Estructuras de medias y covarianzas...

Estructuras de medias y covarianzas (MACS)

Modelo de estructuras de medias y covarianzas

x y

ε

β*

*

*

Modelo Estructuras de Covarianzas (SEM)

y = βx + ε

Vector de parámetros: θ = (β, σxx, σεε )’

Modelo Estructuras de medias y covarianzas (MACS)

y = α + βx + ε

E(y) = α + βE(x) µy = α + β µx

Vector de parámetros: θ = (α, µx, β, σxx, σεε )’

Nuevos parámetros: “Intersección” (constante) de las Variables dependientes (α)

Medias de las variables independientes (µx)


• ¿Cómo introducimos en el modelo los nuevos parámetros?

y = α1 + βx + ε

x = µx1 + xd

Donde: 1 = “Variable” independiente que toma el mismo valor para todas las observaciones (no varianza y no covarianzas) α = coeficiente de regresión asociado a la variable “constante” 1

Donde: 1 = “Variable” independiente que toma el mismo valor para todas las observaciones (no varianza y no covarianzas) Xd = Desviación respecto de la media (σxd = σx)


• Implementación en EQS 1 = V999 Variable independiente sin varianza y no covarianzas con otras variables generada por el programa

1

y

x ε

α

β

µx

xd

V999

V2

V1 E2

*

*

*

E1 Vector de parámetros: θ = (α, µx, β, σxx, σεε )’ *

*


• Especificación del modelo con EQS:

/SPECIFICATIONS ANALYSIS = MOMENTS (MOM); MATRIX = RAW; COV; (+ /MATRIX y /MEANS) COR ( + /MATRIX ; /MEANS y /STA DEV)

/PRINT EFFECT = YES; COVARIANCE = YES;

Analizar momentos de 1er y 2º orden


Ejemplo de regresión simple:

Matriz de datos

x y 6.066 21.074 6.946 25.819 8.800 35.267

10.788 46.569

x y

x 4.391 y 23.589 126.793

Matriz de covarianzas muestral (S)

Vector de medias muestrales (M)

mx my

8.150 32.182


/TITLE Mean and covariance model: simple regression with mean EQS manual /SPECIFICATIONS VARIABLES= 2; CASES= 4; METHOD=ML; MATRIX = COV; ANALYSIS = MOMENTS; /EQUATIONS V1 = 8*V999 + E1; V2 = -10*V999 + 5*V1 + E2; /VARIANCES E1 = 5*; E2 = 1*; /COVARIANCES /PRINT EFFECT = YES; COVARIANCE = YES; /MATRIX 4.391 23.589 126.793 /MEANS 8.150 32.182 /END

COVARIANCE/MEAN MATRIX TO BE ANALYZED: 2 VARIABLES (SELECTED FROM 2 VARIABLES), BASED ON 4 CASES.

V1 V2 V999 V1 4.391 V2 23.589 126.793 V999 8.150 32.182 1.000

RESIDUAL COVARIANCE/MEAN MATRIX (S-SIGMA) :

V1 V2 V999 V1 0.000 V2 0.000 0.000 V999 0.000 0.000 0.000

GOODNESS OF FIT SUMMARY FOR METHOD = ML

CHI-SQUARE = 0.000 BASED ON 0 DEGREES OF FREEDOM


MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @.

V1 =V1 = 8.150*V999 + 1.000 E1 1.210 6.737@

V2 =V2 = 5.372*V1 - 11.601*V999 + 1.000 E2 .073 .613 73.723@ -18.918@

VARIANCES OF INDEPENDENT VARIABLES ----------------------------------

E --- E1 - V1 4.391*I 3.585 I 1.225 I I E2 - V2 .070*I .057 I 1.225 I I

Media de V1 (µx)

Intercept eq. V2 (α)

¿Media de V2? V999

V2

V1 E2

-11.601

5.372

8.150

E1 4.391

0.077


µy = α + β µx ^ ^ ^ ̂

/PRINT EFFECTS = YES;

