Introduction to Hierarchical Models.

Lluís Coromina (Universitat de Girona)

Barcelona, 06/06/2005

N=1371.

Introduction

Observed Variables

M1T1 Frequency of contact / face-to-face

M1T2 Feeling of closeness / face-to-face

M1T3 Feeling of importance / face-to-face

M1T4 Frequency of the alter upsetting to ego / face-to-face

M2T1 Frequency of contact / telephone

M2T2 Feeling of closeness / telephone

M2T3 Feeling of importance / telephone

M2T4 Frequency of the alter upsetting to ego / telephone

Introduction

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.1 2 3 4 5Not 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.1 2 3 4 5Not important Very important

4. How often does this person upset you?1 Never.2 Rarely.3 Sometimes.4 Often.

Introduction

ModelYij = tij Ti + eij (1)

where:• Yij : response or measured variable “i” measured by method “j”.• Ti : unobserved variable of interest (trait). Related to validity.• eij : random error, which is related to lack of reliability.

Model

Modeltitle: CLAS 2X4. TRAIT LOADS EQUAL. 1 nivell.RAW DATA FROM FILE dadesmodel.PSFLATENT VARIABLEST1 T2 T3 T4 RELATIONSHIPSM1T1 = 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 FREESET THE ERROR VARIANCE OF M2T1 FREESET THE ERROR VARIANCE OF M1T2 FREESET THE ERROR VARIANCE OF M2T2 FREESET THE ERROR VARIANCE OF M1T3 FREESET THE ERROR VARIANCE OF M2T3 FREESET THE ERROR VARIANCE OF M1T4 FREESET THE ERROR VARIANCE OF M2T4 FREESET THE VARIANCE OF T1 FREESET THE VARIANCE OF T2 FREESET THE VARIANCE OF T3 FREESET THE VARIANCE OF T4 FREET2 = T1 T4T3 = T1 T4LET T1 AND T4 CORRELATELET T2 AND T3 CORRELATELET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T2 -> M2T2LET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T3 -> M2T3LET THE PATH T1 -> M2T1 BE EQUAL TO THE PATH T4 -> M2T4OPTIONS ND=3 sc RSPATH DIAGRAMEND OF PROBLEM

Model

Figure I : Path diagram for the MTMM model

Table I: Decomposition variance components

T1M1 T2M1 T3M1 T4M1 T1M2 T2M2 T3M2 T4M2

trait variance 87% 79% 83% 74% 87% 82% 85% 78%

error variance 13% 21% 17% 26% 13% 18% 15% 22%

Model Structural Equations T2 = 0.376*T1 - 0.00203*T4, Errorvar.= 0.490 , R² = 0.220 (0.0245) (0.0322) (0.0244) 15.388 -0.0629 20.030 T3 = 0.439*T1 + 0.0656*T4, Errorvar.= 0.566 , R² = 0.269 (0.0261) (0.0344) (0.0278) 16.795 1.906 20.323 Error Covariance for T3 and T2 = 0.533 (0.0242) 22.013

Lisrel Output in latent growth curve

Var (Yij) = tij2Var (Ti) + Var (eij) (2)

The highest level: group level = egos = gThe lowest level: individual level = alters = k

Multilevel model

Multilevel analysis. Two-level model.

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 gth ego. • Ygk is the score on the name interpreter of the kth alter chosen by the gth 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.

Y

gY

YYY gkgkT

YYY ggB

ggkWgk YYY

Multilevel model

Sample covariance matrices:

Multilevel model

GN

YYYY ggkggk

nG

)')((SW=

1

)')((

G

YYYYn gg

G

SB=

1

)')((

N

YYYY gkgk

nG

ST = SB + SW =

(5) (6)

(7)

Population covariance matrices: T = B + W (8)

Yij = tBijTBi + eBij + twijTwi + ewij (9)

YBij YWij

Härnqvist MethodSeparate analysis for SB and SW

Group measuresSw is the ML estimator of ΣW

SB is the ML estimator of ΣW+cΣB (10)

Multilevel model

Model estimated by Maximum Likelihood (ML).

