: Percentages of variance at within level form M1 and M2 Results and interpretation T1M1T1M1 T2M1T2M1 T3M1T3M1 T4M1T4M1 T1M2T1M2 T2M2T2M2 T3M2T3M2 T4M2T4M2 Var(eijw)/ Var(Yij)

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