^

PARAMETER INDIRECT EFFECTS --------------------------

V2 =V2 = 43.783*V999 + 5.372 E1 6.526 .073 6.709@ 73.723@

β µx = 8.150 X 5.372 = 43.782 ^

PARAMETER TOTAL EFFECTS -----------------------

V1 =V1 = 8.150*V999 + 1.000 E1

V2 =V2 = 5.372*V1 + 32.182*V999 + 5.372 E1 + 1.000 E2 .073 6.501 .073 73.723@ 4.950@ 73.723@

µy = 43.782 - 11.601 = 32.181

^

^


/PRINT COVARIANCES = YES;

MODEL COVARIANCE MATRIX FOR MEASURED AND LATENT VARIABLES

V1 V2 V999 V1 4.391 V2 23.589 126.793 V999 8.150 32.182 1.000


• Identificación del modelo: Cálculo de los grados de libertad – Estructura de medias:

• # parámetros de intercepts del modelo = 2 • # elementos en el vector de medias = 2 • grados de libertad = 0 (Estructura de medias saturada)

– Estructura de medias y covarianzas: • # parámetros del modelo = 5 • # elementos en S y X = 5 • grados de libertad = 0 (Modelo de estructuras de medias y covarianzas saturado)


Ej. estructuras de medias con variables latentes:

Ejemplo: McArdle and Epstein (1987)

Datos: R-T. Osborne y colaboradores.

V1 – V4 = Wechsler Intelligence Scale scores

Medias= % de aciertos en el test

4 momentos del tiempo (6 a 11 años)

N = 204 niños

/MEANS 18.034 25.819 35.255 46.593 /STANDARD DEVIATIONS 6.374 7.319 7.796 10.386 /MATRIX 1.000 .809 1.000 .806 .850 1.000 .765 .831 .867 1.000

Modelos de estructuras de medias y covarianzas

/TITLE Growth in WISC scores Osborne data (MCArdle & Epstein, 1987, p.113) (Example in EQS manual) /SPECIFICATIONS CASES = 204; VAR = 4; ANALYSIS=MOMENT; MATRIX = CORRELATIONS; /EQUATIONS V1 = F1 + E1 - 3*V999; V2 = 1*F1 + E2; V3 = F2 + E3; V4 = 1*F2 + E4; F1 = 21*V999 + D1; F2 = 1*F1 + 8*V999 + D2; /VARIANCES D1 = 30*; D2 = 3*; E1 TO E3 = 8*; E4 = 16*; /PRINT EFFECT = YES; COVARIENCE = YES; /MEANS 18.034 25.819 35.255 46.593 /STANDARD DEVIATIONS 6.374 7.319 7.796 10.386 /MATRIX 1.000 .809 1.000 .806 .850 1.000 .765 .831 .867 1.000 /END


• Ejercicio: Cálculo de los grados de libertad: – Estructura de medias:

• # elementos en el vector de medias = • # de intercepts del modelo = • grados de libertad =

– Estructura de covarianzas: • # elementos en el vector de medias = • # parámetros de intercepts del modelo = • grados de libertad =

– Estructura de medias y covarianzas: • grados de libertad =


• Cálculo de los grados de libertad: – Estructura de medias:

• # elementos en el vector de medias = 4 • # de intercepts del modelo = 3 • grados de libertad = 1

– Estructura de medias y covarianzas: • # elementos en el vector de medias = 10 • # parámetros de intercepts del modelo = 9 • grados de libertad = 1

– Estructura de medias y covarianzas: • grados de libertad = 2



V1 =V1 = 1.000 F1 - 3.040*V999 + 1.000 E1 1.123 -2.706@

V2 =V2 = 1.225*F1 + 1.000 E2 .063 19.325@

V3 =V3 = 1.000 F2 + 1.000 E3

V4 =V4 = 1.320*F2 + 1.000 E4 .010 126.650@

CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @.