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 0.00001 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

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

CLAS 2X4. TRAIT LOADS EQUAL. BETWEEN SIMPLIFICAT GROUP 1: BETWEEN LISREL Estimates (Maximum Likelihood) Measurement Equations M1T2 = 1.000*T2, Errorvar.= 0.0321, R² = 0.689 M1T3 = 1.000*T3, Errorvar.= 0.0362, R² = 0.750 M2T2 = 1.000*T2, Errorvar.= 0.0257, R² = 0.734 M2T3 = 1.000*T3, Errorvar.= 0.0287, R² = 0.791 M1T1 = 1.000*T1, Errorvar.= 0.0175, R² = 0.913 M1T4 = 1.000*T4, Errorvar.= 0.000, R² = 1.00 M2T1 = 1.000*T1, Errorvar.= 0.0331, R² = 0.847 M2T4 = 1.000*T4, Errorvar.= 0.0683, R² = 0.653 Structural Equations T2 = - 0.0513*T1 - 0.224*T4, Errorvar.= 0.0634, R² = 0.107 T3 = 0.152*T1 - 0.160*T4, Errorvar.= 0.103, R² = 0.0560 Error Covariance for T3 and T2 = 0.0870 (0.0)

Multilevel model

Lisrel Output in latent growth curve

GROUP 2: WITHIN

LISREL Estimates (Maximum Likelihood) Measurement Equations

M1T2 = 1.000*T2, Errorvar.= 0.184, R² = 0.759 M1T3 = 1.000*T3, Errorvar.= 0.193, R² = 0.782 M2T2 = 0.950*T2, Errorvar.= 0.151, R² = 0.775 M2T3 = 0.950*T3, Errorvar.= 0.151, R² = 0.804 M1T1 = 1.000*T1, Errorvar.= 0.154, R² = 0.842 M1T4 = 1.000*T4, Errorvar.= 0.225, R² = 0.684 M2T1 = 0.950*T1, Errorvar.= 0.167, R² = 0.815 M2T4 = 0.950*T4, Errorvar.= 0.181, R² = 0.708 Structural Equations T2 = 0.474*T1 + 0.0361*T4, Errorvar.= 0.386, R² = 0.334 T3 = 0.502*T1 + 0.114*T4, Errorvar.= 0.448, R² = 0.350 Error Covariance for T3 and T2 = 0.420 (0.0)

Multilevel model

Interpretation:

To analyse each component separately:

Yij = tBijTBi + eBij + twijTwi + ewij (11)

YBij YWij

Decompose the variance:Var (Yij) = tij

2wVar (TiW) + tij

2BVar (TiB) + (12)

Var (eijw) + Var (eijB)

Multilevel model

Table II: Decomposition into 4 variance components.

T1M1 T2M1 T3M1 T4M1 T1M2 T2M2 T3M2 T4M2

trait variance within 0.82 0.58 0.69 0.49 0.76 0.52 0.62 0.44

error variance within 0.16 0.16 0.18 0.22 0.14 0.13 0.13 0.17

trait variance between 0.18 0.07 0.11 0.13 0.18 0.07 0.11 0.13

error variance between* 0.02 0.03 0.03 0.00 0.04 0.02 0.02 0.06

* Boldfaced for small non-significant variances constrained to zero.

Results and interpretation

Table III: Percentages of decomposition into 4 variance components*

T1M1 T2M1 T3M1 T4M1 T1M2 T2M2 T3M2 T4M2

trait variance within 0.70 0.69 0.69 0.58 0.67 0.70 0.71 0.55

error variance within 0.13 0.19 0.17 0.26 0.13 0.18 0.15 0.21

trait variance between 0.15 0.9 0.11 0.16 0.17 0.10 0.12 0.16

error variance between* 0.2 0.3 0.3 0.0 0.3 0.2 0.2 0.8

* Boldfaced for small non-significant variances constrained to zero.

Results and interpretation

tij2

wVar(Tiw)/ [tij2

wVar(Tiw) + tij2

BVar(TiB)]

T1M1 T2M1 T3M1 T4M1 T1M2 T2M2 T3M2 T4M2

0.82 0.89 0.86 0.79 0.80 0.88 0.85 0.77

Table IV: Percentages of variance at within level form M1 and M2

Results and interpretation

T1M1 T2M1 T3M1 T4M1 T1M2 T2M2 T3M2 T4M2

Var(eijw)/ Var(Yij) 0.13 0.19 0.17 0.26 0.13 0.17 0.15 0.21

For further information and contact:

http://www.udg.es/fcee/professors/llcoromina