F1 =F1 = 21.074*V999 + 1.000 D1 1.168 18.037@

F2 =F2 = 1.278*F1 + 8.334*V999 + 1.000 D2 .065 1.214 19.532@ 6.862@


V1 V4 V3 V2

V999

1 1.225 1 1.320

8.334 21.074

1.278

-3.040

*

*

F1 F2 D1 * D2

E1 * E3 * E4 * E2


• Ejercicio: Cálculo de las medias:

(V1,V999) =

(V2,V999) =

(V3,V999) =

(V4,V999) =

(F1,V999) =

(F2,V999) =


• Cálculo de las medias:

(V1,V999) = (21.074 * 1) – 3.040 = 18.034 (V2,V999) = (21.074 * 1.225) = 25.819 (V3,V999) = (8.334 * 1) + (21.074 * 1.278) = 35.267 (V4,V999) = (8.334 * 1.320) + (21.074 * 1.278) = 46.569 (F1,V999) = 21.074 (F2,V999) = (21.074 * 1.278) + 8.334 = 35.267



V1 =V1 = 1.000 F1 + 18.034*V999 + 1.000 E1 + 1.000 D1 .447 40.311@

V2 =V2 = 1.225*F1 + 25.819 V999 + 1.000 E2 + 1.225 D1 .063 .514 .063 19.325@ 50.262@ 19.325@

V3 =V3 = 1.278 F1 + 1.000 F2 + 35.267 V999 + 1.000 E3 .065 .543 19.532@ 64.963@

+ 1.278 D1 + 1.000 D2 .065 19.532@

V4 =V4 = 1.688 F1 + 1.320*F2 + 46.569 V999 + 1.000 E4 .087 .010 .735 19.486@ 126.650@ 63.337@

+ 1.688 D1 + 1.320 D2 .087 .010 19.486@ 126.650@

F1 =F1 = 21.074*V999 + 1.000 D1

F2 =F2 = 1.278*F1 + 35.267*V999 + 1.278 D1 + 1.000 D2 .065 .543 .065 19.532@ 64.963@ 19.532@


MODEL COVARIANCE MATRIX FOR MEASURED AND LATENT VARIABLES

V1 V2 V3 V4 V999 F1 F2 V1 40.628 V2 37.741 53.568 V3 39.370 48.235 59.920 V4 51.987 63.692 70.479 110.102 V999 18.034 25.819 35.267 46.569 1.000 F1 30.805 37.741 39.370 51.987 21.074 30.805 F2 39.370 48.235 53.374 70.479 35.267 39.370 53.374

Modelos de estructuras de medias

V1 V4 V3 V2

V999

* E1 * E3 * E4 * E2

* * * *

Modelo nulo:

Estructura de covarianzas saturada

Restricción:

( µ4 – µ3 ) = (µ3 - µ2 );

/MEANS 18.034 25.819 35.255 46.593



CHI-SQUARE = 12.689 BASED ON 1 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS .00037

/TITLE Mean structures with free covariances (Example in EQS manual) /SPECIFICATIONS CASES = 204; VAR = 4; ANALYSIS=MOMENT; MATRIX = CORRELATIONS; /EQUATIONS V1 = 18*V999 + E1; V2 = 25*V999 + E2; V3 = 35*V999 + E3; V4 = 45*V999 + E4; /VARIANCES E1 TO E4 = 80*; /COVARIANCES E1 TO E4 = *; /CONSTRAINTS (V4,V999) - 2(V3,V999) + (V2,V999) = 0; /PRINT EFFECT = YES; COVARIENCE = YES; /MEANS 18.034 25.819 35.255 46.593 /STANDARD DEVIATIONS 6.374 7.319 7.796 10.386 /MATRIX 1.000 .809 1.000 .806 .850 1.000 .765 .831 .867 1.000 /END

Modelo nulo

Modelos de estructuras de medias RESIDUAL COVARIANCE/MEAN MATRIX (S-SIGMA) :

V1 V2 V3 V4 V999 V1 -.079 V2 -.188 -.448 V3 .027 .065 -.009 V4 -.292 -.695 .101 -1.079 V999 .281 .669 -.097 1.039 .000

STANDARDIZED RESIDUAL MATRIX:

V1 V2 V3 V4 V999 V1 -.002 V2 -.004 -.008 V3 .001 .001 .000 V4 -.004 -.009 .001 -.010 V999 .044 .091 -.012 .100 .000



V1 =V1 = 17.753*V999 + 1.000 E1 .441 40.295@

V2 =V2 = 25.150*V999 + 1.000 E2 .479 52.479@

V3 =V3 = 35.352*V999 + 1.000 E3 .547 64.686@

V4 =V4 = 45.554*V999 + 1.000 E4 .670 67.989@

/MEANS 18.034 25.819 35.255 46.593


• Uso de las estructuras de medias: – Modelizar el cambio en niveles en el tiempo – Análisis de muestra múltiple – Análisis de nieveles múltiples – Modelos con datos ausentes “missing data” – Análisis con datos experimentales

Ilustración

Objetivo … • Se disponen de dos muestras de 162 clientes de

concesionarios de automóviles en España y de 109 clientes de Estados Unidos.

• Para cada cliente se ha obtenido información sobre la calidad percibida del servicio y su lealtad con el concesionario.

• La calidad percibida se ha medido mediante una escala formada por 7 items.

• La lealtad se ha medido mediante una escala de 5 items.

• El objetivo de la investigación es determinar si la calidad percibida afecta a la lealtad y si dicho efecto es igual en ambas muestras.

Modelo

Calidad

Q2

Q3

Q4

Q5

Q7

Q1

Q6

Lealtad

L2

L3

L4

L5

L1

Ejemplos • Calcular:

– Un modelo de medias y covarianzas, donde la estructura de medias es saturada

• ¿Cuáles son los grados de libertad? – En el modelo anterior, contrastar la hipótesis de

igualdad de medias de los indicadores en ambos constructos

– La media de la Calidad Percibida y la Satisfacción • ¿Cuáles son los grados de libertad?

– En el modelo anterior, calcular: • Calcular las medias de los indicadores de ambos constructos • Contrastar la hipótesis de igualdad de medias entre Q y L.

Modelo con medias

Calidad

Q2

Q3

Q4

Q5

Q7

Q1

Q6

Lealtad

L2

L3

L4

L5

L1

1

Muestra España (medias) /TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = *V999 + 1F1 + E1; V2 = *V999 + *F1 + E2; V3 = *V999 + *F1 + E3; V4 = *V999 + *F1 + E4; V5 = *V999 + *F1 + E5; V6 = *V999 + *F1 + E6; V7 = *V999 + *F1 + E7; V8 = *V999 + 1F2 + E8; V9 = *V999 + *F2 + E9; V10 = *V999 + *F2 + E10; V11 = *V999 + *F2 + E11; V12 = *V999 + *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /lmtest /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END

Muestra España (medias) GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST

ROBUST INDEPENDENCE MODEL CHI-SQUARE = 1104.110 ON 66 DEGREES OF FREEDOM

INDEPENDENCE AIC = 972.11041 INDEPENDENCE CAIC = 703.14894 MODEL AIC = -14.70280 MODEL CAIC = -230.68701

SATORRA-BENTLER SCALED CHI-SQUARE = 91.2972 ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS .00085

RESIDUAL-BASED TEST STATISTIC = 121.970 PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS .00000

YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = 69.210 PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS .06668

YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = 1.549 DEGREES OF FREEDOM = 53, 107 PROBABILITY VALUE FOR THE F-STATISTIC IS .02863

FIT INDICES (BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) ----------- BENTLER-BONETT NORMED FIT INDEX = .917 BENTLER-BONETT NON-NORMED FIT INDEX = .954 COMPARATIVE FIT INDEX (CFI) = .963 BOLLEN (IFI) FIT INDEX = .964 MCDONALD (MFI) FIT INDEX = .887 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = .067 90% CONFIDENCE INTERVAL OF RMSEA ( .043, .090)

El mismo ajuste que en el modelo sin medias!!!

Muestra España (medias) MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @. (ROBUST STATISTICS IN PARENTHESES)

Q1 =V1 = 4.881*V999 + 1.000 F1 + 1.000 E1 .130 37.570@ ( .130) ( 37.570@

Q2 =V2 = 4.837*V999 + 1.220*F1 + 1.000 E2 .130 .125 37.189@ 9.761@ ( .130) ( .127) ( 37.189@ ( 9.644@

Q3 =V3 = 5.106*V999 + 1.197*F1 + 1.000 E3 .124 .120 41.154@ 10.011@ ( .124) ( .135) ( 41.154@ ( 8.856@

Q4 =V4 = 5.244*V999 + 1.295*F1 + 1.000 E4 .129 .125 40.805@ 10.402@ ( .129) ( .139) ( 40.805@ ( 9.346@

Medias de las V

Muestra España (medias) STANDARDIZED SOLUTION: R-SQUARED

Q1 =V1 = .000*V999 + .686 F1 + .728 E1 .470 Q2 =V2 = .000*V999 + .836*F1 + .549 E2 .699 Q3 =V3 = .000*V999 + .860*F1 + .511 E3 .739 Q4 =V4 = .000*V999 + .898*F1 + .440 E4 .807 Q5 =V5 = .000*V999 + .770*F1 + .637 E5 .594 Q6 =V6 = .000*V999 + .777*F1 + .630 E6 .604 Q7 =V7 = .000*V999 + .777*F1 + .629 E7 .604 L1 =V8 = .928 F2 + .000*V999 + .372 E8 .862 L2 =V9 = .650*F2 + .000*V999 + .760 E9 .423 L3 =V10 = .907*F2 + .000*V999 + .420 E10 .823 L4 =V11 = .918*F2 + .000*V999 + .396 E11 .843 L5 =V12 = .852*F2 + .000*V999 + .523 E12 .726 F2 =F2 = .829*F1 + .559 D2 .687

En la solución estandarizada, los intercepts son = 0 !!

Muestra España: igualdad medias /TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = *V999 + 1F1 + E1; V2 = *V999 + *F1 + E2; V3 = *V999 + *F1 + E3; V4 = *V999 + *F1 + E4; V5 = *V999 + *F1 + E5; V6 = *V999 + *F1 + E6; V7 = *V999 + *F1 + E7; V8 = *V999 + 1F2 + E8; V9 = *V999 + *F2 + E9; V10 = *V999 + *F2 + E10; V11 = *V999 + *F2 + E11; V12 = *V999 + *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /CONSTRAINTS (V1,V999) = (V2,V999); (V1,V999) = (V3,V999); (V1,V999) = (V4,V999); (V1,V999) = (V5,V999); (V1,V999) = (V6,V999); (V1,V999) = (V7,V999); /lmtest /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END

GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST


INDEPENDENCE MODEL HAS BEEN MODIFIED TO INCLUDE 6 CONSTRAINTS FROM THE SPECIFIED MODEL.

INDEPENDENCE AIC = 1037.42264 INDEPENDENCE CAIC = 744.01013 MODEL AIC = 35.41420 MODEL CAIC = -205.02106





FIT INDICES (BASED ON MODIFIED INDEPENDENCE MODEL, AND ----------- BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) BENTLER-BONETT NORMED FIT INDEX = .913 BENTLER-BONETT NON-NORMED FIT INDEX = .939 COMPARATIVE FIT INDEX (CFI) = .955 BOLLEN (IFI) FIT INDEX = .956 MCDONALD (MFI) FIT INDEX = .856 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = .100 90% CONFIDENCE INTERVAL OF RMSEA ( .081, .119)

Muestra España: medias de los factores

Calidad

Q2

Q3

Q4

Q5

Q7

Q1

Q6

Lealtad

L2

L3

L4

L5

L1

1

mx a


/TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 “VARIABLE DEPENDIENTE” /COVARIANCES /CONSTRAINTS /lmtest /PRINT EFFECT = YES; EIS; FIT=ALL; TABLE=EQUATION; /END


GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST








La diferencia de d.f. es igual al #V – 2 !!


CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @. (ROBUST STATISTICS IN PARENTHESES)

F1 =F1 = 4.852*V999 + 1.000 D1 .135 35.968@ ( .137) ( 35.543@

F2 =F2 = 1.085*F1 - .061*V999 + 1.000 D2 .073 .349 14.879@ -.176 ( .062) ( .322) ( 17.459@ ( -.190)

Media de F1

Intercept


DECOMPOSITION OF EFFECTS WITH NONSTANDARDIZED VALUES STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @.


F1 =F1 = 4.852*V999 + 1.000 D1

F2 =F2 = 1.085*F1 + 5.203*V999 + 1.085 D1 + 1.000 D2

.073 .138 .073 14.879@ 37.787@ 14.879@ ( .062) ( .652) ( .062) ( 17.459@ ( 7.978@ ( 17.459@

Media F2


STANDARDIZED SOLUTION: R-SQUARED

Q1 =V1 = .722 F1 + .692 E1 .521 Q2 =V2 = .804*F1 + .595 E2 .646 Q3 =V3 = .853*F1 + .522 E3 .728 Q4 =V4 = .877*F1 + .481 E4 .769 Q5 =V5 = .817*F1 + .576 E5 .668 Q6 =V6 = .801*F1 + .598 E6 .642 Q7 =V7 = .766*F1 + .643 E7 .587 L1 =V8 = .937 F2 + .349 E8 .878 L2 =V9 = .657*F2 + .754 E9 .431 L3 =V10 = .904*F2 + .428 E10 .817 L4 =V11 = .915*F2 + .402 E11 .838 L5 =V12 = .821*F2 + .571 E12 .674 F1 =F1 = .000*V999 +1.000 D1 .000 F2 =F2 = .827*F1 + .000*V999 + .562 D2 .684

¡ Los intercepts son 0 en la solución estandarizada!

Multi-group mean and covariance structures

MG-MACS


V1 = SINÓNIMOS V2 = ANTÓNIMOS PRE-TEST = 213 NIÑOS DE 11 AÑOS ALEATORIZACIÓN GRUPO CONTROL = 105 NIÑOS GRUPO EXPERIMENTAL = 108 NIÑOS FORMACIÓN ADICIONAL


MODELO PARA DATOS EXPERIMENTALES

Multi-group mean and covariance structures /TITLE

Olsson's data (Sorbom, 1978) control group is first (Example in EQS manual) /SPECIFICATIONS CASES = 105; VAR = 4; ANALYSIS=MOMENT; MATRIX = COV; GROUPS = 2; /LABELS V1 = synonym1; V2 = opposit1; V3 = synonym2; V4 = opposit2; F1 = ABILITY1; F2 = ABILITY2; /EQUATIONS V1 = 20*V999 + 1F1 + E1; V2 = 20*V999 + 0.9*F1 + E2; V3 = 20*V999 + 1F2 + E3; V4 = 20*V999 + 0.9*F2 + E4; F1 = 0V999 + D1; F2 = 0V999 + 0.9*F1 + D2; /VARIANCES D1 = 30*; D2 = 5*; E1 TO E4 = 10*; /COVARIANCES E2,E4 = 5*; /MEANS 18.381 20.229 20.400 21.343 /MATRIX 37.626 24.933 34.680 26.639 24.236 32.013 23.649 27.760 23.565 33.443 /END

/TITLE Olsson's data (Sorbom, 1978)

/TITLE Olsson's data (Sorbom, 1978) Experimental group (Example in EQS manual) /SPECIFICATIONS CASES = 108; VAR = 4; ANALYSIS=MOMENT; MATRIX = COV; GROUPS = 2; /LABELS V1 = synonym1; V2 = opposit1; V3 = synonym2; V4 = opposit2; F1 = ABILITY1; F2 = ABILITY2; /EQUATIONS V1 = 20*V999 + 1F1 + E1; V2 = 20*V999 + 0.9*F1 + E2; V3 = 20*V999 + 1F2 + E3; V4 = 20*V999 + 0.9*F2 + E4; F1 = 2*V999 + D1; F2 = 2*V999 + 0.9*F1 + D2; /VARIANCES D1 = 30*; D2 = 5*; E1 TO E4 = 10*; /COVARIANCES E2,E4 = 5*; /MEANS 20.556 21.241 25.667 25.870 /MATRIX 50.084 42.373 49.872 40.760 36.094 51.237 37.343 40.396 39.890 53.641 /CONSTRAINTS SET = GVV, BVF, BFF; /LMTEST /TECHNICAL ITR = 100; /END



CHI-SQUARE = 3.952 BASED ON 5 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS .55641

CONSTRAINTS FROM GROUP 2

CONSTR: 1 (1,V1,V999)-(2,V1,V999)=0; CONSTR: 2 (1,V2,V999)-(2,V2,V999)=0; CONSTR: 3 (1,V3,V999)-(2,V3,V999)=0; CONSTR: 4 (1,V4,V999)-(2,V4,V999)=0; CONSTR: 5 (1,V2,F1)-(2,V2,F1)=0; CONSTR: 6 (1,V4,F2)-(2,V4,F2)=0; CONSTR: 7 (1,F2,F1)-(2,F2,F1)=0;


MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS

SYNONYM1=V1 = 1.000 F1 +18.619*V999 + 1.000 E1 .597 31.205@

OPPOSIT1=V2 = .878*F1 +19.910*V999 + 1.000 E2 .051 .544 17.286@ 36.603@

SYNONYM2=V3 = 1.000 F2 +20.383*V999 + 1.000 E3 .538 37.882@

OPPOSIT2=V4 = .907*F2 +21.203*V999 + 1.000 E4 .053 .534 17.031@ 39.719@

CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS

ABILITY1=F1 = 1.000 D1

ABILITY2=F2 = .895*F1 + 1.000 D2 .052 17.145@

MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS

SYNONYM1=V1 = 1.000 F1 +18.619*V999 + 1.000 E1 .597 31.205@

OPPOSIT1=V2 = .878*F1 +19.910*V999 + 1.000 E2 .051 .544 17.286@ 36.603@

SYNONYM2=V3 = 1.000 F2 +20.383*V999 + 1.000 E3 .538 37.882@

OPPOSIT2=V4 = .907*F2 +21.203*V999 + 1.000 E4 .053 .534 17.031@ 39.719@


ABILITY1=F1 = 1.875*V999 + 1.000 D1 .899 2.085@

ABILITY2=F2 = .895*F1 + 3.628*V999 + 1.000 D2 .052 .480 17.145@ 7.558@

Control Group Experimental Group

EL GRUPO EXPERIMENTAL TIENE UN FACTOR DE HABILIDAD VERBAL (F1) SIGNIFICATIVAMENTE MAYOR EN EL PRE-TEST EL GRUPO EXPERIMENTAL TIENE UN FACTOR DE HABILIDAD VERBAL (F2) SIGNIFICATIVAMENTE MAYOR EN EL POST-TEST


VARIANCES OF INDEPENDENT VARIABLES ---------------------------------- E D --- --- E1 -SYNONYM1 9.582*I D1 -ABILITY1 29.794*I 2.153 I 4.999 I 4.450@I 5.960@I I I E2 -OPPOSIT1 12.033*I D2 -ABILITY2 1.031*I 2.151 I 1.362 I 5.594@I .757 I I I E3 -SYNONYM2 5.835*I I 1.732 I I 3.368@I I I I E4 -OPPOSIT2 12.502*I I 2.150 I I 5.816@I I I I

COVARIANCES AMONG INDEPENDENT VARIABLES --------------------------------------- E D --- --- E4 -OPPOSIT2 6.393*I I E2 -OPPOSIT1 1.835 I I 3.483@I I I I

Control Group Experimental Group

VARIANCES OF INDEPENDENT VARIABLES --------------------------------- E D --- --- E1 -SYNONYM1 2.548*I D1 -ABILITY1 47.333*I 1.873 I 6.991 I 1.361 I 6.770@I I I E2 -OPPOSIT1 12.359*I D2 -ABILITY2 8.822*I 2.219 I 2.342 I 5.569@I 3.767@I I I E3 -SYNONYM2 7.451*I I 2.372 I I 3.140@I I I I E4 -OPPOSIT2 17.210*I I 2.953 I I 5.828@I I I I

COVARIANCES AMONG INDEPENDENT VARIABLES ---------------------------------------

E D --- --- E4 -OPPOSIT2 7.304*I I E2 -OPPOSIT1 1.890 I I 3.865@I I I I

VARIANZA D1 Y D2 ES MAYOR EN EL GRUPO EXPERIMENTAL



SYNONYM1=V1 = .870 F1 + .000*V999 + .493 E1 .757 OPPOSIT1=V2 = .810*F1 + .000*V999 + .586 E2 .656 SYNONYM2=V3 = .900 F2 + .000*V999 + .436 E3 .810 OPPOSIT2=V4 = .788*F2 + .000*V999 + .616 E4 .621 ABILITY1=F1 = 1.000 D1 .000 ABILITY2=F2 = .979*F1 + .203 D2 .959


SYNONYM1=V1 = .974 F1 + .000*V999 + .226 E1 .949 OPPOSIT1=V2 = .864*F1 + .000*V999 + .503 E2 .747 SYNONYM2=V3 = .929 F2 + .000*V999 + .371 E3 .863 OPPOSIT2=V4 = .831*F2 + .000*V999 + .556 E4 .691 ABILITY1=F1 = .000*V999 + 1.000 D1 .000 ABILITY2=F2 = .901*F1 + .000*V999 + .434 D2 .811

Control Group

Experimental Group

Ilustración

Ejemplos • A partir de los datos sobre concesionarios

de automóviles, estimar un modelo de muestra múltiple para determinar la igualdad de la calidad percibida media y la satisfacción media de los usuarios de concesionarios españoles y americanos

¿Cuáles son las principales conclusiones del modelo?

Multi-group MACS

Calidad

Q2

Q3

Q4

Q5

Q7

Q1

Q6

Lealtad

L2

L3

L4

L5

L1

1

Mx =

a =

=

Iguales entre grupos

Multi-group MACS /TITLE Model built by EQS 6 for Windows in Group 1 /SPECIFICATIONS DATA='E:\Raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; GROUPS=2; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 VARIABLE DEPENDIENTE /COVARIANCES /END

/TITLE Model built by EQS 6 for Windows in Group 2 /SPECIFICATIONS DATA='E:\Raw_data\usa_auto.ESS'; VARIABLES=12; CASES=109; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 VARIABLE DEPENDIENTE /COVARIANCES /PRINT FIT=ALL; TABLE=EQUATION; /LMTEST PROCESS=SIMULTANEOUS; SET=PVV,PFV,PFF,PDD,GVV,GVF,GFV,GFF, BVF,BFF; /CONSTRAINTS (1,F2,F1)=(2,F2,F1); !same slope across groups (1,F1,V999) = (2,F1,V999); !same mean across groups (1,F2,V999) = (2,F2,V999); !same intercept across groups /END

Across-group constraints

Multisample MACS GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST








Multisample MACS CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST

STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH

@. (ROBUST STATISTICS IN PARENTHESES)

F1 =F1 = 5.062*V999 + 1.000 D1 .103 49.170@ ( .105) ( 48.402@

F2 =F2 = 1.086*F1 - .217*V999 + 1.000 D2 .059 .294 18.542@ -.740 ( .052) ( .271) ( 21.069@ ( -.801)


STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED WITH @.

(ROBUST STATISTICS IN PARENTHESES)

F1 =F1 = 5.062*V999 + 1.000 D1 .103 49.170@ ( .105) ( 48.402@

F2 =F2 = 1.086*F1 - .217*V999 + 1.000 D2

.059 .294 18.542@ -.740 ( .052) ( .271) ( 21.069@ ( -.801)

¡¡¡Same means, intercepts and slopes across groups!!!

Estructuras de medias y covarianzas...

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